LMIs in Control/pages/LMI for Schur Stabilization

The System

We consider the following system:

${\begin{aligned}x(k+1)=Ax(k)+Bu(k)\end{aligned}}$

where ${\begin{aligned}x\in \mathbb {R} ^{n}{\text{and}}u\in \mathbb {R} ^{r}\end{aligned}}$ , are the state vector and the input vector, respectively. Moreover, the state feedback control law is defined as follows:

${\begin{aligned}u(k)=Kx(k)\end{aligned}}$

Thus, the closed-loop system is given by:

${\begin{aligned}x(k+1)=(A+BK)x(k)\end{aligned}}$

The Data

{\begin{aligned}{\text{Given matrices}}\quad A\in \mathbb {R} ^{n\times n}{\text{,}}\quad B\in \mathbb {R} ^{n\times r}\quad {\text{, and the scalar}}\quad 0<\gamma \leq 1.\end{aligned}}

The Optimization Problem

Find a matrix ${\begin{aligned}K\in \mathbb {R} ^{r\times n}\end{aligned}}$ such that,

${\begin{aligned}||A+BK||_{2}<\gamma \end{aligned}}$

According to the definition of the spectral norms of matrices, this condition becomes equivalent to:

${\begin{aligned}(A+BK)^{T}(A+BK)<\gamma ^{2}I\end{aligned}}$

One can use the Lemma 1.2 in [1] page 14, the aforementioned inequality can be converted into:

${\begin{aligned}{\begin{bmatrix}-\gamma I&(A+BK)\\(A+BK)^{T}&-\gamma I\end{bmatrix}}<0\\\end{aligned}}$

The LMI: LMI for Schur stabilization

Title and mathematical description of the LMI formulation.

{\begin{aligned}{\text{min}}\;\quad \gamma :&\\{\text{s.t.}}\quad {\begin{bmatrix}-\gamma I&(A+BK)\\(A+BK)^{T}&-\gamma I\end{bmatrix}}<0\\\end{aligned}}

Conclusion:

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

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

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

Related LMIs

LMI for Hurwitz stability

External Links

A list of references documenting and validating the LMI.

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