LMIs in Control/Print version
Basic Matrix Theory
editBasic Matrix Notation
editConsider the complex matrix .
Transpose of a Matrix
The transpose of , denoted as or is:
Adjoint of a Matrix
The adjoint or hermitian conjugate of , denoted as is:
Where is the complex conjugate of matrix element .
Notice that for a real matrix , .
Important Properties of Matricies
editHermitian, Self-Adjoint, and Symmetric Matricies
A square matrix is called Hermitian or self-adjoint if .
If is Hermitian then it is called symmetric.
Unitary Matricies
A square matrix is called unitary if or .
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Notion of Matrix Positivity
editNotation of Positivity
editA symmetric matrix is defined to be:
positive semidefinite, , if for all .
positive definite, , if for all .
negative semidefinite, .
negative definite, .
indefinite if is neither positive semidefinite nor negative semidefinite.
Properties of Positive Matricies
edit- For any matrix , .
- Positive definite matricies are invertible and the inverse is also positive definite.
- A positive definite matrix has a square root, , such that .
- For a positive definite matrix and invertible , .
- If and , then .
- If then for any scalar .
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
KYP Lemma (Bounded Real Lemma)
editKYP Lemma (Bounded Real Lemma)
The Kalman–Popov–Yakubovich (KYP) Lemma is a widely used lemma in control theory. It is sometimes also referred to as the Bounded Real Lemma. The KYP lemma can be used to determine the norm of a system and is also useful for proving many LMI results.
The System
editwhere , , , at any .
The Data
editThe matrices are known.
The Optimization Problem
editThe following optimization problem must be solved.
The LMI: The KYP or Bounded Real Lemma
editSuppose is the system. Then the following are equivalent.
Conclusion:
editThe KYP Lemma can be used to find the bound on the norm of a system. Note from the (1,1) block of the LMI we know that is Hurwitz.
Implementation
editSince the KYP lemma shown above is nonlinear in gamma, in order to implement it in MATLAB we must first linearize it by using the Schur Complement to arrive at the dual formulation below:
- .
This dual KYP LMI is now linear in both and .
This implementation requires the use of Yalmip and Sedumi.
https://github.com/eoskowro/LMI/blob/master/KYP_Lemma_LMI.m
Related LMIs
editExternal Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Positive Real Lemma
editPositive Real Lemma
The Positive Real Lemma is a variation of the Kalman–Popov–Yakubovich (KYP) Lemma. The Positive Real Lemma can be used to determine if a system is passive (positive real).
The System
editwhere , , , at any .
The Data
editThe matrices are known.
The LMI: The Positive Real Lemma
editSuppose is the system. Then the following are equivalent.
Conclusion:
editThe Positive Real Lemma can be used to determine if the system is passive. Note from the (1,1) block of the LMI we know that is Hurwitz.
Implementation
editThis implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/Positive_Real_Lemma.m
Related LMIs
editKYP Lemma (Bounded Real Lemma)
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
KYP Lemma for QSR Dissipative Systems
editThe Concept
editIn systems theory the concept of dissipativity was first introduced by Willems which describes dynamical systems by input-output properties. Considering a dynamical system described by its state , its input and its output , the input-output correlation is given a supply rate . A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function such that , and
As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate .
The physical interpretation is that is the energy stored in the system, whereas is the energy that is supplied to the system.
This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.
Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by Vasile M. Popov, Jan Camiel Willems, D.J. Hill, and P. Moylan. In the case of linear invariant systems, this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems.Dissipative systems are still an active field of research in systems and control, due to their important applications.
The System
editConsider a contiuous-time LTI system, , with minimal state-space realization (A, B, C, D), where and .
The Data
editThe matrices and which defines the state space of the system
The Optimization Problem
editThe system is QSR disipative if
where is the input to is the output of and .
LMI : KYP Lemma for QSR Dissipative Systems
editThe system is also QSR dissipative if and only if there exists where such that
Conclusion:
editIf there exist a positive definite for the the selected Q,S and R matrices then the system is QSR dissipative.
Implementation
editCode for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs
editReferences
edit1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational
Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac-
tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
KYP Lemma witout Feedthrough
editThe Concept
editIt is assumed in the Lemma that the state-space representation (A, B, C, D) is minimal. Then Positive Realness (PR) of the transfer function C(SI − A)-1B + D is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (A, B, C, D)=(−α, 0, 0, 1), with α > 0. The transfer function is H(s) = 0 that is PR
The System
editConsider a contiuous-time LTI system, , with minimal state-space relization (A, B, C, 0), where and .
The Data
editThe matrices The matrices and
LMI : KYP Lemma without Feedthrough
editThe system is positive real (PR) under either of the following equivalet necessary and sufficient conditions.
- 1. There exists where such that
- 2. There exists where such that
This is a special case of the KYP Lemma for QSR dissipative systems with Q = 0, Q = 0.5 and R = 0.
The system is strictly positive real (SPR) under either of the following equivalet necessary and sufficient conditions.
- 1. There exists where such that
- 2. There exists where such that
This is a special case of the KYP Lemma for QSR dissipative systems with Q = ε1, Q = 0.5 and R = 0. where ε
Conclusion:
editIf there exist a positive definite for the the selected Q,S and R matrices then the system is Positive Real.
Implementation
editCode for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs
editKYP Lemma
State Space Stability
Discrete Time KYP Lemma with Feedthrough
References
edit1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational
Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac-
tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London
KYP Lemma for Descriptor Systems
editThe Concept
editDescriptor system descriptions frequently appear when solving computational problems in the analysis and design of standard linear systems. The numerically reliable solution of many standard control problems like the solution of Riccati equations, computation of system zeros, design of fault detection and isolation filters (FDI), etc. relies on using descriptor system techniques.
Many algorithm for standard systems as for example stabilization techniques, factorization methods, minimal realization, model reduction, etc. have been extended to the more general descriptor system descriptions. An important application of these algorithms is the numerically reliable computation with rational and polynomial matrices via equivalent descriptor representations. Recall that each rational matrix R(s) can be seen as the transfer-function matrix of a continuous- or discrete-time descriptor system. Thus, each R(s) can be equivalently realized by a descriptor system quadruple (A-sE, B, C, D) satisfying R(S)= C(SE-A)-1B+D
Many operations on standard matrices (e.g., finding the rank, determinant, inverse or generalized inverses), or the solution of linear matrix equations can be performed for rational matrices as well using descriptor system techniques. Other important applications of descriptor techniques are the computation of inner-outer and spectral factorisations, or minimum degree and normalized coprime factorisations of polynomial and rational matrices. More explanation can be found in the website of Institute of System Dynamics and control
The System
editConsider a square, contiuous-time linear time-invariant (LTI) system, , with minimal state-space relization (E, A, B, C, D), where and .
The Data
editThe matrices The matrices and
LMI : KYP Lemma for Descriptor Systems
editThe system is extended strictly positive real (ESPR) if and only if there exists and such that
The system is also ESPR if there exists such that
Conclusion:
editIf there exist a X and W matrix satisfying above LMIs then the system is Extended Strictly Positive Real.
Implementation
editCode for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs
editKYP Lemma
State Space Stability
Discrete Time KYP Lemma with Feedthrough
References
edit1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational
Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac-
tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London
5. Numerical algorithms and software tools for analysis and modelling of descriptor systems. Prepr. of 2nd IFAC Workshop on System Structure and Control, Prague, Czechoslovakia, pp. 392-395, 1992.
Generealized KYP (GKYP) Lemma for conic Sectors
editThe Concept
editThe conic sector theorem is a powerful input-output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis.
The System
editConsider a square, contiuous-time linear time-invariant (LTI) system, , with minimal state-space relization (A, B, C, D), where and .
Also consider , which is defined as
- ,
where and .
The Data
editThe matrices The matrices and . The values of a and b
LMI : Generalized KYP (GKYP) Lemma for Conic Sectors
editThe following generalized KYP Lemmas give conditions for to be inside the cone within finite frequency bandwidths.
- 1. (Low Frequency Range) The system is inside the cone for all , where and , if there exist and , where , such that
- .
- If and Q = 0, then the traditional Conic Sector Lemma is recovered. The parameter is incuded in the above LMI to effectively transform into the strict inequality
- 2. (Intermediate Frequency Range) The system is inside the cone for all , where and , if there exist and and where and , such that
- .
- The parameter is incuded in the above LMI to effectively transform into the strict inequality .
- 3. (High Frequency Range) The system is inside the cone for all , where and , if there exist , where , such that
- .
If (A, B, C, D) is a minimal realization, then the matrix inequalities in all of the above LMI, then it can be nostrict.
Conclusion:
editIf there exist a positive definite matrix satisfying above LMIs for the given frequency bandwidths then the system is inside the cone [a,b].
Implementation
editCode for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs
editKYP Lemma
State Space Stability
Exterior Conic Sector Lemma
Modified Exterior Conic Sector Lemma
References
edit1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational
Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac-
tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Bridgeman, Leila Jasmine, and James Richard Forbes. "The exterior conic sector lemma." International Journal of Control 88.11 (2015): 2250-2263.
Discrete time Bounded Real Lemma
editDiscrete-Time Bounded Real Lemma
A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.
Discrete-Time Bounded Real Lemma or the H∞ norm can be found by solving a LMI.
The System
editDiscrete-Time LTI System with state space realization
The Data
editThe matrices: System .
The Optimization Problem
editThe following feasibility problem should be optimized:
is minimized while obeying the LMI constraints.
The LMI:
editDiscrete-Time Bounded Real Lemma
The LMI formulation
H∞ norm <
Conclusion:
editThe H∞ norm is the minimum value of that satisfies the LMI condition. If is the minimal realization then the inequalities can be non-strict.
Implementation
editA link to CodeOcean or other online implementation of the LMI
MATLAB Code
Related LMIs
edit[1] - Continuous time KYP_Lemma_(Bounded_Real_Lemma)
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Discrete Time KYP Lemma for QSR Dissipative System
editThe Concept
editIn systems theory the concept of dissipativity was first introduced by Willems which describes dynamical systems by input-output properties. Considering a dynamical system described by its state , its input and its output , the input-output correlation is given a supply rate . A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function such that , and
As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate .
The physical interpretation is that is the energy stored in the system, whereas is the energy that is supplied to the system.
This notion has a strong connection with Lyapunov stability, where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.
Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by Vasile M. Popov, Jan Camiel Willems, D.J. Hill, and P. Moylan. In the case of linear invariant systems, this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems.Dissipative systems are still an active field of research in systems and control, due to their important applications.
The System
editConsider a discrete-time LTI system, , with minimal state-space relization , where and .
The Data
editThe matrices and
The Optimization Problem
editThe system is QSR disipative if
where is the input to is the output of and .
LMI : Discrete-Time KYP Lemma for QSR Dissipative Systems
editThe system is also QSR dissipative if and only if there exists where such that
Conclusion:
editIf there exist a positive definite for the the selected Q,S and R matrices then the system is QSR dissipative.
Implementation
editCode for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs
editKYP Lemma
KYP Lemma for continous Time QSR Dissipative system
References
edit1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational
Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac-
tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
Discrete Time KYP Lemma with Feedthrough
editThe Concept
editIt is assumed in the Lemma that the state-space representation (Ad, Bd, Cd, Dd) is minimal. Then Positive Realness (PR) of the transfer function Cd(SI − Ad)-1Bd + Dd is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (Ad, Bd, Cd, Dd)=(−α, 0, 0, 1), with α > 0. The transfer function is H(s) = 0 that is PR
The System
editConsider a discrete-time LTI system, , with minimal state-space relization , where and .
The Data
editThe matrices and
LMI : Discrete-Time KYP Lemma with Feedthrough
editThe system is positive real (PR) under either of the following equivalet necessary and sufficient conditions.
- 1. There exists where such that
- 2. There exists where such that
- 3. There exists where such that
- 4. There exists where such that
This is a special case of the KYP Lemma for QSR dissipative systems with Q = 0, Q = 0.5 and R = 0.
The system is strictly positive real (SPR) under either of the following equivalet necessary and sufficient conditions.
- 1. There exists where such that
- 2. There exists where such that
- 3. There exists where such that
- 4. There exists where such that
This is a special case of the KYP Lemma for QSR dissipative systems with Q = ε1, Q = 0.5 and R = 0. where ε
Conclusion:
editIf there exist a positive definite for the the selected Q,S and R matrices then the system is Positive Real.
Implementation
editCode for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs
editKYP Lemma
State Space Stability
KYP Lemma without Feedthrough
References
edit1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational
Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac-
tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London
Schur Complement
editAn important tool for proving many LMI theorems is the Schur Compliment. It is frequently used as a method of LMI linearization.
The Schur Compliment
editConsider the matricies , , and where and are self-adjoint. Then the following statements are equivalent:
- and both hold.
- and both hold.
- is satisfied.
More concisely:
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
LMI for Eigenvalue Minimization
editLMI for Minimizing Eigenvalue of a Matrix
Synthesizing the eigenvalues of a matrix plays an important role in designing controllers for linear systems. The eigenvalues of the state matrix of a linear time-invariant system determine if the system is stable or not. The system is stable if all the eigenvalues of the state matrix are located in the left half of the complex plane. Thus, we may desire to minimize the maximal eigenvalue of the state matrix such that the minimized eigenvalue is placed in the left half-plane, which guarantees that the system is stable.
The System
editAssume that we have a matrix function of variables :
where are symmetric matrices.
The Data
editThe symmetric matrices () are given.
The Optimization Problem
editThe optimization problem is to find the variables to minimize the following cost function:
where is the cost function and indicates the maximim eigenvalue of a matrix.
According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent
where is defined as the maximim eigenvalue of the matrix .
The LMI: LMI for eigenvalue minimization
editThis optimization problem can be converted to an LMI problem.
The mathematical description of the LMI formulation can be written as follows:
Conclusion:
editAs a result, the variables after solving this LMI problem.
Moreover, we obtain the maximum eigenvalue, , of the matrix .
Implementation
editA link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Minimizing-the-Maximum-Eigenvalue-of-Matrix
Related LMIs
editLMI for Generalized Eigenvalue Problem
LMI for Matrix Norm Minimization
LMI for Maximum Singular Value of a Complex Matrix
External Links
edit- [2] - LMI in Control Systems Analysis, Design and Applications
- Eigenvalues and Eigenvectors of a Matrix
LMI for Matrix Norm Minimization
editLMI for Matrix Norm Minimization
This problem is a slight generalization of the eigenvalue minimization problem for a matrix. Calculating norm of a matrix is necessary in designing an or an optimal controller for linear time-invariant systems. In those cases, we need to compute the norm of the matrix of the closed-loop system. Moreover, we desire to design the controller so as to minimize the closed-loop matrix norm.
The System
editAssume that we have a matrix function of variables :
where are symmetric matrices.
The Data
editThe symmetric matrices () are given.
The Optimization Problem
editThe optimization problem is to find the variables in order to minimize the following cost function:
where is the cost function and indicates the norm of the matrix function .
According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent:
The LMI: LMI for matrix norm minimization
editThis optimization problem can be converted to an LMI problem.
The mathematical description of the LMI formulation can be written as follows:
Conclusion:
editAs a result, the variables after solving this LMI problem and we obtain that is the norm of matrix function .
Implementation
editA link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Matrix-Norm-Minimization
Related LMIs
editLMI for Matrix Norm Minimization
LMI for Generalized Eigenvalue Problem
LMI for Maximum Singular Value of a Complex Matrix
External Links
editA list of references documenting and validating the LMI.
- [3] - LMI in Control Systems Analysis, Design and Applications
LMI for Generalized Eigenvalue Problem
editLMI for Generalized Eigenvalue Problem
Technically, the generalized eigenvalue problem considers two matrices, like and , to find the generalized eigenvector, , and eigenvalues, , that satisfies . If the matrix is an identity matrix with the proper dimension, the generalized eigenvalue problem is reduced to the eigenvalue problem.
The System
editAssume that we have three matrice functions which are functions of variables as follows:
where are , , and () are the coefficient matrices.
The Data
editThe , , and are matrix functions of appropriate dimensions which are all linear in the variable and , , are given matrix coefficients.
The Optimization Problem
editThe problem is to find such that:
, , and are satisfied and is a scalar variable.
The LMI: LMI for Schur stabilization
editA mathematical description of the LMI formulation for the generalized eigenvalue problem can be written as follows:
Conclusion:
editThe solution for this LMI problem is the values of variables such that the scalar parameter, , is minimized. In practical applications, many problems involving LMIs can be expressed in the aforementioned form. In those cases, the objective is to minimize a scalar parameter that is involved in the constraints of the problem.
Implementation
editA link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Schur-Stability
Related LMIs
editLMI for Generalized Eigenvalue Problem
LMI for Matrix Norm Minimization
LMI for Maximum Singular Value of a Complex Matrix
External Links
edit- [4] - LMI in Control Systems Analysis, Design and Applications
LMI for Linear Programming
editLMI for Linear Programming
Linear programming has been known as a technique for the optimization of a linear objective function subject to linear equality or inequality constraints. The feasible region for this problem is a convex polytope. This region is defined as a set of the intersection of many finite half-spaces which are created by the inequality constraints. The solution for this problem is to find a point in the polytope of existing solutions where the objective function has its extremum (minimum or maximum) value.
The System
editWe define the objective function as:
and constraints of the problem as:
.
.
.
The Data
editSuppose that , , and are given parameters where and . Moreover, is an vector of positive variables.
The Optimization Problem
editThe optimization problem is to minimize the objective function, when the aforementioned linear constraints are satisfied.
The LMI: LMI for linear programming
editThe mathematical description of the optimization problem can be readily written in the following LMI formulation:
Conclusion:
editSolving this problem results in the values of variables which minimize the objective function. It is also worthwhile to note that if , the computational cost for solving this problem would be .
There does not exist an analytical formulation to solve a general linear programming problem. Nonetheless, there are some efficient algorithms, like the Simplex algorithm, for solving a linear programming problem.
Implementation
editA link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Linear-Programming
Related LMIs
editExternal Links
edit- [5] - LMI in Control Systems Analysis, Design and Applications
LMI for Feasibility Problem
editLMI for Feasibility Problem in Optimization
The feasibility problem is to find any feasible solutions for an optimization problem without regard to the objective value. This problem can be considered as a special case of an optimization problem where the objective value is the same for all the feasible solutions. Many optimization problems have to start from a feasible point in the range of all possible solutions. One way is to add a slack variable to the problem in order to relax the feasibility condition. By adding the slack variable the problem any start point would be a feasible solution. Then, the optimization problem is converted to find the minimum value for the slack variable until the feasibility is satisfied.
The System
editAssume that we have two matrices as follows:
which are matrix functions of variables .
The Data
editSuppose that the matrices and are given.
The Optimization Problem
editThe optimization problem is to find variables such that the following constraint is satisfied:
The LMI: LMI for Feasibility Problem
editThis optimization problem can be converted to a standard LMI problem by adding a slack variable, .
The mathematical description for this problem is to minimize in the following form of the LMI formulation:
Conclusion:
editIn this problem, and are decision variables of the LMI problem.
As a result, these variables are determined in the optimization problem such that the minimum value of is found while the inequality constraint is satisfied.
Implementation
editA link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Feasibility-Problem-of-Convex-Optimization
Related LMIs
editExternal Links
edit- [6] - LMI in Control Systems Analysis, Design and Applications
Structured Singular Value
editLMIs in Control/Print version
The LMI can be used to find a that belongs to the set of scalings . Minimizing allows to minimize the squared norm of .
The System
editThe Data
editThe matrices .
The Optimization Problem
edit
The LMI:
editConclusion:
edit
Implementation
edithttps://github.com/mcavorsi/LMI
Related LMIs
editExternal Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
Eigenvalue Problem
editLMIs in Control/Print version
The maximum eigenvalue of a matrix is going to have the most impact on system performance. This LMI allows for minimization of the maximum eigenvalue by minimizing .
The System
editThe Data
editThe matrices .
The Optimization Problem
edit
The LMI:
editConclusion:
editThe eigenvalue problem can be utilized to minimize the maximum eigenvalue of a matrix that depends affinely on a variable.
Implementation
edithttps://github.com/mcavorsi/LMI
Related LMIs
editExternal Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
LMI for Minimizing Condition Number of Positive Definite Matrix
editLMIs in Control/Print version
The System:
editA related problem is minimizing the condition number of a positive-defnite matrix that depends affinely on the variable , subject to the LMI constraint > 0. This problem can be reformulated as the GEVP.
The Optimization Problem:
editThe GEVP can be formulated as follows:
minimize
subject to > 0;
>0;
< < .
We can reformulate this GEVP as an EVP as follows. Suppose,
= + , = +
The LMI:
editDefining the new variables = , = we can express the previous formulation as the EVP with variables and :
miminize
subject to + >0; < + <
Conclusion:
editThe LMI is feasible.
Implementation
editReferences
edit- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Continuous Quadratic Stability
editLMIs in Control/Print version
To study stability of a LTI system, we first ask whether all trajectories of system converge to zero as . A sufficient condition for this is the existence of a quadratic function , that decreases along every nonzero trajectory of system . If there exists such a P, we say the system is quadratically stable and we call a quadratic Lyapunov function.
The System
editThe Data
editThe system coefficient matrix takes the form of
where is a known matrix, which represents the nominal system matrix, while is the system matrix perturbation, where
- are known matrices, which represent the perturbation matrices.
- which represent the uncertain parameters in the system.
- is the uncertain parameter vector, which is often assumed to be within a certain compact and convex set : : that is
The LMI: Continuous-Time Quadratic Stability
editThe uncertain system is quadratically stable if and only if there exists , where such that
The following statements can be made for particular sets of perturbations.
Case 1: Regular Polyhedron
editConsider the case where the set of perturbation parameters is defined by a regular polyhedron as
The uncertain system is quadratically stable if and only if there exists , where such that
Case 2: Polytope
editConsider the case where the set of perturbation parameters is defined by a polytope as
The uncertain system is quadratically stable if and only if there exists , where such that
Conclusion:
editIf feasible, System is Quadratically stable for any
Implementation
edithttps://github.com/Ricky-10/coding107/blob/master/PolytopicUncertainities
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
Exterior Conic Sector Lemma
editThe Concept
editThe conic sector theorem is a powerful input-output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis.
The System
editConsider a square, contiuous-time linear time-invariant (LTI) system, , with minimal state-space relization (A, B, C, D), where and .
The Data
editThe matrices The matrices and
LMI : Exterior Conic Sector Lemma
editThe system is in the exterior cone of radius r centered at c (i.e. exconer(c)), where and , under either of the following equivalent necessary and sufficient conditions.
- 1. There exists P , where P , such that
- 2. There exists P , where P , such that
Proof, Applying the Schur complement lemma to the terms in (1) gives (2).
Conclusion:
editIf there exist a positive definite matrix satisfying above LMIs then the system is in the exterior cone of radius r centered at c.
Implementation
editCode for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs
editKYP Lemma
State Space Stability
References
edit1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational
Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transactions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Bridgeman, Leila Jasmine, and James Richard Forbes. "The exterior conic sector lemma." International Journal of Control 88.11 (2015): 2250-2263.
Modified Exterior Conic Sector Lemma
editThe Concept
editThe conic sector theorem is a powerful input-output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis.
The System
editConsider a square, contiuous-time linear time-invariant (LTI) system, , with minimal state-space relization (A, B, C, D), where and .
The Data
editThe matrices The matrices and
LMI : Modified Exterior Conic Sector Lemma
editThe system is in the exterior cone of radius r centered at c (i.e. exconer(c)), where and , under either of the following sufficient conditions.
- 1. There exists P , where P , such that
- Proof. The term in the Actual Exterior Conic Sector Lemma makes the matrix inequality more neagtive definite.
Therefore,
- 2. There exists P , where P , such that
- Proof. Applying the Schur complement lemma to the terms in (1) gives (2).
Conclusion:
editIf there exist a positive definite matrix satisfying above LMIs then the system is in the exterior cone of radius r centered at c.
Implementation
editCode for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs
editKYP Lemma
State Space Stability
Exterior Conic Sector Lemma
References
edit1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational
Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac-
tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Bridgeman, Leila Jasmine, and James Richard Forbes. "The exterior conic sector lemma." International Journal of Control 88.11 (2015): 2250-2263.
DC Gain of a Transfer Matrix
editThe continuous-time DC gain is the transfer function value at the frequency s = 0.
The System
editConsider a square continuous time Linear Time invariant system, with the state space realization and
The Data
edit
The LMI: LMI for DC Gain of a Transfer Matrix
editThe transfer matrix is given by
The DC Gain of the system is strictly less than if the following LMIs are satisfied.
OR
Conclusion
editThe DC Gain of the continuous-time LTI system, whose state space realization
is give by (), is
- Upon implementation we can see that the value of obtained from the LMI approach and the value of obtained from the above formula are the same
Implementation
editA link to the Matlab code for a simple implementation of this problem in the Github repository:
https://github.com/yashgvd/LMI_wikibooks
External Links
edit- [7] - LMI in Control Systems Analysis, Design and Applications
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- [8] -Mathworks reference to DC Gain
Discrete Time H2 Norm
editDiscrete-Time H2 Norm
A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.
Discrete-Time LTI systems' H2 norm can be found by solving a LMI.
The System
editDiscrete-Time LTI System with state space realization
The Data
editThe matrices: System .
The Optimization Problem
editThe following feasibility problem should be optimized:
is minimized while obeying the LMI constraints.
The LMI:
editDiscrete-Time Bounded Real Lemma
The LMI formulation
H2 norm <
Conclusion:
editThe H2 norm is the minimum value of that satisfies the LMI condition.
Implementation
editA link to CodeOcean or other online implementation of the LMI
MATLAB Code
Related LMIs
edit[9] - Continuous time H2 norm.
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- [10] - MATLAB function for the H2 norm.
Discrete Time Minimum Gain Lemma
editThe Concept
editThe output of the system y(t) is fed back through a sensor measurement F to a comparison with the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.
This is called a single-input-single-output (SISO) control system; MIMO (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions).
If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations:
Solving for Y(s) in terms of R(s) gives
The expression is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If , i.e., it has a large norm with each value of s, and if , then Y(s) is approximately equal to R(s) and the output closely tracks the reference input. This page gives an LMI to reduce the gain so that the output closely tracks the reference input.
The System
editConsider a discrete-time LTI system, , with minimal state-space relization , where and .
The Data
editThe matrices and
LMI : Discrete-Time Minimum Gain Lemma
editThe system has minimium gain γ under any of the following equivalent sufficient conditions.
- 1. There exists and γ where such that
- 2. There exists and where such that
: Applying the Schur complement lemma to the γ2 term in equation 1 gives equation 2.
Conclusion:
editIf there exist a positive definite for the system , then the minimum gain of the system is γ can be obtaied from above defined LMIs.
Implementation
editCode for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs
editKYP Lemma
State Space Stability
KYP Lemma without Feedthrough
References
edit1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational
Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac-
tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London
Modified Discrete Time Minimum Gain Lemma
editThe Concept
editThe output of the system y(t) is fed back through a sensor measurement F to a comparison with the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.
This is called a single-input-single-output (SISO) control system; MIMO (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions).
If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations:
Solving for Y(s) in terms of R(s) gives
The expression is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If , i.e., it has a large norm with each value of s, and if , then Y(s) is approximately equal to R(s) and the output closely tracks the reference input. This page gives an LMI to reduce the gain so that the output closely tracks the reference input.
The System
editConsider a discrete-time LTI system, , with minimal state-space relization , where and .
The Data
editThe matrices and
LMI : Discrete-Time Modified Minimum Gain Lemma
editThe system has minimium gain γ under any of the following equivalent sufficient conditions.
- 1. There exists and γ where such that
. The term in Discrete Time Minimum Gain Lemma makes the matrix inequality more negative definite. Therefore,
- 2. There exists and where such that
: Applying the Schur complement lemma to the γ2 term in equation 1 gives equation 2.
Conclusion:
editIf there exist a positive definite for the system , then the minimum gain of the system γ can be obtaied from above defined LMIs.
Implementation
editCode for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs
editKYP Lemma
State Space Stability
KYP Lemma without Feedthrough
Discrete Time Minimum Gain Lemma
References
edit1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational
Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac-
tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London
Discrete-Time Algebraic Riccati Equation
editLMIs in Control/Print version
The System
editConsider a Discrete-Time LTI system
Consider Ad ∈ nxn ; Bd ∈ nxm
The Data
editThe Matrices Ad , Bd , Cd , Q, R are given
Q and R should necessarily be Hermitian Matrices.
A square matrix is Hermitian if it is equal to its complex conjugate transpose.
The Optimization Problem
editOur aim is to find
P - Unique solution to the discrete-time algebraic Riccati equation, returned as a matrix.
K - State-feedback gain, returned as a matrix.
The algorithm used to evaluate the State-feedback gain is given by
L - Closed-loop eigenvalues, returned as a matrix.
The LMI: Discrete-Time Algebraic Riccati Inequality (DARE)
editAn algebraic Riccati equation is a type of nonlinear equation that arises in the context of infinite-horizon optimal control problems in continuous time or discrete time
The Discrete-Time Algebraic Riccati Inequality is given by
P , Q ∈ n and R ∈ m where P > 0, Q ≥ 0, R > 0
P is the unknown n by n symmetric matrix and A, B, Q, R are known real coefficient matrices.
The above equation can be rewritten using the Schur Complement Lemma as :
Conclusion:
editAlgebraic Riccati Inequalities play a key role in LQR/LQG control, H2- and H∞ control and Kalman filtering. We try to find the unique stabilizing solution, if it exists. A solution is stabilizing, if controller of the system makes the closed loop system stable.
Equivalently, this Discrete-Time algebraic Riccati Inequality is satisfied under the following necessary and sufficient condition:
Implementation
edit( X in the output corresponds to P in the LMI )
A link to the Matlab code for a simple implementation of this problem in the Github repository:
https://github.com/yashgvd/LMI_wikibooks
Related LMIs
editLMI for Continuous-Time Algebraic Riccati Inequality
LMI for Schur Stabilization
External Links
editA list of references documenting and validating the LMI.
- [11] - LMI in Control Systems Analysis, Design and Applications
Deduced LMI Conditions for H-infinity Index
editH-infinity Index Deduced LMI
Although the KYP Lemma, also known as the Bounded Real Lemma, is a basic condition to evaluate an upper bound on the H∞, the verification of the bound on the H∞-gain of the system can be done via the deduced condition.
The System
editA state-space representation of a linear system as given below:
where , and are the system state, output, and the disturbance vector respectively. The transfer function of such a system can be evaluated as:
The Data
editNumber of states n, number of outputs m and number of external noise channels r need to be known. Moreover, the system matrices A,B,C,D are also required to be known.
The Feasibility LMI
editFor an arbitrary , the transfer function G(s) satisfies
if and only if there exists a symmetric matrix X > 0 and a matrix such that:
where:
The above LMI can be combined with the bisection method to find minimum to find the minimum upper bound on the H∞ gain of .
Conclusion:
editIf there is a feasible solution to the aforementioned LMI, then the upper bounds the infinity norm of the system G(s).
Implementation
editTo solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:
https://github.com/smhassaan/LMI-Examples/blob/master/Deduced_hinf_example.m
Related LMIs
editExternal Links
editA list of references documenting and validating the LMI.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 5.2.2 pp. 153–156.
Deduced LMI Conditions for H2 Index
editH2 Index Deduced LMI
Although there are ways to evaluate an upper bound on the H2, the verification of the bound on the H2-gain of the system can be done via the deduced condition.
The System
editWe consider the generalized Continuous-Time LTI system with the state space realization of
where , and are the system state, output, and the input vectors respectively.
The transfer function of such a system can be evaluated as:
The Data
editThe system matrices are known.
The Optimization Problem
editFor an arbitrary (a given scalar), the transfer function satisfies
The H2-norm condition on Transfer function holds only when the matrix A is stable. And this can be conveniently converted to an LMI problem
if and only if
1. There exists a symmetric matrix such that:
,
2. There exists a symmetric matrix such that:
,
The LMI - Deduced Conditions for H2-norm
editThese deduced condition can be derived from the above equations. According to this
For an arbitrary (a given scalar), the transfer function satisfies
if and only if there exists symmetric matrices and ; and a matrix such that
The above LMI can be combined with the bisection method to find minimum to find the minimum upper bound on the H2 gain of .
Conclusion:
editIf there is a feasible solution to the aforementioned LMI, then the upper bounds the norm of the system G(s).
Implementation
editTo solve the feasibility LMI, YALMIP toolbox is required for setting up the problem, and SeDuMi or MOSEK is required to solve the problem. The following link showcases an example of the problem:
https://github.com/yashgvd/ygovada
Related LMIs
editBounded Real Lemma
Deduced LMIs for H-infinity index
External Links
editA list of references documenting and validating the LMI.
- [12] - LMI in Control Systems Analysis, Design and Applications
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
Dissipativity of Systems
editDissipativity of Systems
The dissipativity of systems is associated with their supply function. In general, a linear system is dissipative if the accumulated sum (integration) of the supply function is non-negative over all the duration of .
The System
editA state-space representation of a linear system as given below:
where , and are the system state, output, and the input vector respectively. A, B, C and D are system coefficient matrices of appropriate dimensions. The control input u is restricted to be a piece-wise continuous vector function defined of .
The transfer function of such a system can be evaluated as:
For such a system, a general quadratic supply function is defined as:
where Q is a real symmetric matrix of (m+r) dimensions. Q need not be either symmetric positive or negative definite.
The Data
editNumber of states n, number of outputs m and number of control inputs r need to be known. Moreover, the system matrices A,B,C,D are also required to be known. The system should also be controllable.
The Feasibility LMI
editThe system defined can be evaluated to be dissipative with respect to a supply function iff there exist and a (defining ) such that the following is feasible:
Conclusion:
editIf there is a feasible solution to the aforementioned LMI, then there exists a supply function for which the system is dissipative. Since the assumption of the system being controllable is required for it to be dissipative, this check can be used of as a sufficient condition to check the controllability of the linear system, just like the feasibility for Lyapunov stability.
Implementation
editTo solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:
https://github.com/smhassaan/LMI-Examples/blob/master/Dissipativity_example.m
Related LMIs
editContinuous_Time_Lyapunov_Inequality
External Links
editA list of references documenting and validating the LMI.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.3 pp. 178–184.
D-Stabilization
edit-Stabilization
There are a wide variety of control design problems that are addressed in a wide variety of different ways. One of the most important control design problem is that of state feedback stabilization. One such state feedback problem, which will be the main focus of this article, is that of -Stabilization, a form of -Stabilization where the closed-loop poles are located on the left-half of the complex plane.
The System
editFor this particular problem, suppose that we were given a linear system in the form of:
where , , and represents either the differential operator (in the continuous-time case) or the one-step forward operator (for the discrete-time system case). Then the LMI for determining the -stabilization case would be obtained as described below.
The Data
editIn order to obtain the LMI, we need the following 2 matrices: .
The Optimization Problem
editSuppose - for the linear system given above - we were asked to design a state-feedback control law where such that the closed-loop system:
is -stable, then the system would be stabilized as follows.
The LMI: -Stabilization
editFrom the given pieces of information, it is clear that the optimization problem only has a solution if there exists a matrix and a symmetric matrix that satisfies the following:
Conclusion:
editGiven the resulting controller matrix , it can be observed that the matrix is -stable.
Implementation
edit- Example Code - A GitHub link that contains code (titled "DStability.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.
Related LMIs
edit- H stabilization - Equivalent LMI for -stabilization.
- Continuous Time D-Stability Controller - LMI for deriving a Controller using D-Stability.
- Continuous Time D-Stability Observer - LMI for deriving an Observer using D-Stability.
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - A book co-authored by Guang-Ren Duan and Hai-Hua Yu.
H-Stabilization
edit-Stabilization
There are a wide variety of control design problems that are addressed in a wide variety of different ways. One of the most important control design problem is that of state feedback stabilization. One such state feedback problem, which will be the main focus of this article, is that of -Stabilization, a form of -Stabilization where the real components are located on the left-half of the complex plane.
The System
editFor this particular problem, suppose that we were given a linear system in the form of:
where and . Then the LMI for determining the -stabilization case would be obtained as described below.
The Data
editIn order to obtain the LMI, we need the following 2 matrices: .
The Optimization Problem
editSuppose - for the linear system given above - we were asked to design a state-feedback control law where such that the closed-loop system:
is stable, then the system would be stabilized as follows.
The LMI: -Stabilization
editFrom the given pieces of information, it is clear that the optimization problem only has a solution if there exists a matrix and a symmetric matrix that satisfy the following:
Conclusion:
editGiven the resulting controller matrix , it can be observed that the matrix is -stable.
Implementation
edit- Example Code - A GitHub link that contains code (titled "HStability.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.
Related LMIs
edit- D stabilization - Equivalent LMI for -stabilization.
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - A book co-authored by Guang-Ren Duan and Hai-Hua Yu.
H-2 Norm of the System
edit-norm of System
The -norm is conceptually identical to the Frobenius (aka Euclidean) norm on a matrix. It can be used to determine whether the system representation can be reduced to its simplest form, thereby allowing its use in performing effective block-diagram algebra.
The System
editSuppose we define the state-space system if:
where , , , and for any . Then the -norm of the system can be determined as described below.
The Data
editIn order to determine the -norm of the system, we need the matrices , , and .
The Optimization Problem
editSuppose we wanted to to infer properties of the system behaviour (which is represented in the form (A,B,C,D)). Then it becomes necessary to ensure that the overall system forms an algebra, as the standard use of block-diagram algebra would otherwise be invalid. The only way this is possible is by calculating and/or -norms - both of which are signal norms that (in a certain sense) measure the size of the transfer function.
The LMI: The Norm
editAssuming that , this means that the following are equivalent:
Conclusion:
editThe LMI can be used to minimize the -norm of the system. It is worth noting that a finite -norm does not guarantee finite -norm, and that in order for the block diagram algebra to be valid, -norm must be finite.
Implementation
edit- Example Code - A GitHub link that contains code (titled "H2Norm.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.
Related LMIs
edit- -Filtering - LMI for -Filtering
- Discrete-Time Norm - LMI for -norm in the Discrete-Time case.
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Algebraic Riccati Equation
editAlgebraic Riccati Equations are particularly significant in Optimal Control, filtering and estimation problems. The need to solve such equations is common in the analysis and linear quadratic Gaussian control along with general Control problems. In one form or the other, Riccati Equations play significant roles in optimal control of multivariable and large-scale systems, scattering theory, estimation, and detection processes. In addition, closed forms solution of Riccti Equations are intractable for two reasons namely; one, they are nonlinear and two, are in matrix forms.
The System
editThe Data
editFollowing matrices are needed as Inputs:.
- .
The Optimization Problem
editIn control systems theory, many analysis and design problems are closely related to Riccati algebraic equations or inequalities. Find
The LMI: Algebraic Riccati Inequality
editTitle and mathematical description of the LMI formulation.
Conclusion:
editIf the solution exists, LMIs give a unique, stabilizing, symmetric matrix P.
Implementation:
editMatlab code for this LMI in the Github repository:
- REDIRECT [[13]]- CODE
External links
edit- [14]-Optimal Solution to Matrix Riccati Equation
- https://https://arxiv.org/abs/1903.08599/ LMI Properties and Applications in Systems, Stability, and Control Theory.- - A List of LMIs by Ryan Caverly and James Forbes
System Zeros without feedthrough
editLet's say we have a transfer function defined as a ratio of two polynomials: Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting and solving for s.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Similarly, the system zeros are either real or appear in complex conjugate pairs. In the case of system zeros without feedthrough, we take the assumption that .
The System
editConsider a continuous-time LTI system, , with minimal statespace representation
The Data
editThe matrices:
The LMI: System Zeros without feedthrough
editThe transmission zeros of are the eigenvalues of , where . Therefore , is a minimum phase if and only if there exists , where such that
Conclusion:
editIf P exists, it ensures non-minimum phase. Eigenvalues of NAM then gives the zeros of the system.
Implementation
edithttps://github.com/Ricky-10/coding107/blob/master/Systemzeroswithoutfeedthrough
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- Control_Systems/Poles_and_Zeros
System zeros with feedthrough
editLet's say we have a transfer function defined as a ratio of two polynomials: Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting and solving for s.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Similarly, the system zeros are either real or appear in complex conjugate pairs. In the case of system zeros with feedthrough, we take as full rank.
The System
editConsider a continuous-time LTI system, , with minimal statespace representation
The Data
editThe matrices needed as inputs are:
In this case,
The LMI: System Zeros with feedthrough
editThe transmission zeros of are the eigenvalues of . Therefore , is a minimum phase if and only if there exists , where such that
Conclusion:
editIf P exists, it ensures non-minimum phase. Eigenvalues of then gives the zeros of the system.
Related LMIs
editLMIs_in_Controls/pages/systemzeroswithoutfeedthrough
Implementation
edithttps://github.com/Ricky-10/coding107/blob/master/systemzeroswithfeedthrough
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- Control_Systems/Poles_and_Zeros
Negative Imaginary Lemma
editLMIs in Control/Print version Positive real systems are often related to systems involving energy dissipation. But the standard Positive real theory will not be helpful in establishing closed-loop stability. However transfer functions of systems with a degree more than one can be satisfied with the negative imaginary conditions for all frequency values and such systems are called "systems with negative imaginary frequency response" or "negative imaginary systems".
The System
editConsider a square continuous time Linear Time invariant system, with the state space realization
The Data
edit
The LMI: LMI for Negative Imaginary Lemma
editAccording to the Lemma, The aforementioned system is negative imaginary under either of the following equivalent necessary and sufficient conditions
- There exists a ∈ n,where , such that,
- There exists a ∈ n,where , such that,
Conclusion
editThe system is strictly negative if det() 0 and either of the above LMIs are feasible with resulting or
Implementation
editA link to the Matlab code for a simple implementation of this problem in the Github repository:
https://github.com/yashgvd/ygovada
Related LMIs
editPositive Real Lemma
External Links
edit- [15] - LMI in Control Systems Analysis, Design and Applications
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- Xiong, Junlin & Petersen, Ian & Lanzon, Alexander. (2010). A Negative Imaginary Lemma and the Stability of Interconnections of Linear Negative Imaginary Systems.
Small Gain Theorem
editLMIs in Control/Matrix and LMI Properties and Tools/Small Gain Theorem
The Small Gain Theorem provides a sufficient condition for the stability of a feedback connection.
Theorem
editSuppose is a Banach Algebra and . If , then exists, and furthermore,
Proof
editAssuming we have an interconnected system :
and,
The above equations can be represented in matrix form as
Making the subject, we then have:
If is well-behaved, then the interconnection is stable.
For to be well-behaved, must be finite.
Hence, we have
and for the higher exponents of to converge to
Conclusion
editIf , then this implies stability, since the higher exponents of in the summation of will converge to , instead of blowing up to infinity.
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Tangential Nevanlinna-Pick Interpolation
editTangential Nevanlinna-Pick
editThe Tangential Nevanlinna-Pick arises in multi-input, multi-output (MIMO) control theory, particularly robust and optimal control.
The problem is to try and find a function which is analytic in and satisfies
with
The System
editis a set of matrix valued Nevanlinna functions. The matrix valued function H({\lambda}) analytic on the open upper half plane is a Nevanlinna function if .
The Data
editGiven:
Initial sequence of data points on real axis with ,
And two sequences of row vectors containing distinct target points with , and with .
The LMI: Tangential Nevanlinna- Pick
editProblem (1) has a solution if and only if the following is true:
Nevanlinna-Pick Approach
edit
Lyapunov Approach
editN can also be found using the Lyapunov equation:
where
The tangential Nevanlinna-Pick problem is then solved by confirming that .
Conclusion:
editIf is positive (semi)-definite, then there exists a norm-bounded analytic function, which satisfies with
Implementation
editImplementation requires YALMIP and a linear solver such as sedumi. [16] - MATLAB code for Tangential Nevanlinna-Pick Problem.
Related LMIs
editNevalinna-Pick Interpolation with Scaling
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- Generalized Interpolation in by Donald Sarason.
- Tangential Nevanlinna-Pick Interpolation Problem With Boundary Nodes in the Nevanlinna Class And The Related Moment Problem by Yong Jian Hu and Xiu Ping Zhang.
Nevanlinna-Pick Interpolation with Scaling
editNevanlinna-Pick Interpolation with Scaling
editThe Nevanlinna-Pick problem arises in multi-input, multi-output (MIMO) control theory, particularly robust and optimal controller synthesis with structured perturbations.
The problem is to try and find such that is analytic in and define the scaling, and finally,
The System
editThe scaling factor is given as a set of block-diagonal matrices with specified block structure. The matrix valued function H({\lambda}) analytic on the open upper half plane is a Nevanlinna function if . The Nevanlinna LMI matrix is defined as . The matrix is a diagonal matrix of the given sequence of data points
The Data
editGiven:
Initial sequence of data points in the complex plane with .
Two sequences of row vectors containing distinct target points with , and with .
The LMI: Nevanlinna- Pick Interpolation with Scaling
editFirst, implement a change of variables for and .
From this substitution it can be concluded that is the smallest positive such that there exists a such that the following is true:
,
,
Conclusion:
editIf the LMI constraints are met, then there exists a norm-bounded optimal gain which satisfies the scaled Nevanlinna-Pick interpolation objective defined above in Problem (1).
Implementation
editImplementation requires YALMIP and Mosek. [17] - MATLAB code for Nevanlinna-Pick Interpolation.
Related LMIs
editExternal Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- Generalized Interpolation in .
Generalized Norm
editGeneralized Norm
editThe norm characterizes the average frequency response of a system. To find the H2 norm, the system must be strictly proper, meaning the state space represented matrix must equal zero. The H2 norm is frequently used in optimal control to design a stabilizing controller which minimizes the average value of the transfer function, as much as possible. This optimal control problem is also called the Linear Quadratic Gaussian.
The System
editConsider a continuous-time, linear, time-invariant system with state space realization where , , , amd is Hurwitz. The generalized norm of is:
The Data
editThe transfer function , and system matrices , , are known and is Hurwitz.
The LMI: Generalized Norm LMIs
editThe inequality holds under the following conditions:
1. There exists and where such that:
- .
- .
2. There exists and where such that:
- .
- .
3. There exists and where such that:
- .
- .
Conclusion:
editThe generalized norm of is the minimum value of that satisfies the LMIs presented in this page.
Implementation
editThis implementation requires Yalmip and Sedumi.
Generalized Norm - MATLAB code for Generalized Norm.
Related LMIs
editExternal Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.
Passivity and Positive Realness
editThis section deals with passivity of a system.
The System
editGiven a state-space representation of a linear system
are the state, output and input vectors respectively.
The Data
editare system matrices.
Definition
editThe linear system with the same number of input and output variables is called passive if
-
()
hold for any arbitrary , arbitrary input , and the corresponding solution of the system with . In addition, the transfer function matrix
-
()
of system is called is positive real if it is square and satisfies
-
()
LMI Condition
editLet the linear system be controllable. Then, the system is passive if an only if there exists such that
-
()
Implementation
editThis implementation requires Yalmip and Mosek.
Conclusion
editThus, it is seen that passivity and positive-realness describe the same property of a linear system, one gives the time-domain feature and the other provides frequency-domain feature of this property.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013
Non-expansivity and Bounded Realness
editThis section studies the non-expansivity and bounded-realness of a system.
The System
editGiven a state-space representation of a linear system
are the state, output and input vectors respectively.
The Data
editare system matrices.
Definition
editThe linear system with the same number of input and output variables is called non-expansive if
-
()
hold for any arbitrary , arbitrary input , and the corresponding solution of the system with . In addition, the transfer function matrix
-
()
of system is called is positive real if it is square and satisfies
-
()
LMI Condition
editLet the linear system be controllable. Then, the system is bounded-real if an only if there exists such that
-
()
and
-
()
Implementation
editThis implementation requires Yalmip and Mosek.
Conclusion:
editThus, it is seen that passivity and positive-realness describe the same property of a linear system, one gives the time-domain feature and the other provides frequency-domain feature of this property.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013
Change of Subject
editLMIs in Control/Matrix and LMI Properties and Tools/Change of Subject
A Bilinear Matrix Inequality (BMI) can sometimes be converted into a Linear Matrix Inequality (LMI) using a change of variables. This is a basic mathematical technique of changing the position of variables with respect to equal signs and the inequality operators. The change of subject will be demonstrated by the example below.
Example
editConsider , and , where .
The matrix inequality given by:
is bilinear in the variables and .
Defining a change of variable as to obtain
,
which is an LMI in the variables and .
Once this LMI is solved, the original variable can be recovered by .
Conclusion
editIt is important that a change of variables is chosen to be a one-to-one mapping in order for the new matrix inequality to be equivalent to the original matrix inequality. The change of variable from the above example is a one-to-one mapping since is invertible, which gives a unique solution for the reverse change of variable .
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.
Congruence Transformation
editLMIs in Control/Matrix and LMI Properties and Tools/Congruence Transformation
This methods uses change of variable and some matrix properties to transform Bilinear Matrix Inequalities to Linear Matrix Inequalities. This method preserves the definiteness of the matrices that undergo the transformation.
Theorem
editConsider , where . The matrix inequality is satisfied if and only if or equivalently, .
Example
editConsider and , where and . The matrix inequality given by
is linear in variable and bilinear in the variable pair . Choose the matrix to obtain the equivalent BMI given by
Using a change of variable and , the above equation becomes
which is an LMI of variables and . The original variable is recovered by doing a reverse change of variable .
Conclusion
editA congruence transformation preserves the definiteness of a matrix by ensuring that and are equivalent. A congruence transformation is related, but not equivalent to a similarity transformation , which preserves not only the definiteness, but also the eigenvalues of a matrix. A congruence transformation is equivalent to a similarity transformation in the special case when .
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.
Finsler's Lemma
editLMIs in Control/Matrix and LMI Properties and Tools/Finsler's Lemma
This method It states equivalent ways to express the positive definiteness of a quadratic form Q constrained by a linear form L. It is equivalent to other lemmas used in optimization and control theory, such as Yakubovich's S-lemma, Finsler's lemma and it is wedely used in Linear Matrix Inequalities
Theorem
editConsider and . There exists such that
if and only if there exists such that
Alternative Forms of Finsler's Lemma
editConsider and . If there exists such that
holds for all ≠ satisfying , then there exists such that
Modified Finsler's Lemma
editConsider and , where is less that on equal to , and . There exists such that
there exists such that
Conclusion
editIn summary, a number of identical methods have been stated above to determine the positive definiteness of LMIs.
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.
D-Stability
editTime-Delay Systems
editParametric, Norm-Bounded Uncertain System Quadratic Stability
editLMIs in Control/Print version
Given a system with matrices A,M,N,Q the quadratic stability of the system with parametric, norm-bounded uncertainty can be determined by the following LMI. The feasibility of the LMI tells if the system is quadratically stable or not.
The System
editThe Data
editThe matrices .
The LMI:
editConclusion:
editThe system above is quadratically stable if and only if there exists some mu >= 0 and P > 0 such that the LMI is feasible.
Implementation
edithttps://github.com/mcavorsi/LMI
Related LMIs
editStability of Structured, Norm-Bounded Uncertainty
Stability under Arbitrary Switching
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
Stability of Structured, Norm-Bounded Uncertainty
editLMIs in Control/Print version
Given a system with matrices A,M,N,Q with structured, norm-bounded uncertainty, the stability of the system can be found using the following LMI. The LMI takes variables P and and checks for a feasible solution.
The System
editThe Data
editThe matrices .
The LMI:
editConclusion:
edit
Implementation
edithttps://github.com/mcavorsi/LMI
Related LMIs
editParametric, Norm-Bounded Uncertain System Quadratic Stability
Stability under Arbitrary Switching
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
Stability under Arbitrary Switching
editLMIs in Control/Stability Analysis/Continuous Time/Stability under Arbitrary Switching
Using the LMI below, find a P matrix that fits the constraints. If there exists one, then the system can switch between subsystems and arbitrarily and remain stable.
The System
editThe Data
editThe matrices .
The LMI
editConclusion
editThe switched system is stable under arbitrary switching if there exists some P > 0 such that the LMIs hold.
Implementation
edithttps://github.com/mcavorsi/LMI
Related LMIs
editParametric, Norm-Bounded Uncertain System Quadratic Stability
Stability of Structured, Norm-Bounded Uncertainty
External links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
Quadratic Stability Margins
editLMIs in Control/Print version
The System
editThe Data
editThe matrices .
The Optimization Problem
edit
The LMI:
editConclusion:
editIf there exists an then the system is quadratically stable, and the stability margin is the largest such .
Implementation
edithttps://github.com/mcavorsi/LMI
Related LMIs
editParametric, Norm-Bounded Uncertain System Quadratic Stability
Stability of Structured, Norm-Bounded Uncertainty
Stability under Arbitrary Switching
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Stability of Linear Delayed Differential Equations
editThe System
editwhere and .
The Data
editThe matrices .
The LMI:
editSolve the following LMIP
Implementation
edithttps://github.com/mkhajenejad/Mohammad-Khajenejad/commit/50fc71737b69f2cf57d15634f2f19d091bf37d02
Conclusion
editThe stability of the above linear delayed differential equation is proved, using Lyapunov functionals of the form , if the provided LMIP is feasible.
Remark
editThe techniques for proving stability of norm-bound LDIs [Boyd, ch.5] can also be used.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
H infinity Norm for Affine Parametric Varying Systems
editThe System
editwhere and depend affinity on parameter .
The Data
editThe matrices .
The Optimization Problem:
editSolve the following semi-definite program
Implementation
edithttps://github.com/mkhajenejad/Mohammad-Khajenejad/commit/5462bc1dc441bc298d50a2c35075e9466eba8355
Conclusion
editThe value function of the above semi-definite program returns the norm of the system.
Remark
editIt is assumed that is stable and is controllable and the semi-infinite convex constraint for all , is converted into a finite-dimensional convex LMI.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Entropy Bond for Affine Parametric Varying Systems
editThe System
editwhere and depend affinely on parameter .
The Data
editThe matrices .
The Optimization Problem:
editSolve the following semi-definite program
Implementation
edithttps://github.com/mkhajenejad/Mohammad-Khajenejad/commit/02f31a2d7a22b2464dfe9212eb76409bda9439b1
Conclusion
editThe value function of the above semi-definite program returns a bound for -entropy of the system, which is defined as
Remark
editWhen it is finite, is given by where , is asymmetric matrix with the smallest possible maximum singular value among all solutions of a Riccati equation.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Dissipativity of Affine Parametric Varying Systems
editThe System
editwhere and depend affinely on parameter .
The Data
editThe matrices .
The Optimization Problem:
editSolve the following semi-definite program
Implementation
edithttps://github.com/mkhajenejad/Mohammad-Khajenejad/commit/b6cd6b81f75be4a2052ba3fa76cad1a2f9c49caa
Conclusion
editThe dissipativity of (see [Boyd,eq:6.59]) exceeds if and only if the above LMI holds and the value function returns the minimum provable dissipativity.
Remark
editIt is worth noticing that passivity corresponds to zero dissipativity.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Hankel Norm of Affine Parameter Varying Systems
editThe System
editwhere and depend affinely on parameter .
The Data
editThe matrices .
The Optimization Problem:
editSolve the following semi-definite program
where is the controllability Gramian, i.e., .
Implementation
edithttps://github.com/mkhajenejad/Mohammad-Khajenejad/commit/0faedcdd9fba92bc27a318d80159c04a0b342f35
Conclusion
editThe Hanakel norm (i.e., the square root of the maximum eigenvalue) of is less than if and only if the above LMI holds and the value function returns the maximum provable Hankel norm.
Remark
editis assumed to be zero.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Positive Orthant Stabilizability
editPositive Orthant Stabilizability
The positive orthant stability of a linear system refers to the property of the system states being real and positive for all and decaying down to zero over time. In this section, the feasibility problem for systems to be positive orthant stable, and the stabilizability conditions to make the system positive orthant stable will be covered.
The System
editConsider a linear state-space representation of a system as:
where and are the system state and the input vector respectively. A and B are system coefficient matrices of appropriate dimensions.
The Data
editNumber of states n and number of control inputs r need to be known. Moreover, the system matrices A,B are also required to be known.
The Feasibility LMI
editAn LTI system is positive orthant stable if implies that . Moreover, as , . This is possible if and only if the following conditions hold:
The above LMI feasibility is the positive orthant stability criteria. To convert it into a positive orthant stabilizability check, the problem can be modified so that we check if is positive orthant stable. As is also a design variable here, the second inequality in the above LMI will result in bilinearity. A simple change of variables can overcome that to result in the following LMI feasibility problem for checking positive orthant stabilizability of the LTI system:
If the above LMI is feasible, the LTI system is stabilizable with controller .
Conclusion:
editThe feasibility of the above LMIs guarantees that the system is positive orthant stable if the first LMI is feasible or stabilizable with a controller if the second LMI holds.
Implementation
editTo solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:
https://github.com/smhassaan/LMI-Examples/blob/master/Positive_Orthant_LMI.m
External Links
editA list of references documenting and validating the LMI.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
LMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control
editLMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control
The LMI in this page gives the feasibility conditions which, if satisfied, imply that the correstponding system can be stabilized.
The System
editwhere is the state, is the control input.
For the system given as above, its symmetrical saturated control form can be derived by following the procedure in the original article. The new system will have the form:
where
The Data
editThe system matrices , the saturation bounds of the control inputs. Positive scalars .
The LMI: The Stabilization Feasibility Condition
editHere is a diagonal matrix with a component either 0 or 1, and and
Conclusion:
editThe feasibility of the given LMI implies that the system is stabilizable with control gains .
Implementation
editA link to CodeOcean or other online implementation of the LMI
Related LMIs
editExternal Links
edit- Stabilization of Systems with Unsymmetrical Saturated Control: An LMI Approach Link to the original article.
LMI Condition For Exponential Stability of Linear Systems With Interval Time-Varying Delays
editLMI Condition For Exponential Stability of Linear Systems With Interval Time-Varying Delays
For systems experiencing time-varying delays where the delays are bounded, the feasibility LMI in this section can be used to determine if the system is -exponentially stable.
The System
editwhere is the state, are the matrices of delay dynamics, and is the initial function with norm and it is continuously differentiable function on . The tyime-varying delay function satisfies:
The Data
editThe matrices are known, as well as the bounds of the time-varying delay.
The Optimization Problem
editFor a given , the zero solution of the system described above is -exponentially stable if there exists a positive number such that every solution satisfies the following condition:
The LMI: -Stability Condition
editThe following feasibility LMI can be used to check if the system is -exponentially stable or not for a given :
The above LMI can be combined with the bisection method to find .
Conclusion:
editFor systems with time-varying delays with intervals, the LMI in this section can be used to check if the system is exponentially stable with a certain . The bisection algorithm can be additionally used to compute .
Implementation
editTo solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:
https://github.com/smhassaan/LMI-Examples/blob/master/Intervaled_Delay_Sys_Stability_example.m
External Links
editA list of references documenting and validating the LMI.
- LMI approach to exponential stability of linear systems with interval time-varying delays - Original Article by Phat et al.
Return to Main Page:
editConic Sector Lemma
editConic Sector Lemma
For general input-output systems, sector conditions are formulated to verify or enforce the feedback stability. One of these sector conditions is the conic sector lemma, and the problem that designs the feedback controller is the conic sector theorem.
The System
editConsider a square, contiuous-time linear time-invariant (LTI) system, , with minimal state-space relization , where and . The state-space representation is:
where , and are the system state, output, and the input vector respectively.
The Data
editThe system coefficient matrices are required. Optionally, the parameters to define a cone, either in the form of where or a radius and ceter .
The Feasibility LMI
editThe system is inside the given cone if the following is feasible:
The above LMI can be used to also determine the cone parameters by setting as a variable along with the condition , and use the bisection method to find .
If the given cone is represented by a center and radius , then the following feasibility problem can be evaluated to check if is inside the given cone:
In order to also find the cone parameters, substituting as a decision variable with additional constraint and then solving for via the bisection method will give the cone in which the system resides if the problem is feasible.
Conclusion:
editThe aforementioned LMIs can be utilized to either check if is in the specified cone or not, or can be used to check the stability of by finding if a feasible cone can be obtained that encloses . An important point to note here is that the Conic Sector Lemma is a special case of the KYP Lemma for QSR dissipative systems with:
.
Implementation
editTo solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:
https://github.com/smhassaan/LMI-Examples/blob/master/Conic_sector_example.m
Related LMIs
editExternal Links
editA list of references documenting and validating the LMI.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
Return to Main Page:
editPolytopic Quadratic Stability
editAn important result to determine the stability of the system with uncertainties
The System:
editConsider the system with Affine Time-Varying uncertainty (No input)
where
where
lies in either the intervals
or the simplex
where and
The Data
editThe matrix A and are known
The Optimization
editThe Definitions: Quadratic Stability for Dynamic Uncertainty
The system
is Quadraticallly Stable over if there exists a P > 0
Theorem
is quadratically stable over if and only if
there exists a P > 0 such that
The theorem says the LMI only needs to hold at the EXTREMAL POINTS or VERTICES of the polytope.
- Quadratic Stability MUST be expressed as an LMI
The LMI
editConclusion:
editQuadratic Stability Implies Stability of trajectories for any with for all
Quadratic Stability is CONSERVATIVE.
There are Stable System which are not Quadratically stable.
Quadratic Stability is sometimes referred to as an "infinite-dimensional LMI"
- Meaning it represents an infinite number of LMI constraints.
- One for each possible value with
- Also called a parameterized LMI
- Such LMIs are not tractable.
- For polytopic sets, the LMI can be made finite.
Implementation
editA link to implementation of the LMI
https://github.com/JalpeshBhadra/LMI/blob/master/polytopicstability.m
Related LMIs
editExternal Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editMu Analysis
editMu Synthesis. The technique of synthesis extends the methods of synthesis to design a robust controller for an uncertain plant. You can perform synthesis on plants with parameter uncertainty, dynamic uncertainty, or both using the "musyn" command in MATLAB. analysis is an extremely powerful multivariable technique which has been applied to many problems in the almost every industry including Aerospace, process industry etc.
The System:
editConsider the continuous-time generalized LTI plant with minimal states-space realization
where it is assumed that is Invertible.
The Data
editThe matrices needed as inputs are only, and .
The LMI: - Analysis
editThe inequality holds if and only if there exist and , where , satisfying:
Conclusion:
editThe inequality holds for where X satisfies the above Inequality.
Implementation
editExternal links
edit- [18]-MATLAB -synthesis Implementation
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
Optimization Over Affine Family of Linear Systems
editOptimization over an Affine Family of Linear Systems
editPresented in this page is a general framework for optimizing various convex functionals for a system which depends affinely, or linearly, on a parameter using linear matrix inequalities. The optimization problem presented on this page generalizes an LMI which can be applied to various problems within linear systems and control. Some examples of these applications are finding the and norms, entropy, dissipativity, and the Hankel norm of an affinely parametric system.
The System
editConsider a family of linear systems
with state space realization where and depend affinely on the parameter .
We assume is stable and is controllable.
The transfer function, depends affinely on .
The Data
editThe transfer function , and system matrices , , , are known. represents the convex functionals, and represent some auxiliary variables dependent on the problem being solved.
The LMI:Generalized Optimization for Affine Linear Systems
editSeveral control theory problems, mentioned earlier, take the following form:
minimize
subject to
Problems of this nature can be formulated as an LMI by representing as an LMI in and possibly such that
Thus, the general optimization problem to be applied to an affine family of linear systems is as follows:
minimize
subject to
Conclusion:
editThe LMI for this generalized optimization problem may be extended to various convex functionals for affine parametric systems. For extensions of this LMI, see the related LMIs section.
Implementation
editImplementation of LMI's of this form require Yalmip and a linear solver such as Sedumi or SDPT3.
Norm for Affine Parametric Systems - MATLAB code for an extension of this generalized LMI.
Entropy Bond for Affine Parametric Systems - MATLAB code for an extension of this generalized LMI.
LMI can be applied to other extensions in stability and controller analysis. Please see the related LMI pages in the section below.
Related LMIs
editNorm for Affine Parametric Systems
Entropy Bond for Affine Parametric Systems
Dissipativity for Affine Parametric Systems
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.
Return to Main Page:
editLMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control
Hurwitz Stabilizability
editThis section studies the stabilizability properties of the control systems.
The System
editGiven a state-space representation of a linear system
Where represents the differential operator ( when the system is continuous-time) or one-step forward shift operator ( Discrete-Time system). are the state, output and input vectors respectively.
The Data
editare system matrices.
Definition
editThe system , or the matrix pair is Hurwitz Stabilizable if there exists a real matrix such that is Hurwitz Stable. The condition for Hurwitz Stabilizability of a given matrix pair (A,B) is given by the PBH criterion:
-
()
The PBH criterion shows that the system is Hurwitz stabilizable if all uncontrollable modes are Hurwitz stable.
LMI Condition
editThe system, or matrix pair is Hurwitz stabilizable if and only if there exists symmetric positive definite matrix and such that:
-
()
Following definition of Hurwitz Stabilizability and Lyapunov Stability theory, the PBH criterion is true if and only if , a matrix and a matrix satisfying:
-
()
Letting
-
()
Putting (4) in (3) gives us (2).
Implementation
editThis implementation requires Yalmip and Mosek.
Conclusion
editCompared with the second rank condition, LMI has a computational advantage while also maintaining numerical reliability.
References
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013
Return to Main Page:
editQuadratic Hurwitz Stabilization for Polytopic Systems
editThis section studies the Quadratic Hurwitz stabilization for polytopic systems.
The System
editGiven a state-space representation of a linear system
-
()
LMI Condition
editWith , the quadratic Hurwitz Stabilization problem has a solution if and only if there exists a symmetric positive definite matrix and a matrix satisfying the below LMI :
-
()
In this case, a solution to the problem is given by
-
()
Conclusion
editStability of a system does not guarantee quadratic stability. Since quadratic stability can represent infinite LMI constraints, it is not tractable. Therefore, to make it feasible and tractable, polytopic sets are helpful.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013
Discrete-Time Lyapunov Stability
editDiscrete-Time Lyapunov Stability
A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.
Stability of DT LTI systems can be determined by solving Lyapunov Inequality.
The System
editDiscrete-Time System
The Data
editThe matrices: System .
The Optimization Problem
editThe following feasibility problem should be optimized:
Find P obeying the LMI constraints.
The LMI:
editDiscrete-Time Bounded Real Lemma
The LMI formulation
Conclusion:
editIf there exists a satisfying the LMI then, and the equilibrium point of the system is Lyapunov stable.
Implementation
editA link to CodeOcean or other online implementation of the LMI
MATLAB Code
Related LMIs
editContinuous_Time_Lyapunov_Inequality - Lyapunov_Inequality
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editLMI for Schur Stabilization
editLMI for Schur Stabilization
Similar to the stability of continuous-time systems, one can analyze the stability of discrete-time systems. A discrete-time system is said to be stable if all roots of its characteristic equation lie in the open unit disk. This provides a condition for the stability of discrete-time linear systems and a linear time-invariant system with this property is called a Schur stable system.
The System
editWe consider the following system:
where the matrices , , , and are the state matrix, input matrix, state vector, and the input vector, respectively.
Moreover, represents time in the discrete-time system and is the next time step.
The state feedback control law is defined as follows:
where is the controller gain. Thus, the closed-loop system is given by:
The Data
editThe matrices and are given.
We define the scalar as with the range of .
The Optimization Problem
editThe optimization problem is to find a matrix such that:
According to the definition of the spectral norms of matrices, this condition becomes equivalent to:
Using the Lemma 1.2 in LMI in Control Systems Analysis, Design and Applications (page 14), the aforementioned inequality can be converted into:
The LMI: LMI for Schur stabilization
editThe LMI for Schur stabilization can be written as minimization of the scalar, , in the following constraints:
Conclusion:
editAfter solving the LMI problem, we obtain the controller gain and the minimized parameter . This problem is a special case of Intensive Disk Region Design (page 230 in [1]). This problem may not have a solution even when the system is stabilizable. In other words, once there exists a solution, the solution is robust in the sense that when there are parameter perturbations, the closed-loop system's eigenvalues are not easy to go outside of a circle region within the unit circle [1].
Implementation
editA link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Schur-Stability
Related LMIs
editExternal Links
edit- [19] - LMI in Control Systems Analysis, Design and Applications
Return to Main Page
editL2-Gain of Systems with Multiplicative noise
editThe System
editwhere , are independent, identically distributed random variables with and is independent of the process .
The Data
editThe matrices .
The LMI:
editImplementation
edithttps://github.com/mkhajenejad/Mohammad-Khajenejad/commit/a34713575cd8ae9831cb5b7eb759d0fd45a8c37f
Conclusion
editThe optimal returns an upper bound on the gain of the system. .
Remark
editIt is straightforward to apply scaling method [Boyd, sec.6.3.4] to obtain component-wise results.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editDiscrete-Time Quadratic Stability
editDiscrete-Time Quadratic Stability
editStability is an important property, stability analysis is necessary for control theory. For robust control, this criterion is applicable for the uncertain discrete-time linear system. It is based on the Discrete Time Lyapunov Stability.
The System
edit
The Data
editThe matrices .
The Optimization Problem
editThe following feasibility problem should be solved:
Where .
In case of polytopic uncertainty:
Conclusion:
editThis LMI allows us to investigate stability for the robust control problem in the case of polytopic uncertainty and gives on the controller for this case
Implementation:
edit- [20] - Matlab implementation using the YALMIP framework and Mosek solver
Related LMIs: =
edit- - Discrete Time Stabilizability
- Polytopic stability for continuous time case
- Quadratic polytopic stabilization
- Discrete Time Lyapunov Stability
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes. (3.20.2 page 64)
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editStability of Lure's Systems
editThe System
editThe Data
editThe matrices .
The LMI: The Lure's System's Stability
editThe following feasibility problem should be solved as sufficient condition for the stability of the above Lur'e system.
Implementation
edithttps://codeocean.com/capsule/0232754/tree
Conclusion
editIf the feasibility problem with LMI constraints has solution, then the Lure's system is stable.
Remark
editThe LMI is only a sufficient condition for the existence of a Lur’e Lyapunov function that proves stability of Lur'e system . It is also necessary when there is only one nonlinearity, i.e., when .
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editL2 Gain of Lure's Systems
editThe System
editThe Data
editThe matrices .
The Optimization Problem:
editThe following semi-definite problem should be solved.
Implementation
edithttps://github.com/mkhajenejad/Mohammad-Khajenejad/commit/12a7039f9e3d966e24b43fd58a3cce3725282ed2
Conclusion
editThe value function returns the square of the smallest provable upper bound on the gain of the Lure's system.
Remark
editThe Lyapunov function which is used to proof is similar to the one for the systems with unknown parameters.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editOutput Energy Bound for Lure's Systems
editThe System
editThe Data
editThe matrices .
The Optimization Problem:
editThe following optimization problem should be to find the tightest upper bound for the output energy of the above Lur'e system.
Implementation
editConclusion
editThe value function returns the the lowest bound for the energy function of the Lure's systems, i.e., with initial conditions .
Remark
editThe key step in the proof is to satisfy , where is Lyapunov function in a special form.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editStability of Quadratic Constrained Systems
editThe System
editThe Data
editThe matrices .
The LMI:
editThe following feasibility problem should be solved.
Implementation
edithttps://github.com/mkhajenejad/Mohammad-Khajenejad/commit/38f3b55ca7060a1260384a96e9dc31142af07a9a
Conclusion
editThe integral quadratic constrained system is stable if the provided LMI is feasible
Remark
editThe key point of the proof is to satisfy whenever , using -procedure.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editConic Sector Lemma
editConic Sector Lemma
For general input-output systems, sector conditions are formulated to verify or enforce the feedback stability. One of these sector conditions is the conic sector lemma, and the problem that designs the feedback controller is the conic sector theorem.
The System
editConsider a square, contiuous-time linear time-invariant (LTI) system, , with minimal state-space relization , where and . The state-space representation is:
where , and are the system state, output, and the input vector respectively.
The Data
editThe system coefficient matrices are required. Optionally, the parameters to define a cone, either in the form of where or a radius and ceter .
The Feasibility LMI
editThe system is inside the given cone if the following is feasible:
The above LMI can be used to also determine the cone parameters by setting as a variable along with the condition , and use the bisection method to find .
If the given cone is represented by a center and radius , then the following feasibility problem can be evaluated to check if is inside the given cone:
In order to also find the cone parameters, substituting as a decision variable with additional constraint and then solving for via the bisection method will give the cone in which the system resides if the problem is feasible.
Conclusion:
editThe aforementioned LMIs can be utilized to either check if is in the specified cone or not, or can be used to check the stability of by finding if a feasible cone can be obtained that encloses . An important point to note here is that the Conic Sector Lemma is a special case of the KYP Lemma for QSR dissipative systems with:
.
Implementation
editTo solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:
https://github.com/smhassaan/LMI-Examples/blob/master/Conic_sector_example.m
Related LMIs
editExternal Links
editA list of references documenting and validating the LMI.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
Return to Main Page:
editState Feedback
edit- H-infinity
- H-2
- Mixed
- Stabilization of Second-Order Systems
- LQ Regulation via H2 Control
- Controller to achieve the desired Reachable set; Polytopic uncertainty
- Controller to achieve the desired Reachable set; Norm bound uncertainty
- Controller to achieve the desired Reachable set; Diagonal Norm-bound uncertainty
D-Stability
editOptimal State Feedback
editOutput Feedback
editStatic Output Feedback
editOptimal Output Feedback
editStabilizability LMI
editStabilizability LMI
A system is stabilizable if all unstable modes of the system are controllable. This implies that if the system is controllable, it will also be stabilizable. Thus, stabilizability is a essentially a weaker version of the controllability condition. The LMI condition for stabilizability of pair is shown below.
The System
editwhere , , at any .
The Data
editThe matrices necessary for this LMI are and . There is no restriction on the stability of A.
The LMI: Stabilizability LMI
editis stabilizable if and only if there exists such that
- ,
where the stabilizing controller is given by
- .
Conclusion:
editIf we are able to find such that the above LMI holds it means the matrix pair is stabilizable. In words, a system pair is stabilizable if for any initial state an appropriate input can be found so that the state asymptotically approaches the origin. Stabilizability is a weaker condition than controllability in that we only need to approach as whereas controllability requires that the state must reach the origin in a finite time.
Implementation
editThis implementation requires Yalmip and Sedumi.
https://github.com/eoskowro/LMI/blob/master/Stabilizability_LMI.m
Related LMIs
editExternal Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.1.1 and Table 6.1 pp. 166–170, 192.
- A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Section 2.2.3, pp. 71-73.
Return to Main Page:
editLMI for the Controllability Grammian
editLMI to Find the Controllability Gramian
Being able to adjust a system in a desired manor using feedback and sensors is a very important part of control engineering. However, not all systems are able to be adjusted. This ability to be adjusted refers to the idea of a "controllable" system and motivates the necessity of determining the "controllability" of the system. Controllability refers to the ability to accurately and precisely manipulate the state of a system using inputs. Essentially if a system is controllable then it implies that there is a control law that will transfer a given initial state and transfer it to a desired final state . There are multiple ways to determine if a system is controllable, one of which is to compute the rank "controllability Gramian". If the Gramian is full rank, the system is controllable and a state transferring control law exists.
The System
editwhere , , at any .
The Data
editThe matrices necessary for this LMI are and . must be stable for the problem to be feasible.
The LMI: LMI to Determine the Controllability Gramian
editis controllable if and only if is the unique solution to
- ,
where is the Controllability Gramian.
Conclusion:
editThe LMI above finds the controllability Gramian of the system . If the problem is feasible and a unique can be found, then we also will be able to say the system is controllable. The controllability Gramian of the system can also be computed as: , with control law that will transfer the given initial state to a desired final state .
Implementation
editThis implementation requires Yalmip and Sedumi.
https://github.com/eoskowro/LMI/blob/master/Controllability_Gram_LMI.m
Related LMIs
editExternal Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.1.1 and Table 6.1 pp. 166–170, 192.
- A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Section 2.2.3, pp. 71-73.
Return to Main Page:
editLMI for Decentralized Feedback Control
editLMI for Decentralized Feedback Control
In large-scale systems like a multi-agent robotic system, national economies, or chemical refineries, an actuator should act based on local information, which necessitates a decentralized or distributed control strategy. In a decentralized control framework, the controllers are distributed and each controller has only access to a subset of local measurements. We describe LMI formulations for a general decentralized control framework and then provide an illustrative example of a decentralized control design.
The System
editIn a decentralized controller design, the state feedback controller can be divided into sub-controllers .
The Data
editA general state space representation of a linear time-invariant system is as follows:
where is a vector of state variables, is the input matrix, is the output matrix, and is called the feedforward matrix. We assume that all the four matrices, , , , and are given.
The Optimization Problem
editWe aim to solve the -optimal full-state feedback control problem using a controller .
In a decentralized fashion, the control input can be divided into sub-controllers and can be written as .
For instance, let and . Thus, there are three control inputs , , and . We also assume that u_{1} only depends on the first and the second states, while and only depend on thrid to sixth states. For this example, the controller gain matrix can be described by:
Thus, the decentralized controller gain consists of sub-matrices of gains.
The LMI: LMI for decentralized feedback controller
editThe mathematical description of the LMI formulation for a decentralised optimal full-state feedback controller can be described by:
where is a positive definite matrix and such that the aforemtntioned constraints in LMIs are satisfied.
Conclusion:
editThe controller gain matrix is defined as:
where can be found after solving the LMIs and obtaining the variables matrices and . Thus,
.
Implementation
editA link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI_for_decentralized_feedback_controller/tree/master
Related LMIs
editExternal Links
editA list of references documenting and validating the LMI.
- [21] - LMI in Control Systems Analysis, Design and Applications
Return to Main Page
editLMI for Mixed Output Feedback Controller
editLMI for Mixed Output Feedback Controller
The mixed output feedback control has been known as an example of a multi-objective optimal control problem. In this problem, the control feedback should respond properly to several specifications. In the controller, the channel is used to improve the robustness of the design while the channel guarantees good performance of the system.
The System
editWe consider the following state-space representation for a linear system:
where , , , and are the state matrix, input matrix, output matrix, and feedforward matrix, respectively.
These are the system (plant) matrices that can be shown as .
The Data
editWe assume that all the four matrices of the plant, , are given.
The Optimization Problem
editIn this problem, we use an LMI to formulate and solve the optimal output-feedback problem to minimize both the <> and <> norms. Giving equal weights to each of the norms, we will have the optimization problem in the following form:
The LMI: LMI for mixed /
editMathematical description of the LMI formulation for a mixed / optimal output-feedback problem can be written as follows:
where and are defined as the and norm of the system:
Moreover, , , , , , and are variable matrices with appropriate dimensions that are found after solving the LMIs.
Conclusion:
editThe calculated scalars and are the and norms of the system, respectively. Thus, the norm of mixed / is defined as . The results for each individual norm and norms of the system show that a bigger value of norms are found in comparison with the case they are solved separately.
Implementation
editA link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI_for_Mixed_H2_Hinf_Output_Feedback_Controller
Related LMIs
editExternal Links
edit- [22] - LMI in Control Systems Analysis, Design and Applications
Return to Main Page
editQuadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty
editLMIs in Control/Print version
If the system is quadratically stable, then there exists some and such that the LMI is feasible. The and matrices can also be used to create a quadratically stabilizing controller.
The System
editThe Data
editThe matrices .
The LMI:
editConclusion:
editThere exists a controller for the system with where is the quadratically stabilizing controller, if the above LMI is feasible.
Implementation
edithttps://github.com/mcavorsi/LMI
Related LMIs
editH-infinity Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty
Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty
Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
Return to Main Page:
editH-inf Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty
editLMIs in Control/Print version
If there exists some , and such that the LMI holds, then the system satisfies There also exists a controller with
The System
editThe Data
editThe matrices .
The Optimization Problem
editMinimize subject to the LMI constraints below.
The LMI:
editConclusion:
editThe controller gains, K, are calculated by .
Implementation
edithttps://github.com/mcavorsi/LMI
Related LMIs
editQuadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty
Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty
Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
Return to Main Page:
editStabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty
editLMIs in Control/Print version
The System
editThe Data
editThe matrices .
The LMI:
editConclusion:
editIf the LMI is feasible, the controller, K, is calculated by .
Implementation
edithttps://github.com/mcavorsi/LMI
Related LMIs
editQuadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty
H-infinity Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty
Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
Return to Main Page:
editOptimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty
editLMIs in Control/Print version
The System
editThe Data
editThe matrices .
The Optimization Problem
editsubject to the LMI constraints.
The LMI:
editConclusion:
editThe controller is .
Implementation
edithttps://github.com/mcavorsi/LMI
Related LMIs
editQuadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty
H-infinity Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty
Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
Return to Main Page:
editOptimal Output Controllability for Systems With Transients
editOptimal Output Controllability for Systems With Transients
This LMI provides an optimal output controllability problem to check if such controllers for systems with unknown exogenous disturbances and initial conditions can exist or not.
The System
editwhere is the state, is the exogenous input, is the control input, is the measured output and is the regulated output.
The Data
editSystem matrices need to be known. It is assumed that . are matrices with their columns forming the bais of kernels of and respectively.
The Optimization Problem
editFor a given , the following condition needs to be fulfilled:
The LMI: Output Feedback Controller for Systems With Transients
editConclusion:
editSolution of the above LMI gives a check to see if an optimal output controller for systems with transients can exist or not.
Implementation
editA link to CodeOcean or other online implementation of the LMI
Related LMIs
editLinks to other closely-related LMIs
External Links
edit- LMI-based -optimal control with transients Link to the original article.
Return to Main Page:
editQuadratic Polytopic Stabilization
editA Quadratic Polytopic Stabilization Controller Synthesis can be done using this LMI, requiring the information about , , and matrices.
The System
editwhere , , at any .
The system consist of uncertainties of the following form
where ,, and
The Data
editThe matrices necessary for this LMI are , , and
The Optimization and LMI:LMI for Controller Synthesis using the theorem of Polytopic Quadratic Stability
editThere exists a K such that
is quadratically stable for if and only if there exists some P>0 and Z such that
Conclusion:
editThe Controller gain matrix is extracted as
Note that here the controller doesn't depend on
- If you want K to depend on , the problem is harder.
- But this would require sensing in real-time.
Implementation
editThis implementation requires Yalmip and Sedumi. https://github.com/JalpeshBhadra/LMI/blob/master/quadraticpolytopicstabilization.m
Related LMIs
editQuadratic Polytopic Controller
Quadratic Polytopic Controller
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Quadratic D-Stabilization
editContinuous-Time D-Stability Controller
This LMI will let you place poles at a specific location based on system performance like rising time, settling time and percent overshoot, while also ensuring the stability of the system.
The System
editSuppose we were given the continuous-time system
whose stability was not known, and where , , , and for any .
Adding uncertainty to the system
The Data
editIn order to properly define the acceptable region of the poles in the complex plane, we need the following pieces of data:
- matrices , , ,
- rise time ()
- settling time ()
- percent overshoot ()
Having these pieces of information will now help us in formulating the optimization problem.
The Optimization Problem
editUsing the data given above, we can now define our optimization problem. In order to do that, we have to first define the acceptable region in the complex plane that the poles can lie on using the following inequality constraints:
Rise Time:
Settling Time:
Percent Overshoot:
Assume that is the complex pole location, then:
This then allows us to modify our inequality constraints as:
Rise Time:
Settling Time:
Percent Overshoot:
which not only allows us to map the relationship between complex pole locations and inequality constraints but it also now allows us to easily formulate our LMIs for this problem.
The LMI: An LMI for Quadratic D-Stabilization
editSuppose there exists and such that
for
Conclusion:
editGiven the resulting controller , we can now determine that the pole locations of satisfies the inequality constraints , and for all
Implementation
editThe implementation of this LMI requires Yalmip and Sedumi https://github.com/JalpeshBhadra/LMI/blob/master/quadraticDstabilization.m
Related LMIs
edit- Continuous Time D-Stability Observer - Equivalent D-stability LMI for a continuous-time observer.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editQuadratic Polytopic Full State Feedback Optimal Control
editQuadratic Polytopic Full State Feedback Optimal Control
editFor a system having polytopic uncertainties, Full State Feedback is a control technique that attempts to place the system's closed-loop system poles in specified locations based off of performance specifications given. methods formulate this task as an optimization problem and attempt to minimize the norm of the system.
The System
editConsider System with following state-space representation.
where , , , , , , , , , , , , , for any .
Add uncertainty to system matrices
New state-space representation
The Optimization Problem:
editRecall the closed-loop in state feedback is:
This problem can be formulated as optimal state-feedback, where K is a controller gain matrix.
The LMI:
editAn LMI for Quadratic Polytopic Optimal
State-Feedback Control
Conclusion:
editThe Optimal State-Feedback Controller is recovered by
Controller will determine the bound on the norm of the system.
Implementation:
edithttps://github.com/JalpeshBhadra/LMI/tree/master
Related LMIs
editFull State Feedback Optimal Controller
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Quadratic Polytopic Full State Feedback Optimal Control
editLMIs in Control/Print version
Quadratic Polytopic Full State Feedback Optimal Control
editFor a system having polytopic uncertainties, Full State Feedback is a control technique that attempts to place the system's closed-loop system poles in specified locations based on performance specifications given, such as requiring stability or bounding the overshoot of the output. By minimizing the norm of this system we are minimizing the effect noise has on the system as part of the performance specifications.
The System
editConsider System with following state-space representation.
where , , , , , , , , , , , , , for any .
Add uncertainty to system matrices
New state-space representation
The Data
editThe matrices necessary for this LMI are
The Optimization Problem:
editRecall the closed-loop in state feedback is:
This problem can be formulated as optimal state-feedback, where K is a controller gain matrix.
The LMI: An LMI for Quadratic Polytopic Optimal
editState-Feedback Control
Conclusion:
editThe Optimal State-Feedback Controller is recovered by
Implementation:
edithttps://github.com/JalpeshBhadra/LMI/blob/master/H2_optimal_statefeedback_controller.m
Related LMIs
editOptimal State-Feedback Controller
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Continuous-Time Static Output Feedback Stabilizability
editLMIs in Control/Print version
In view of applications, static feedback of the full state is not
feasible in general: only a few of the state variables (or a linear
combination of them,
, called the output) can be
actually measured and re-injected into the system.
So, we are led to the notion of static output feedback
The System
editConsider the continuous-time LTI system, with generalized state-space realization
The Data
editThe Optimization Problem
editThis system is static output feedback
stabilizable (SOFS) if there exists a matrix
F such that the closed-loop system
(obtained by replacing which means applying static output feedback)
is asymptotically stable at the origin
The LMI: LMI for Continuous Time - Static Output Feedback Stabilizability
editThe system is static output feedback stabilizable if and only if it satisfies any of the following conditions:
- There exists a and , where , such that
- There exists a and , where , such that
- There exists a and , where , such that
- There exists a and , where , such that
Conclusion
editOn implementation and optimization of the above LMI using YALMIP and MOSEK (or) SeDuMi we get 2 output matrices one of which is the Symmeteric matrix (or ) and
Implementation
editA link to the Matlab code for a simple implementation of this problem in the Github repository:
https://github.com/yashgvd/LMI_wikibooks
Related LMIs
editDiscrete time Static Output Feedback Stabilizability
Static Feedback Stabilizability
External Links
edit- [23] - LMI in Control Systems Analysis, Design and Applications
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- D. d. S. Madeira and J. Adamy, "Static output feedback: An LMI condition for stabilizability based on passivity indices," 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, 2016, pp. 960-965.
Return to Main Page:
editMulti-Criterion LQG
editLMIs in Control/Print version
The Multi-Criterion Linear Quadratic Gaussian (LQG) linear matrix inequality will allow one to form an optimized controller, similar to that in an LQR framework, for a state space system with gaussian noise based on several different criterions defined in the Q and R matrices, that are optimized as a part of the arbitrary cost function. Just like traditional LQR, the cost matrices must be tuned in much a similar fashion as traditional gains in classical control. In the LQR and LQG framework however, the gains are more intuitive as each correlates directly to a state or an input.
The System
editThe system is a linear time-invariant system, that can be represented in state space as shown below:
where represent the state vector, the measured output vector, and the output vector of interest, respectively, is the disturbance vector, and are the system matrices of appropriate dimension. To further define: is and is the state vector, is and is the state matrix, is and is the input matrix, is and is the exogenous input, is and are the output matrices, and and are and are the output and the output of interest, respectively.
and , and the system is controllable and observable.
The Data
editThe matrices and the noise signals .
The Optimization Problem
editIn the Linear Quadratic Gaussian (LQG) control problem, the goal is to minimize a quadratic cost function while the plant has random initial conditions and suffers white noise disturbance on the input and measurement.
There are multiple outputs of interest for this problem. They are defined by
For each of these outputs of interest, we associate a cost function:
Additionally, the matrices and must be found as the solutions to the following Riccati equations:
The optimization problem is to minimize over u subject to the measurability condition and the constraints . This optimization problem can be formulated as:
over , with:
The LMI: Multi-Criterion LQG
editover , subject to the following constraints:
Conclusion:
editThe result of this LMI is the solution to the aforementioned Ricatti equations:
Implementation
editThis implementation requires Yalmip and Sedumi.
https://github.com/rezajamesahmed/LMImatlabcode/blob/master/multicriterionquadraticproblems.m
Related LMIs
editExternal Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editInverse Problem of Optimal Control
editLMIs in Control/Print version
In some cases, it is needed to solve the inverse problem of optimal control within an LQR framework. In this inverse problem, a given controller matrix needs to be verified for the system by assuring that it is the optimal solution to some LQR optimization problem that is controllable and detectable. In other words: in this inverse problem, the controller is known and the LQR gain matrices are to be calculated such that the controller is the optimal solution.
The System
editThe system is a linear time-invariant system, that can be represented in state space as shown below:
where represent the state vector, the measured output vector, and the output vector of interest, respectively, is the disturbance vector, and are the system matrices of appropriate dimension. To further define: is and is the state vector, is and is the state matrix, is and is the input matrix, is and is the exogenous input, is and are the output matrices, and and are and are the output and the output of interest, respectively.
The Data
editThe matrices that define the system, and a given controller for which the inverse problem is to be solved.
The Optimization Problem
editIn this LMI, the following cost function is to be minimized for a given controller K by finding an optimal input:
the solution of the problem can be formulated as a state feedback controller given as:
The LMI: Inverse Problem of Optimal Control
editthe inverse problem of optimal control is the following: Given a matrix , determine if there exist and , such that is detectable and is the optimal control for the corresponding LQR problem. Equivalently, we seek and such that there exist nonnegative and positive-definite satisfying
Conclusion:
editIf the solution exists, then is the optimal controller for the LQR optimization on the matrices and
Implementation
editThis implementation requires Yalmip and Sedumi.
https://github.com/rezajamesahmed/LMImatlabcode/blob/master/inverseprob.m
Related LMIs
editExternal Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editNonconvex Multi-Criterion Quadratic Problems
editLMIs in Control/Print version
The Non-Concex Multi-Criterion Quadratic linear matrix inequality will allow one to form an optimized controller, similar to that in an LQR framework, for a non-convex state space system based on several different criterions defined in the Q and R matrices, that are optimized as a part of the arbitrary cost function. Just like traditional LQR, the cost matrices must be tuned in much a similar fashion as traditional gains in classical control. In the LQR and LQG framework however, the gains are more intuitive as each correlates directly to a state or an input.
The System
editThe system for this LMI is a linear time invariant system that can be represented in state space as shown below:
where the system is assumed to be controllable.
where represents the state vector, respectively, is the disturbance vector, and are the system matrices of appropriate dimension. To further define: is and is the state vector, is and is the state matrix, is and is the input matrix, is and is the exogenous input.
for any input, we define a set cost indices by
Here the symmetric matrices,
- ,
are not necessarily positive-definite.
The Data
editThe matrices .
The Optimization Problem
editThe constrained optimal control problem is:
subject to
The LMI: Nonconvex Multi-Criterion Quadratic Problems
editThe solution to this problem proceeds as follows: We first define
where and for every , we define
then, the solution can be found by:
subject to
Conclusion:
editIf the solution exists, then is the optimal controller and can be solved for via an EVP in P.
Implementation
editThis implementation requires Yalmip and Sedumi.
https://github.com/rezajamesahmed/LMImatlabcode/blob/master/multicriterionquadraticproblems.m
Related LMIs
editExternal Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editStatic-State Feedback Problem
editLMIs in Control/Print version
We are attempting to stabilizing The Static State-Feedback Problem
The System
editConsider a continuous time Linear Time invariant system
The Data
editare known matrices
The Optimization Problem
editThe Problem's main aim is to find a feedback matrix such that the system
and
is stable
Initially we find the matrix such that is Hurwitz.
The LMI: Static State Feedback Problem
editThis problem can now be formulated into an LMI as Problem 1:
From the above equation and we have to find K
The problem as we can see is bilinear in
- The bilinear in X and K is a common paradigm
- Bilinear optimization is not Convex. To Convexify the problem, we use a change of variables.
Problem 2:
where and we find
The Problem 1 is equivalent to Problem 2
Conclusion
editIf the (A,B) are controllable, We can obtain a controller matrix that stabilizes the system.
Implementation
editA link to the Matlab code for a simple implementation of this problem in the Github repository:
https://github.com/yashgvd/ygovada
Related LMIs
editHurwitz Stability
External Links
edit- [24] - LMI in Control Systems Analysis, Design and Applications
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- [25] -Mathworks reference to DC Gain
Return to Main Page:
editMixed H2 Hinf with desired pole location control
editLMI for Mixed with desired pole location Controller
The mixed output feedback control has been known as an example of a multi-objective optimal control problem. In this problem, the control feedback should respond properly to several specifications. In the controller, the channel is used to improve the robustness of the design while the channel guarantees good performance of the system and additional constraint is used to place poles at desired location.
The System
editWe consider the following state-space representation for a linear system:
where
- , are the state vector and the output vectors, respectively
- , are the disturbance vector and the control vector
- , ,, ,,,, and are the system coefficient matrices of appropriate dimensions
The Data
editWe assume that all the four matrices of the plant,, ,, ,,,, and are given.
The Optimization Problem
editFor the system with the following feedback law:
The closed loop system can be obtained as:
the transfer function matrices are and
Thus the performance and the performance requirements for the system are, respectiverly
and
.
For the performance of the system response, we introduce the closed-loop eigenvalue
location requirement. Let
It is a region on the complex plane, which can be used to restrain the closed-loop
eigenvalue locations.
Hence a state feedback control law is designed such that,
- The performance and the performance are satisfied.
- The closed-loop eigenvalues are all located in , that is,
.
The LMI: LMI for mixed / with desired Pole locations
editThe optimization problem discussed above has a solution if there exist two symmetric matrices and a matrix , satisfying
min
s.t
where and are the weighting factors.
Conclusion:
editThe calculated scalars and are the and norms of the system, respectively. The controller is extracted as
Implementation
editA link to Matlab codes for this problem in the Github repository:
Related LMIs
editMixed H2 Hinf with desired pole location for perturbed system
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page
editMixed H2 Hinf with desired pole location control for perturbed systems
editLMI for Mixed with desired pole location Controller for perturbed system case
The mixed output feedback control has been known as an example of a multi-objective optimal control problem. In this problem, the control feedback should respond properly to several specifications. In the controller, the channel is used to improve the robustness of the design while the channel guarantees good performance of the system and additional constraint is used to place poles at desired location.
The System
editWe consider the following state-space representation for a linear system:
where
- , are the state vector and the output vectors, respectively
- , are the disturbance vector and the control vector
- , ,, ,,,, and are the system coefficient matrices of appropriate dimensions.
- and are real valued matrix functions which represent the time varying parameters uncertainities.
Furthermore, the parameter uncertainties and are in the form of
where
- , and are known matrices of appropriate dimensions.
- is a matrix containing the uncertainty, which satisfies
The Data
editWe assume that all the four matrices of the plant,, , ,, ,,,, and are given.
The Optimization Problem
editFor the system with the following feedback law:
The closed loop system can be obtained as:
the transfer function matrices are and
Thus the performance and the performance requirements for the system are, respectiverly
and
.
For the performance of the system response, we introduce the closed-loop eigenvalue
location requirement. Let
It is a region on the complex plane, which can be used to restrain the closed-loop
eigenvalue locations.
Hence a state feedback control law is designed such that,
- The performance and the performance are satisfied.
- The closed-loop eigenvalues are all located in , that is,
.
The LMI: LMI for mixed / with desired Pole locations
editThe optimization problem discussed above has a solution if there exist scalars two symmetric matrices and a matrix , satisfying
min
s.t
where
and are the weighting factors.
Conclusion:
editThe calculated scalars and are the and norms of the system, respectively. The controller is extracted as
Implementation
editA link to Matlab codes for this problem in the Github repository:
Related LMIs
editMixed H2 Hinf with desired poles controller
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page
editRobust H2 State Feedback Control
editRobust State Feedback Control
editFor the uncertain linear system given below, and a scalar . The goal is to design a state feedback control in the form of such that the closed-loop system is asymptotically stable and satisfies.
The System
editConsider System with following state-space representation.
where , , , . For state feedback control
and are real valued matrix functions that represent the time varying parameter uncertainties and of the form
where matrices and are some known matrices of appropriate dimensions, while is a matrix which contains the uncertain parameters and satisfies.
For the perturbation, we obviously have
- , for
- , for
The Problem Formulation:
editThe state feedback control problem has a solution if and only if there exist a scalar , a matrix , two symmetric matrices and satisfying the following LMI's problem.
The LMI:
edit
where is the definition that is need for the above LMI.
Conclusion:
editIn this case, an state feedback control law is given by .
External Links
edit- LMIs in Control Systems Analysis, Design and Applications - Duan and Yu
- A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
LQ Regulation via H2 control
editLQ Regulation via Control
editThe LQR design problem is to build an optimal state feedback controller for the system such that the following quadratic performance index.
is minimized, where
The following assumptions should hold for a traditional solution.
is stabilizable.
is observable, with .
Relation to performance
editFor the system given above an auxiliary system is constructed
where
Where represents an impulse disturbance. Then with state feedback controller the closed loop transfer function from disturbance to output is
Then the LQ problem and the norm of are related as
Then norm minimization leads minimization of .
Data
editThe state-representation of the system is given and matrices are chosen for the optimal LQ problem.
The Problem Formulation:
editLet assumptions and hold, then the state feedback control of the form exists such that if and only if there exist , and . Then can be obtained by the following LMI.
The LMI:
edit
Conclusion:
editIn this case, a feedback control law is given as .
External Links
edit- LMIs in Control Systems Analysis, Design and Applications - Duan and Yu
- A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
State Feedback
editOptimal State Feedback
editOutput Feedback
editStatic Output Feedback
editOptimal Output Feedback
editOptimal Dynamic Output Feedback
editDiscrete Time Stabilizability
editDiscrete-Time Stabilizability
A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.
Discrete-Time LTI systems can be made stable using controller gain K, which can be found using LMI optimization, such that the close loop system is stable.
The System
editDiscrete-Time LTI System with state space realization
The Data
editThe matrices: System .
The Optimization Problem
editThe following feasibility problem should be optimized:
Maximize P while obeying the LMI constraints.
Then K is found.
The LMI:
editDiscrete-Time Stabilizability
The LMI formulation
Conclusion:
editThe system is stabilizable iff there exits a , such that . The matrix is Schur with
Implementation
editA link to CodeOcean or other online implementation of the LMI
MATLAB Code
Related LMIs
edit[26] - Continuous Time Stabilizability
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editQuadratic Schur Stabilization
editLMI for Quadratic Schur Stabilization
A discrete-time system is said to be stable if all roots of its characteristic equation lie in the open unit disk. This provides a condition for the stability of discrete-time linear systems with polytopic uncertainties and a linear time-invariant system with this property is called a Schur stable system.
The System
editConsider discrete time system
where , , at any .
The system consist of uncertainties of the following form
where ,, and
The Data
editThe matrices necessary for this LMI are , , and
The LMI:
editThere exists some X > 0 and Z such that
The Optimization Problem
editThe optimization problem is to find a matrix such that:
According to the definition of the spectral norms of matrices, this condition becomes equivalent to:
Using the Lemma 1.2 in LMI in Control Systems Analysis, Design and Applications (page 14), the aforementioned inequality can be converted into:
Conclusion:
editThe Controller gain matrix is extracted as
It follows that the trajectories of the closed-loop system (A+BK) are stable for any
Implementation
edithttps://github.com/JalpeshBhadra/LMI/blob/master/quadratic_schur_stabilization.m
Related LMIs
editSchur Complement
Schur Stabilization
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMI in Control Systems Analysis, Design and Applications
Generic Insensitive Strip Region Design
editInsensitive Strip Region Design
Suppose if one were interested in robust stabilization where closed-loop eigenvalues are placed in particular regions of the complex plane where the said regions has an inner boundary that is insensitive to perturbations of the system parameter matrices. This would be accomplished with the help of 2 design problems: the insensitive strip region design and insensitive disk region design (see link below for the latter).
The System
editSuppose we consider the following continuous-time linear system that needs to be controlled:
where , , and are the state, output and input vectors respectively. Then the steps to obtain the LMI for insensitive strip region design would be obtained as follows.
The Data
editPrior to obtaining the LMI, we need the following matrices: , , and .
The Optimization Problem
editConsider the above linear system as well as 2 scalars and . Then the output feedback control law would be such that , where:
Letting being the solution to the above problem, then
where
The LMI: Insensitive Strip Region Design
editUsing the above info, we can simplify the problem by setting to for all practical applications. This then simplifies our problem and results in the following LMI:
Conclusion:
editIf the resulting solution from the LMI above produces a negative , then the output feedback controller is Hurwitz-stable. Hoewever, if is a really small positive number, then must be negative for the controller to be Hurwitz-stable.
Implementation
edit- Example Code - A GitHub link that contains code (titled "InsensitiveStripRegion.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.
Related LMIs
edit- Insensitive Disk Region Design - Equivalent LMI for Insensitive Disk Region Design
- Strip Region Design - LMI for Strip Region Design
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - A book co-authored by Guang-Ren Duan and Hai-Hua Yu.
Return to Main Page:
editGeneric Insensitive Disk Region Design
editInsensitive Disk Region Design
Similar to the insensitive strip region design problem, insensitive disk region design is another way with which robust stabilization can be achieved where closed-loop eigenvalues are placed in particular regions of the complex plane where the said regions has an inner boundary that is insensitive to perturbations of the system parameter matrices.
The System
editSuppose we consider the following linear system that needs to be controlled:
where , , and are the state, output and input vectors respectively, and represents the differential operator (in the continuous-time case) or one-step shift forward operator (i.e., ) (in the discrete-time case). Then the steps to obtain the LMI for insensitive strip region design would be obtained as follows.
The Data
editPrior to obtaining the LMI, we need the following matrices: , , and .
The Optimization Problem
editConsider the above linear system as well as 2 positive scalars and . Then the output feedback control law would be designed such that:
Recalling the definition, we have:
and
Letting being the solution to the above problem, then
The LMI: Insensitive Strip Region Design
editUsing the above info, we can convert the given problem into an LMI, which - after using Schur compliment Lemma - results in the following:
Conclusion:
editFor Schur stabilization, we can choose to solve the problem with . Schur stability is achieved when . Alternately, if is greater than (but very close to) 1, then Schur stability is also achieved when .
Implementation
edit- Example Code - A GitHub link that contains code (titled "InsensitiveDiskRegion.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.
Related LMIs
edit- Disk Region Design - LMI for Disk Region Design with minimal gain
- Insensitive Strip Region Design - Equivalent LMI for Insensitive Strip Region Design
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - A book co-authored by Guang-Ren Duan and Hai-Hua Yu.
Return to Main Page:
editDesign for Insensitive Strip Region
editInsensitive Strip Region Design with Minimum Gain
When designing controllers with insensitive region conditions, the aim is to place the closed-loop poles of the system in a particular region defined by its inner boundary. These regions are specified based on their insensitivity to perturbations to the system parameter matrices.
One type of such design is the Insensitive Strip Region Design. In this section, building upon that, optimization problems will be provided that ensure that the conditions for insensitive strip region design are satisfied with some bounds on the gain of the closed-loop system.
The System
editA state-space representation of a linear system as given below:
where , and are the system state, output, and the input vector respectively.
The Data
editTo solve the design optimization problem, the linear system matrices A,B,C are required. Furthermore, to define the strip region on the eigenvalue-space, two parameters and are required.
The Optimization Problem
editThe problem of designing an optimal controller that results in the closed loop system insensitive to a certain strip region involves two sub-problems:
- Finding a control gain such that: .
- The conditions for insensitive strip region design for the closed-loop system, as provided in the section Insensitive Strip Region Design are fulfilled.
- The optimization goal is to minimize such that above two hold.
The LMI: Optimal Control Design for Insensitive Strip Region
editThe problem above has a solution if and only if the following optimization problem has a solution :
Conclusion:
editBy using the design problem provided here, an optimal controller is designed to make the closed-loop system robust to perturbations in the system matrices.
Implementation
editTo solve the optimization problem with LMI presented here, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:
https://github.com/smhassaan/LMI-Examples/blob/master/H2_Strip_example.m
Related LMIs
editInsensitive Strip Region Design
External Links
editA list of references documenting and validating the LMI.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 10.1.1 pp. 322–323.
Return to Main Page:
editDesign for Insensitive Disk Region
editInsensitive Disk Region Design with Minimum Gain
Apart from the design for the insensitive strip region with minimum gain, another type of such design is the Insensitive Disk Region Design. In this section, optimization problems will be provided that ensure that the conditions for insensitive disk region design are satisfied with some bounds on the gain of the closed-loop system.
The System
editA state-space representation of a linear system as given below:
where , and are the system state, output, and the input vector respectively. represents the differential operation for continuous time systems, or the one-step shift forward operator for discrete time case.
The Data
editTo solve the design optimization problem, the linear system matrices A,B,C are required. Furthermore, to define the disk region on the eigenvalue-space, its radius is required.
The Optimization Problem
editThe problem of designing an optimal controller that results in the closed loop system insensitive to a certain disk region involves two sub-problems:
- Finding a control gain such that: .
- The conditions for insensitive disk region design for the closed-loop system, as provided in the section Insensitive Disk Region Design are fulfilled.
- The optimization goal is to minimize such that above two hold.
The LMI: Optimal Control Design for Insensitive Disk Region
editThe problem above has a solution if and only if the following optimization problem has a solution :
Conclusion:
editBy using the design problem provided here, an optimal controller is designed to make the closed-loop system robust to perturbations in the system matrices.
Implementation
editTo solve the optimization problem with LMI presented here, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:
https://github.com/smhassaan/LMI-Examples/blob/master/H2_Disk_example.m
Related LMIs
editInsensitive Disk Region Design
External Links
editA list of references documenting and validating the LMI.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 10.1.2 pp. 323–325.
Return to Main Page:
editQuadratic Stability
editLMIs in Control/Print version
The System:
editA TS fuzzy model allows the representation of a non-linear model as a set of local LTI (Linear Time Invariant) models , each one called subsystem. A subsystem is the local representation of the system in the space of premise variables = which are known and could depend on the state variables and input variables.
The Optimization Problem:
editLet consider an autonomous system = with being a constant matrix. If we define the Lyapunov function =, then the system is stable if there exist such that condition is satisfied.
If we have a family of matrices (where is a parameter that is bounded by a polytope ∆) instead of a single matrix A, then the system equation becomes = and condition should be satisfied for all possible values of . If exists such that following condition is satisfied then the system is quadratically stable.
∀ ∈ ∆.
Since there are an infinite number of matrices A(δ(t)) there is also an infinite number of constraints like that for quadratic stability mentioned previously that should be fulfilled. From a practical point of view this makes the problem impossible to be solved. Let consider now that the system can be written in a polytopic form as a Takagi-Sugeno (TS) polytopic system with premise variables and a set of r subsystems for .
.
It can be proven that a polytopic autonomous system is quadratically stable if previous condition is satisfied in the vertices (subsystems) of the polytope. Therefore there is no need to check stability in an infinite number of matrices, but only in subsystems matrices .
∀i = 1, . . . , r.
Stability conditions can be applied to the closed-loop system and the following set of conditions are obtained.
∀i = 1, . . . , r.
∀i, j ∈ {1, . . . , r}, i < j.
where and .
In the special case where matrices Bi are constant (i.e. ), the first set of inequalities are enough to prove stability. Therefore, assuming constant B for all the subsystems, if there exist P > 0 such that conditions are fulfilled, then the polytopic TS model (2.2) with state feedback control is quadratically stable inside the polytope.
∀i = 1, . . . , r.
The assumption of constant B can be achieve using a prefiltering of the input. This change is not restrictive and the main consequence is the addition of some new state variables (the ones from the filter) to the TS model.
The LMI:
editThe design of the controller that stabilizes the closed-loop system boils down to solve the Linear Matrix Inequality (LMI) problem of finding a positive definite matrix P and a set of matrices such that conditions are fulfilled. However, since the constraints should be linear combinations of the unknown variable, the following change of variables is applied: where . The solution of the LMI problem is the set of matrices such that conditions are fulfilled.
. ∀i = 1, . . . , r.
The i-th controller is computed from the solution as =
Conclusion:
editThe LMI is feasible.
Implementation
editReferences
edit- Control, A. (2016). Gain-scheduling Control of a Quadrotor Using the Takagi-Sugeno Approach.
Apkarian Filter and State Feedback
editLMIs in Control/Print version
The System:
editThe number of LMI constraints needed to check quadratic stability is reduced if all the subsystems in the polytopic model has the same matrix . This can be achieved by adding an Apkarian filter in the input of the system.
The Optimization Problem:
editApkarian Filter
Let consider our TS-LIA model. This can be re written in linear form as:
The filter should be such that the equilibrium of the states are the input values and the dynamics should be fast, so we could assume the dynamics of the filter negligible (i.e. the input of the filter is equivalent to the input of the quadrotor). One possible filter is shown , where = −100, = 100 and ∈ is the identity matrix.
- .
When applying the filter, we are imposing that the output of the filter is the new input of the TS-LIA model (i.e. = ). Then, the extended model is:
This prefiltering does not affect the procedure followed to obtain the TS-LIA model, so the premise variables, membership functions and activations functions remains the same.
State Feedback Controller Design
Let consider the state feedback control law for the extended TS-LIA model:, where the state feedback control laws are :, we get the closed loop system :
The LMI:
editThe design of the controller is done by solving an LMI problem involving the quadratic stability constraints. In case we want D- stabilization, the following set of LMI constraints are needed:
- ∀i = 1, . . . , 32.
Conclusion:
editThe LMI is feasible.
Related LMIs
edit- LMI for Natural Frequency in State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Maximum_Natural_Frequency_in_State_Feedback#The_LMI%3A
- LMI for Minimum Decay Rate in State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Minimum_Decay_Rate_in_State_Feedback#The_LMI%3A
References
edit- Control, A. (2016). Gain-scheduling Control of a Quadrotor Using the Takagi-Sugeno Approach.
Minimum Decay Rate in State Feedback
editLMIs in Control/Print version
The System:
editThe number of LMI constraints needed to check quadratic stability is reduced if all the subsystems in the polytopic model has the same matrix . This can be achieved by adding an Apkarian filter in the input of the system.
The Optimization Problem:
editApkarian Filter
Let consider our TS-LIA model. This can be re written in linear form as:
The filter should be such that the equilibrium of the states are the input values and the dynamics should be fast, so we could assume the dynamics of the filter negligible (i.e. the input of the filter is equivalent to the input of the quadrotor). One possible filter is shown , where = −100, = 100 and ∈ is the identity matrix.
- .
When applying the filter, we are imposing that the output of the filter is the new input of the TS-LIA model (i.e. = ). Then, the extended model is:
This prefiltering does not affect the procedure followed to obtain the TS-LIA model, so the premise variables, membership functions and activations functions remains the same.
State Feedback Controller Design
Let consider the state feedback control law for the extended TS-LIA model:, where the state feedback control laws are :, we get the closed loop system :
The LMI:
editThe design of the controller is done by solving an LMI problem involving the quadratic stability constraints. In case we want D- stabilization, the following set of LMI constraints are needed:
- ∀i = 1, . . . , 32.
A pair of conjugate complex poles s of the closed loop system can be written as = − where is the damping ratio, is the undamped natural frequency and is the frequency response defined as .Three different LMI regions have been considered, each one related with a performance specification regarding and :
Minimum Decay Rate:
If we want to set a minimum decay rate α in the closed loop system response, the poles should be inside the LMI region defined in : = [s = x + j y | x < − ].where > 0. In this case L = and M = 1.
Applying condition to the closed-loop system , the LMI condition associated to this LMI region is:
- ∀i = 1, . . . , 32.
Conclusion:
editThe LMI is feasible.
Related LMIs
edit- Apkarian Filter and State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Apkarian_Filter-and_State_Feedback
- Maximum Natural Frequency in State Feedback https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Maximum_Natural_Frequency_in_State_Feedback#The_LMI%3A
References
edit- Control, A. (2016). Gain-scheduling Control of a Quadrotor Using the Takagi-Sugeno Approach.
Maximum Natural Frequency in State Feedback
editLMIs in Control/Print version
The System:
editThe number of LMI constraints needed to check quadratic stability is reduced if all the subsystems in the polytopic model has the same matrix . This can be achieved by adding an Apkarian filter in the input of the system.
The Optimization Problem:
editApkarian Filter
Let consider our TS-LIA model. This can be re written in linear form as:
The filter should be such that the equilibrium of the states are the input values and the dynamics should be fast, so we could assume the dynamics of the filter negligible (i.e. the input of the filter is equivalent to the input of the quadrotor). One possible filter is shown , where = −100, = 100 and ∈ is the identity matrix.
- .
When applying the filter, we are imposing that the output of the filter is the new input of the TS-LIA model (i.e. = ). Then, the extended model is:
This prefiltering does not affect the procedure followed to obtain the TS-LIA model, so the premise variables, membership functions and activations functions remains the same.
State Feedback Controller Design
Let consider the state feedback control law for the extended TS-LIA model:, where the state feedback control laws are :, we get the closed loop system :
The LMI:
editThe design of the controller is done by solving an LMI problem involving the quadratic stability constraints. In case we want D- stabilization, the following set of LMI constraints are needed:
- ∀i = 1, . . . , 32.
A pair of conjugate complex poles s of the closed loop system can be written as = − where is the damping ratio, is the undamped natural frequency and is the frequency response defined as .Three different LMI regions have been considered, each one related with a performance specification regarding and :
Maximizing Natural Frequency:
Natural frequency is related with the maximum frequency response in the undamped case ( = 0). If we want to set a maximum condition, the LMI region associated is = [s = x + jy | |x + jy| < ], ::. Resulting LMI condition is:
- ∀i = 1, . . . , 32.
Conclusion:
editThe LMI is feasible.
Related LMIs
edit- Apkarian Filter and State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Apkarian_Filter-and_State_Feedback
- Minimum Decay Rate in State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Minimum_Decay_Rate_in_State_Feedback#The_LMI%3A
References
edit- Control, A. (2016). Gain-scheduling Control of a Quadrotor Using the Takagi-Sugeno Approach.
Optimal Observer and State Estimation
edit{{:LMIs in Control/Observer Synthesis/Continuous Time/Optimal Observer and State Estimation}
Detectability LMI
editDetectability LMI
editDetectability is a weaker version of observability. A system is detectable if all unstable modes of the system are observable, whereas observability requires all modes to be observable. This implies that if a system is observable it will also be detectable. The LMI condition to determine detectability of the pair is shown below.
The System
editwhere , , at any .
The Data
editThe matrices necessary for this LMI are and . There is no restriction on the stability of .
The LMI: Detectability LMI
editis detectable if and only if there exists such that
- .
Conclusion:
editIf we are able to find such that the above LMI holds it means the matrix pair is detectable. In words, a system pair is detectable if the unobservable states asymptotically approach the origin. This is a weaker condition than observability since observability requires that all initial states must be able to be uniquely determined in a finite time interval given knowledge of the input and output .
Implementation
editThis implementation requires Yalmip and Sedumi.
https://github.com/eoskowro/LMI/blob/master/Detectability_LMI.m
Related LMIs
editExternal Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.1.1 and Table 6.1 pp. 166–170, 192.
- A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Section 2.2.3, pp. 71-73.
Return to Main Page:
editLMI for the Observability Grammian
editLMI for the Observability Grammian
Observability is a system property which says that the state of the system can be reconstructed using the input and output on an interval . This is necessary when knowledge of the full state is not available. If observable, estimators or observers can be created to reconstruct the full state. Observability and controllability are dual concepts. Thus in order to investigate the observability of a system we can study the controllability of the dual system. Although system observability can be determined with multiple methods, one is to compute the rank of the observability grammian.
The System
editwhere , , at any .
The Data
editThe matrices necessary for this LMI are and .
The LMI:LMI to Determine the Observability Grammian
editis observable if and only if is the unique solution to
- ,
where is the observability grammian.
Conclusion:
editThe above LMI attempts to find the observability grammian of the system . If the problem is feasible and a unique is found, then the system is also observable. The observability grammian can also be computed as: . Due to the dual nature of observability and controllability this LMI can be determined by determining the controllability of the dual nature, which results in the above LMI. The Observability and Controllability matricies are written as and respectively. They are related as follows:
Hence is observable if and only if is controllable. Please refer to the section on controllability grammians.
Implementation
editThis implementation requires Yalmip and Sedumi.
https://github.com/eoskowro/LMI/blob/master/Observability_Gram_LMI.m
Related LMIs
editExternal Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.1.1 and Table 6.1 pp. 166–170, 192.
- A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Section 2.2.3, pp. 71-73.
Return to Main Page:
editH-infinity filtering
editLMIs in Control/Print version
For systems that have disturbances, filtering can be used to reduce the effects of these disturbances. Described on this page is a method of attaining a filter that will reduce the effects of the disturbances as completely as possible. To do this, we look to find a set of new coefficient matrices that describe the filtered system. The process to achieve such a new system is described below. The H-infinity-filter tries to minimize the maximum magnitude of error.
The System
editFor the application of this LMI, we will look at linear systems that can be represented in state space as
where represent the state vector, the measured output vector, and the output vector of interest, respectively, is the disturbance vector, and and are the system matrices of appropriate dimension.
To further define: is and is the state vector, is and is the state matrix, is and is the input matrix, is and is the exogenous input, is and is the output matrix, and are and are feedthrough matrices, and and are and are the output and the output of interest, respectively.
The Data
editThe data are (the disturbance vector), and and (the system matrices). Furthermore, the matrix is assumed to be stable
The Optimization Problem
editWe need to design a filter that will eliminate the effects of the disturbances as best we can. For this, we take a filter of the following form:
where is the state vector, is the estimation vector of z, and are the coefficient matrices of appropriate dimensions.
Note that the combined complete system can be represented as
where is the estimation error,
is the state vector of the system, and are the coefficient matrices, defined as:
In other words, for the system defined above we need to find such that
where is a positive constant, and
The LMI: H-inf Filtering
editThe solution can be obtained by finding matrices that obey the following LMIs:
Conclusion:
editTo find the corresponding filter, use the optimized matrices from the solution to find:
These matrices can then be used to produce to construct the filter described above, that will best eliminate the disturbances of the system.
Implementation
editThis implementation requires Yalmip and Sedumi.
https://github.com/rezajamesahmed/LMImatlabcode/blob/master/hinf_filtering.m
Related LMIs
editExternal Links
editThis LMI comes from
- [27] - "LMIs in Control Systems: Analysis, Design and Applications" by Guang-Ren Duan and Hai-Hua Yu
Other resources:
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
References
editDuan, G. (2013). LMIs in control systems: analysis, design and applications. Boca Raton: CRC Press, Taylor & Francis Group.
Return to Main Page:
editH2 filtering
editLMIs in Control/Print version
For systems that have disturbances, filtering can be used to reduce the effects of these disturbances. Described on this page is a method of attaining a filter that will reduce the effects of the disturbances as completely as possible. To do this, we look to find a set of new coefficient matrices that describe the filtered system. The process to achieve such a new system is described below. The H2-filter tries to minimize the average magnitude of error.
The System
editFor the application of this LMI, we will look at linear systems that can be represented in state space as
where represent the state vector, the measured output vector, and the output vector of interest, respectively, is the disturbance vector, and and are the system matrices of appropriate dimension. To further define: is and is the state vector, is and is the state matrix, is and is the input matrix, is and is the exogenous input, is and is the output matrix, and are and are feedthrough matrices, and and are and are the output and the output of interest, respectively.
The Data
editThe data are (the disturbance vector), and and (the system matrices). Furthermore, the matrix is assumed to be stable
The Optimization Problem
editWe need to design a filter that will eliminate the effects of the disturbances as best we can. For this, we take a filter of the following form:
where is the state vector, is the estimation vector, and are the coefficient matrices of appropriate dimensions.
Note that the combined complete system can be represented as
where is the estimation error,
is the state vector of the system, and are the coefficient matrices, defined as:
In other words, for the system defined above we need to find such that
where is a positive constant, and
The LMI: H-2 Filtering
editFor this LMI, the solution exists if one of the following sets of LMIs hold:
Matrices exist that obey the following LMIs:
or
Matrices exist that obey the following LMIs:
Conclusion:
editTo find the corresponding filter, use the optimized matrices from the first solution to find:
Or the second solution to find:
These matrices can then be used to produce to construct the final filter below, that will best eliminate the disturbances of the system.
Implementation
editThis implementation requires Yalmip and Sedumi.
https://github.com/rezajamesahmed/LMImatlabcode/blob/master/H2_Filtering.m
Related LMIs
editExternal Links
editThis LMI comes from
- [28] - "LMIs in Control Systems: Analysis, Design and Applications" by Guang-Ren Duan and Hai-Hua Yu
Other resources:
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
References
editDuan, G. (2013). LMIs in control systems: analysis, design and applications. Boca Raton: CRC Press, Taylor & Francis Group.
Return to Main Page:
editH2 Optimal Observer
editState observer is a system that provides estimates of internal states of a given real system, from measurements of the inputs and outputs of the real system.The goal of -optimal state estimation is to design an observer that minimizes the norm of the closed-loop transfer matrix from w to z. Kalman filter is a form of Optimal Observer.
The System
editConsider the continuous-time generalized plant with state-space realization
The Data
editThe matrices needed as input are .
The Optimization Problem
editThe task is to design an observer of the following form:
The LMI: Optimal Observer
editLMIs in the variables are given by:
Conclusion:
editThe -optimal observer gain is recovered by and the norm of T(s) is
Implementation
edithttps://github.com/Ricky-10/coding107/blob/master/H2%20Optimal%20Observer
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- https://onlinelibrary.wiley.com/doi/abs/10.1002/rnc.1310
Return to Main Page:
editHInf Optimal Observer
edit-Optimal observers yield robust estimates of some or all internal plant states by processing measurement data. Robust observers are increasingly demanded in industry as they may provide state and parameter estimates for monitoring and diagnosis purposes even in the presence of large disturbances such as noise etc. It is there where Kalman filters may tend to fail. State observer is a system that provides estimates of internal states of a given real system, from measurements of the inputs and outputs of the real system. The goal of -optimal state estimation is to design an observer that minimizes the norm of the closed-loop transfer matrix from w to z.
The System
editConsider the continuous-time generalized plant with state-space realization
The Data
editThe matrices needed as input are .
The Optimization Problem
editThe observer gain is to be designed such that the of the transfer matrix from w to z, given by
is minimized. The form of the observer would be:
The LMI: Optimal Observer
editThe -optimal observer gain is synthesized by solving for , and that minimize subject to and
Conclusion:
editThe -optimal observer gain is recovered by and the norm of T(s) is .
Implementation
editLink to the MATLAB code designing - Optimal Observer
https://github.com/Ricky-10/coding107/blob/master/HinfinityOptimalobserver
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- [29]
- [30]
Return to Main Page:
editMixed H2 HInf Optimal Observer
editThe goal of mixed -optimal state estimation is to design an observer that minimizes the norm of the closed-loop transfer matrix from to , while ensuring that the norm of the closed-loop transfer matrix from to is below a specified bound.
The System
editConsider the continuous-time generalized plant with state-space realization
where it is assumed that is detectable.
The Data
editThe matrices needed as input are .
The Optimization Problem
editThe observer gain L is to be designed to minimize the norm of the closed-loop transfer matrix from the exogenous input to the performance output while ensuring the norm of the closed-loop transfer matrix from the exogenous input to the performance output is less than , where
is minimized. The form of the observer would be:
is to be designed, where is the observer gain.
The LMI: Optimal Observer
editThe mixed -optimal observer gain is synthesized by solving for , and that minimize subject to ,
Conclusion:
editThe mixed -optimal observer gain is recovered by , the norm of is less than and the norm of T(s) is less than .
Implementation
editLink to the MATLAB code designing - Optimal Observer
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Related LMIs
edit
Return to Main Page:
editH2 Optimal Filter
editOptimal filtering is a means of adaptive extraction of a weak desired signal in the presence of noise and interfering signals. Optimal filters normally are free from stability problems. There are simple operational checks on an optimal filter when it is being used that indicate whether it is operating correctly. Optimal filters are probably easier to make adaptive to parameter changes than suboptimal filters.The goal of optimal filtering is to design a filter that acts on the output of the generalized plant and optimizes the transfer matrix from w to the filtered output.
The System:
editConsider the continuous-time generalized LTI plant with minimal states-space realization
where it is assumed that is Hurwitz.
The Data
editThe matrices needed as inputs are .
The Optimization Problem:
editAn -optimal filter is designed to minimize the norm of in following equation.
To ensure that has a finite norm, it is required that , which results in
The LMI: - Optimal filter
editSolve for , and that minimize subject to .
Conclusion:
editThe filter is recovered by and .
Implementation
editExternal links
edit- [31]- Optimal Filtering
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- [http://users.cecs.anu.edu.au/~john/papers/BOOK/B02.PDF}- Optimal Filtering by Brian D.O. Anderson
HInf Optimal Filter
editOptimal filtering is a means of adaptive extraction of a weak desired signal in the presence of noise and interfering signals. The goal of optimal filtering is to design a filter that acts on the output of the generalized plant and optimizes the transfer matrix from w to the filtered output.
The System:
editConsider the continuous-time generalized LTI plant with minimal states-space realization
where it is assumed that is Hurwitz.
The Data
editThe matrices needed as inputs are .
The Optimization Problem:
editAn -optimal filter is designed to minimize the norm of in following equation.
The LMI: - Optimal filter
editSolve for , and that minimize subject to .
Conclusion:
editThe filter is recovered by and .
Implementation
editExternal links
edit- [32]- Optimal Filtering
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
FDI Filter Design For Systems With Sensor Faults: an LMI
editFDI Filter Design For Systems With Sensor Faults: an LMI
Systems with faulty sensors are a very common type of systems. In many cases, redundancy is added in the form of additional sensors, but in certain cases it could be a costly solution. For general linear system models, the LMI in this section can be utilized to design state estimators which can detect and isolate faulty sensor readings in order to mitigate their effects.
The System
editwhere is the state, is the control input, is the process noise, is the output and is the measurement noise.
The Data
editThe state space matrices are required to be known.
The Optimization LMI
editThe following LMI is used to design the Fault Detection and Isolation (FDI) filter:
Then the filter is .
Conclusion:
editThe LMI designed in this section is used to design filters that can work on systems that are prone to sensors getting damaged or faulty.
Implementation
editTo solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:
https://github.com/smhassaan/LMI-Examples/blob/master/FDI_Filter_example.m
Related LMIs
editExternal Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editH2 Optimal State estimation
editLMIs in Control/Print version
The H2 norm of a stable system H is the root-mean-square of the impulse response of the system. The H2 norm measures the steady-state covariance (or power) of the output response to unit noise input. In this module, the goal of H2 optimal state estimation is to design an observer that minimizes the H2 norm of the closed loop transfer matrix
The System
editConsider the continuous-time generalized plant P with state-space realization
where it is assumed that (A,C2) is detectable. An observer of the form
The Data
edit- ∈ n, ∈ l , ∈ m are respectively the state vector, the measured
output vector, and the output vector of interests
- ∈ p and ∈ r are the disturbance vector and the control vector,
respectively
- A, B1, B2, C1, C2, D1, and D2 are the system coefficient matrices of
appropriate dimensions
The Optimization Problem
editGiven the system and a positive scalar we have to find the matrix L such that
||||2 <
An observer of the form
is to be designed, where L is the observer gain.
Defining the error state as
The break dynamics are found to be
For the system we introduce a full state observer in the following form:
are the observation vector and the observer gain.
The transfer function for this case is
and thus the problem of state observer design is to find L such that
||
The LMI: LMI for H2 Observer estimation
editThe H2 state observer problem has a solution if and only if there exists a matrix , a symmetric matrix and a symmetric matrix such that
and from the solution of the above LMIs we can obtain the observer matrix as
Conclusion
editThus by formulation, we have converted the problem of H2 state observer design into an LMI feasibility problem by optimizing the above LMIs. In application we are often concerned with the problem of finding the minimal attenuation level
On implementation and optimization of the above LMI using YALMIP and MOSEK (or) SeDuMi we get 3 matrices as output, and also which is used to calculate which is the H2 norm of the system.
There exists another set of LMIs which holds true for the same optimization problem as above.
When a minimal is obtained, the minimal attenuation level is
Implementation
editA link to the Matlab code for a simple implementation of this problem in the Github repository:
https://github.com/yashgvd/ygovada
Related LMIs
editH State Observer Design
Discrete time H2 State Observer Design
External Links
edit- [33] - LMI in Control Systems Analysis, Design and Applications
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
Return to Main Page:
editHurwitz Detectability
editLMIs in Control/Print version
Hurwitz Detectability
editHurwitz detectability is a dual concept of Hurwitz stabilizability and is defined as the matrix pair , is said to be Hurwitz detectable if there exists a real matrix such that is Hurwitz stable.
The System
editwhere , , , at any .
The Data
edit- The matrices are system matrices of appropriate dimensions and are known.
The Optimization Problem
editThere exist a symmetric positive definite matrix and a matrix satisfying
There exists a symmetric positive definite matrix satisfying
with being the right orthogonal complement of .
There exists a symmetric positive definite matrix such that
for some scalar
The LMI:
editMatrix pair , is Hurwitz detectable if and only if following LMI holds
Conclusion:
editThus by proving the above conditions we prove that the matrix pair is Hurwitz Detectable.
Implementation
editFind the MATLAB implementation at this link below
Hurwitz detectability
Related LMIs
editLinks to other closely-related LMIs
LMI for Hurwitz stability
LMI for Schur stability
Schur Detectability
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editFull-Order State Observer
editLMIs in Control/Print version
Full-Order State Observer
editThe problem of constructing a simple full-order state observer directly follows from the result of Hurwitz detectability LMI's, Which essentially is the dual of Hurwitz stabilizability. If a feasible solution to the first LMI for Hurwitz detectability exist then using the results we can back out a full state observer such that is Hurwitz stable.
The System
editwhere , , , at any .
The Data
edit- The matrices are system matrices of appropriate dimensions and are known.
The Optimization Problem
editThe full-order state observer problem essential is finding a positive definite such that the following LMI conclusions hold.
The LMI:
edit1) The full-order state observer problem has a solution if and only if there exist a symmetric positive definite Matrix and a matrix that satisfy
Then the observer can be obtained as
2) The full-state state observer can be found if and only if there is a symmetric positive definite Matrix that satisfies the below Matrix inequality
In this case the observer can be reconstructed as . It can be seen that the second relation can be directly obtained by substituting in the first condition.
Conclusion:
editHence, both the above LMI's result in a full-order observer such that is Hurwitz stable.
External Links
editA list of references documenting and validating the LMI.
- LMIs in Control Systems Analysis, Design and Applications - Duan and Yu
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editFull-Order H-infinity State Observer
editIn this section, we design full order H- state observer.
The System
editGiven a state-space representation of a linear system
- are the state vector, measured output vector and output vectors of interest.
- are the disturbance vector and control vector respectively.
The Data
editare system matrices
Definition
editFor the system , a full order state observer of the form of equation (1) is introduced and the estimate of interested output is given by .
-
()
The estimate of interested output is
-
()
Given the system and a positive scalar , L is found such that
-
()
LMI Condition
editThe state observers problem has a solution if and only if there exists a symmetric positive definite matrix and a matrix satisfying the below LMI
-
()
When such a pair of matrics is found, the solution is
-
()
Implementation
editThis implementation requires Yalmip and Mosek.
Conclusion
editThus, an state observer is designed such that the output vectors of interest are accurately estimated.
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013
Reduced-Order State Observer
editLMIs in Control/Print version
Reduced Order State Observer
editThe Reduced Order State Observer design paradigm follows naturally from the design of Full Order State Observer.
The System
editwhere , , , at any .
The Data
edit- The matrices are system matrices of appropriate dimensions and are known.
The Problem Formulation
editGiven a State-space representation of a system given as above. First an arbitrary matrix is chosen such that the vertical augmented matrix given as
is nonsingular, then
Furthermore, let
then the matrix pair is detectable if and only if is detectable, then let
then a new system of the form given below can be obtained
once an estimate of is obtained the the full state estimate can be given as
the the reduced order observer can be obtained in the form.
Such that for arbitrary control and arbitrary initial system values, There holds
The value for can be obtain by solving the following LMI.
The LMI:
editThe reduced-order observer exists if and only if one of the two conditions holds.
1) There exist a symmetric positive definite Matrix and a matrix that satisfy
Then
2) There exist a symmetric positive definite Matrix that satisfies the below Matrix inequality
Then .
By using this value of we can reconstruct the observer state matrices as
Conclusion:
editHence, we are able to form a reduced-order observer using which we can back of full state information as per the equation given at the end of the problem formulation given above.
External Links
editA list of references documenting and validating the LMI.
- LMIs in Control Systems Analysis, Design and Applications - Duan and Yu
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editOptimal Observer; Mixed
editLMIs in Control/Print version
In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize both H2 and Hinf norms, to minimize both the average and the maximum error of the observer.
The System
editwhere and is the state vector, and is the state matrix, and is the input matrix, and is the exogenous input, and is the output matrix, and is the feedthrough matrix, and is the output, and it is assumed that is detectable.
The Data
editThe matrices .
The Optimization Problem
editAn observer of the form:
is to be designed, where is the observer gain.
Defining the error state , the error dynamics are found to be
,
and the performance output is defined as
.
The observer gain is to be designed to minimize the norm of the closed loop transfer matrix from the exogenous input to the performance output is less than , where
The LMI: Discrete-Time Mixed H2-Hinf-Optimal Observer
editThe discrete-time mixed--optimal observer gain is synthesized by solving for , , , and that minimize J subject to ,
where refers to the trace of a matrix.
Conclusion:
editThe mixed--optimal observer gain is recovered by , the norm of is less than , and the norm of is less than . This result gives us a matrix of observer gains that allow us to optimally observe the states of the system indirectly as:
Implementation
editThis implementation requires Yalmip and Sedumi.
https://github.com/rezajamesahmed/LMImatlabcode/blob/master/mixedh2hinfobsdiscretetime.m
Related LMIs
editDiscrete-Time_Hinfinity-Optimal_Observer
Discrete-Time_H2-Optimal_Observer
External Links
editThis LMI comes from Ryan Caverly's text on LMI's (Section 5.3.2):
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
Other resources:
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editOptimal Observer; H2
editLMIs in Control/Print version
In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize the H2 norm, which conceptually is minimizing the average magnitude of error in the observer.
The System
editwhere and is the state vector, and is the state matrix, and is the input matrix, and is the exogenous input, and is the output matrix, and is the feedthrough matrix, and is the output, and it is assumed that is detectable.
The Data
editThe matrices .
The Optimization Problem
editAn observer of the form:
is to be designed, where is the observer gain.
Defining the error state , the error dynamics are found to be
,
and the performance output is defined as
.
The observer gain is to be designed such that the of the transfer matrix from to , given by
is minimized.
The LMI: Discrete-Time H2-Optimal Observer
editThe discrete-time -optimal observer gain is synthesized by solving for , , , and that minimize subject to ,
where refers to the trace of a matrix.
Conclusion:
editThe -optimal observer gain is recovered by and the norm of is . The matrix is the observer gains that can be used to form the optimal observer:
Implementation
editThis implementation requires Yalmip and Sedumi.
Related LMIs
editMixed H2-Hinfinity discrete time observer
Discrete-Time_Hinfinity-Optimal_Observer
External Links
editThis LMI comes from Ryan Caverly's text on LMI's (Section 5.1.2):
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
Other resources:
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editOptimal Observer; Hinf
editLMIs in Control/Print version
In many applications, perhaps even most, the state of the system cannot be directly known. In this case, you will need to strategically to measure key system outputs that will make the system states indirectly observable. Observers need to converge much faster than the system dynamics in order for their estimations to be accurate. Optimal observer synthesis is therefore advantageous. In this LMI, we seek to optimize the H-infinity norm, which conceptually is minimizing the maximum magnitude of error in the observer.
The System
editThe system needed for this LMI is a discrete-time LTI plant , which has the state space realization:
where and is the state vector, and is the state matrix, and is the input matrix, and is the exogenous input, and is the output matrix, and is the feedthrough matrix, and is the output, and it is assumed that is detectable.
The Data
editThe matrices .
The Optimization Problem
editAn observer of the form:
is to be designed, where is the observer gain.
Defining the error state , the error dynamics are found to be
,
and the performance output is defined as
.
The observer gain is to be designed such that the of the transfer matrix from to , given by
is minimized.
The LMI: Discrete-Time Hinf-Optimal Observer
editThe discrete-time -optimal observer gain is synthesized by solving for , , and that minimize J subject to , and
Conclusion:
editThe -optimal observer gain is recovered by and the norm of is . This matrix of observer gains can then be used to form the optimal observer formulated by:
Implementation
editThis implementation requires Yalmip and Sedumi.
https://github.com/rezajamesahmed/LMImatlabcode/blob/master/Hinfobsdiscretetime.m
Related LMIs
editMixed H2-Hinfinity discrete time observer
Discrete-Time_H2-Optimal_Observer
External Links
editThis LMI comes from Ryan Caverly's text on LMI's (Section 5.2.2):
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
Other resources:
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editDiscrete Time Detectability
editDiscrete-Time Detectability
A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.
Discrete-Time LTI systems can be made detectable using observer gain L, which can be found using LMI optimization, such that the close loop system is detectable.
The System
editDiscrete-Time LTI System with state space realization
The Data
editThe matrices: System .
The Optimization Problem
editThe following feasibility problem should be optimized:
Maximize P while obeying the LMI constraints.
Then L is found.
The LMI:
editDiscrete-Time Detectability
The LMI formulation
Conclusion:
editThe system is detectabe iff there exits a , such that . The matrix is Schur with
Implementation
editA link to CodeOcean or other online implementation of the LMI
MATLAB Code
Related LMIs
edit[34] - Continuous time Detectability
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editSchur Detectability
editSchur Detectability
Schur detectability is a dual concept of Schur stabilizability and is defined as follows, the matrix pair is said to be Schur detectable if there exists a real matrix such that is Schur stable.
The System
editWe consider the following system:
where the matrices , , ,, , and are the state matrix, input matrix, state vector, and the input vector, respectively.
Moreover, represents time in the discrete-time system and is the next time step.
The state feedback control law is defined as follows:
where is the controller gain. Thus, the closed-loop system is given by:
The Data
edit- The matrices are system matrices of appropriate dimensions and are known.
The Optimization Problem
editThere exist a symmetric matrix and a matrix W satisfying
There exists a symmetric matrix satisfying
with being the right orthogonal complement of .
There exists a symmetric matrix P such that
The LMI:
editThe LMI for Schur detecability can be written as minimization of the scalar, , in the following constraints:
Conclusion:
editThus by proving the above conditions we prove that the matrix pair is Schur Detectable.
Implementation
editA link to Matlab codes for this problem in the Github repository: Schur Detectability
Related LMIs
editLMI for Hurwitz stability
LMI for Schur stability
Hurwitz Detectability
External Links
edit- [35] - LMI in Control Systems Analysis, Design and Applications
Return to Main Page
editRobust Stabilization of Second-Order Systems
editLMIs in Control/Print version
Stabilization is a vastly important concept in controls, and is no less important for second order systems with perturbations. Such a second order system can be conceptualized most simply by the model of a mass-spring-damper, with added perturbations. Velocity and position are of course chosen as the states for this system, and the state space model can be written as it is below. The goal of stabilization in this context is to design a control law that is made up of two controller gain matrices , and . These allow the construction of a stabilized closed loop controller.
The System
editIn this LMI, we have an uncertain second-order linear system, that can be modeled in state space as:
where and are the state vector and the control vector, respectively, and are the derivative output vector and the proportional output vector, respectively, and are the system coefficient matrices of appropriate dimensions.
and are the perturbations of matrices and , respectively, are bounded, and satisfy
or
where and are two sets of given positive scalars, and are the i-th row and j-th collumn elements of matrices and , respectively. Also, the perturbation notations also satisfy the assumption that and .
To further define: is and is the state vector, is and is the state matrix on , is and is the state matrix on , is and is the state matrix on , is and is the input matrix, is and is the input, and are and are the output matrices, is and is the output from , and is and is the output from .
The Data
editThe matrices and perturbations describing the second order system with perturbations.
The Optimization Problem
editFor the system defined as shown above, we need to design a feedback control law such that the following system is Hurwitz stable. In other words, we need to find the matrices and in the below system.
However, to do proceed with the solution, first we need to present a Lemma. This Lemma comes from Appendix A.6 in "LMI's in Control systems" by Guang-Ren Duan and Hai-Hua Yu. This Lemma states the following: if , then the following is also true for the system described above:
The system is hurwitz stable if
- ,
or
the system is hurwitz stable if
, where are the numbers of nonzero elements in matrices respectively.
The LMI: Robust Stabilization of Second Order Systems
editThis problem is solved by finding matrices and that satisfy either of the following sets of LMIs.
or
Conclusion:
editHaving solved the above problem, the matrices and can be substituted into the input as to robustly stabilize the second order uncertain system. The new system is stable in closed loop.
Implementation
editThis implementation requires Yalmip and Sedumi.
https://github.com/rezajamesahmed/LMImatlabcode/blob/master/ROBstab2ndorder.m
Related LMIs
editStabilization of Second-Order Systems
External Links
editThis LMI comes from
- [36] - "LMIs in Control Systems: Analysis, Design and Applications" by Guang-Ren Duan and Hai-Hua Yu
Other resources:
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
References
editDuan, G. (2013). LMIs in control systems: analysis, design and applications. Boca Raton: CRC Press, Taylor & Francis Group.
Return to Main Page:
editRobust Stabilization of Optimal State Feedback Control
editRobust Full State Feedback Optimal Control
editAdditive uncertainty
editFull State Feedback is a control technique which places a given system's closed loop system poles in locations specified by desired performance specifications. One can use methods to turn this into an optimization problem with the goal to minimize the impact of uncertain perturbations in a closed loop system while maintaining system stability. This is done by minimizing the norm of the uncertain closed loop system, which minimizes the worst case effect of the system disturbance or perturbation. This can be done for single-input single-output (SISO) or multiple-input multiple-output (MIMO) systems. Here we consider the case of a MIMO system with additive uncertainties.
The System
editConsider linear system with uncertainty below:
Where is the state, is the output, is the exogenous input or disturbance vector, and is the actuator input or control vector, at any
and are real-valued matrices which represent the time-varying parameter uncertainties in the form:
Where
are known matrices with appropriate dimensions and is the uncertain parameter matrix which satisfies:
For additive perturbations:
Where
are the known system matrices and
are the perturbation parameters which satisfy
Thus, with
The Data
edit, , , , , , , , are known.
The LMI:Full State Feedback Optimal Control LMI
editThere exists and and scalar such that
- .
Where
And .
Conclusion:
editOnce K is found from the optimization LMI above, it can be substituted into the state feedback control law to find the robustly stabilized closed loop system as shown below:
where is the state, is the output, is the exogenous input or disturbance vector, and is the actuator input or control vector, at any
Finally, the transfer function of the system is denoted as follows:
Implementation
editThis implementation requires Yalmip and Sedumi. https://github.com/rubindan/LMIcontrol/blob/master/HinfFilter.m
Related LMIs
editFull State Feedback Optimal H_inf LMI
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.
Return to Main Page:
editLMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control
Robust H inf State Feedback Control
editRobust Full State Feedback Optimal Control
editAdditive uncertainty
editFull State Feedback is a control technique which places a given system's closed loop system poles in locations specified by desired performance specifications. One can use methods to turn this into an optimization problem with the goal to minimize the impact of uncertain perturbations in a closed loop system while maintaining system stability. This is done by minimizing the norm of the uncertain closed loop system, which minimizes the worst case effect of the system disturbance or perturbation. This can be done for single-input single-output (SISO) or multiple-input multiple-output (MIMO) systems. Here we consider the case of a MIMO system with additive uncertainties.
The System
editConsider linear system with uncertainty below:
Where is the state, is the output, is the exogenous input or disturbance vector, and is the actuator input or control vector, at any
and are real-valued matrices which represent the time-varying parameter uncertainties in the form:
Where
are known matrices with appropriate dimensions and is the uncertain parameter matrix which satisfies:
For additive perturbations:
Where
are the known system matrices and
are the perturbation parameters which satisfy
Thus, with
The Data
edit, , , , , , , , are known.
The LMI:Full State Feedback Optimal Control LMI
editThere exists and and scalar such that
- .
Where
And .
Conclusion:
editOnce K is found from the optimization LMI above, it can be substituted into the state feedback control law to find the robustly stabilized closed loop system as shown below:
where is the state, is the output, is the exogenous input or disturbance vector, and is the actuator input or control vector, at any
Finally, the transfer function of the system is denoted as follows:
Implementation
editThis implementation requires Yalmip and Sedumi. https://github.com/rubindan/LMIcontrol/blob/master/HinfFilter.m
Related LMIs
editFull State Feedback Optimal H_inf LMI
External Links
edit- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.
Return to Main Page:
editLMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control
LMI for Time-Delay system on delay Independent Condition
editThe System
editThe problem is to check the stability of the following linear time-delay system
where
is the initial condition
represents the time-delay
is a known upper-bound of
The Data
editThe matrices are known
The LMI: The Time-Delay systems (Delay Independent Condition)
editFrom the given pieces of information, it is clear that the optimization problem only has a solution if there exists two symmetric matrices such that
This LMI has been derived from the Lyapunov function for the system.
By Schur Complement we can see that the above matrix inequality is equivalent to the Riccati inequality
Conclusion:
editWe can now implement these LMIs to do stability analysis for a Time delay system on the delay independent condition
Implementation
editThe implementation of the above LMI can be seen here
https://github.com/yashgvd/LMI_wikibooks
Related LMIs
editTime Delay systems (Delay Dependent Condition)
External Links
edit- [37] - LMI in Control Systems Analysis, Design and Applications
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- D. d. S. Madeira and J. Adamy, "Static output feedback: An LMI condition for stabilizability based on passivity indices," 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, 2016, pp. 960-965.
Return to Main Page:
editLMI for Time-Delay system on delay Dependent Condition
editThe System
editThe problem is to check the stability of the following linear time-delay system on a delay dependent condition
where
is the initial condition
represents the time-delay
is a known upper-bound of
For the purpose of the delay dependent system we rewrite the system as
The Data
editThe matrices are known
The LMI: The Time-Delay systems (Delay Dependent Condition)
editFrom the given pieces of information, it is clear that the optimization problem only has a solution if there exists a symmetric positive definite matrix
and a scalar such that
Here
This LMI has been derived from the Lyapunov function for the system. It follows that the system is asymptotically stable if
This is obtained by replacing with
Conclusion:
editWe can now implement these LMIs to do stability analysis for a Time delay system on the delay dependent condition
Implementation
editThe implementation of the above LMI can be seen here
https://github.com/yashgvd/LMI_wikibooks
Related LMIs
editTime Delay systems (Delay Independent Condition)
External Links
edit- [38] - LMI in Control Systems Analysis, Design and Applications
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- D. d. S. Madeira and J. Adamy, "Static output feedback: An LMI condition for stabilizability based on passivity indices," 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, 2016, pp. 960-965.
Return to Main Page:
editLMI for Stability of Retarded Differential Equation with Slowly-Varying Delay
editLMIs in Control/Print version
This page describes an LMI for stability analysis of a continuous-time system with a time-varying delay. In particular, a delay-independent condition is provided to test uniform asymptotic stability of a retarded differential equation through feasibility of an LMI. The system under consideration pertains a single discrete delay, with the extent of the delay at any time bounded by some known value. Moreover, the delay is assumed to vary only slowly in time, with a temporal derivative bounded by a value less than one. Solving the LMI for a particular value of this bound, uniform asymptotic stability can be shown for any time-delay satisfying this bound.
The System
editThe system under consideration is one of the form:
In this description, and are matrices in . The variable denotes a delay in the state at time , assuming a value no greater than some . Moreover, we assume that the function is differentiable at any time, with the derivative bounded by some value , assuring the delay to be slowly-varying in time.
The Data
editTo determine stability of the system, the following parameters must be known:
The Optimization Problem
editBased on the provided data, uniform asymptotic stability can be determined by testing feasibility of the following LMI:
The LMI: Delay-Independent Uniform Asymptotic Stability for Continuous-Time TDS
editConclusion:
editIf the presented LMI is feasible, the system will be uniformly asymptotically stable for any delay function satisfying . That is, independent of the values of the delays and the starting time :
- For any real number , there exists a real number such that:
- There exists a real number such that for any real number , there exists a time such that:
Here, we let for denote the delayed state function at time . The norm of this function is defined as the maximal value of the vector norm assumed by the state over the delayed time interval, given by:
Obtaining a feasible point for the LMI, this result can be proven using a Lyapunov-Krasovkii functional:
Notably, if matrices prove feasibility of the LMI for the pair , these same matrices will also prove feasibility of the LMI for the pair . As such, feasibility of this LMI proves uniform asymptotic stability of both systems:
Moreover, since the result is independent of the value of the delay, it will also hold for a delay . Hence, if the LMI is feasible, the matrices will be Hurwitz.
Implementation
editAn example of the implementation of this LMI in Matlab is provided on the following site:
Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.
Related LMIs
edit- [39] - Delay-dependent stability LMI for continuous-time TDS
- [40] - Stability LMI for delayed discrete-time system
External Links
editThe presented results have been obtained from:
- Fridman E. 2014. Introduction to Time-Delay Systems, Analysis and Control. Springer. ISBN: 978-3-319-09392-5.
Additional information on LMI's in control theory can be obtained from the following resources:
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editLMI for Robust Stability of Retarded Differential Equation with Norm-Bounded Uncertainty
editLMIs in Control/Print version
This page describes an LMI for stability analysis of an uncertain continuous-time system with a time-varying delay. In particular, a delay-independent condition is provided to test uniform asymptotic stability of a retarded differential equation with uncertain matrices through feasibility of an LMI. The system under consideration pertains a single discrete delay, with the extent of the delay at any time bounded by some known value. The matrices describing the system are assumed to be uncertain, with the norm of the uncertainty bounded by a value of one. In addition, the delay is assumed to vary only slowly in time, with a temporal derivative bounded by a value less than one. Solving the LMI for a particular value of this bound, uniform asymptotic stability can be shown for any time-delay satisfying this bound, independent of the value of the uncertainty function.
The System
editThe system under consideration is one of the form:
In this description, and are matrices in . The variable denotes a delay in the state at time , assuming a value no greater than some . Moreover, we assume that the function is differentiable at any time, with the derivative bounded by some value , assuring the delay to be slowly-varying in time. The uncertainty is also allowed to vary in time, but at any time must satisfy the inequality:
The uncertainty affects the system through matrices and , which are constant in time and assumed to be known.
The Data
editTo determine stability of the system, the following parameters must be known:
The Optimization Problem
editBased on the provided data, uniform asymptotic stability can be determined by testing feasibility of the following LMI:
The LMI: Delay-Independent Robust Uniform Asymptotic Stability for Continuous-Time TDS
editConclusion:
editIf the presented LMI is feasible, the system will be uniformly asymptotically stable for any delay function satisfying , and any uncertainty satisfying . That is, independent of the values of the delays , uncertainties , and the starting time :
- For any real number , there exists a real number such that:
- There exists a real number such that for any real number , there exists a time such that:
Here, we let for denote the delayed state function at time . The norm of this function is defined as the maximal value of the vector norm assumed by the state over the delayed time interval, given by:
The proof of this result relies on the fact that the following inequality holds for any value and constant matrices of appropriate dimensions:
Using this inequality with and , the described LMI can then be derived from that presented in [41], corresponding to a situation without uncertainty.
Implementation
editAn example of the implementation of this LMI in Matlab is provided on the following site:
Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.
Related LMIs
edit- [42] - Stability LMI for continuous-time RDE with slowly-varying delay without uncertainty
- [43] - LMI for quadratic stability of continuous-time system with norm-bounded uncertainty
- [44] - Stability LMI for delayed discrete-time system
External Links
editThe presented results have been obtained from:
- Fridman E. 2014. Introduction to Time-Delay Systems, Analysis and Control. Springer. ISBN: 978-3-319-09392-5.
Additional information on LMI's in control theory can be obtained from the following resources:
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editBounded Real Lemma under Slowly-Varying Delay
editLMIs in Control/Print version
This page describes a bounded real lemma for a continuous-time system with a time-varying delay. In particular, a condition is provided to obtain a bound on the -gain of a retarded differential system through feasibility of an LMI. The system under consideration pertains a single discrete delay, with the extent of the delay at any time bounded by some known value. This delay is only present in the state, with no direct delay in the effects of exogenous inputs on the state. In addition, the delay is assumed to vary only slowly in time, with a temporal derivative bounded by a value less than one, although results can also be attained if no bound is known. Solving the LMI for a particular value of the bound, while minimizing a scalar variable, an upper limit on the -gain of the system can be shown for any time-delay satisfying this bound.
The System
editThe system under consideration is one of the form:
In this description, and are constant matrices in . In addition, is a constant matrix in , and are constant matrices in where denote the number of exogenous inputs and regulated outputs respectively. The variable denotes a delay in the state at time , assuming a value no greater than some . Moreover, we assume that the function is differentiable at any time, with the derivative bounded by some value , assuring the delay to be slowly-varying in time.
The Data
editTo obtain a bound on the -gain of the system, the following parameters must be known:
The Optimization Problem
editBased on the provided data, we can obtain a bound on the -gain of the system by testing feasibility of an LMI. In particular, the bounded real lemma states that if the LMI presented below is feasible for some , the -gain of the system is less than or equal to this . To attain a bound that is as small as possible, we minimize the value of while solving the LMI:
The LMI: L2-gain for TDS with Slowly-Varying Delay
editIn this notation, the symbols are used to indicate appropriate matrices to assure the overall matrix is symmetric.
Conclusion:
editIf the presented LMI is feasible for some , the system is internally stable, and will have an -gain less than . That is, independent of the values of the delays :
It should be noted that this result is conservative. That is, even when minimizing the value of , there is no guarantee that the bound obtained on the -gain is sharp.
In a scenario where no bound on the change in the delay is known, the above LMI can still be used to obtain a bound on the -gain. In particular, setting in the above LMI, a bound can be attained independent of the value of the derivative of the delay.
Implementation
editAn example of the implementation of this LMI in Matlab is provided on the following site:
Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.
Related LMIs
edit- [45] - Bounded real lemma for continuous-time system without delay
- [46] - Bounded real lemma for discrete-time system without delay
- [47] - Stability LMI for continuous-time RDE with slowly-varying delay
External Links
editThe presented results have been obtained from:
- Fridman E. 2014. Introduction to Time-Delay Systems, Analysis and Control. Springer. ISBN: 978-3-319-09392-5.
Additional information on LMI's in control theory can be obtained from the following resources:
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editLMI for L2-Optimal State-Feedback Control under Time-Varying Input Delay
editLMIs in Control/Print version
This page describes a method for constructing a full-state-feedback controller for a continuous-time system with a time-varying input delay. In particular, a condition is provided to obtain a bound on the -gain of closed-loop system under time-varying delay through feasibility of an LMI. The system under consideration pertains a single discrete delay in the actuator input, with the extent of the delay at any time bounded by some known value. Moreover, the delay is assumed to vary only slowly in time, with a temporal derivative bounded by a value less than one, although results may also be attained if no bound is known. Solving the LMI for a particular value of the bound, while minimizing a scalar variable, an upper limit on the -gain of the system can be shown for any time-delay satisfying this bound.
The System
editThe system under consideration is one of the form:
In this description, and are constant matrices in . In addition, is a constant matrix in , and is a constant matrix in , where denote the number of exogenous and actuator inputs respectively. Finally, and are constant matrices in and respectively, where denotes the number of regulated outputs. The variable denotes a delay in the actuator input at time , assuming a value no greater than some . Moreover, we assume that the function is differentiable at any time, with the derivative bounded by some value , assuring the delay to be slowly-varying in time.
The Data
editTo construct an -optimal controller of the system, the following parameters must be known:
In addition to these parameters, a tuning scalar is also implemented in the LMI.
The Optimization Problem
editBased on the provided data, we can construct an -optimal full-state-feedback controller of the system by testing feasibility of an LMI. In particular, we note that if the LMI presented below is feasible for some and matrices and , implementing the state-feedback with , the -gain of the closed-loop system will be less than or equal to . To attain a bound that is as small as possible, we minimize the value of while solving the LMI:
The LMI: L2-Optimal Full-State-Feedback for TDS with Slowly-Varying Input Delay
editIn this notation, the symbols are used to indicate appropriate matrices to assure the overall matrix is symmetric.
Conclusion:
editIf the presented LMI is feasible for some , implementing the full-state-feedback controller , the closed-loop system will be asymptotically stable, and will have an -gain less than . That is, independent of the values of the delays , the system:
with:
will satisfy:
Here we note that is guaranteed to exist as is positive definite, and thus nonsingular.
It should be noted that the obtained result is conservative. That is, even when minimizing the value of , there is no guarantee that the bound obtained on the -gain is sharp, meaning that the actual -gain of the closed-loop can be (significantly) smaller than .
In a scenario where no bound on the change in the delay is known, or this bound is greater than one, the above LMI may still be used to construct a controller. In particular, if the presented LMI is feasible with , the closed-loop system imposing will be internally exponentially stable with an -gain less than independent of the value of .
Implementation
editAn example of the implementation of this LMI in Matlab is provided on the following site:
Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.
Related LMIs
edit- [48] - Bounded real lemma for continuous-time system with slowly-varying delay
- [49] - LMI for Hinf-optimal full-state-feedback control in a non-delayed continuous-time system
- [50] - LMI for Hinf-optimal output-feedback control in a non-delayed continuous-time system
External Links
editThe presented results have been obtained from:
- Fridman E. 2014. Introduction to Time-Delay Systems, Analysis and Control. Springer. ISBN: 978-3-319-09392-5.
Additional information on LMI's in control theory can be obtained from the following resources:
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editDiscrete Time
editLMIs in Control/Print version
This page describes an LMI for stability analysis of a discrete-time system with a time-varying delay. In particular, a delay-dependent condition is provided to test asymptotic stability of a discrete-delay system through feasibility of an LMI. The system under consideration pertains a single discrete delay, with the extent of the delay at any time bounded by some known value. Solving the LMI for different values of this bound, a limit on the delay can be attained for which the system remains asymptotically stable.
The System
editThe system under consideration is one of the form:
In this description, and are matrices in . The variable denotes a delay in the state at discrete time , assuming a value no greater than some .
The Data
editTo determine stability of the system, the following parameters must be known:
The Optimization Problem
editBased on the provided data, asymptotic stability can be determined by testing feasibility of the following LMI:
The LMI: Asymptotic Stability for Discrete-Time TDS
editIn this notation, the symbols are used to indicate appropriate matrices to assure the overall matrix is symmetric.
Conclusion:
editIf the presented LMI is feasible, the system will be asymptotically stable for any sequence of delays within the interval . That is, independent of the values of the delays at any time:
- For any real number , there exists a real number such that:
Obtaining a feasible point for the LMI, this result can be proven using a Lyapunov-Krasovkii functional:
where:
Implementation
editAn example of the implementation of this LMI in Matlab is provided on the following site:
Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.
Related LMIs
edit- TDSDC – Delay-dependent stability LMI for continuous-time TDS
- LMI for Stability of Retarded Differential Equation with Slowly-Varying Delay – Delay-independent stability LMI for continuous-time TDS
- Discrete Time Lyapunov Stability – Stability LMI for non-delayed discrete-time system
External Links
editThe presented results have been obtained from:
- Fridman E. 2014. Introduction to Time-Delay Systems, Analysis and Control. Springer. ISBN: 978-3-319-09392-5.
Additional information on LMI's in control theory can be obtained from the following resources:
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Return to Main Page:
editLMI for Attitude Control of Nonrotating Missiles, Pitch Channel
editLMI for Attitude Control of Nonrotating Missles, Pitch Channel
The dynamic model of a missile is very complicated and a simplified model is used. To do so, we consider a simplified attitude system model for the pitch channel in the system. We aim to achieve a non-rotating motion of missiles. It is worthwhile to note that the attitude control design for the pitch channel and the yaw/roll channel can be solved exactly in the same way while representing matrices of the system are different.
The System
editThe state-space representation for the pitch channel can be written as follows:
where , , , and are the state variable, control input, output, and disturbance vectors, respectively. The paprameters , , , , , , and stand for the attack angle, pitch angular velocity, the elevator deflection, the input actuator deflection, the overload on the side direction, the sideslip angle, and the yaw angular velocity, respectively.
The Data
editIn the aforementioned pitch channel system, the matrices and are given as:
where and are the system parameters. Moreover, is the speed of the missle and , , and are the rotary inertia of the missle corresponding to the body coordinates.
The Optimization Problem
editThe optimization problem is to find a state feedback control law such that:
1. The closed-loop system:
is stable.
2. The norm of the transfer function:
is less than a positive scalar value, . Thus:
The LMI: LMI for non-rotating missle attitude control
editUsing Theorem 8.1 in [1], the problem can be equivalently expressed in the following form:
Conclusion:
editAs mentioned, the aim is to attenuate the disturbance on the performance of the missile. The parameter is the disturbance attenuation level. When the matrices and are determined in the optimization problem, the controller gain matrix can be computed by:
Implementation
editA link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Non-rotating-Missle-Attitude-Control
Related LMIs
editLMI for Attitude Control of Nonrotating Missles, Yaw/Roll Channel
External Links
edit- [51] - LMI in Control Systems Analysis, Design and Applications
Return to Main Page
editLMI for Attitude Control of Nonrotating Missiles, Yaw/Roll Channel
editLMI for Attitude Control of Nonrotating Missles, Yaw/Roll Channel
Deriving the exact dynamic modeling of a missile is a very complicated procedure. Thus, a simplified model is used to model the missile dynamics. To do so, we consider a simplified attitude system model for the yaw/roll channel of the system. We aim to achieve a non-rotating motion of missiles. Note that the attitude control design for the yaw/roll channel and the pitch channel can be solved exactly in the same way except for different representing matrices of the system.
The System
editThe state-space representation for the yaw/roll channel can be written as follows:
where , , , and are the state variable, control input, output, and disturbance vectors, respectively. The paprameters , , , , , , and stand for the attack angle, pitch angular velocity, the elevator deflection, the input actuator deflection, the overload on the side direction, the sideslip angle, and the yaw angular velocity, respectively.
The Data
editIn the aforementioned yaw/roll channel system, the matrices and are given as:
where
and
where and are the system parameters. Moreover, is the speed of the missle and , , and are the rotary inertia of the missle corresponding to the body coordinates.
The Optimization Problem
editThe optimization problem is to find a state feedback control law such that:
1. The closed-loop system:
is stable.
2. The norm of the transfer function:
is less than a positive scalar value, . Thus:
The LMI: LMI for non-rotating missle attitude control
editUsing Theorem 8.1 in [1], the problem can be equivalently expressed in the following form:
Conclusion:
editAs mentioned, the aim is to attenuate the disturbance on the performance of the missile. The parameter is the disturbance attenuation level. When the matrices and are determined in the optimization problem, the controller gain matrix can be computed by:
Implementation
editA link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Attitude-Control-Nonrotating-Missle-Yaw-Roll-Channel
Related LMIs
editLMI for Attitude Control of Nonrotating Missles, Pitch Channel
External Links
edit- [52] - LMI in Control Systems Analysis, Design and Applications
Return to Main Page
editLMI for H2/Hinf Polytopic Controller for Robot Arm on a Quadrotor
editLMIs in Control/Print version
The System:
edit
The Optimization Problem:
editGiven a state space system of
where ,, and form the K matrix as defined in below. This, therefore, means that the Regulator system can be re-written as:
With the above 9-matrix representation in mind, the we can now derive the controller needed for solving the problem, which in turn will be accomplished through the use of LMI's. Firstly, we will be taking our /state-feedback control and make some modifications to it. More specifically, since the focus is modeling for worst-case scenario of a given parameter, we will be modifying the LMI's such that the mixed / controller is polytopic.
The LMI:
edit/ Polytopic Controller for Quadrotor with Robotic Arm.
Recall that from the 9-matrix framework , and represent our process and sensor noises respectively and represents our input channel. Suppose we were interested in modeling noise across all three of these channels. Then the best way to model uncertainty across all three cases would be modifying the matrix to , where ( parameters, , and is a constant noise value). This, in turn results in our - matrices to be modifified to -
Using the LMI's given for optimal /-optimal state-feedback controller from Peet Lecture 11 as reference, our resulting polytopic LMI becomes:
+
CD=0
where i=1,..,k,& and and:
After solving for both the optimal and gain ratios as well as , we can then construct our worst-case scenario controller by setting our matrix (and consequently our matrices) to the highest value. This results in the controller:
which is constructed by setting:
where:
Conclusion:
editThe LMI is feasible and the resulting controller is found to be stable under normal noise disturbances for all states.
Implementation
editReferences
edit1. An LMI-Based Approach for Altitude and Attitude Mixed H2/Hinf-Polytopic Regulator Control of a Quadrotor Manipulator by Aditya Ramani and Sudhanshu Katarey.
An LMI for the Kalman Filter
editLMIs in Control/Print version
This is a An LMI for the Kalman Filter. The Kalman Filter is one of the most widely used state-estimation techniques. It has applications in multiple aspects of navigation (inertial, terrain-aided, stellar.)
The System
editContinuous Time:
The process and sensor noises are given by and respectively.
Discrete Time:
The process and sensor noises are given by and respectively.
The Data
editThe data required for the Kalman Filter include a model of the system that the states are trying to be output and a measurement that is the output of the system dynamics being estimated.
The Filter
editThe Filter and Estimator equations can be written as:
Continuous Time
Discrete Time
The Error
editThe error dynamics evolve according to the following expression
Continuous Time
Discrete Time
The Optimization Problem
editThe Kalman Filtering (or LQE) problem is a Dual to the LQR problem. Replace the matrices from LQR with
The Kalman Filter chooses to minimize the cost This cost can be thought of as the covariance of the state error between the actual and estimated state. When the state error covariance is low the filter has converged and the estimate is good.
The Luenberger or Kalman gain can be computed from
The process and measurement noise covariances for the Kalman filter are given by
The matrix satisfies the following equality
We also cover the discrete Kalman Filter formulation which is more useful for real-life computer implementations.
The discrete Kalman filter chooses the gain where the PSDs of the process and sensor noises are given by
The steady-state covariance of the error in the estimated state is given by and satisfies the following Riccati equation.
- Objective: State Estimate Error Covariance
- Variables: Observer Gains
- Constraints: Dynamics of System to be Estimated
The LMI: H2-Optimal Control Full-State Feedback to LQR to Kalman Filter
editThe Kalman Filter is a dual to the LQR problem which has been shown to be equivalent to a special case of H2-static state feedback.
Start with the H2-Optimal Control Full-State Feedback.
The following are equivalent
To solve the LQR problem using H2 optimal state-feedback control the following variable substitutions are required.
Then
This results in the following LMI.
To solve the Kalman Filtering problem using the LQR LMI replace with and This results in the following LMI.
The discrete-time Kalman Filtering LMI is saved for another page as it requires derivation of the Discrete-Time LQR LMI problem which was not covered in class.
Conclusion:
editThe LMI for the Kalman Filter allows us to calculate the optimal gain for state estimation. It is shown that it can be found as a special case of the H2-optimal state feedback with the appropriate substitution of matrices. The LMI gives us a different way of computing the optimal Kalman gain.
Implementation
editA link to CodeOcean or other online implementation of the LMI
Related LMIs
editLinks to other closely-related LMIs
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd
- [53] - Applied Optimal Estimation, Arthur Gelb (Kalman Filtering and Optimal Estimation Classic)
- [54] - All Source Position, Navigation and Timing. New textbook that has a good development and description of the Kalman Filter and some of its very practical uses and implementations by Boeing Senior Technical Fellow Ken Li.
Return to Main Page:
editHinf Optimal Model Reduction
editGiven a full order model and an initial estimate of a reduced order model it is possible to obtain a reduced order model optimal in sense. This methods uses LMI techniques iteratively to obtain the result.
The System
editGiven a state-space representation of a system and an initial estimate of reduced order model .
Where and . Where are full order, reduced order, number of inputs and number of outputs respectively.
The Data
editThe full order state matrices and the reduced model order .
The Optimization Problem
editThe objective of the optimization is to reduce the norm distance of the two systems. Minimizing with respect to .
The LMI: The Lyapunov Inequality
editObjective: .
Subject to::
It can be seen from the above LMI that the second matrix inequality is not linear in . But making constant it is linear in . And if are constant it is linear in . Hence the following iterative algorithm can be used.
(a) Start with initial estimate obtained from techniques like Hankel-norm reduction/Balanced truncation.
(b) Fix and optimize with respect to .
(c) Fix and optimize with respect to .
(d) Repeat steps (b) and (c) until the solution converges.
Conclusion:
editThe LMI techniques results in model reduction close to the theoretical limits set by the largest removed hankel singular value. The improvements are often not significant to that of Hankel-norm reduction. Due to high computational load it is recommended to only use this algorithm if optimal performance becomes a necessity.
External Links
editA list of references documenting and validating the LMI.
- Model order Reduction using LMIs - A conference paper by Helmersson, Anders, Proceedings of the 33rd IEEE Conference on Decision and Control, 1994, p. 3217-3222 vol.4
Return to Main Page:
editAn LMI for Multi-Robot Systems
editAn LMI for Multi-Robot Systems
Helicopter Inner Loop LMI
editLMIs in Control/Print version
This is a Helicopter Inner Loop LMI. Optimization methods and optimal control have had difficulty gaining traction in the rotorcraft control law community. However, this LMI derived in the referenced paper attempts to address the issues with a LMI for Robust, Optimal Control.
The System
editContinuous Time:
The Helicopter model is given by knowledge of the stability and control derivatives which populate the elements of the matrices in the dynamic equations above.
The state vector is given by the typical elements of a rigid 6-DOF body model. . The input vector is given by which pertain to the main rotor collective, longitudinal/lateral cyclic and tail rotor collective blade angles in radians.
The gust disturbance is denoted by and is assumed to be random in nature. The stability and control derivative matrices are modeled with uncertainty as follows:
The terms represent the uncertainties in the helicopter system model.
The Data
editThe Data required for this LMI are the stability and control derivatives that populate the A and B-matrices of the system above which can be obtained from linearizing non-linear models. It can also be obtained from experimental methods such as step responses and swept sines (System Identification.)
The Control Architecture
editA control architecture for the inner loop of the helicopter model mentioned above is designed using a state feedback control law.
The objective for the inner loop control is to design a full state feedback law such that the closed-loop helicopter system satisfies the following 3 performance specifications.
The Optimization Problem
editObjective 1: The closed-loop system is internally stable for any admissible uncertainty.
Objective 2: Poles of the close-loop system lie within the disk with center and radius , for any admissible uncertainty.
Objective 3: Given gust disturbance suppression index , for any admissible uncertainty, the effect of the gust disturbance to selected flight states and control input is in the given level, i.e.
where and are weighting matrices with appropriate dimensions and
It can be shown that the inner loop performance specifications listed in Objectives 1-3 can be met with a state feedback control law if the LMI described in the following section is true.
- Objective: Objectives listed above
- Variables: Controller Gains
- Constraints: Rotorcraft Dynamics and Modeled Actuator Limits
The LMI: H-Inf Inner Loop D-Stabilization Optimization
editThe paper derives and LMI of the form below and asserts that the if there exists a constant , matrix with appropriate dimensions and a symmetric positive matrix , such that
where,
This LMI is shown to satisfy Objectives 1, 2,3, and the control law is given by
Conclusion:
editThe LMI for Helicopter Inner Loop Control design provides an optimization-based approach towards achieving Level 1 Handling Qualities per ADS-33E. This is an interesting way to approach a very difficult problem that has usually been approached through classical control methods and with extensive piloted simulation and flight test.
Implementation
editA link to CodeOcean or other online implementation of the LMI
Related LMIs
editLinks to other closely-related LMIs
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- [55] - Multi-Mode Flight Control for an Unmanned Helicopter Based on Robust H∞ D-Stabilization and PI Tracking Configuration
Return to Main Page:
editHinf LMI Satellite Attitude Control
editLMIs in Control/Print version
This is a LMI for Satellite Attitude Control. Satellite attitude control is necessary to allow satellites in orbit accomplish their mission. Poor satellite attitude control results in poor pointing performance which can result in increased cost, delayed service, and reduced lifetime of the satellite.
The System
editThe full derivation of the system from first principles is accomplished in the companion LMI for Satellite Attitude Control. The link to that page is at the bottom with the references.
Continuous Time:
The above model was derived by substituting satellite attitude kinematics into the attitude dynamics of a satellite. The following are definitions of the variables above:
- Moments of inertia about the corresponding axis:
- Euler Angles:
- Disturbance Torques (flywheel, gravitational, and disturbance):
- Rotational-angular velocity of the Earth:
The state-space representation of the system can be found by the following steps. Let
Introduce the notations
where stand for any element in . Then the state-space system is:
where the matrices in the above state-space representation are defined as follows:
The Data
editData required for this LMI include moments of inertia of the satellite being controlled and the angular velocity of the earth. Any knowledge of the disturbance torques would also facilitate solution of the problem.
The Optimization Problem
editThe idea is to design a state feedback control law for the previous satellite state-space system of the form
This control law is designed so that the closed-loop system is stable and the transfer function matrix from disturbance to output
satisfies
for a minimal positive scalar which represents the minimum attenuation level.
The idea here is to attenuate the disturbances as much as possible while still maintaining the ability of the satellite to track. This minimum attenuation level is found from the LMI in the following section.
- Objective: Hinf norm
- Variables: Controller Gains
- Constraints: Satellite Attitude Dynamics and Kinematics. Maximum safe rotational rate of Satellite, maximum jet pulse thrust
The LMI: Feedback Control of the Satellite System
editDuan and Yu approach the satellite system as follows. The minimum attenuation level from disturbance to output can be found by solving the following LMI optimization problem.
which is the same as Theorem 8.1 in Duan and Yu's Book, the solution to the problem.
Conclusion:
editThe Duan and Yu textbook takes as typical values of the satellite moment of inertias as:
They then proceed to solve the optimization problem to find a controller gain that yields an attenuation level of 0.0010. Though this value is very small and represents very good attenuation the optimized controller pushes the poles of the closed loop system very close to the imaginary axis, resulting in slow oscillatory behavior with a very long settling time.
To address this a second approach was used by the authors which involves modifying the final LMI in the expression above and requiring that it be constrained as follows
These results are planned verified in the linked code implementation using YALMIP, whereas the authors took advantage of the MATLAB LMI Toolbox to achieve their results.
Implementation
editA link to CodeOcean or other online implementation of the LMI
Related LMIs
editLinks to other closely-related LMIs
- H2_LMI_SatelliteAttitudeControl
- Optimal_Output_Feedback_Hinf_LMI
- Full-State_Feedback_Optimal_Control_Hinf_LMI
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- [56] - LMIs in Control Systems: Analysis, Design, and Applications - Duan and Yu
H2 LMI Satellite Attitude Control
editLMIs in Control/Print version
This is a H2 LMI for Satellite Attitude Control
Satellite attitude control is important for military, civil, and scientific activities. Attitude control of a satellite involves fast maneuvering and accurate pointing in the presence of all kinds of disturbances and parameter uncertainties.
The System
editThe satellite state-space formulation is given in the LMI page for Satellite Attitude Control which is also in the applications section of this WikiBook. This section discusses the derivation of that state-space formulation based on first principles.
The attitude dynamics of a satellite in an inertial coordinate system can be described in terms of the time rate of change of its angular momentum and the sum of the external torques and moments acting on the system. That is:
where the following variables are defined as follows:
- are the flywheel torque, the gravitational torque, and the disturbance torque.
- is the total momentum/torque acting on the satellite
- is the inertia matrix/tensor for the satellite
- is the angular velocity vector of the satellite.
The time derivative of the total angular momentum in an arbitrary rotating reference frame (such as the body frame of the satellite) is given by:
which takes into the account of the angular velocity of the rotating reference frame relative to the inertial reference frame where Newton's laws are valid.
Combining equations, collecting terms and choosing the principle axes of the spacecraft so that the Inertia Tensor is diagonalized yields the following equations of motion:
Using the small angle approximation, the angular velocity of the satellite in the inertial coordinate system represented in the body coordinate system can be written as
These equations form the basis of the state-space representation used in the H-inf LMI for satellite attitude control. For clarity, they are repeated below.
The Data
editData required for this LMI include moments of inertia of the satellite being controlled and the angular velocity of the earth. Any knowledge of the disturbance torques would also facilitate solution of the problem.
The Optimization Problem
editThe optimization problem seeks to minimize the H2 norm of the transfer function from disturbance to output. Thus, we expect slightly different results than the H-inf case. Deriving the H2 control problem and setup also serves for useful setup for the mixed H-inf/H2 optimization that the book follows up with later.
- Objective: H2 norm
- Variables: Controller Gains
- Constraints: Satellite Attitude Dynamics and Kinematics. Maximum safe rotational rate of Satellite, maximum jet pulse thrust
The LMI: H-2 Satellite Attitude Control
editDuan and Yu use the following H-2 Satellite Attitude Control LMI to minimize the attenuation level from disturbance to output. Note that in the H2-case we are minimizing the integral of the magnitude of the bode plot transfer function whereas in the H-inf case the optimization is minimizing the maximum value of the bode plot magnitude.
To design an optimizing controller of the form
such that the closed-loop system is stable and the transfer function matrix
satisfies
for a minimal positive scalar .
This scalar is found from the solution of the following LMI
and the controller is given by
Conclusion:
editThe LMI for H-2 Satellite Attitude Control comes up with a different attenuation value for the disturbance vs the H-inf problem which is expected. It also serves for good preparation for the mixed H2/H-inf problem that Duan and Yu cover in a later section. Though no implementation is included for the mixed H2/H-inf optimization problem it is interesting to compare the results of all three cases for the satellite attitude control problem.
Implementation
editA link to CodeOcean or other online implementation of the LMI
Related LMIs
editLinks to other closely-related LMIs
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- [57] - LMIs in Control Systems: Analysis, Design, and Applications - Duan and Yu
Problem of Space Rendezvous and LMI Approaches
editLMIs in Control/Print version
This is a Problem of Space Rendezvous and LMI Approaches
In Section 12.4 of their book LMIs in Control Systems: Analysis, Design, and Applications, Duan and Yu discuss the problem of space rendezvous and how it can be formulated into an LMI problem. Modeling and simulating space rendezvous is of importance because it is used for any cargo or passenger spacecraft traveling to and from earth-orbiting space stations and also for satellites servicing aging in-orbit satellites, and for potential missions to mine asteroids.
The System
editThough Duan and Yu first mention space rendezvous in Example 7.14 of their book. In this example, they show that the relative orbital dynamic model of spacecraft rendezvous can be described by the famous Clohessy-Wiltshire equations.
where
- are the components of the relative position between chaser and target
- [rad/h] is the orbital angular velocity of the target satellite
- is the mass of the chaser
- is the i-th component of the control input force acting on the relative motion dynamics
- is the i-th component of the external disturbance
The C-W equations give a first-order approximation of the chaser's motion in a target-centered coordinate system and is often used in planning space rendezvous problems (ISS, Salyut, and Tiangong space stations are just some examples.)
With appropriate definitions of states and variables the dynamic equations of motion for space-rendezvous can be converted into standard state-space form for LMI optimization as follows:
where the vectors in the above state-space representation are defined as follows:
and the matrices in the above state-space representation are defined as follows:
The Data
editThe data required are the mass properties of both the target and chaser vehicles for space rendezvous. Also required is the orbital angular velocities of the target and chasers and measurements of relative kinematics between the two.
The Optimization Problem
editThe optimization problem is trying to attenuate the disturbance to output transfer function using either the H-inf or H2 norm.
- Objective: Hinf or H2 norm
- Variables: Controller Gains
- Constraints: Relative Dynamics/Kinematics between Chaser and Target in Orbit
The LMI: Space Rendezvous LMI Optimization
editThe space rendezvous problem can be approached with either H-inf or H-2 optimization formulations. Both formulations can achieve closed-loop stability which ensures that rendezvous occurs because the relative distance between target and chaser eventually approaches zero. The LMIs for the H-inf and H2 optimization problem are shown below which are easily solvable because the matrices for the space rendezvous problem are available above in standard form.
Duan and Yu approach the . The minimum attenuation level from disturbance to output can be found by solving the following LMI optimization problem.
which is the same as Theorem 8.1 in Duan and Yu's Book, the solution to the problem.
Conclusion:
editThe LMI for Space Rendezvous is a useful and interesting method to model and simulate practical problems in spacecraft engineering. Space Rendezvous usually requires very good vision-based navigation or an exceptional human operator that can close the gap for final mating of the two docking adapters.
Implementation
editA link to CodeOcean or other online implementation of the LMI
Related LMIs
editLinks to other closely-related LMIs
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- [58] - LMIs in Control Systems: Analysis, Design, and Applications - Duan and Yu
Template
editThis methods uses LMI techniques iteratively to obtain the result.
The System
editGiven a state-space representation of a system and an initial estimate of reduced order model .
Where and .
The Data
editThe full order state matrices .
The Optimization Problem
editThe objective of the optimization is to reduce the norm .
The LMI: The Lyapunov Inequality
editObjective: .
Subject to::
Conclusion:
editThe LMI techniques results in model reduction close to the theoretical bounds.
External Links
editA list of references documenting and validating the LMI.
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.