LMIs in Control/Click here to continue/Applications of Non-Linear Systems/Stability of Switching Systems - Arbitrary Switching
Stability of Switching Systems - Quadratic Stability Under Arbitrary Switching
[edit | edit source]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
[edit | edit source]The state space formulation of each subsystem is given as follows:
Where i = 1,2,...,n for each of the subsystems in the switching system.
The Data
[edit | edit source]For a switching system with multiple subsystems, the A matrix for each is defined by
The LMI:Quadratic Stability Under Arbitrary Switching
[edit | edit source]The switched system is stable under arbitrary switching if there exists some P > 0 such that
and
Conclusion:
[edit | edit source]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
[edit | edit source]This implementation requires Yalmip and Sedumi.
Quadratic Stability Under Arbitrary Switching
Related LMIs
[edit | edit source]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
[edit | edit source]- 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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