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.
