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Stability of Switching Systems - Quadratic Stability Under Arbitrary Switching

For gain scheduled systems, stability of each subsystem {A₁,A₂} does not guarantee stability under arbitrary switching. Additionally, smart switching can stabilize two unstable systems.

The System

The state space formulation of each subsystem is given as follows:

${\begin{aligned}{\dot {x}}(t)=A_{i}(t)+B_{i}u(t)\\y(t)=C_{i}x(t)+D_{i}u(t)\end{aligned}}$

Where i = 1,2,...,n for each of the subsystems in the switching system.

The Data

For a switching system with multiple subsystems, the A matrix for each is defined by

$A_{i}=A_{1},A_{2},...A_{n}$

The LMI:Quadratic Stability Under Arbitrary Switching

The switched system ${\dot {x}}(t)\in {A_{1}x(t),A_{2}x(t)}$ is stable under arbitrary switching if there exists some P > 0 such that

$A_{1}^{T}P+PA_{1}<0$ and

$A_{2}^{T}P+PA_{2}<0$

Conclusion:

This implies that both A₁ and A₂ are Hurwitz.

There is not necessarily a common quadratic Lyapunov function for both A₁ and A₂.

This quadratic stability condition under arbitrary switching is a useful condition to use when designing controllers for switching systems. This LMI does not provide information on how the controller is designed, but is to be used as an additional condition to stabilize a switching system.

Implementation

This implementation requires Yalmip and Sedumi.

Quadratic Stability Under Arbitrary Switching

Related LMIs

Lyapunov Stability of a System with Polynomial Dynamics/

Global Minimum of Polynomial via SOS Method/

Local Minimum of Polynomial via SOS Method/

Global Lyapunov Function Search for System with Polynomial Dynamics/

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.

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