Application of Linear Matrix Inequalities (LMI) in Control Theory

Table of contents

(1) LMI processing toolkit in Matlab

(2) Why has LMI become an important tool in the field of control theory?

(3) The relationship between LMI and Lyapunov inequality

(1) Linear matrix inequality

(2) Linear matrix inequality system

(3) Schur supplement

(4) Common lemmas in LMI

Lemma 2 (Generalized KYP Lemma [4])

Corollary 1 (Generalized KYP Lemma Corollary [4])

Theorem 3 (projection theorem [1])

Lemma 4 (Jensen’s inequality [5,6]

Lemma 5 (Finsler's Lemma[7]):

references

(1) LMI processing toolkit in Matlab

There is YALMIP, a toolkit for solving linear matrix inequalities in matlab . It can be downloaded and installed on the official website . Please refer to the yalmip installation tutorial . yalmip only provides some basic LMI solving methods. If you have more complex inequality solving needs, you can install the cplex tool package. For information on how to use the yalmip toolkit, please refer to the yalmip code writing tutorial and the LMI toolbox tutorial and tutorial documents . MONSK installation

(2) Why has LMI become an important tool in the field of control theory?

Linear matrix inequality (LMI) technology is an important tool for analyzing and synthesizing control systems, especially in the field of robust control . The main factors are the following three [1]:

For classic control methods, the advantage of LMI technology is its simplicity of operation. Before the emergence of LMI technology, people designed optimal controllers by solving the Ricaati equation , but it is difficult to solve the Ricaati equation. LMI technology requires only a small number of concepts and basic principles to develop practical tools (now we can easily solve LMI problems using the YALMIP toolkit).
LMI technology offers a broad perspective on control problems, including robustness analysis, nominal H∞, H2 and robust control synthesis, multi-objective synthesis, linear parameter variation synthesis, some of which cannot be solved in the classical control domain.
LMI technology is a powerful and effective numerical tool that utilizes convex optimization, and adds effective software tools to the theoretical system.

(3) The relationship between LMI and Lyapunov inequality

(1) Linear matrix inequality

Consider the linear matrix inequality expression as follows [2]:

In the above formula, $x_1,....x_m$ is the decision variable. In particular, the function in the above general form

is a real symmetric matrix. F(x)<0 in the above formula means that the matrix F(x) is negative definite, that is, for all non-zero vectors $v\epsilon R^m$ , $v^TF(x)v<0$ or the maximum eigenvalue of F(x) is less than 0.

In many system and control problems, the variables of the problem appear in the form of matrices, such as Lyapunov matrix inequality:

Among them, the matrix

is a known constant matrix with appropriate dimensions, and Q is a known symmetric matrix,

is a symmetric matrix variable. Assuming a set of basis $E_1,E_2,....,E_M$ in $R^n$ , then for any symmetry

exists $x_1,x_2,....,x_M$ such that

So there are:

Through this transformation, a more general expression of linear matrix inequality is obtained.

(2) Linear matrix inequality system

Suppose there are multiple matrix inequalities:

The overall composition is called a linear matrix inequality system. introduced $F(x)=diag\left \{F_1(x),....F_k(x) \right \}$ , then $F_1(x)<0,...F_k(x)<0$ it is established simultaneously if and only if $F(x)<0$ . Therefore a system of linear matrix inequalities can also be represented by a single linear matrix inequality.

(3) Schur supplement

In many problems that convert nonlinear matrix inequalities into linear matrix inequalities, we often use the Schur complement property of matrices. Consider a matrix and block S:

Which $S_{11}$ is $r \times r$ dimensional. Assuming $S_{11}$ it is non-singular, it is called " $S_{11}$ Sehar's complement in S". The following lemma gives the Schur complement property of matrices.

Lemma 1 Schur complement property

For a given symmetric matrix:

Which $S_{11}$ is $r \times r$ dimensional. The following 3 conditions are equivalent:

（1） $S<0$

（2） $S_{11}<0,S_{22}-S{12}^TS_{11}^{-1}S_{12}<0$

（3） $S_{22}<0,S_{11}-S{12}S_{22}^{-1}S_{12}^T<0$

(For the proof method, please refer to page 8 of Yu Li's "Robust Control - Linear Matrix Inequality Processing Method" )

In some control problems, quadratic matrix inequality [3] is often encountered:

where $A,B,Q=Q^T>0,R=R^T>0$ is a given constant matrix of appropriate dimensions and $P$ is a symmetric matrix variable, then applying Lemma 1, the feasibility problem of the above matrix inequality can be transformed into an equivalent matrix inequality:

feasibility problem, and the latter is a linear matrix inequality with respect to the matrix variable P.

Therefore, in control problems, we often need to design the Lyapunov function V(t). In order to ensure the stability of the system $\dot{V}(t)<0$ , we can convert $\dot{V}(t)<0$ this inequality into the form of a linear matrix inequality and directly solve it using YALMIP in matlab. .

(4) Common lemmas in LMI

Lemma 2 (Generalized KYP Lemma [4])

Given a matrix $\Theta$ , $F$ and $\Phi$ , $\Psi$ , and $N_w$ represents $T_wF$ the null space, where $T_w=\left [ \begin{matrix} I & -jwI \end{matrix} \right ]$ , then the inequality

$N^{*}_w\Theta N_w<0,w\epsilon \left [ \begin{matrix} \varpi _1 & \varpi _2 \end{matrix} \right$

holds if and only if there is a symmetric matrix $P$ and $Q>0$ , such that holds, where

where * represents the conjugate transpose of the matrix, j is the imaginary unit, which $\Phi \otimes P$ represents the right Kronecker product, that is

$\Phi \otimes P= \left [ \begin{matrix} 0 & P \\ P & 0 \end{matrix} \right ]$

Corollary 1 (Generalized KYP Lemma Corollary [4])

For a linear system , $G(jw)$ it is the transfer function from disturbance to control output of the system. Then for a given symmetric matrix $\Pi$ , the following two statements are equivalent:

1) Finite frequency domain inequality

established.

2) There are symmetric matrices Р and Q that satisfy Q>0, such that

established, among which

And represents $\Pi$ the upper right block and lower right block of the matrix, and * in the matrix represents the transpose of its corresponding block.

Theorem 3 (projection theorem [1])

For a given scalar $\Gamma ,\Lambda ,\Theta$ , the matrix $F$ satisfies $\Gamma F\Lambda +(\Gamma F\Lambda )^T+\Theta <0$ if and only if the following two conditions hold:

Lemma 3 (Reflection Theorem [1]): $P$ is a given positive definite symmetric matrix, and the inequality is $\Psi +S+S^T<0$ equivalent to the following linear matrix inequality (LMI) solution problem:

The symbol in the formula $[W]_s$ is used to represent the $W$ sum of the matrix and its transpose, that is $[W]_s=W+W^T$ .

Lemma 4 (Jensen’s inequality [5,6]

For any positive definite symmetric constant matrix $M\epsilon R^{n \times n}$ , the scalar $r$ satisfies $r>0$ , and there is a vector $w:[0,r]\rightarrow R^n$ , then the following inequality holds:

Lemma 5 (Finsler's Lemma[7]):

Let $x\epsilon R^n$ , $p\epsilon S^n$ , and $H\epsilon R^{m \times n}$ satisfy the rank of H less than n, rank(H) = r <n. Then the following two formulas are equivalent:

references

【1】Apkarian P,Tuan H D,Bernussou J.Continuous-Time analysis,eigenstructure as-signment,and H2synthesis with enhanced Linear Matrix Inequalities(LMI)char-acterizations[J].IEEE Transactions on Automatic Control,2001,42(12):1941–1946.

【2】"Robust Control Theory and Application" Wang Juan, Zhang Tao and Xu Guokai

[3] "Robust Control - Linear Matrix Inequality Processing Method" Yu Li

【4】Iwasaki T,Hara S.Generalized KYP Lemma:uniﬁed frequency domain inequal-ities with design applications[J].IEEE Transactions on Automatic Control,2005,50(1):41–59.

【5】Wu J,Chen X,Gao H.H∞ﬁltering with stochastic sampling[J].Signal Proces-siong,2010,90(4):1131–1145.

【6】Gao H,Wu J,Shi P.Robust sampled-data H∞control with stochastic sampling[J].Automatica,2009,45(7):1729–1736.

【7】Qiu J,Feng G,Yang J.New results on robust energy-to-peakﬁltering for discrete-time switched polytopic linear systems with time-varying delay[J].IET ControlTheory and Applications,2008,2(9):795–806.