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

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

${\begin{aligned}{\text{If there exists some }}\Theta \in P\Theta ,P>0,{\text{ and }}Z{\text{ such that the LMI is feasible, then the system satisfies }}||y||_{L_{2}}\leq \gamma ||u||_{L_{2}}.{\text{ There also exists a controller with }}u(t)=Kx(t).\end{aligned}}$

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)+Mp(t)+B_{2}w(t),&&p(t)=\Delta (t)q(t),\\q(t)&=Nx(t)+D_{12}u(t),&&\Delta \in {\bf {{\Delta }\;,||\Delta ||\leq 1}}\\y(t)&=Cx(t)+D_{22}u(t)\\\end{aligned}}$

The Data

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

The Optimization Problem

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

The LMI:

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

Conclusion:

The controller is $K=ZP^{-1}$.

Implementation

https://github.com/mcavorsi/LMI

Related LMIs

Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

H-infinity Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.