User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Parametric, Norm-Bounded Uncertain System Quadratic Stability

Given a system with matrices A,M,N,Q the quadratic stability of the system with parametric, norm-bounded uncertainty can be determined by the following LMI. The feasibility of the LMI tells if the system is quadratically stable or not.

The System edit

{\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 edit

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

The LMI: edit

{\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: edit

The system above is quadratically stable if and only if there exists some mu >= 0 and P > 0 such that the LMI is feasible.

Implementation edit

https://github.com/mcavorsi/LMI

Related LMIs edit

Stability of Structured, Norm-Bounded Uncertainty

Stability under Arbitrary Switching

Quadratic Stability Margins

External Links edit

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