STABILITY under CONSTRAINED SWITCHING ;

SWITCHED SYSTEMS with INPUTS and OUTPUTS

Daniel Liberzon

Coordinated Science Laboratory andDept. of Electrical & Computer Eng.,Univ. of Illinois at Urbana-Champaign

IAAC Workshop, Herzliya, Israel, June 1, 2009

TWO BASIC PROBLEMS

• Stability for arbitrary switching

• Stability for constrained switching

MULTIPLE LYAPUNOV FUNCTIONS

Useful for analysis of state-dependent switching

– GAS

– respective Lyapunov functions

is GAS

MULTIPLE LYAPUNOV FUNCTIONS

decreasing sequence

decreasing sequence

[DeCarlo, Branicky]

GAS

DWELL TIME

The switching times satisfy

dwell time– GES

– respective Lyapunov functions

DWELL TIME

– GES

Need:

The switching times satisfy

DWELL TIME

– GES

Need:

The switching times satisfy

DWELL TIME

– GES

Need:

must be 1

The switching times satisfy

AVERAGE DWELL TIME

# of switches on average dwell time

– dwell time: cannot switch twice if

AVERAGE DWELL TIME

Theorem: [Hespanha ‘99] Switched system is GAS if

Lyapunov functions s.t. • .

•

•

Useful for analysis of hysteresis-based switching logics

# of switches on average dwell time

MULTIPLE WEAK LYAPUNOV FUNCTIONS

Theorem: is GAS if

• .

•

•

•

– milder than ADT

Extends to nonlinear switched systems as before

observable for each

s.t. there are infinitely many

switching intervals of length

For every pair of switching times

s.t.

have

APPLICATION: FEEDBACK SYSTEMS (Popov criterion)

Corollary: switched system is GAS if

• s.t. infinitely many switching intervals of length

• For every pair of switching times at

which we have

linear system observable

positive real

See also invariance principles for switched systems in: [Lygeros et al., Bacciotti–Mazzi, Mancilla-Aguilar, Goebel–Sanfelice–Teel]

Weak Lyapunov functions:

STATE-DEPENDENT SWITCHING

But switched system is stable for (many) other

Switched system

unstable for some

no common

switch on the axes

is a Lyapunov function

STATE-DEPENDENT SWITCHING

But switched system is stable for (many) other

level sets of level sets of

Switched system

unstable for some

no common

Switch on y-axis

GAS

STABILIZATION by SWITCHING

– both unstable

Assume: stable for some

STABILIZATION by SWITCHING

[Wicks et al. ’98]

– both unstable

Assume: stable for some

So for each

either or

UNSTABLE CONVEX COMBINATIONS

Can also use multiple Lyapunov functions

Linear matrix inequalities

SWITCHED SYSTEMS with INPUTS and OUTPUTS

• Background

• Input-to-state stability (ISS)

• Main results

• ISS under ADT switching

• Invertibility of switched systems

Outline:

INPUT-TO-STATE STABILITY (ISS)

classNonlinear gain functions:

Equivalent Lyapunov characterization [Sontag–Wang]:

without loss of generality,can replace by

ISS [Sontag ’89]:

classclass , e.g.

(means: pos. def., rad. unbdd.)

ISS under ADT SWITCHING

eachsubsystem

is ISS

[Vu–Chatterjee–L, Automatica, Apr 2007]

If has average dwell time

• .

•

•

class functions and constants

such that :

Suppose functions

then switched system is ISS

SKETCH of PROOF

1

1 Let be switching times on

Consider

Recall ADT definition:

2

3

SKETCH of PROOF

12

3

2

1

3

Special cases:

• GAS when

• ISS under arbitrary switching if (common )

• ISS without switching (single )

– ISS

VARIANTS

• Output-to-state stability (OSS) [M. Müller]

• Stochastic versions of ISS for randomly switched systems [D. Chatterjee]

• Some subsystems not ISS [Müller, Chatterjee]

finds application in switching adaptive control

• Integral ISS:

[Vu–L, Automatica, Apr 2008; Tanwani–L, CDC 2008]

SWITCHED SYSTEMS with INPUTS and OUTPUTS

• Background

• Input-to-state stability (ISS)

• Main results

• ISS under ADT switching

• Invertibility of switched systems

Outline:

PROBLEM FORMULATION

Invertibility problem: recover uniquely from for given

• Desirable: fault detection (in power systems)

Related work: [Sundaram–Hadjicostis, Millerioux–Daafouz]; [Vidal et al., Babaali et al., De Santis et al.]

• Undesirable: security (in multi-agent networked systems)

MOTIVATING EXAMPLE

because

Guess:

INVERTIBILITY of NON-SWITCHED SYSTEMS

Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham]

INVERTIBILITY of NON-SWITCHED SYSTEMS

Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham]

Nonlinear: [Hirschorn, Isidori–Moog, Nijmeijer, Respondek, Singh]

INVERTIBILITY of NON-SWITCHED SYSTEMS

Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham]

Nonlinear: [Hirschorn, Isidori–Moog, Nijmeijer, Respondek, Singh]

SISO nonlinear system affine in control:

Suppose it has relative degree at :

Then we can solve for :

Inverse system

BACK to the EXAMPLE

We can check that each subsystem is invertible

For MIMO systems, can use nonlinear structure algorithm

– similar

SWITCH-SINGULAR PAIRS

Consider two subsystems and

is a switch-singular pair if such that

|||

FUNCTIONAL REPRODUCIBILITY

SISO system affine in control with relative degree at :

For given and , that produces this output

if and only if

CHECKING for SWITCH-SINGULAR PAIRS

is a switch-singular pair for SISO subsystems

with relative degrees if and only if

MIMO systems – via nonlinear structure algorithm

Existence of switch-singular pairs is difficult to check in general

For linear systems, this can be characterized by a

matrix rank condition

MAIN RESULT

Theorem:

Switched system is invertible at over output set

if and only if each subsystem is invertible at and

there are no switched-singular pairs

Idea of proof:

The devil is in the details

no switch-singular pairs can recover

subsystems are invertible can recover

BACK to the EXAMPLE

Let us check for switched singular pairs:

Stop here because relative degree

For every , and with

form a switch-singular pair

Switched system is not invertible on the diagonal

OUTPUT GENERATION

Recall our example again:

Given and , find (if exist) s. t.

may be unique for some but not all

OUTPUT GENERATION

Recall our example again:

switch-singular pair

Given and , find (if exist) s. t.

may be unique for some but not all

Solution from :

OUTPUT GENERATION

Recall our example again:

switch-singular pair

Given and , find (if exist) s. t.

may be unique for some but not all

Solution from :

OUTPUT GENERATION

Recall our example again:

Case 1: no switch at

Then up to

At , must switch to 2

But then

won’t match the given output

Given and , find (if exist) s. t.

may be unique for some but not all

OUTPUT GENERATION

Recall our example again:

Case 2: switch at

Given and , find (if exist) s. t.

may be unique for some but not all

No more switch-singular pairs

OUTPUT GENERATION

Recall our example again:

Given and , find (if exist) s. t.

may be unique for some but not all

Case 2: switch at

No more switch-singular pairs

OUTPUT GENERATION

Recall our example again:

Given and , find (if exist) s. t.

We also obtain from

We see how one switch can helprecover an earlier “hidden” switch

may be unique for some but not all

Case 2: switch at

No more switch-singular pairs