LMIs in Control/LMIs for Controller Synthesis/Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

The System edit

{\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 \mathbf {\Delta } \;:=\{\Delta \in \mathbb {R} ^{n\times n}:\|\Delta \|\leq 1\}\\\end{aligned}}

The Data edit

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

The LMI: The Lyapunov Inequality edit

{\begin{aligned}{\text{Find}}\;&P>0,\mu \geq 0,{\text{ and }}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}}+\mu {\begin{bmatrix}MM^{T}&MQ^{T}\\QM^{T}&QQ^{T}-I\end{bmatrix}}<0\\\end{aligned}}

Conclusion: edit

${\begin{aligned}{\text{If the system is quadratically stable, then there exists some }}\mu \geq 0,P>0,{\text{ and }}Z{\text{ such that the above LMI is feasible.}}\\{\text{ Also there exists a controller for the system with }}u(t)=Kx(t){\text{ where }}K=ZP^{-1}{\text{ is the quadratically stabilizing controller.}}\end{aligned}}$