IFAC AIRTC, Budapest, October 2000 On the Dynamic Instability of a Class of Switching System Robert Noel Shorten Department of Computer Science National.

IFAC AIRTC, Budapest, October 2000

On the Dynamic Instability of a Class of Switching System

Robert Noel Shorten Department of Computer Science

National University of IrelandMaynooth, Ireland

Fiacre Ó CairbreDepartment of Mathematics

National University of IrelandMaynooth, Ireland

Paul CurranDepartment of Electrical Engineering

National University of IrelandDublin, Ireland

The switching system: MotivationThe switching system: Motivation• We are interested in the asymptotic stability of linear switching systems:

where x(t)Rn, and where A(t)Rnn belongs to the finite set {A1,A2,…AM}.

• Switching is now common place in control engineering practice.– Gain scheduling.– Fuzzy and Hybrid control.– Multiple models, switching and tuning.

• Recent work by Douglas Leith (VB families) has shown a dynamic equivalence between classes of linear switching systems and non-linear systems.

]1[,)( xtAx

So what is the problem: Asymptotic stabilitySo what is the problem: Asymptotic stability

• The dynamic system, A Rnn, is asymptotically stable if the matrix A is Hurwitz (i(A), i {1,…,n} C -).

• If A is Hurwitz, the solution of the Lyapunov equation,

is P = PT>0, for all Q=QT>0.

•

,QPAPAT

,: AxxA

.:for function Lyapunov quadratic a is )( AxxPxxxV A

T

Switching systems: The issue of stabilitySwitching systems: The issue of stability

The car in the desert scenario!

Periodic oscillations and instability

Periodic oscillations and instability

Instability due to switching

Periodic orbit due to switching

Traditional approach to analysisTraditional approach to analysis

• Stability: Most results in the literature pertain to stability

– Lyapunov

– Input-output

– Slowly varying systems ….

– Conservatism is well documented

• Instability: Few results concern instability

– Describing functions

– Chattering (sliding modes)

– Routes to instability (chaos): Potentially much tighter conditions

Overview of talkOverview of talk

• Some background discussion and definitions.

• Some geometric observations

• Main theorem and proof.

• Consequences of main theorem.

• Extensions

• Concluding remarks

Hurwitz matricesHurwitz matrices

Hurwitz matrices: The matrix A,

is said to be Hurwitz if its eigenvalues lie

in the open left-half of the complex plane.

A matrix A is said to be not-Hurwitz if some

of its eigenvalues lie in the open right-half of

the complex plane.

nn

iRA,C)A(

,RA nn

Asymptotic stability of the originAsymptotic stability of the origin

>0

InstabilityInstability

The switching system [1] is unstable if some switching sequence exists such that as time increases the magnitude of the solution to [1], x(t) is unbounded:.

)(tx

Matrix pencilsMatrix pencils

A matrix pencil is defined as:

If the eigenvalues of [Ai,M] are in C-for all

non-negative i, then [Ai,M] is referred to as a

Hurwitz pencil.

.0,0,:],[11

M

iii

M

iiii AMA

Common quadratic Lyapunov function (CQLF)Common quadratic Lyapunov function (CQLF)

• V(x) = xTPx is said to be a common quadratic Lyapunov function (CQLF) for the dynamic systems,

if

and

},,...,1{,: MixAx iAi

},,...,1{, MiQPAPA ii

T

i

}.,...,1{,0 MiQQ T

ii

A geometric observationA geometric observation

A local observation (at a point)

Theorem 1: An instability resultTheorem 1: An instability result

Outline of the proof Outline of the proof

• We consider a periodic switching sequence for [1].

• We use known instability conditions for periodic systems using Floquet theory.

• We show that Theorem 1 implies instability for such systems.

Proof: A sufficient condition for instability (Floquet theory)

Proof: A sufficient condition for instability (Floquet theory)

unity.an greater th is magnitude whose

eigenvaluean has )(matrix that theis

y instabilitfor condition sufficientA

,)(

)()(

: system periodic heConsider t

10

02211

T

tTxT

xeeeTx

T)A(tA(t)

M

i

tAtAtA MM

Proof: A sufficient condition for instabilityProof: A sufficient condition for instability

.in

analytic is and series,power convergent

absolutely ofproduct theis )( that Note

],[

)()(

series,

power a as expanded becan )(

2

21

T

MT

TMAI

TKTAIT

T

i

M

iii

Proof: an approximationProof: an approximation

• So, for T small enough, the effect of the higher order terms become negligible, and we have,

1)(

],[)(

KJTIK

TMAIT i

j

1

)(Tj

Complex plane

j

Proof: an approximationProof: an approximation

• So, for T small enough, the effect of the higher order terms become negligible, and we have,

j

1

)(Tj

Complex plane

j

j

1

)(Tj

Complex plane

j1)(

],[)(

KJTIK

TMAIT i

Proof: A theorem by Kato and LancasterProof: A theorem by Kato and Lancaster

General switching systems:The existence of CQLF

General switching systems:The existence of CQLF

• It has long been known that a necessary condition for the existence of a CQLF is that the matrix pencil:

is Hurwitz. In general, this is a very conservative condition.

• Now we know that this conditions is necessary for stability of the system [1].

.0,0,:],[11

M

iii

M

iiii AMA

Equivalence of stability and CQLF for low order systems

Equivalence of stability and CQLF for low order systems

• Necessary and sufficient conditions for the existence of a CQLF for two second order systems

is that the matrix pencils are both Hurwitz.

• Non-existence of CQLF implies that one of the dual switching systems is unstable.

22121

2221

},,{)(,)(

},,{)(,)(

RAAAtAxtAx

RAAAtAxtAx

i

i

]1,0[,)1(

]1,0[,)1(1

21

21

AA

AA

xAx

xAx

A

A

2

1

:

:

2

1

Pair-wise triangular switching systems

Pair-wise triangular switching systems

ngular.upper tria

},{ thesuch that exist

matricessingular -non ofset a and

stableally asymptotic are thewhere

},,...,{)(,)(

system switching heconsider t We

11

1

ijjijijiijij

i

NN

iM

TATTATT

A

RAAAtAxtAx

Pair-wise triangular switching systems: Comments

Pair-wise triangular switching systems: Comments

• A single T implies the existence of a CQLF for each of the component systems. Is this a robust result?

• Pair-wise triangularisability and some extra conditions imply global attractivity.

• Are general pairwise triangularisable systems stable?

Robustness of triangular systemsRobustness of triangular systems

)0,1(

)1,0(

)1

,1(L

)1,1

(L Eigenvectors of A 2

Eigenvectors of A 1

11

11

,10

0,1

0

01

22

21

L

LMML

KMA

L

KA

Robustness of triangular systemsRobustness of triangular systems

11

11

,10

0,1

0

01

22

21

L

LMML

KMA

L

KA

21

21

limlim (b)

s.eigenvalue identical have and (a)

:properties

following esatisfy th matrices The

AA

AA

LL

Robustness of triangular systemsRobustness of triangular systems

• Consider the periodic switching system with duty cycle 0.5 with:

• As L increases A1 and A2 become more and

more triangularisable. However, for K>4,L>2, an unstable switching sequence always exists.

223

34

22

1 11

11

,10

0

LLK

LLK

LLK

LK

AL

KA

Pairwise triangularisabilityPairwise triangularisability

Pairwise triangularisabilityPairwise triangularisability

ConclusionsConclusions

• Looked at a local stability theorem.

• Presented a formal proof.

• Used theorem to answer some open questions.

• Presented some extensions to the work.

• Gained insights into conservatism (or non-conservatism) of the CQLF.