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:

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

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

The Data[edit | edit source]

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[edit | edit source]

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:[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.