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

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

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

The System

$\begin{array}{cccc}\dot{x}(t)& =Ax(t)+B_{p}p(t),& & p^{T}p\le {\alpha }^2{x}^T{C}_q^T{C}_{q}x\\ \end{array}$

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:

$\begin{array}{cc}\text{Find}& P,\lambda ,\text{ and }\beta ={\alpha }^2:\\ \left[\begin{array}{cc}{A}^{T}P+PA+\beta \lambda C_q^T{C}_q& P{B}_p\\ B_p^{T}P& -\lambda I\end{array}\right]<0\\ \end{array}$

Conclusion:

If there exists an $\alpha \ge 0$ 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

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.