LMIs in Control/Stability Analysis/Continuous Time/Quadratic Stability Margins

${\begin{aligned}{\text{The quadratic stability margin of the system is defined as the largest }}\alpha \geq 0{\text{ for which the system is quadratically stable.}}{\text{This LMI applies for systems with norm-bounded uncertainty.}}\end{aligned}}$

The System

{\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t),&&p^{T}p\leq \alpha ^{2}x^{T}C_{q}^{T}C_{q}x\\\end{aligned}}

The Data

The matrices $A,B_{p},C_{q}$ .

The Optimization Problem

${\text{Maximize }}\beta =\alpha ^{2}{\text{ subject to the LMI constraint.}}$

The LMI:

{\begin{aligned}{\text{Find}}\;&P,\lambda ,{\text{ and }}\beta =\alpha ^{2}:\\{\begin{bmatrix}A^{T}P+PA+\beta \lambda C_{q}^{T}C_{q}&PB_{p}\\B_{p}^{T}P&-\lambda I\end{bmatrix}}<0\\\end{aligned}}

Conclusion:

If there exists an $\alpha \geq 0$ then the system is quadratically stable, and the stability margin is the largest such $\alpha$ .

Implementation

https://github.com/mcavorsi/LMI

Stability of Structured, Norm-Bounded Uncertainty

Stability under Arbitrary Switching

External Links

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

LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.