LMIs in Control/Controller Synthesis/Continuous Time/Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

LMIs in Control/Controller Synthesis/Continuous Time/Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

{\begin{aligned}{\text{The system is quadratically stable if and only if there exists some }}\Theta \in P\Theta ,P>0,{\text{ and }}Z{\text{ such that the LMI is feasible.}}{\text{ Furthermore, there exists a quadratically stabilizing controller with }}u(t)=Kx(t).\end{aligned}} The System

{\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)+Mp(t),&&p(t)=\Delta (t)q(t),\\q(t)&=Nx(t)+Qp(t)+D_{12}u(t),&&\Delta \in {\bf {{\Delta }\;,||\Delta ||\leq 1}}\\\end{aligned}} The Data

The matrices $A,B,M,N,Q,D_{12}$ .

The LMI:

{\begin{aligned}{\text{Find}}\;&P>0,Z:\\{\begin{bmatrix}AP+BZ+PA^{T}+Z^{T}B^{T}&PN^{T}+Z^{T}D_{12}^{T}\\NP+D_{12}Z&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:

If the LMI is feasible, the controller, K, is calculated by $K=ZP^{-1}$ .