Stability Analysis of Continuous-Time Switched Systems: A Variational Approach. Michael Margaliot School of EE-Systems Tel Aviv University, Israel. Joint work with: Michael S. Branicky (CWRU) Daniel Liberzon (UIUC). Overview. Switched systems
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Stability Analysis of Continuous-Time Switched Systems: A Variational Approach
School of EE-Systems Tel Aviv University, Israel
Joint work with: Michael S. Branicky (CWRU) Daniel Liberzon (UIUC)
Systems that can switch between
several modes of operation.
Switched power converter
A multi-controller scheme
Switched controllers are “stronger” than regular controllers.
For more details, see:
- Introduction to hybrid systems, Branicky
- Basic problems in stability and design of
switched systems, Liberzon & Morse
Driving: use mode 1 (wheels)
Braking: use mode 2 (legs)
The advantage: no compromise
“Switched systems are more than the
sum of their subsystems.“
A solution is an absolutely continuous function satisfying (DI) for almost all t.
Definition The differential inclusion
is called GAS if for any solution
The closed-loop system:
A is Hurwitz, so CL is asym. stable for
Absolute Stability Problem
For CL is asym. stable for any
Absolute Stability ProblemFind
This implies that
Although both and are
stable, is not stable.
Instability requires repeated switching.
Problem Find a control maximizing
Write as the bilinear control
is the worst-case switching law (WCSL).
Analyze the corresponding trajectory
Theorem (Pyatnitsky) If then:
(1) The function
is finite, convex, positive, and homogeneous (i.e.).
(2) For every initial condition there exists a solution such that
is a functional:
1. Hamilton-Jacobi-Bellman (HJB)
2. Maximum Principle.
Find such that
An upper bound for ,
obtained for the maximizing Eq. (HJB).
Margaliot & Langholz (2003) derived an
explicit solution for when n=2.
This yields an easily verifiable necessary and sufficient condition for stability of second-order switched linear systems.
The function is a first integral of if
We know that so
Thus, is a concatenation of two first integrals and
→ an explicit expression for V (and an explicit solution of the HJB).
More on the Planar Case
Theorem For a planar bilinear control system
[Margaliot & Branicky, 2009]
Corollary GAS of 2nd-order positive
linear switched systems.
where are GAS.
Problem Find a sufficient condition guaranteeing GAS of (NLDI).
For simplicity, consider the linear
Suppose that A and B commute, i.e.
Definition The Lie bracket of Ax and Bx is [Ax,Bx]:=ABx-BAx.
Hence, [Ax,Bx]=0 implies GAS.
This is why we can park our car.
The term is the reason this takes
Definitionk’th order nilpotency:
all Lie brackets involving k+1 terms vanish.
1st order nilpotency: [A,B]=0
2nd order nilpotency: [A,[A,B]]=[B,[A,B]]=0
Q: Does k’th order nilpotency imply GAS?
Linear switched systems:
Nonlinear switched systems:
Theorem(Margaliot & Liberzon, 2004)
2nd order nilpotency implies GAS.
Proof By the PMP, the WCSL satisfies
2nd order nilpotency
up to a single switch in the WCSL.
1st order nilpotency
Differentiating again yields:
If m(t)0, the Maximum Principle
does not necessarily provide enough
information to characterize the WCSL.
Singularity can be ruled out using
thenotion ofstrong extremality
3rd Order Nilpotency
In this case:
further differentiation cannot be carried out.
Theorem (Sharon & Margaliot, 2007)
3rd order nilpotency implies
(1) Hall-Sussmann canonical system;
(2) A second-order MP
A natural and powerful idea is to
consider the “most unstable” trajectory.
More info on the variational approach:
“Stability analysis of switched systems using variational principles: an introduction”, Automatica 42(12): 2059-2077, 2006.