# LMIs in Control/pages/Quadratic polytopic stabilization

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

## The System

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

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

There exists a K such that

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

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

{\displaystyle {\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 ${\displaystyle K=ZP^{-1}}$
Note that here the controller doesn't depend on ${\displaystyle \Delta }$

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