1. INTRODUCTION to SWITCHED SYSTEMS ; STABILITY under ARBITRARY SWITCHING
Daniel Liberzon
Coordinated Science Laboratory and
Dept. of Electrical & Computer Eng.,
Univ. of Illinois at Urbana-Champaign
IAAC Workshop, Herzliya, Israel, June 1, 2009

2. SWITCHED and HYBRID SYSTEMS
Hybrid systems combine continuous and discrete dynamics
Which practical systems are hybrid?
electric circuit
thermostat
cell division
walking
stick shift
Picking the most relevant examples for Israel 
Walking: robots or humans
On the last point, can mention a bouncing ball
Which practical systems are not hybrid?
More tractable models of continuous phenomena

3. MODELS of HYBRID SYSTEMS…
[Van der Schaft–Schumacher ’00]
[Proceedings of HSCC]
Mention jumps
The difference is in whether the emphasis is more on continuous, or on discrete, or equally on both
Mention a more “balanced” hybrid model in terms of guards and location (see HYCON sheet)
continuous
discrete
[Nešić–L ‘05]

4. SWITCHED vs. HYBRID SYSTEMS
Switched system:
is a family of systems
is a switching signal
Switching can be:
State-dependent or time-dependent
Autonomous or controlled
Assuming common equilibrium
Sigma is piecewise constant
Hybrid systems give rise to classes of switching signals
Measurable switching – related to control system (Filippov’s lemma), more on this later
Details of discrete behavior are “abstracted away”
Discrete dynamics classes of switching signals
Properties of the continuous state
: stability and beyond

5. STABILITY ISSUE Asymptotic stability of each subsystem is
And vice versa: can stabilize, even if subsystems are unstable
This is valid in dimensions 2 and higher
unstable
Asymptotic stability of each subsystem is
not sufficient for stability

6. TWO BASIC PROBLEMS Stability for arbitrary switching
Stability for constrained switching
Second is more relevant for control
First leads to second

7. TWO BASIC PROBLEMS Stability for arbitrary switching
Stability for constrained switching
In this talk we’ll mainly deal with the first problem, except for towards the end

8. GLOBAL UNIFORM ASYMPTOTIC STABILITY
GUAS is: Lyapunov stability
plus asymptotic convergence
Reduces to standard GAS for non-switched systems
Explain uniformity w.r.t. sigma
GUES:

9. COMMON LYAPUNOV FUNCTION
where is positive definite
is GUAS if (and only if) s.t.
quadratic is GUES

10. OUTLINE Stability criteria to be discussed:
Commutation relations (Lie algebras)
Feedback systems (absolute stability)
Observability and LaSalle-like theorems
Last bullet: WEAK Lyapunov function
Common Lyapunov functions will play a central role

11. COMMUTING STABLE MATRICES => GUES
(commuting Hurwitz matrices)
“In order to elucidate the role of commutation relations…”
For subsystems – similarly

12. COMMUTING STABLE MATRICES => GUES
Alternative proof:
quadratic common Lyapunov function
[Narendra–Balakrishnan ’94]
Alternative to LMI solvers for commuting case
Last equation is a hyperlink .
is a common Lyapunov function

13. NILPOTENT LIE ALGEBRA => GUES
Lie bracket:
Nilpotent means sufficiently high-order Lie brackets are 0
For example:
(2nd-order nilpotent)
Recall: in commuting case
Gurvits proved it for second-order nilpotent
In 2nd-order nilpotent case
Hence still GUES [Gurvits ’95]

14. SOLVABLE LIE ALGEBRA => GUES
Larger class containing all nilpotent Lie algebras
Suff. high-order brackets with certain structure are 0
Lie’s Theorem: is solvable triangular form
Example:
This is a more general result
exponentially fast
exp fast
quadratic common Lyap fcn
diagonal
[Kutepov ’82, L–Hespanha–Morse ’99]

15. SUMMARY: LINEAR CASE Lie algebra commuting subsystems:
w.r.t.
Assuming GES of all modes, GUES is guaranteed for:
commuting subsystems:
nilpotent Lie algebras (suff. high-order Lie brackets are 0)
e.g.
solvable Lie algebras (triangular up to coord. transf.)
solvable + compact (purely imaginary eigenvalues)
Quadratic common Lyapunov function exists in all these cases
No further extension based on Lie algebra only [Agrachev–L ’01]

16. SWITCHED NONLINEAR SYSTEMS
Lie bracket of nonlinear vector fields:
Reduces to earlier notion for linear vector fields
(modulo the sign)

17. SWITCHED NONLINEAR SYSTEMS
Commuting systems
GUAS
Can prove by trajectory analysis [Mancilla-Aguilar ’00]
or common Lyapunov function [Shim et al. ’98, Vu–L ’05]
Linearization (Lyapunov’s indirect method)
Lie bracket is defined for differentiable vector fields, but commutativity of the flows is enough
Shim – for ES subsystems, Linh – for AS
Comment on the iterative construction of common Lyapunov function
“Common Lyapunov function” (actually, a transparent rectangle behind it) is a hyperlink
Global results beyond commuting case – ?
[Unsolved Problems in Math. Systems & Control Theory ’04]

18. SPECIAL CASE globally asymptotically stable Want to show: is GUAS
Will show: differential inclusion
is GAS
Nilpotent of order 2 – covered by known results in linear case
Some technical conditions are needed (see paper)

19. OPTIMAL CONTROL APPROACH
Associated control system:
where
(original switched system )
Worst-case control law [Pyatnitskiy, Rapoport, Boscain, Margaliot]:
fix and small enough
Idea appeared before, but not pursued for nonlinear case. MM proposed to use it in this case, this is joint work with him
Final time small for solutions to exist – but will show aposteriori that it could be arbitrary
The optimal control problem is well defined (Filippov, Lin-Sontag-Wang)
Explain how this will give us GUAS

20. MAXIMUM PRINCIPLE Optimal control: is linear in (unless )
(along optimal trajectory)
Optimal control:
is linear in
\lambda is costate, \varphi is switching function (defines optimal control)
This is of course where Lie brackets come in
Strong version of bang-bang theorem
GAS is easy to prove from definition, and intuitively clear
Note: commuting case covered in same way (worst-case law is just to follow one subsystem)
To generalize, need to make sure terms multiplying u are 0
(unless )
at most 1 switch
GAS

21. GENERAL CASE [Margaliot–L ’06, Sharon–Margaliot ’07]
Want: polynomial of degree
(proof – by induction on )
bang-bang with switches
These \varphi define the optimal control
Uniform bound on degree – guaranteed by suitable Lie bracket conditions (on next slide)
Singularity is handled as before
Explain difference with bang-bang theorems of Sussmann and Agrachev-Vakhrameev
GAS
[Margaliot–L ’06, Sharon–Margaliot ’07]

22. REMARKS on LIE-ALGEBRAIC CRITERIA
Checkable conditions
In terms of the original data
Independent of representation
Not robust to small perturbations
Can use tools from Lie algebras (such as Killing form) to check
No need to look for a specific basis
(related to the previous point) They reveal the fact that what matters is the structure of the Lie algebra, and not its representation
Strong negative emotions are well justified 
Theoretically appealing (but not very practical – unless we can measure “closeness”)
In any neighborhood of any pair of matrices
there exists a pair of matrices generating the entire
Lie algebra [Agrachev–L ’01]
How to measure closeness to a “nice” Lie algebra?

23. FEEDBACK SYSTEMS: ABSOLUTE STABILITY
controllable
Circle criterion: quadratic common Lyapunov function
is strictly positive real (SPR):
Common way in which a switched system arises
Switched system is GUES
Discuss passivity a little more, mention KYP (see notes)
More on Popov later
For this reduces to SPR (passivity)
Popov criterion not suitable: depends on

24. FEEDBACK SYSTEMS: SMALL-GAIN THEOREM
controllable
Small-gain theorem:
quadratic common Lyapunov function
Again, switched system is GUES
Can handle norms other than 1 (clear)

25. OBSERVABILITY and ASYMPTOTIC STABILITY
Barbashin-Krasovskii-LaSalle theorem:
is GAS if s.t.
(weak Lyapunov function)
is not identically zero along any nonzero solution
(observability with respect to )
observable
=> GAS
Example:
Will spend some time on observability, but let me give motivation first
V satisfies the usual properties of candidate Lyapunov function
Remember to clarify the meaning of “weak” verbally
The connection between LaSalle and observability is not always made explicitly, but has been discussed by several people (Artstein)
C gives fictitious output

26. SWITCHED LINEAR SYSTEMS
[Hespanha ’04]
Theorem (common weak Lyapunov function):
Switched linear system is GAS if
s.t
observable for each
infinitely many switching intervals of length
The condition on sigma: prevents sliding/chattering in the kernel of the output matrix (assume same output for subsystems). We’ll see the reason for this assumption later (we’ll need a piece of y of length tau to say anything about x). The counterexample is in Joao’s paper. Fast switching, state is bounded but doesn’t converge.
This is not GUAS – have uniformity over p-dwell (but only in the linear case)
Detectability is not enough (unless no switching in unobservable modes) – doesn’t decay enough in \tau time units
Observability is the only one of the three conditions that is linear in nature (weak Lyap fcn – already had the nonlinear version, last one – on sigma only)
To handle nonlinear switched systems and
non-quadratic weak Lyapunov functions,
need a suitable nonlinear observability notion

27. SWITCHED NONLINEAR SYSTEMS
Theorem (common weak Lyapunov function):
Switched system is GAS if
s.t.
infinitely many switching intervals of length
Each system
is norm-observable:
V has the usual properties of a candidate Lyapunov function
\tau used in the obs condition is the \tau from the last one (which we don’t know – but can have a slow switching version)
It’s better to first list hypotheses and then give conclusion?
Will come back to this result in the next lecture
[Hespanha–L–Sontag–Angeli ’05]