# LMIs in Control/pages/Quadratic polytopic stabilization

A Quadratic Polytopic Stabilization Controller Synthesis can be done using this LMI, requiring the information about $A$ ,$\Delta _{A(t)}$ ,$B$ and $\Delta _{B(t)}$ matrices.

## The System

{\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\x(0)&=x_{0},\end{aligned}}

where $x(t)\in \mathbb {R} ^{n}$ , $u(t)\in \mathbb {R} ^{m}$ , 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 Optimization and LMI:LMI for Controller Synthesis using the theorem of Polytopic Quadratic Stability

There exists a K such that

{\begin{aligned}{\dot {x}}(t)&=(A+\Delta _{A}+(B+\Delta _{B})K)x(t)\\\end{aligned}}

is quadratically stable for $(\Delta _{A},\Delta _{B})\in C_{0}((A_{1},B_{2}),...,(A_{k},B_{k}))$  if and only if there exists some P>0 and Z such that

{\begin{aligned}(A+A_{i})P+P(A+A_{i})^{T}+(B+B_{i})Z+Z^{T}(B+B_{i})^{T}<0\quad for\quad i=1,...k.\end{aligned}}

## Conclusion:

The Controller gain matrix is extracted as $K=ZP^{-1}$
Note that here the controller doesn't depend on $\Delta$

• If you want K to depend on $\Delta$  , the problem is harder.
• But this would require sensing $\Delta$  in real-time.