LMIs in Control/Stability Analysis/Continuous Time/Quadratic Hurwitz Stabilization for Polytopic Systems

This section studies the Quadratic Hurwitz stabilization for polytopic systems.

The System

Given a state-space representation of a linear system

${\begin{aligned}\ {\dot {x}}(t)=A(\delta (t)x(t)+B\delta (t)u(t)\\A(\delta (t))=A_{0}+\delta _{1}(t)A_{1}+\delta _{2}(t)A_{2}\\B(\delta (t))=B_{0}+\delta _{1}(t)B_{1}+\delta _{2}(t)B_{2}\end{aligned}}$

(1)

LMI Condition

With $\Delta =\Delta _{p}$ , the quadratic Hurwitz Stabilization problem has a solution if and only if there exists a symmetric positive definite matrix $P$ and a matrix $W$ satisfying the below LMI :

${\begin{aligned}(A_{0}+A_{i})P+P(A_{0}+A_{i})^{T}+(B_{0}+B_{i})W+W(B_{0}+B_{i})^{T}<0,i&=1,2,....k\end{aligned}}$

(2)

In this case, a solution to the problem is given by

${\begin{aligned}K=WP^{-1}\\\end{aligned}}$

(3)

Conclusion

Stability of a system does not guarantee quadratic stability. Since quadratic stability can represent infinite LMI constraints, it is not tractable. Therefore, to make it feasible and tractable, polytopic sets are helpful.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013