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:

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

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

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

The state feedback control law is defined as follows:

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

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

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

The Data

The matrices ${\displaystyle A}$ and ${\displaystyle B}$ are given.

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

The Optimization Problem

The optimization problem is to find a matrix ${\displaystyle {\begin{aligned}K\in \mathbb {R} ^{r\times n}\end{aligned}}}$ such that:

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

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

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

Using the Lemma 1.2 in LMI in Control Systems Analysis, Design and Applications (page 14), the aforementioned inequality can be converted into:

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

The LMI: LMI for Schur stabilization

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

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

Conclusion:

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

• - LMI in Control Systems Analysis, Design and Applications

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