LMIs in Control/Print version

Basic Matrix TheoryEdit

Basic Matrix NotationEdit

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 MatriciesEdit

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 LinksEdit

Notion of Matrix PositivityEdit

Notation of PositivityEdit

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 MatriciesEdit

  • 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 LinksEdit

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 SystemEdit

where , , , at any .

The DataEdit

The matrices are known.

The Optimization ProblemEdit

The following optimization problem must be solved.

The LMI: The KYP or Bounded Real LemmaEdit

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.

ImplementationEdit

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 LMIsEdit

Positive Real Lemma

External LinksEdit

A list of references documenting and validating the LMI.


Positive Real LemmaEdit

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 SystemEdit

where , , , at any .

The DataEdit

The matrices are known.

The LMI: The Positive Real LemmaEdit

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.

ImplementationEdit

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/Positive_Real_Lemma.m

Related LMIsEdit

KYP Lemma (Bounded Real Lemma)

External LinksEdit

A list of references documenting and validating the LMI.


KYP Lemma for QSR Dissipative SystemsEdit

The ConceptEdit

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 SystemEdit

Consider a contiuous-time LTI system, , with minimal state-space realization (A, B, C, D), where and .

The DataEdit

The matrices and which defines the state space of the system

The Optimization ProblemEdit

The system is QSR disipative if

where is the input to is the output of and .


LMI : KYP Lemma for QSR Dissipative SystemsEdit

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.

ImplementationEdit

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIsEdit

KYP Lemma

ReferencesEdit

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 FeedthroughEdit

The ConceptEdit

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 SystemEdit

Consider a contiuous-time LTI system, , with minimal state-space relization (A, B, C, 0), where and .

The DataEdit

The matrices The matrices and

LMI : KYP Lemma without FeedthroughEdit

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.

ImplementationEdit

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIsEdit

KYP Lemma
State Space Stability
Discrete Time KYP Lemma with Feedthrough

ReferencesEdit

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 SystemsEdit

The ConceptEdit

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 SystemEdit

Consider a square, contiuous-time linear time-invariant (LTI) system, , with minimal state-space relization (E, A, B, C, D), where and .

The DataEdit

The matrices The matrices and

LMI : KYP Lemma for Descriptor SystemsEdit

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.

ImplementationEdit

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIsEdit

KYP Lemma
State Space Stability
Discrete Time KYP Lemma with Feedthrough

ReferencesEdit

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 SectorsEdit

The ConceptEdit

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 SystemEdit

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 DataEdit

The matrices The matrices and . The values of a and b

LMI : Generalized KYP (GKYP) Lemma for Conic SectorsEdit

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].

ImplementationEdit

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIsEdit

KYP Lemma
State Space Stability
Exterior Conic Sector Lemma
Modified Exterior Conic Sector Lemma

ReferencesEdit

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 LemmaEdit

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 SystemEdit

Discrete-Time LTI System with state space realization

The DataEdit

The matrices: System .

The Optimization ProblemEdit

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.

ImplementationEdit

A link to CodeOcean or other online implementation of the LMI
MATLAB Code

Related LMIsEdit

[1] - Continuous time KYP_Lemma_(Bounded_Real_Lemma)

External LinksEdit

A list of references documenting and validating the LMI.


Discrete Time KYP Lemma for QSR Dissipative SystemEdit

The ConceptEdit

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 SystemEdit

Consider a discrete-time LTI system, , with minimal state-space relization , where and .

The DataEdit

The matrices and

The Optimization ProblemEdit

The system is QSR disipative if

where is the input to is the output of and .


LMI : Discrete-Time KYP Lemma for QSR Dissipative SystemsEdit

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.

ImplementationEdit

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIsEdit

KYP Lemma
KYP Lemma for continous Time QSR Dissipative system

ReferencesEdit

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 FeedthroughEdit

The ConceptEdit

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 SystemEdit

Consider a discrete-time LTI system, , with minimal state-space relization , where and .

The DataEdit

The matrices and

LMI : Discrete-Time KYP Lemma with FeedthroughEdit

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.

ImplementationEdit

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIsEdit

KYP Lemma
State Space Stability
KYP Lemma without Feedthrough

ReferencesEdit

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 ComplementEdit

An important tool for proving many LMI theorems is the Schur Compliment. It is frequently used as a method of LMI linearization.

The Schur ComplimentEdit

Consider the matricies , , and where and are self-adjoint. Then the following statements are equivalent:

  1. and both hold.
  2. and both hold.
  3. is satisfied.

More concisely:

External LinksEdit

LMI for Eigenvalue MinimizationEdit

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 SystemEdit

Assume that we have a matrix function of variables :

where are symmetric matrices.

The DataEdit

The symmetric matrices () are given.

The Optimization ProblemEdit

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 minimizationEdit

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 .

ImplementationEdit

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 LMIsEdit

LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

LMI for Matrix Positivity

External LinksEdit

  • [2] - LMI in Control Systems Analysis, Design and Applications
  • Eigenvalues and Eigenvectors of a Matrix


LMI for Matrix Norm MinimizationEdit

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 SystemEdit

Assume that we have a matrix function of variables :

where are symmetric matrices.

The DataEdit

The symmetric matrices () are given.

The Optimization ProblemEdit

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 minimizationEdit

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 .

ImplementationEdit

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Matrix-Norm-Minimization

Related LMIsEdit

LMI for Matrix Norm Minimization

LMI for Generalized Eigenvalue Problem

LMI for Maximum Singular Value of a Complex Matrix

LMI for Matrix Positivity

External LinksEdit

A list of references documenting and validating the LMI.

  • [3] - LMI in Control Systems Analysis, Design and Applications

LMI for Generalized Eigenvalue ProblemEdit

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 SystemEdit

Assume that we have three matrice functions which are functions of variables as follows:

where are , , and () are the coefficient matrices.

The DataEdit

The , , and are matrix functions of appropriate dimensions which are all linear in the variable and , , are given matrix coefficients.

The Optimization ProblemEdit

The problem is to find such that:

, , and are satisfied and is a scalar variable.

The LMI: LMI for Schur stabilizationEdit

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.

ImplementationEdit

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Schur-Stability

Related LMIsEdit

LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

External LinksEdit

  • [4] - LMI in Control Systems Analysis, Design and Applications

LMI for Linear ProgrammingEdit

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 SystemEdit

We define the objective function as:

and constraints of the problem as:

.

.

.

The DataEdit

Suppose that , , and are given parameters where and . Moreover, is an vector of positive variables.

The Optimization ProblemEdit

The optimization problem is to minimize the objective function, when the aforementioned linear constraints are satisfied.

The LMI: LMI for linear programmingEdit

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.

ImplementationEdit

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Linear-Programming

Related LMIsEdit

LMI for Feasibility Problem

External LinksEdit

  • [5] - LMI in Control Systems Analysis, Design and Applications

LMI for Feasibility ProblemEdit

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 SystemEdit

Assume that we have two matrices as follows:

which are matrix functions of variables .

The DataEdit

Suppose that the matrices and are given.

The Optimization ProblemEdit

The optimization problem is to find variables such that the following constraint is satisfied:

The LMI: LMI for Feasibility ProblemEdit

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.

ImplementationEdit

A link to Matlab codes for this problem in the Github repository:

https://github.com/asalimil/LMI-for-Feasibility-Problem-of-Convex-Optimization

Related LMIsEdit

LMI for Linear Programming

External LinksEdit

  • [6] - LMI in Control Systems Analysis, Design and Applications

Structured Singular ValueEdit

LMIs in Control/Print version

The LMI can be used to find a that belongs to the set of scalings . Minimizing allows to minimize the squared norm of .

The SystemEdit

The DataEdit

The matrices .

The Optimization ProblemEdit

The LMI:Edit

Conclusion:Edit

ImplementationEdit

https://github.com/mcavorsi/LMI

Related LMIsEdit

Eigenvalue Problem

External LinksEdit

Eigenvalue ProblemEdit

LMIs in Control/Print version

The maximum eigenvalue of a matrix is going to have the most impact on system performance. This LMI allows for minimization of the maximum eigenvalue by minimizing .

The SystemEdit

The DataEdit

The matrices .

The Optimization ProblemEdit

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.

ImplementationEdit

https://github.com/mcavorsi/LMI

Related LMIsEdit

Structured Singular Value

External LinksEdit

LMI for Minimizing Condition Number of Positive Definite MatrixEdit

LMIs in Control/Print version


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.

ImplementationEdit

ReferencesEdit

Continuous Quadratic StabilityEdit

LMIs in Control/Print version

To study stability of a LTI system, we first ask whether all trajectories of system converge to zero as . A sufficient condition for this is the existence of a quadratic function , that decreases along every nonzero trajectory of system . If there exists such a P, we say the system is quadratically stable and we call a quadratic Lyapunov function.

The SystemEdit

The DataEdit

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 StabilityEdit

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 PolyhedronEdit

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: PolytopeEdit

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

ImplementationEdit

https://github.com/Ricky-10/coding107/blob/master/PolytopicUncertainities

External LinksEdit


Exterior Conic Sector LemmaEdit

The ConceptEdit

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 SystemEdit

Consider a square, contiuous-time linear time-invariant (LTI) system, , with minimal state-space relization (A, B, C, D), where and .

The DataEdit

The matrices The matrices and