LMIs in Control/pages/LMI for Schur Stabilization

LMIs in Control/pages/LMI for Schur Stabilization

The System

We consider the following system:

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

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

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

Thus, the closed-loop system is given by:

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

The Data

$\begin{array}{c}\text{Given matrices}A\in {ℝ}^{n\times n}\text{,}B\in {ℝ}^{n\times r}\text{, and the scalar}0<\gamma \le 1.\end{array}$

The Optimization Problem

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

$\begin{array}{c}||A+BK|{|}_2<\gamma \end{array}$

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

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

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

$\begin{array}{c}\left[\begin{array}{cc}-\gamma I& (A+BK)\\ (A+BK{)}^T& -\gamma I\end{array}\right]<0\\ \end{array}$

The LMI: LMI for Schur stabilization

Title and mathematical description of the LMI formulation.

$\begin{array}{cc}\text{min}\gamma :& \\ \text{s.t.}\left[\begin{array}{cc}-\gamma I& (A+BK)\\ (A+BK{)}^T& -\gamma I\end{array}\right]<0\\ \end{array}$

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