INPUT-TO-STATE STABILITY of

SWITCHED SYSTEMS

Debasish Chatterjee, Linh Vu, Daniel Liberzon

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

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• Stability margin

• Integral ISS (with stability margin)

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

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

• Some subsystems not ISS [Müller, Chatterjee, Yang]

finds application in switching adaptive control

INVERTIBILITY of SWITCHED SYSTEMS

Aneel Tanwani, Linh Vu, Daniel Liberzon

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

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 switch-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

CONCLUSIONS

• Showed how results on stability under slow switching

extend in a natural way to external stability (ISS)

• Studied new invertibility problem: recovering both the

input and the switching signal

• Both problems have applications in control design

• General motivation/application: analysis and design

of complex interconnected systems