Stability Analysis of Switched Systems: A Variational Approach

1
Stability Analysis of Switched Systems A
Variational Approach

• Michael Margaliot
• School of EE-Systems Tel Aviv University
• Joint work with Daniel Liberzon (UIUC)

2
Overview

• Switched systems
• Stability
• Stability analysis
• A control-theoretic approach
• A geometric approach
• An integrated approach
• Conclusions

3
Switched Systems

• Systems that can switch between
• several modes of operation.

Mode 1
Mode 2
4
Example 1
server
5
Example 2

• Switched power converter

50v
100v
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Example 3

• A multi-controller scheme

plant

controller1
controller2
switching logic
Switched controllers are stronger than regular
controllers.
7
More Examples

• Air traffic control
• Biological switches
• Turbo-decoding

8
Synthesis of Switched Systems

• Driving use mode 1 (wheels)
• Braking use mode 2 (legs)
• The advantage no compromise

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Mathematical Modeling with Differential Inclusions
easier ANALYSIS harder
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The Gestalt Principle

• Switched systems are more than the
• sum of their subsystems.
• ? theoretically interesting
• ? practically promising

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Differential Inclusions
A solution is an absolutely continuous function
satisfying (DI) for all t. Example
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Stability

• The differential inclusion
• is called GAS if for any solution
• (i)
• (ii)

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The Challenge

• Why is stability analysis difficult?
• A DI has an infinite number of solutions for each
  initial condition.
• The gestalt principle.

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Absolute Stability

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Problem of Absolute Stability
The closed-loop system
A is Hurwitz, so CL is asym. stable for any
The Problem of Absolute Stability Find
For CL is asym. stable for any
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Absolute Stability and Switched Systems
The Problem of Absolute Stability Find
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Example
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Trajectory of the Switched System

This implies that
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• Although both and are
  
• stable, is not stable.
• Instability requires repeated switching.
• This presents a serious problem in
• multi-controller schemes.

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Optimal Control Approach

• Write as a control
  system
• Fix Define
• Problem Find the control that
  maximizes

is the worst-case switching law
(WCSL). Analyze the corresponding trajectory
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Optimal Control Approach
Consider
as
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Optimal Control Approach

• 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

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Solving Optimal Control Problems

• is a functional
• Two approaches
• The Hamilton-Jacobi-Bellman (HJB) equation.
• The Maximum Principle.

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The HJB Equation

• Find such that
• Integrating
• or
• An upper bound for ,
• obtained for the maximizing Eq. (HJB).

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The HJB for a LDI Hence, In general,
finding is difficult.
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The Maximum Principle

• Let Then,
• Differentiating we get
• A differential equation for with a
• boundary condition at

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• Summarizing,
• The WCSL is the maximizing
• that is,
• We can simulate the optimal solution
• backwards in time.

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The Case n2

• Margaliot Langholz (2003) derived an
• explicit solution for when n2.
• This yields an easily verifiable necessary and
  sufficient condition for stability of
  second-order switched linear systems.

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The Basic Idea
The function is a first integral of
if We know that so
Thus, is a concatenation of two first
integrals and
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Example
where and
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Thus, so we have an explicit expression for V
(and an explicit solution of HJB).
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Nonlinear Switched Systems

• where are
  GAS.
• Problem Find a sufficient condition
  guaranteeing GAS of (NLDI).

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Lie-Algebraic Approach

• For the sake of simplicity, consider
• the LDI
• so

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Commutation and GAS

• Suppose that A and B commute,
• ABBA, then
• Definition The Lie bracket of Ax and
  Bx is Ax,BxABx-BAx.
• Hence, Ax,Bx0 implies GAS.

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Lie Brackets and Geometry

• Consider
• Then

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Geometry of Car Parking

• This is why we can park our car.
• The term is the reason it takes
• so long.

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Nilpotency

• Definition kth order nilpotency -
• all Lie brackets involving k1
  terms vanish.
• 1st order nilpotency A,B0
• 2nd order nilpotency A,A,BB,A,B0
• Q Does kth order nilpotency imply GAS?

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Some Known Results

• Switched linear systems
• k 2 implies GAS (Gurvits,1995).
• kth order nilpotency implies GAS (Liberzon,
  Hespanha, and Morse, 1999).(The proof is based
  on Lies Theorem)
• Switched nonlinear systems
• k 1 implies GAS.
• An open problem higher orders of k? (Liberzon,
  2003)

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A Partial Answer

• Thm. 1 (Margaliot Liberzon, 2004)
• 2nd order nilpotency implies GAS.
• Proof Consider the WCSL
• Define the switching function

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• Differentiating m(t) yields
• 1st order nilpotency ? ?
• ? no switching in the WCSL.
• Differentiating again, we get
• 2nd order nilpotency ? ?
• ? up to a single switch in the WCSL.

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Handling Singularity

• If m(t)?0, then the Maximum Principle
• does not necessarily provide enough
• information to characterize the WCSL.
• Singularity can be ruled out using
• the notion of strong extermality
• (Sussmann, 1979).

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3rd order Nilpotency
In this case
further differentiation cannot be carried out.
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3rd order Nilpotency
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.
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Hall-Sussmann System
Consider the case A,B0.
Guess the solution
Then
so
and
(HS system)
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Hall-Sussmann System
If two controls u, v yield the same values for
then they yield the same
value for
does not depend on u,
Since
we conclude that any
and
measurable control can be replaced with a
bang-bang control with a single switch
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3rd order Nilpotency
In this case,
The HS system
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Conclusions

• Switched systems and differential inclusions
  are important in various scientific fields,
  and pose interesting theoretical questions.

• Stability analysis is difficult.
• A natural and useful idea is to
• consider the most unstable trajectory.

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For more information, see the survey
paper Stability analysis of switched systems
using variational principles an introduction,
Automatica 42(12), 2059-2077, 2006. Available
online www.eng.tau.ac.il/michaelm