LMIs in Control/pages/Schur Stabilization

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:

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

where the matrices $A\in {ℝ}^{n\times n}$, $B\in {ℝ}^{n\times r}$, $x\in {ℝ}^n$, and $u\in {ℝ}^r$ 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:

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

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

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

The Data

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

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

The Optimization Problem

The optimization problem is to 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:

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

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

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

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

Conclusion:

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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