LMIs in Control/Parametric, Norm-Bounded Uncertain System Quadratic Stability (dual version)

LMIs in Control/Parametric, Norm-Bounded Uncertain System Quadratic Stability (dual version)

The System

$\begin{array}{cccc}\dot{x}(t)& =Ax(t)+Mp(t),& & p(t)=\mathrm{\Delta }(t)q(t),\\ q(t)& =Nx(t)+Qp(t),& & \mathrm{\Delta }\in \mathrm{\Delta }:=\{\mathrm{\Delta }\in {ℝ}^{n\times n}:\Vert \mathrm{\Delta }\Vert \le 1\}\\ \end{array}$

The Data

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

The LMI: The Lyapunov Inequality

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

Conclusion:

The system above is quadratically stable if and only if there exists some mu >= 0 and P > 0 such that the LMI is feasible. This LMI is a good way to determine quadratic stability of a system with parametric, norm-bounded uncertainty.

Implementation

A link to CodeOcean or other online implementation of the LMI

Related LMIs

Links to other closely-related LMIs

External Links

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