LMIs in Control/pages/LMI for Generalized eigenvalue problem

LMI for Generalized Eigenvalue Problem

Technically, the generalized eigenvalue problem considers two matrices, like $A$ and $B$, to find the generalized eigenvector, $x$, and eigenvalues, $\lambda$, that satisfies $Ax=\lambda Bx$. If the matrix $B$ is an identity matrix with the proper dimension, the generalized eigenvalue problem is reduced to the eigenvalue problem.

The System

Assume that we have three matrice functions which are functions of variables $x=[x_1{x}_2...x_n{]}^{\text{T}}\in {ℝ}^n$ as follows:

$A(x)=A_0+A_1{x}_1+...+A_n{x}_n$

$B(x)=B_0+B_1{x}_1+...+B_n{x}_n$

$C(x)=C_0+C_1{x}_1+...+C_n{x}_n$

where are $A_i$, $B_i$, and $C_i$ ($i=1,2,...,n$) are the coefficient matrices.

The Data

The $A(x)$, $B(x)$, and $C(x)$ are matrix functions of appropriate dimensions which are all linear in the variable $x$ and $A_i$, $B_i$, $C_i$ are given matrix coefficients.

The Optimization Problem

The problem is to find $\begin{array}{c}x=[x_1{x}_2...x_n]\end{array}$ such that:

$A(x)<\lambda B(x)$, $B(x)>0$, and $C(x)<0$ are satisfied and $\lambda$ is a scalar variable.

The LMI: LMI for Schur stabilization

A mathematical description of the LMI formulation for the generalized eigenvalue problem can be written as follows:

$\begin{array}{cc}& \text{min}\lambda \\ & \text{s.t.}A(x)<\lambda B(x)\\ & B(x)>0\\ & C(x)<0\end{array}$

Conclusion:

The solution for this LMI problem is the values of variables $x$ such that the scalar parameter, $\lambda$, 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

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

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

Related LMIs

LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

External Links

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

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