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Stability Analysis of Switched Systems: A Variational Approach. Michael Margaliot School of EE-Systems Tel Aviv University Joint work with Daniel Liberzon (UIUC). Overview. Switched systems Stability Stability analysis: A control-theoretic approach A geometric approach
School of EE-Systems Tel Aviv University
Joint work with Daniel Liberzon (UIUC)
Systems that can switch between
several modes of operation.
A multi-controller scheme
Switched controllers are “stronger” than
Driving: use mode 1 (wheels)
Braking: use mode 2 (legs)
The advantage: no compromise
weakerMathematical Modeling with Differential Inclusions
easier ANALYSIS harder
“Switched systems are more than the
sum of their subsystems.“
A solution is an absolutely continuous function satisfying (DI) for all t.
The differential inclusion
is called GAS if for any solution
The closed-loop system:
A is Hurwitz, so CL is asym. stable for
The Problem of Absolute Stability:
For CL is asym. stable for any
The Problem of Absolute Stability: Find
This implies that
stable, is not stable.
Instability requires repeated switching.
This presents a serious problem in
Write as a control system:
Problem: Find the control that maximizes
is the worst-case switching law (WCSL).
Analyze the corresponding trajectory
Thm. 1 (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:
Find such that
An upper bound for ,
obtained for the maximizing Eq. (HJB).
In general, finding is difficult.
Differentiating we get
A differential equation for with a
boundary condition at
The WCSL is the maximizing
We can simulate the optimal solution
backwards in time.
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
so we have an explicit expression for V (and an explicit solution of HJB).
where are GAS.
Problem: Find a sufficient condition guaranteeing GAS of (NLDI).
For the sake of simplicity, consider
Suppose that A and B commute,
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 it takes
Definition: k’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?
Switched linear systems:
Switched nonlinear systems:
Thm. 1 (Margaliot & Liberzon, 2004)
2nd order nilpotency implies GAS.
Proof: Consider the WCSL
Define the switching function
1st order nilpotency
no switching in the WCSL.
Differentiating again, we get
2nd order nilpotency
up to a single switch in the WCSL.
If m(t)0, then the Maximum Principle
does not necessarily provide enough
information to characterize the WCSL.
Singularity can be ruled out using
thenotion of strong extermality
In this case:
further differentiation cannot be carried out.
Thm. 2 (Sharon & Margaliot, 2005)
3rd order nilpotency implies
The proof is based on using: (1) the Hall-Sussmann canonical system; and (2) the second-order Agrachev-Gamkrelidze MP.
Consider the case [A,B]=0.
Guess the solution:
If two controls u, v yield the same values for
then they yield the same
does not depend on u,
we conclude that any
measurable control can be replaced with a
bang-bang control with a single switch:
In this case,
The HS system:
A natural and useful idea is to
consider the most unstable trajectory.
“Stability analysis of switched systems using variational principles: an introduction”, Automatica 42(12), 2059-2077, 2006.