LMIs in Control/Stability Analysis/Stability of Structured, Norm-Bounded Uncertainty

LMIs in Control/Stability Analysis/Stability of Structured, Norm-Bounded Uncertainty

The System

$\begin{array}{cccc}\dot{x}(t)& =Ax(t)+Mp(t),& & p(t)=\mathrm{\Delta }(t)q(t),\\ q(t)& =Nx(t)+Qp(t),& & \mathrm{\Delta }\in \mathrm{\Delta },\mathbf{|}\mathbf{|}\mathrm{\Delta }\mathbf{|}\mathbf{|}\mathbf{\le }\mathrm{𝟏}\\ \end{array}$

The Data

The matrices $A,M,N,Q$.

The LMI: The Lyapunov Inequality

$\begin{array}{cc}\text{Find}& P>0:\\ \left[\begin{array}{cc}AP+P{A}^T& P{N}^T\\ NP& 0\end{array}\right]+\left[\begin{array}{cc}M\mathrm{\Theta }{M}^T& M\mathrm{\Theta }{Q}^T\\ Q\mathrm{\Theta }{M}^T& Q\mathrm{\Theta }{Q}^T-\mathrm{\Theta }\end{array}\right]<0\\ \end{array}$

Conclusion:

$\begin{array}{c}\text{The system above is quadratically stable if and only if there exists some }\mathrm{\Theta }\in P\mathrm{\Theta }\text{ and }P>0\text{ such that the LMI is feasible.}\end{array}$

Implementation

https://github.com/mcavorsi/LMI

Related LMIs

Parametric, Norm-Bounded Uncertain System Quadratic Stability

Stability under Arbitrary Switching

Quadratic Stability Margins

External Links

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