On the Dynamic Instability of a Class of Switching System

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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
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The switching system Motivation

• We are interested in the asymptotic stability
  of linear switching systems
• 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.

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So what is the problem Asymptotic stability

• The dynamic system, A? Rn?n,
  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 PTgt0, for all QQTgt0.

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Switching systems The issue of stability
The car in the desert scenario!
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Periodic oscillations and instability
Instability due to switching
Periodic orbit due to switching
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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

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Overview of talk

• Some background discussion and definitions.
• Some geometric observations
• Main theorem and proof.
• Consequences of main theorem.
• Extensions
• Concluding remarks

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Hurwitz 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.

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Asymptotic stability of the origin

gt0
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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.

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Matrix 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.

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Common quadratic Lyapunov function (CQLF)

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

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A geometric observation
A local observation (at a point)
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Theorem 1 An instability result
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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.

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Proof A sufficient condition for instability
(Floquet theory)
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Proof A sufficient condition for instability
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Proof an approximation

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

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Proof an approximation

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

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Proof A theorem by Kato and Lancaster
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General switching systemsThe 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.

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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.

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Pair-wise triangular switching systems
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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?

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Robustness of triangular systems
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Robustness of triangular systems
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Robustness 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 Kgt4,Lgt2, an
  unstable switching sequence always exists.

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Pairwise triangularisability
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Pairwise triangularisability
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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
  non-conservatism) of the CQLF.