Stability and stabilization of hybrid systemsMikael Johansson
School of Electrical Engineering
KTH
Example
Question: Does GUAS of switched linear system imply existence of a
common quadratic Lyapunov function?
Answer: No, the system given by
is GUAS, but does not admit any common quadratic Lyapunov function since
satisfy the infeasibility condition.
(there is, however, a common piecewise quadratic Lyapunov function)
Example
Sample trajectories of switched system
(under two different switching strategies)
Even if solutions are very different, all motions are asymptotically stable
P2: Stabilization
Problem: given matrices Ai, find switching rule (x,i) such that
is asymptotically stable.
Stabilization of switched linear systems
Stabilizing switching rules (I)
Thus, for each x, at least one mode satisfies
This implies, in turn, that the switching rule
is well-defined for all x and that it generates
globally asymptotically stable motions. State-dependent switching strategy designed from Lyapunov function for Aeq
Solve Lyapunov equality . It follows that
Stabilizing switching rules (II)
Alternative switching strategy: activate mode i fraction i of the time, e.g.,
(strategy repeats after duty cycle of T seconds). “Average dynamics” is
and for sufficiently small T the spectral radius of
is less than one (i.e., state at beginning of each duty cycle will tend to zero)
Example
Consider the two subsystems given by
Both subsystems are unstable, but the matrix Aeq=0.5A1+0.5A2 is stable.
State-dependent switching: set Q=I, solve Lyapunov equation to find
Time-dependent switching: choose duty cycle T such that spectral radius of
is less than one. Alternate between modes each T/2 seconds.
Example cont’d
Time-driven switching State-dependent switching
P3: Stability for a given switching strategy
Problem: how can we verify that the switched system
is globally asymptotically stable?
Stability for given switching strategy
For simplicity, consider a system with two modes, and assume that
are globally asymptotically stable with Lyapunov functions Vi
Even if there is no common Lyapunov function, stability follows if
where tk denote the switching times.
Reason: Vi is continuous Lyapunov function for the switched system.
Multiple Lyapunov function approach
Note: need to know switching times very hard to apply (more later).
Multiple Lyapunov function approach
Weaker versions exist:
No need to require that submodels are stable, sufficient to require that all submodels admit Lyapunov-like functions:
where Xi contains all x for which submodel fi can be activated.
Can weaken requirement that Vi should decrease along trajectories of fi
See the references for details and precise statements.
Summary
A whirlwind tour:
selected results on stability and stabilization of hybrid systems
Three specific problems
Guaranteeing stability independent of switching signal
Design a stabilizing switching strategy (stabilizability)
Prove stability for a given switching strategy
Focus has been on Lyapunov-function techniques
Alternative approaches exist!
Strong theoretical results, but hard to apply in practice
Can be overcome by developing automated numerical techniques (Lecture 2!)
References
R. A. DeCarlo, M. S. Branicky, S. Pettersson and B. Lennartsson, “Perspectives and results on the stability and stabilizability of hybrid systems”, Proceedings of the IEEE, Vol. 88, No. 7, July 2000.
J. P. Hespanha, “Stabilization through hybrid control”, UNESCO Encyclopedia of Life Support Systems”, 2005.
M. Johansson, “Piecewise linear control systems – a compuational approach”, Springer Lecture Notes in Control and Information Sciences no. 284, 2002.
J. Goncalves, ”Constructive Global Analysis of Hybrid Systems”, Ph.D. Thesis, Massachusetts Institute of Technology, September 2000.
Automatic Control Group Signals, Sensors and Systems Royal Institute of Technology SE-10044 Stockholm, Sweden
Email : mikaelj@s3.kth.se Phone: +46-8-7907436 WWW: www.s3.kth.se/~mikaelj Mikael Johansson Associate Professor
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