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 $K=ZP^{-1}$.

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

Optimal 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.