LMIs in Control/Stability Analysis/Parametric, Norm-Bounded Uncertain System Quadratic Stability

LMIs in Control/Stability Analysis/Parametric, Norm-Bounded Uncertain System Quadratic Stability

The System

${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Mp(t),&&p(t)=\Delta (t)q(t),\\q(t)&=Nx(t)+Qp(t),&&\Delta \in \mathbf {\Delta } \;:=\{\Delta \in \mathbb {R} ^{n\times n}:\|\Delta \|\leq 1\}\\\end{aligned}}$

The Data

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

The LMI: The Lyapunov Inequality

${\begin{aligned}{\text{Find}}\;&P>0,\mu \geq 0:\\{\begin{bmatrix}AP+PA^{T}&PN^{T}\\NP&0\end{bmatrix}}+\mu {\begin{bmatrix}MM^{T}&MQ^{T}\\QM^{T}&QQ^{T}-I\end{bmatrix}}<0\\\end{aligned}}$

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

https://github.com/mcavorsi/LMI

Related LMIs

Stability of Structured, Norm-Bounded Uncertainty

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.