LMIs in Control/Click here to continue/LMIs in system and stability Theory/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

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 :

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

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.