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LMI for Schur Stabilization

Similar to the stability of continuous-time systems, one can analyze the stability of discrete-time systems. A discrete-time system is said to be stable if all roots of its characteristic equation lie in the open unit disk. This provides a condition for the stability of discrete-time linear systems and a linear time-invariant system with this property is called a Schur stable system.

The System

We consider the following system:

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

where the matrices $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times r}$, $x\in \mathbb {R} ^{n}$, and $u\in \mathbb {R} ^{r}$ are the state matrix, input matrix, state vector, and the input vector, respectively.

Moreover, $k$ represents time in the discrete-time system and $k+1$ is the next time step.

The state feedback control law is defined as follows:

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

where $K\in \mathbb {R} ^{n\times r}$ is the controller gain. Thus, the closed-loop system is given by:

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

The Data

The matrices $A$ and $B$ are given.

We define the scalar as $\gamma$ with the range of $0<\gamma \leq 1$.

The Optimization Problem

The optimization problem is to 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:

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

Using the Lemma 1.2 in LMI in Control Systems Analysis, Design and Applications (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

The LMI for Schur stabilization can be written as minimization of the scalar, $\gamma$, in the following constraints:

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

Conclusion:

After solving the LMI problem, we obtain the controller gain $K$ and the minimized parameter $\gamma$. 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

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

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