LMIs in Control/Click here to continue/Controller synthesis/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.

Implementation

This implementation requires Yalmip and Sedumi. https://github.com/JalpeshBhadra/LMI/blob/master/quadraticpolytopicstabilization.m

Related LMIs

Quadratic Polytopic $H_{\infty }$ Controller
Quadratic Polytopic $H_{2}$ Controller

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.