User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Quadratic Stability Margins

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

The System


\begin{align} \dot x(t)&=Ax(t)+B_pp(t),&&p^Tp \leq \alpha ^2 x^TC_q^TC_qx\\ \end{align}$$

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{align} \text{Find} \; &P,\lambda, \text{ and } \beta = \alpha ^2:\\ \begin{bmatrix} A^TP+PA+ \beta \lambda C_q^TC_q & PB_p \\ B_p^TP & -\lambda I \end{bmatrix} < 0\\ \end{align}$$

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

Related LMIs
Parametric, Norm-Bounded Uncertain System Quadratic Stability

Stability of Structured, Norm-Bounded Uncertainty

Stability under Arbitrary Switching