LMIs in Control/Stability Analysis/Continuous Time/Stability of Structured, Norm-Bounded Uncertainty

LMIs in Control/Stability Analysis/Continuous Time/Stability of Structured, Norm-Bounded Uncertainty

Given a system with matrices A,M,N,Q with structured, norm-bounded uncertainty, the stability of the system can be found using the following LMI. The LMI takes variables P and $\Theta$ and checks for a feasible solution.

The System

{\begin{aligned}{\dot {x}}(t)&=Ax(t)+Mp(t),&&p(t)=\Delta (t)q(t),\\q(t)&=Nx(t)+Qp(t),&&\Delta \in {\bf {{\Delta }\;,||\Delta ||\leq 1}}\\\end{aligned}} The Data

The matrices $A,M,N,Q$ .

The LMI:

{\begin{aligned}{\text{Find}}\;&P>0:\\{\begin{bmatrix}AP+PA^{T}&PN^{T}\\NP&0\end{bmatrix}}+{\begin{bmatrix}M\Theta M^{T}&M\Theta Q^{T}\\Q\Theta M^{T}&Q\Theta Q^{T}-\Theta \end{bmatrix}}<0\\\end{aligned}} Conclusion:

{\begin{aligned}{\text{The system above is quadratically stable if and only if there exists some }}\Theta \in P\Theta {\text{ and }}P>0{\text{ such that the LMI is feasible.}}\end{aligned}} 