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

{\displaystyle {\begin{aligned}x_{k+1}=Ax_{k}+Bu_{k},\\\end{aligned}}}

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

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

The Data

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

The LMI:

There exists some X > 0 and Z such that

{\displaystyle {\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 {\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}}}

Conclusion:

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

{\displaystyle {\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 ${\displaystyle \,\Delta \,\in \,C_{0}(\Delta _{1},...,\Delta _{k})}$