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

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

If the system is quadratically stable, then there exists some $\mu \geq 0,P>0,$ and $Z$ such that the LMI is feasible. The $Z$ and $P$ matrices can also be used to create a quadratically stabilizing controller.

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

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

The LMI:

{\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:

There exists a controller for the system with $u(t)=Kx(t)$ where $K=ZP^{-1}$ is the quadratically stabilizing controller, if the above LMI is feasible.