# LMIs in Control/pages/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{aligned}x_{k+1}=Ax_{k}+Bu_{k},\\\end{aligned}}

where $x_{k}\in \mathbb {R} ^{n}$ , $u_{k}\in \mathbb {R} ^{m}$ , at any $t\in \mathbb {R}$ .
The system consist of uncertainties of the following form

{\begin{aligned}\Delta _{A(t)}=A_{1}\delta _{1}(t)+....+A_{k}\delta _{k}(t)\\\Delta _{B(t)}=B_{1}\delta _{1}(t)+....+B_{k}\delta _{k}(t)\\\end{aligned}}

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

## The Data

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

## The LMI:

There exists some X > 0 and Z such that

{\begin{aligned}{\begin{bmatrix}X&&AX+BZ\\(AX+BZ)^{T}&&X\end{bmatrix}}+{\begin{bmatrix}0&&A_{i}X+B_{i}Z\\(A_{i}X+B_{i}Z)^{T}&&0\end{bmatrix}}>0\quad i=1,......,k\end{aligned}}

## 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}}

## Conclusion:

The Controller gain matrix is extracted as $F=ZX^{-1}$
$u_{k}=Fx_{k}$

{\begin{aligned}x_{k+1}=Ax_{k}+Bu_{k},\\\quad \quad \quad =Ax_{k}+BFx_{k}\\\quad \quad =(A+BF)x_{k}\end{aligned}}

It follows that the trajectories of the closed-loop system (A+BK) are stable for any $\,\Delta \,\in \,C_{0}(\Delta _{1},...,\Delta _{k})$