Transcript On the Dynamic Instability of a Class of Switching System

On the Dynamic Instability of a
Class of Switching System
Robert Noel Shorten
Department of Computer Science
National University of Ireland
Maynooth, Ireland
Fiacre Ó Cairbre
Department of Mathematics
National University of Ireland
Maynooth, Ireland
Paul Curran
Department of Electrical Engineering
National University of Ireland
Dublin, Ireland
IFAC AIRTC, Budapest,
October 2000
The switching system: Motivation
• We are interested in the asymptotic stability of linear switching systems:
x  A(t ) x,
[1]
where x(t)Rn, and where A(t)Rnn 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.
So what is the problem: Asymptotic stability
nn
• The dynamic system,  A : x  Ax , A R , 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,
AT P  PA  Q,
T
T
is P = P >0, for all Q=Q >0.
• V ( x)  xT Px is a quadratic Lyapunov function for  A : x  Ax.
Switching systems: The issue of stability
The car in the desert scenario!
Periodic oscillations and instability
Instability due to switching
Periodic orbit due to switching
Traditional 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 talk
• Some background discussion and definitions.
• Some geometric observations
• Main theorem and proof.
• Consequences of main theorem.
• Extensions
• Concluding remarks
Hurwitz matrices
Hurwitz matrices: The matrix A, A  R nn ,
is said to be Hurwitz if its eigenvalues lie
in the open left-half of the complex plane.
 ( A ) C ,

i
A  R n n
A matrix A is said to be not-Hurwitz if some
of its eigenvalues lie in the open right-half of
the complex plane.
Asymptotic stability of the origin
>0
Instability
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:.
x(t )  
Matrix pencils
A matrix pencil is defined as:
 [ Ai , M ] :  i Ai ,  i  0,
M
i 1
 i
M
i 1
 0.
If the eigenvalues of [Ai,M] are in C-for all
non-negative i, then [Ai,M] is referred to as a
Hurwitz pencil.
Common quadratic Lyapunov function (CQLF)
• V(x) = xTPx is said to be a common quadratic
Lyapunov function (CQLF) for the dynamic systems,
 A : x  Ai x, i  {1,..., M },
i
if
AiT P  PAi  Qi , i  {1,..., M },
and
Qi  QiT  0, i  {1,..., M }.
A geometric observation
A local observation (at a point)
Theorem 1: An instability result
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)
Consider t he periodic system A(t)  A(t  T) :
x(T )  (e A t e A t e A t ) x0
1 1
2 2
  (T ) x0 ,
M M
M
T   ti
1
A sufficient condition for instabilit y
is that the matrix  (T ) has an eigenvalue
whose magnitude is greater th an unity.
Proof: A sufficient condition for instability
 (T ) can be expanded as a power
series,
 (T )

I  (   i Ai )T  K 2T 2  

I   [ Ai , M ]T  
M
i 1
Note that  (T ) is the product of M absolutely
convergent power series, and is analytic
in T .
Proof: an approximation
• So, for T small enough, the effect of the higher order terms
become negligible, and we have,
Com plex plane
j
(T )

I   [ Ai , M ]T

K ( I  JT ) K 1
 j (T )
j
1
Proof: an approximation
• So, for T small enough, the effect of the higher order terms
become negligible, and we have,
Com plex plane
j
(T )

I   [ Ai , M ]T

K ( I  JT ) K 1
 j (T )
j
1
Proof: A theorem by Kato and Lancaster
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:
 [ Ai , M ] :  i Ai ,  i  0,
M
i 1
 i
M
i 1
 0.
is Hurwitz. In general, this is a very conservative
condition.
• Now we know that this conditions is necessary for
stability of the system [1].
Equivalence of stability and
CQLF for low order systems
• Necessary and sufficient conditions for the existence of a CQLF for two
second order systems
 A : x  A1 x
1
 A : x  A2 x
2
is that the matrix pencils are both Hurwitz.
A1  (1   ) A2 ,
 [0,1]
A1  (1   ) A21 ,
 [0,1]
• Non-existence of CQLF implies that one of the dual switching systems
is unstable.
x  A(t ) x, A(t ) { A1 , A2 }, Ai  R 22
x  A(t ) x, A(t ) { A1 , A21}, Ai  R 22
Pair-wise triangular
switching systems
We consider t he switching system
x  A(t ) x, A(t ) { A1 ,..., AM }, Ai  R N  N
where the Ai are asymptotic ally stable
and a set of non - singular matrices
Tij exist such that the {Tij AiTij1 , Tij AjTij1}
upper tria ngular.
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 systems
(0,1)
1
( ,1)
L
Eigenvectors of A2
1
(1, )
L
Eigenvectors of A1
 K
A1  
0

0 
 K
1 , A M
 2 2
0



L
(1,0)

1
0 

1  M 1 , M  
 2
1

L

L
1
L

1

Robustness of triangular systems
 K
A1  
0

0 
 K
1 , A M
 2 2
0



L

0 
1
1
1 M ,M 
 2
1

L

L
The matrices satisfy th e following
properties :
(a) A1 and A2 have identical eigenvalue s.
(b) lim L A1  lim L A2
1
L

1

Robustness of triangular systems
• Consider the periodic switching system with
duty cycle 0.5 with:
 K
A1  
 0

1


K

0 

4
L

1 ,A 
 2 2  K 1
  3
L
 L L
K 1
 3
L L 
K 1
 2
2
L L
• As L increases A1 and A2 become more and
more triangularisable. However, for K>4,L>2,
an unstable switching sequence always exists.
Pairwise triangularisability
Pairwise triangularisability
Conclusions
• 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 nonconservatism) of the CQLF.