LMIs in Control/pages/MatrixEigenValueMinimization

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
Assume that we have a matrix function of variables $$x$$:

$$ \begin{align} A(x) = A_{0}+A_{1}x_{1}+ ... + A_{n}x_{n} \end{align}$$

where $$ \begin{align} A_{i}, \quad i=1, 2, ..., n \end{align}$$ are symmetric matrices.

The Data
The symmetric matrices $$A_{i}$$ ($$ \begin{align} A_{0}, A_{1}, ..., A_{n}\end{align}$$) are given.

The Optimization Problem
The optimization problem is to find the variables $$\begin{align} x = [x_{1} \quad x_{2} ... x_{n}] \end{align}$$ to minimize the following cost function:

$$\begin{align} J(x) = \lambda_{\text{max}}(A(x)) \end{align}$$

where $$J(x)$$ is the cost function and $$\lambda_{\text{max}}(.)$$ 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

$$\begin{align} \lambda_{max}(A(x)) \leq t \iff A(x)-tI \leq 0 \end{align}$$

where $$t$$ is defined as the maximim eigenvalue of the matrix $$A$$.

The LMI: LMI for eigenvalue minimization
This optimization problem can be converted to an LMI problem.

The mathematical description of the LMI formulation can be written as follows:

$$ \begin{align} &\text{min} \quad t\\ &\text{s.t.} \quad A(x)-tI \leq 0 \end{align}$$

Conclusion:
As a result, the variables $$ x_{i}, \quad i =1, 2, ..., n $$ after solving this LMI problem. Moreover, we obtain the maximum eigenvalue, $$ t $$, of the matrix $$A(x)$$.

Implementation
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
LMI for Generalized Eigenvalue Problem

LMI for Matrix Norm Minimization

LMI for Maximum Singular Value of a Complex Matrix

LMI for Matrix Positivity

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