LMIs in Control/pages/Quadratic Schur Stabilization

LMI for Quadratic Schur Stabilization

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 with polytopic uncertainties and a linear time-invariant system with this property is called a Schur stable system.

The System

Consider discrete time system

$\begin{array}{c}{x}_{k+1}=A{x}_k+B{u}_k,\\ \end{array}$

where $x_k\in {ℝ}^n$, $u_k\in {ℝ}^m$, at any $t\in ℝ$.
The system consist of uncertainties of the following form

$\begin{array}{c}{\mathrm{\Delta }}_{A(t)}=A_1{\delta }_1(t)+....+A_k{\delta }_k(t)\\ {\mathrm{\Delta }}_{B(t)}=B_1{\delta }_1(t)+....+B_k{\delta }_k(t)\\ \end{array}$

where $x\in {ℝ}^m$,$u\in {ℝ}^n$,$A\in {ℝ}^{mxm}$ and $B\in {ℝ}^{mxn}$

The Data

The matrices necessary for this LMI are $A$,${\mathrm{\Delta }}_{A(t)}ie{A}_i$ ,$B$ and ${\mathrm{\Delta }}_{B(t)}ie{B}_i$

The LMI:

There exists some X > 0 and Z such that

$\begin{array}{c}\left[\begin{array}{ccc}X& & AX+BZ\\ (AX+BZ{)}^T& & X\end{array}\right]+\left[\begin{array}{ccc}0& & A_{i}X+B_{i}Z\\ (A_{i}X+B_{i}Z{)}^T& & 0\end{array}\right]>0i=1,......,k\end{array}$

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:

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

Conclusion:

The Controller gain matrix is extracted as $F=Z{X}^{-1}$
$u_k=F{x}_k$

$\begin{array}{c}{x}_{k+1}=A{x}_k+B{u}_k,\\ =A{x}_k+BF{x}_k\\ =(A+BF)x_k\end{array}$

It follows that the trajectories of the closed-loop system (A+BK) are stable for any $\mathrm{\Delta }\in C_0({\mathrm{\Delta }}_1,...,{\mathrm{\Delta }}_k)$

Implementation

https://github.com/JalpeshBhadra/LMI/blob/master/quadratic_schur_stabilization.m

Related LMIs

Schur Complement
Schur Stabilization

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMI in Control Systems Analysis, Design and Applications