User:ShakespeareFan00/Sandbox
Basic Matrix Theory edit
Basic Matrix Notation edit
Consider 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 edit
Hermitian, 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 edit
Notation of Positivity edit
A 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) edit
KYP 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 edit
where , , , at any .
The Data edit
The matrices are known.
The Optimization Problem edit
The following optimization problem must be solved.
The LMI: The KYP or Bounded Real Lemma edit
Suppose is the system. Then the following are equivalent.
Conclusion: edit
The 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 edit
Since 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 edit
External Links edit
A 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 edit
Positive 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 edit
where , , , at any .
The Data edit
The matrices are known.
The LMI: The Positive Real Lemma edit
Suppose is the system. Then the following are equivalent.
Conclusion: edit
The 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 edit
This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/Positive_Real_Lemma.m
Related LMIs edit
KYP Lemma (Bounded Real Lemma)
External Links edit
A 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 edit
The Concept edit
In 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 edit
Consider a contiuous-time LTI system, , with minimal state-space realization (A, B, C, D), where and .
The Data edit
The matrices and which defines the state space of the system
The Optimization Problem edit
The system is QSR disipative if
where is the input to is the output of and .
LMI : KYP Lemma for QSR Dissipative Systems edit
The system is also QSR dissipative if and only if there exists where such that
Conclusion: edit
If there exist a positive definite for the the selected Q,S and R matrices then the system is QSR dissipative.
Implementation edit
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs edit
References edit
1. 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 edit
The Concept edit
It 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 edit
Consider a contiuous-time LTI system, , with minimal state-space relization (A, B, C, 0), where and .
The Data edit
The matrices The matrices and
LMI : KYP Lemma without Feedthrough edit
The 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: edit
If there exist a positive definite for the the selected Q,S and R matrices then the system is Positive Real.
Implementation edit
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs edit
KYP Lemma
State Space Stability
Discrete Time KYP Lemma with Feedthrough
References edit
1. 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 edit
The Concept edit
Descriptor 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 edit
Consider a square, contiuous-time linear time-invariant (LTI) system, , with minimal state-space relization (E, A, B, C, D), where and .
The Data edit
The matrices The matrices and
LMI : KYP Lemma for Descriptor Systems edit
The 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: edit
If there exist a X and W matrix satisfying above LMIs then the system is Extended Strictly Positive Real.
Implementation edit
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs edit
KYP Lemma
State Space Stability
Discrete Time KYP Lemma with Feedthrough
References edit
1. 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 edit
The Concept edit
The 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 edit
Consider 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 edit
The matrices The matrices and . The values of a and b
LMI : Generalized KYP (GKYP) Lemma for Conic Sectors edit
The 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: edit
If there exist a positive definite matrix satisfying above LMIs for the given frequency bandwidths then the system is inside the cone [a,b].
Implementation edit
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs edit
KYP Lemma
State Space Stability
Exterior Conic Sector Lemma
Modified Exterior Conic Sector Lemma
References edit
1. 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 edit
Discrete-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 edit
Discrete-Time LTI System with state space realization
The Data edit
The matrices: System .
The Optimization Problem edit
The following feasibility problem should be optimized:
is minimized while obeying the LMI constraints.
The LMI: edit
Discrete-Time Bounded Real Lemma
The LMI formulation
H∞ norm <
Conclusion: edit
The 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 edit
A 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 edit
A 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 edit
The Concept edit
In 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 edit
Consider a discrete-time LTI system, , with minimal state-space relization , where and .
The Data edit
The matrices and
The Optimization Problem edit
The system is QSR disipative if
where is the input to is the output of and .
LMI : Discrete-Time KYP Lemma for QSR Dissipative Systems edit
The system is also QSR dissipative if and only if there exists where such that
Conclusion: edit
If there exist a positive definite for the the selected Q,S and R matrices then the system is QSR dissipative.
Implementation edit
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs edit
KYP Lemma
KYP Lemma for continous Time QSR Dissipative system
References edit
1. 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 edit
The Concept edit
It 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 edit
Consider a discrete-time LTI system, , with minimal state-space relization , where and .
The Data edit
The matrices and
LMI : Discrete-Time KYP Lemma with Feedthrough edit
The 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: edit
If there exist a positive definite for the the selected Q,S and R matrices then the system is Positive Real.
Implementation edit
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs edit
KYP Lemma
State Space Stability
KYP Lemma without Feedthrough
References edit
1. 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 edit
An important tool for proving many LMI theorems is the Schur Compliment. It is frequently used as a method of LMI linearization.
The Schur Compliment edit
Consider 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 edit
LMI 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 edit
Assume that we have a matrix function of variables :
where are symmetric matrices.
The Data edit
The symmetric matrices () are given.
The Optimization Problem edit
The 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 edit
This optimization problem can be converted to an LMI problem.
The mathematical description of the LMI formulation can be written as follows:
Conclusion: edit
As a result, the variables after solving this LMI problem.
Moreover, we obtain the maximum eigenvalue, , of the matrix .
Implementation edit
A 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 edit
LMI 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 edit
LMI 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 edit
Assume that we have a matrix function of variables :
where are symmetric matrices.
The Data edit
The symmetric matrices () are given.
The Optimization Problem edit
The 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 edit
This optimization problem can be converted to an LMI problem.
The mathematical description of the LMI formulation can be written as follows:
Conclusion: edit
As a result, the variables after solving this LMI problem and we obtain that is the norm of matrix function .
Implementation edit
A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Matrix-Norm-Minimization
Related LMIs edit
LMI for Matrix Norm Minimization
LMI for Generalized Eigenvalue Problem
LMI for Maximum Singular Value of a Complex Matrix
External Links edit
A list of references documenting and validating the LMI.
- [3] - LMI in Control Systems Analysis, Design and Applications
LMI for Generalized Eigenvalue Problem edit
LMI 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 edit
Assume that we have three matrice functions which are functions of variables as follows:
where are , , and () are the coefficient matrices.
The Data edit
The , , and are matrix functions of appropriate dimensions which are all linear in the variable and , , are given matrix coefficients.
The Optimization Problem edit
The problem is to find such that:
, , and are satisfied and is a scalar variable.
The LMI: LMI for Schur stabilization edit
A mathematical description of the LMI formulation for the generalized eigenvalue problem can be written as follows:
Conclusion: edit
The 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 edit
A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Schur-Stability
Related LMIs edit
LMI 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 edit
LMI 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 edit
We define the objective function as:
and constraints of the problem as:
.
.
.
The Data edit
Suppose that , , and are given parameters where and . Moreover, is an vector of positive variables.
The Optimization Problem edit
The optimization problem is to minimize the objective function, when the aforementioned linear constraints are satisfied.
The LMI: LMI for linear programming edit
The mathematical description of the optimization problem can be readily written in the following LMI formulation:
Conclusion: edit
Solving 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 edit
A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Linear-Programming
Related LMIs edit
External Links edit
- [5] - LMI in Control Systems Analysis, Design and Applications
LMI for Feasibility Problem edit
LMI 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 edit
Assume that we have two matrices as follows:
which are matrix functions of variables .
The Data edit
Suppose that the matrices and are given.
The Optimization Problem edit
The optimization problem is to find variables such that the following constraint is satisfied:
The LMI: LMI for Feasibility Problem edit
This 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: edit
In 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 edit
A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Feasibility-Problem-of-Convex-Optimization
Related LMIs edit
External Links edit
- [6] - LMI in Control Systems Analysis, Design and Applications
Structured Singular Value edit
User:ShakespeareFan00/Sandbox
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 edit
The Data edit
The matrices .
The Optimization Problem edit
The LMI: edit
Conclusion: edit
Implementation edit
https://github.com/mcavorsi/LMI
Related LMIs edit
External Links edit
- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
Eigenvalue Problem edit
User:ShakespeareFan00/Sandbox
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 edit
The Data edit
The matrices .
The Optimization Problem edit
The LMI: edit
Conclusion: edit
The eigenvalue problem can be utilized to minimize the maximum eigenvalue of a matrix that depends affinely on a variable.
Implementation edit
https://github.com/mcavorsi/LMI
Related LMIs edit
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 for Minimizing Condition Number of Positive Definite Matrix edit
User:ShakespeareFan00/Sandbox
The System: edit
A 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: edit
The GEVP can be formulated as follows:
minimize
subject to > 0;
>0;
< < .
We can reformulate this GEVP as an EVP as follows. Suppose,
= + , = +
The LMI: edit
Defining the new variables = , = we can express the previous formulation as the EVP with variables and :
miminize
subject to + >0; < + <
Conclusion: edit
The LMI is feasible.
Implementation edit
References edit
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
Continuous Quadratic Stability edit
User:ShakespeareFan00/Sandbox
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 edit
The Data edit
The 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 edit
The 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 edit
Consider 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 edit
Consider 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: edit
If feasible, System is Quadratically stable for any
Implementation edit
https://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 edit
The Concept edit
The 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 edit
Consider a square, contiuous-time linear time-invariant (LTI) system, , with minimal state-space relization (A, B, C, D), where and .
The Data edit
The matrices The matrices and
LMI : Exterior Conic Sector Lemma edit
The 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: edit
If there exist a positive definite matrix satisfying above LMIs then the system is in the exterior cone of radius r centered at c.
Implementation edit
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs edit
KYP Lemma
State Space Stability
References edit
1. 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 edit
The Concept edit
The 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 edit
Consider a square, contiuous-time linear time-invariant (LTI) system, , with minimal state-space relization (A, B, C, D), where and .
The Data edit
The matrices The matrices and
LMI : Modified Exterior Conic Sector Lemma edit
The 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: edit
If there exist a positive definite matrix satisfying above LMIs then the system is in the exterior cone of radius r centered at c.
Implementation edit
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs edit
KYP Lemma
State Space Stability
Exterior Conic Sector Lemma
References edit
1. 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 edit
The continuous-time DC gain is the transfer function value at the frequency s = 0.
The System edit
Consider 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 edit
The transfer matrix is given by
The DC Gain of the system is strictly less than if the following LMIs are satisfied.
OR
Conclusion edit
The 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 edit
A 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 edit
Discrete-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 edit
Discrete-Time LTI System with state space realization
The Data edit
The matrices: System .
The Optimization Problem edit
The following feasibility problem should be optimized:
is minimized while obeying the LMI constraints.
The LMI: edit
Discrete-Time Bounded Real Lemma
The LMI formulation
H2 norm <
Conclusion: edit
The H2 norm is the minimum value of that satisfies the LMI condition.
Implementation edit
A link to CodeOcean or other online implementation of the LMI
MATLAB Code
Related LMIs edit
[9] - Continuous time H2 norm.
External Links edit
A 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 edit
The Concept edit
The 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 ouput closely tracks the reference input.
The System edit
Consider a discrete-time LTI system, , with minimal state-space relization , where and .
The Data edit
The matrices and
LMI : Discrete-Time Minimum Gain Lemma edit
The 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: edit
If 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 edit
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs edit
KYP Lemma
State Space Stability
KYP Lemma without Feedthrough
References edit
1. 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 edit
The Concept edit
The 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 ouput closely tracks the reference input.
The System edit
Consider a discrete-time LTI system, , with minimal state-space relization , where and .
The Data edit
The matrices and
LMI : Discrete-Time Modified Minimum Gain Lemma edit
The 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: edit
If there exist a positive definite for the system , then the minimum gain of the system γ can be obtaied from above defined LMIs.
Implementation edit
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs edit
KYP Lemma
State Space Stability
KYP Lemma without Feedthrough
Discrete Time Minimum Gain Lemma
References edit
1. 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 edit
User:ShakespeareFan00/Sandbox
The System edit
Consider a Discrete-Time LTI system
Consider Ad ∈ nxn ; Bd ∈ nxm
The Data edit
The 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 edit
Our 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) edit
An 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: edit
Algebraic 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 edit
LMI for Continuous-Time Algebraic Riccati Inequality
LMI for Schur Stabilization
External Links edit
A list of references documenting and validating the LMI.
- [11] - LMI in Control Systems Analysis, Design and Applications
Deduced LMI Conditions for H-infinity Index edit
H-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 edit
A 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 edit
Number 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 edit
For 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: edit
If there is a feasible solution to the aforementioned LMI, then the upper bounds the infinity norm of the system G(s).
Implementation edit
To 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 edit
External Links edit
A 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 edit
H2 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 edit
We 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 edit
The system matrices are known.
The Optimization Problem edit
For 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 edit
These 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: edit
If there is a feasible solution to the aforementioned LMI, then the upper bounds the norm of the system G(s).
Implementation edit
To 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 edit
Bounded Real Lemma
Deduced LMIs for H-infinity index
External Links edit
A 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 edit
Dissipativity 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 edit
A 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 edit
Number 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 edit
The 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: edit
If 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 edit
To 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 edit
Continuous_Time_Lyapunov_Inequality
External Links edit
A 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 edit
For 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 edit
In order to obtain the LMI, we need the following 2 matrices: .
The Optimization Problem edit
Suppose - 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 edit
From 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: edit
Given 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 edit
A 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 edit
For 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 edit
In order to obtain the LMI, we need the following 2 matrices: .
The Optimization Problem edit
Suppose - 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 edit
From 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: edit
Given 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 edit
A 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 edit
Suppose 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 edit
In order to determine the -norm of the system, we need the matrices , , and .
The Optimization Problem edit
Suppose 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 edit
Assuming that , this means that the following are equivalent:
Conclusion: edit
The 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 edit
A 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 edit
Algebraic 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 edit
The Data edit
Following matrices are needed as Inputs:.
- .
The Optimization Problem edit
In control systems theory, many analysis and design problems are closely related to Riccati algebraic equations or inequalities. Find
The LMI: Algebraic Riccati Inequality edit
Title and mathematical description of the LMI formulation.
Conclusion: edit
If the solution exists, LMIs give a unique, stabilizing, symmetric matrix P.
Implementation: edit
Matlab 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 edit
Let'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 edit
Consider a continuous-time LTI system, , with minimal statespace representation
The Data edit
The matrices:
The LMI: System Zeros without feedthrough edit
The transmission zeros of are the eigenvalues of , where . Therefore , is a minimum phase if and only if there exists , where such that
Conclusion: edit
If P exists, it ensures non-minimum phase. Eigenvalues of NAM then gives the zeros of the system.
Implementation edit
https://github.com/Ricky-10/coding107/blob/master/Systemzeroswithoutfeedthrough
External Links edit
A 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 edit
Let'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 edit
Consider a continuous-time LTI system, , with minimal statespace representation
The Data edit
The matrices needed as inputs are:
In this case,
The LMI: System Zeros with feedthrough edit
The transmission zeros of are the eigenvalues of . Therefore , is a minimum phase if and only if there exists , where such that
Conclusion: edit
If P exists, it ensures non-minimum phase. Eigenvalues of then gives the zeros of the system.
Related LMIs edit
LMIs_in_Controls/pages/systemzeroswithoutfeedthrough
Implementation edit
https://github.com/Ricky-10/coding107/blob/master/systemzeroswithfeedthrough
External Links edit
A 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 edit
User:ShakespeareFan00/Sandbox 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 edit
Consider a square continuous time Linear Time invariant system, with the state space realization
The Data edit
The LMI: LMI for Negative Imaginary Lemma edit
According 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 edit
The system is strictly negative if det() 0 and either of the above LMIs are feasible with resulting or
Implementation edit
A link to the Matlab code for a simple implementation of this problem in the Github repository:
https://github.com/yashgvd/ygovada
Related LMIs edit
Positive 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 edit
LMIs 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 edit
Suppose is a Banach Algebra and . If , then exists, and furthermore,
Proof edit
Assuming 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 edit
If , then this implies stability, since the higher exponents of in the summation of will converge to , instead of blowing up to infinity.
External Links edit
A 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 edit
Tangential Nevanlinna-Pick edit
The 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 edit
is 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 edit
Given:
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 edit
Problem (1) has a solution if and only if the following is true:
Nevanlinna-Pick Approach edit
Lyapunov Approach edit
N can also be found using the Lyapunov equation:
where
The tangential Nevanlinna-Pick problem is then solved by confirming that .
Conclusion: edit
If is positive (semi)-definite, then there exists a norm-bounded analytic function, which satisfies with
Implementation edit
Implementation requires YALMIP and a linear solver such as sedumi. [16] - MATLAB code for Tangential Nevanlinna-Pick Problem.
Related LMIs edit
Nevalinna-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 edit
Nevanlinna-Pick Interpolation with Scaling edit
The 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 edit
The 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 edit
Given:
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 edit
First, 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: edit
If 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 edit
Implementation requires YALMIP and Mosek. [17] - MATLAB code for Nevanlinna-Pick Interpolation.
Related LMIs edit
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 .
Generalized Norm edit
Generalized Norm edit
The 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 edit
Consider a continuous-time, linear, time-invariant system with state space realization where , , , amd is Hurwitz. The generalized norm of is:
The Data edit
The transfer function , and system matrices , , are known and is Hurwitz.
The LMI: Generalized Norm LMIs edit
The 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: edit
The generalized norm of is the minimum value of that satisfies the LMIs presented in this page.
Implementation edit
This implementation requires Yalmip and Sedumi.
Generalized Norm - MATLAB code for Generalized Norm.
Related LMIs edit
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.
Passivity and Positive Realness edit
This section deals with passivity of a system.
The System edit
Given a state-space representation of a linear system
are the state, output and input vectors respectively.
The Data edit
are system matrices.
Definition edit
The 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 edit
Let the linear system be controllable. Then, the system is passive if an only if there exists such that
-
(
)
Implementation edit
This implementation requires Yalmip and Mosek.
Conclusion edit
Thus, 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 edit
This section studies the non-expansivity and bounded-realness of a system.
The System edit
Given a state-space representation of a linear system
are the state, output and input vectors respectively.
The Data edit
are system matrices.
Definition edit
The 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 edit
Let the linear system be controllable. Then, the system is bounded-real if an only if there exists such that
-
(
)
and
-
(
)
Implementation edit
This implementation requires Yalmip and Mosek.
Conclusion: edit
Thus, 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 edit
LMIs 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 edit
Consider , 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 edit
It 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 edit
A 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 edit
LMIs 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 edit
Consider , where . The matrix inequality is satisfied if and only if or equivalently, .
Example edit
Consider 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 edit
A 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 edit
A 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 edit
LMIs 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 edit
Consider and . There exists such that
if and only if there exists such that
Alternative Forms of Finsler's Lemma edit
Consider and . If there exists such that
holds for all ≠ satisfying , then there exists such that
Modified Finsler's Lemma edit
Consider and , where is less that on equal to , and . There exists such that
there exists such that
Conclusion edit
In summary, a number of identical methods have been stated above to determine the positive definiteness of LMIs.
External Links edit
A 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 edit
Time-Delay Systems edit
Parametric, Norm-Bounded Uncertain System Quadratic Stability edit
User:ShakespeareFan00/Sandbox
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 edit
The Data edit
The matrices .
The LMI: edit
Conclusion: edit
The system above is quadratically stable if and only if there exists some mu >= 0 and P > 0 such that the LMI is feasible.
Implementation edit
https://github.com/mcavorsi/LMI
Related LMIs edit
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.
Stability of Structured, Norm-Bounded Uncertainty edit
User:ShakespeareFan00/Sandbox
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 edit
The Data edit
The matrices .
The LMI: edit
Conclusion: edit
Implementation edit
https://github.com/mcavorsi/LMI
Related LMIs edit
Parametric, 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 edit
LMIs 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 edit
The Data edit
The matrices .
The LMI edit
Conclusion edit
The switched system is stable under arbitrary switching if there exists some P > 0 such that the LMIs hold.
Implementation edit
https://github.com/mcavorsi/LMI
Related LMIs edit
Parametric, 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 edit
User:ShakespeareFan00/Sandbox
The System edit
The Data edit
The matrices .
The Optimization Problem edit
The LMI: edit
Conclusion: edit
If there exists an then the system is quadratically stable, and the stability margin is the largest such .
Implementation edit
https://github.com/mcavorsi/LMI
Related LMIs edit
Parametric, 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 edit
The System edit
where and .
The Data edit
The matrices .
The LMI: edit
Solve the following LMIP
Implementation edit
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/50fc71737b69f2cf57d15634f2f19d091bf37d02
Conclusion edit
The stability of the above linear delayed differential equation is proved, using Lyapunov functionals of the form , if the provided LMIP is feasible.
Remark edit
The 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 edit
The System edit
where and depend affinity on parameter .
The Data edit
The matrices .
The Optimization Problem: edit
Solve the following semi-definite program
Implementation edit
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/5462bc1dc441bc298d50a2c35075e9466eba8355
Conclusion edit
The value function of the above semi-definite program returns the norm of the system.
Remark edit
It 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 edit
The System edit
where and depend affinely on parameter .
The Data edit
The matrices .
The Optimization Problem: edit
Solve the following semi-definite program
Implementation edit
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/02f31a2d7a22b2464dfe9212eb76409bda9439b1
Conclusion edit
The value function of the above semi-definite program returns a bound for -entropy of the system, which is defined as
Remark edit
When 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 edit
The System edit
where and depend affinely on parameter .
The Data edit
The matrices .
The Optimization Problem: edit
Solve the following semi-definite program
Implementation edit
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/b6cd6b81f75be4a2052ba3fa76cad1a2f9c49caa
Conclusion edit
The 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 edit
It 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 edit
The System edit
where and depend affinely on parameter .
The Data edit
The matrices .
The Optimization Problem: edit
Solve the following semi-definite program
where is the controllability Gramian, i.e., .
Implementation edit
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/0faedcdd9fba92bc27a318d80159c04a0b342f35
Conclusion edit
The 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 edit
is 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 edit
Positive 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 edit
Consider 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 edit
Number 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 edit
An 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: edit
The 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 edit
To 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 edit
A 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 edit
LMI 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 edit
where 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 edit
The system matrices , the saturation bounds of the control inputs. Positive scalars .
The LMI: The Stabilization Feasibility Condition edit
Here is a diagonal matrix with a component either 0 or 1, and and
Conclusion: edit
The feasibility of the given LMI implies that the system is stabilizable with control gains .
Implementation edit
A link to CodeOcean or other online implementation of the LMI
Related LMIs edit
External 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 edit
LMI 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 edit
where 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 edit
The matrices are known, as well as the bounds of the time-varying delay.
The Optimization Problem edit
For 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 edit
The 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: edit
For 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 edit
To 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 edit
A 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: edit
Conic Sector Lemma edit
Conic 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 edit
Consider 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 edit
The 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 edit
The 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: edit
The 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 edit
To 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 edit
External Links edit
A 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: edit
Polytopic Quadratic Stability edit
An important result to determine the stability of the system with uncertainties
The System: edit
Consider the system with Affine Time-Varying uncertainty (No input)
where
where
lies in either the intervals
or the simplex
where and
The Data edit
The matrix A and are known
The Optimization edit
The 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 edit
Conclusion: edit
Quadratic 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 edit
A link to implementation of the LMI
https://github.com/JalpeshBhadra/LMI/blob/master/polytopicstability.m
Related LMIs edit
External Links edit
A 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: edit
Mu Analysis edit
Mu 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: edit
Consider the continuous-time generalized LTI plant with minimal states-space realization
where it is assumed that is Invertible.
The Data edit
The matrices needed as inputs are only, and .
The LMI: - Analysis edit
The inequality holds if and only if there exist and , where , satisfying:
Conclusion: edit
The inequality holds for where X satisfies the above Inequality.
Implementation edit
External 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 edit
Optimization over an Affine Family of Linear Systems edit
Presented 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 edit
Consider 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 edit
The 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 edit
Several 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: edit
The 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 edit
Implementation 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 edit
Norm 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: edit
LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control
Hurwitz Stabilizability edit
This section studies the stabilizability properties of the control systems.
The System edit
Given 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 edit
are system matrices.
Definition edit
The 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 edit
The 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 edit
This implementation requires Yalmip and Mosek.
Conclusion edit
Compared 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: edit
Quadratic Hurwitz Stabilization for Polytopic Systems edit
This section studies the Quadratic Hurwitz stabilization for polytopic systems.
The System edit
Given a state-space representation of a linear system
-
(
)
LMI Condition edit
With , 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 edit
Stability 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 edit
Discrete-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 edit
Discrete-Time System
The Data edit
The matrices: System .
The Optimization Problem edit
The following feasibility problem should be optimized:
Find P obeying the LMI constraints.
The LMI: edit
Discrete-Time Bounded Real Lemma
The LMI formulation
Conclusion: edit
If there exists a satisfying the LMI then, and the equilibrium point of the system is Lyapunov stable.
Implementation edit
A link to CodeOcean or other online implementation of the LMI
MATLAB Code
Related LMIs edit
Continuous_Time_Lyapunov_Inequality - Lyapunov_Inequality
External Links edit
A 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: edit
LMI for Schur Stabilization edit
LMI 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 edit
We 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 and are given.
We define the scalar as with the range of .
The Optimization Problem edit
The 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 edit
The LMI for Schur stabilization can be written as minimization of the scalar, , in the following constraints:
Conclusion: edit
After 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 edit
A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Schur-Stability
Related LMIs edit
External Links edit
- [19] - LMI in Control Systems Analysis, Design and Applications
Return to Main Page edit
L2-Gain of Systems with Multiplicative noise edit
The System edit
where , are independent, identically distributed random variables with and is independent of the process .
The Data edit
The matrices .
The LMI: edit
Implementation edit
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/a34713575cd8ae9831cb5b7eb759d0fd45a8c37f
Conclusion edit
The optimal returns an upper bound on the gain of the system. .
Remark edit
It 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: edit
Discrete-Time Quadratic Stability edit
Discrete-Time Quadratic Stability edit
Stability 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 edit
The matrices .
The Optimization Problem edit
The following feasibility problem should be solved:
Where .
In case of polytopic uncertainty:
Conclusion: edit
This 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 edit
A 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: edit
Stability of Lure's Systems edit
The System edit
The Data edit
The matrices .
The LMI: The Lure's System's Stability edit
The following feasibility problem should be solved as sufficient condition for the stability of the above Lur'e system.
Implementation edit
https://codeocean.com/capsule/0232754/tree
Conclusion edit
If the feasibility problem with LMI constraints has solution, then the Lure's system is stable.
Remark edit
The 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: edit
L2 Gain of Lure's Systems edit
The System edit
The Data edit
The matrices .
The Optimization Problem: edit
The following semi-definite problem should be solved.
Implementation edit
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/12a7039f9e3d966e24b43fd58a3cce3725282ed2
Conclusion edit
The value function returns the square of the smallest provable upper bound on the gain of the Lure's system.
Remark edit
The 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: edit
Output Energy Bound for Lure's Systems edit
The System edit
The Data edit
The matrices .
The Optimization Problem: edit
The following optimization problem should be to find the tightest upper bound for the output energy of the above Lur'e system.
Implementation edit
Conclusion edit
The value function returns the the lowest bound for the energy function of the Lure's systems, i.e., with initial conditions .
Remark edit
The 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: edit
Stability of Quadratic Constrained Systems edit
The System edit
The Data edit
The matrices .
The LMI: edit
The following feasibility problem should be solved.
Implementation edit
https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/38f3b55ca7060a1260384a96e9dc31142af07a9a
Conclusion edit
The integral quadratic constrained system is stable if the provided LMI is feasible
Remark edit
The 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: edit
Conic Sector Lemma edit
Conic 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 edit
Consider 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 edit
The 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 edit
The 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: edit
The 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 edit
To 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 edit
External Links edit
A 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.