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

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

If there exists some ${\displaystyle \mu \geq 0,P>0}$, and ${\displaystyle Z}$ such that the LMI holds, then the system satisfies ${\displaystyle \|y\|_{L_{2}}\leq \gamma \|u\|_{L_{2}}.}$ There also exists a controller with ${\displaystyle u(t)=Kx(t).}$

## The System

{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)+Mp(t)+B_{2}w(t),&&p(t)=\Delta (t)q(t),\\q(t)&=Nx(t)+D_{12}u(t),&&\Delta \in \mathbf {\Delta } \;:=\{\Delta \in \mathbb {R} ^{n\times n}:\|\Delta \|\leq 1\}\\y(t)&=Cx(t)+D_{22}u(t)\\\end{aligned}}}

## The Data

The matrices ${\displaystyle A,B,M,B_{2},N,D_{12},C,D_{22}}$.

## The Optimization Problem

Minimize ${\displaystyle \gamma }$ subject to the LMI constraints below.

## The LMI:

{\displaystyle {\begin{aligned}{\text{Find}}\;&P>0,\mu \geq 0,{\text{ and }}Z:\\{\begin{bmatrix}AP+BZ+PA^{T}+Z^{T}B^{T}+B_{2}B_{2}^{T}+\mu MM^{T}&(CP+D_{22}Z)^{T}&PN^{T}+Z^{T}D_{12}^{T}\\CP+D_{22}Z&-\gamma ^{2}I&0\\NP+D_{12}Z&0&-\mu I\end{bmatrix}}<0\\\end{aligned}}}

## Conclusion:

The controller gains, K, are calculated by ${\displaystyle K=ZP^{-1}}$.