LMIs in Control/pages/Quadratic polytopic stabilization

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

The System

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+Bu(t),\\ x(0)& =x_0,\end{array}$

where $x(t)\in {ℝ}^n$, $u(t)\in {ℝ}^m$, at any $t\in ℝ$.
The system consist of uncertainties of the following form

$\begin{array}{c}{\mathrm{\Delta }}_{A(t)}=A_1{\delta }_1(t)+....+A_k{\delta }_k(t)\\ {\mathrm{\Delta }}_{B(t)}=B_1{\delta }_1(t)+....+B_k{\delta }_k(t)\\ \end{array}$

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

The Data

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

The Optimization and LMI:LMI for Controller Synthesis using the theorem of Polytopic Quadratic Stability

There exists a K such that

$\begin{array}{cc}\dot{x}(t)& =(A+{\mathrm{\Delta }}_A+(B+{\mathrm{\Delta }}_B)K)x(t)\\ \end{array}$

is quadratically stable for $({\mathrm{\Delta }}_A,{\mathrm{\Delta }}_B)\in C_0((A_1,B_2),...,(A_k,B_k))$ if and only if there exists some P>0 and Z such that

$\begin{array}{c}(A+A_i)P+P(A+A_i{)}^T+(B+B_i)Z+Z^T(B+B_i{)}^T<0fori=1,...k.\end{array}$

Conclusion:

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

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

Implementation

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

Related LMIs

Quadratic Polytopic $H_{\mathrm{\infty }}$ 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.