LMIs in Control/Stability Analysis/LMI for Schur Stabilization

The System
We consider the following system:

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

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

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

Thus, the closed-loop system is given by:

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

The Data


\begin{align} \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{align}$$

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

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

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

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

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

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

The LMI: LMI for Schur stabilization
Title and mathematical description of the LMI formulation.



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

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