1. STABILITY under CONSTRAINED SWITCHING ; SWITCHED SYSTEMS with INPUTS and OUTPUTS
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. TWO BASIC PROBLEMS Stability for arbitrary switching
Stability for constrained switching

3. MULTIPLE LYAPUNOV FUNCTIONS
– GAS
– respective Lyapunov functions
is GAS
No GUAS => no common Lyapunov function
Useful for analysis of state-dependent switching

4. MULTIPLE LYAPUNOV FUNCTIONS
decreasing sequence
decreasing sequence
[DeCarlo, Branicky]
GAS
Need to have information about solutions

5. DWELL TIME The switching times satisfy – GES dwell time
– respective Lyapunov functions
System is stable if switching is slow enough (dwell time large enough)
Continuous counterpart – stability of slowly time-varying systems

6. DWELL TIME The switching times satisfy – GES Need:
Of course need same thing for V_2

7. DWELL TIME The switching times satisfy – GES Need:
Here is the calculation

8. DWELL TIME The switching times satisfy – GES Need: must be
From this we get a bound on dwell time (just take the log)
must be

9. AVERAGE DWELL TIME – dwell time: cannot switch twice if
# of switches on
average dwell time
– dwell time: cannot switch twice if
Motivate ADT

10. AVERAGE DWELL TIME Theorem: [Hespanha ‘99] Switched system is GAS if
# of switches on
average dwell time
Theorem: [Hespanha ‘99] Switched system is GAS if
Lyapunov functions s.t. .
No proof for now, but will see it later (for the case of inputs)
Useful for analysis of hysteresis-based switching logics

11. MULTIPLE WEAK LYAPUNOV FUNCTIONS
Theorem: is GAS if .
observable for each
s.t. there are infinitely many
switching intervals of length
– milder than ADT
For every pair of switching times
s.t.
have
Extends to nonlinear switched systems as before

12. APPLICATION: FEEDBACK SYSTEMS (Popov criterion)
linear system observable
positive real
Weak Lyapunov functions:
Corollary: switched system is GAS if
s.t. infinitely many switching intervals of length
For every pair of switching times at
which we have
Last condition quite restrictive for control design purposes, but we don’t need equality
See also invariance principles for switched systems in:
[Lygeros et al., Bacciotti–Mazzi, Mancilla-Aguilar, Goebel–Sanfelice–Teel]

13. STATE-DEPENDENT SWITCHING
Switched system
unstable for some
no common
But switched system is stable for (many) other
switch on the axes
(Multiple) Lyapunov functions are useful for this
But not a common one! Only decreases in appropriate regions
I’m just giving examples here, but can search for MLFs systematically by LMIs
is a Lyapunov function

14. STATE-DEPENDENT SWITCHING
Switched system
unstable for some
no common
But switched system is stable for (many) other
Switch on y-axis
level sets of
For this switching signal GAS is pretty clear: with every half-rotation get closer to 0
Need MLFs to prove this
GAS

15. STABILIZATION by SWITCHING
– both unstable
Assume: stable for some

16. STABILIZATION by SWITCHING
– both unstable
Assume: stable for some
So for each
either or
Can verbally mention epsilon-robustification
So stable convex combination implies quadratic stability (single Lyapunov function)
[Wicks et al. ’98]

17. UNSTABLE CONVEX COMBINATIONS
Motivate: stable conv comb difficult to find, may not exist
Difference with previous examples: both systems unstable
Single Lyapunov function – weak one (Feron’s result, see notes)
Reference: my book
Can also use multiple Lyapunov functions
Linear matrix inequalities

18. SWITCHED SYSTEMS with INPUTS and OUTPUTS
Outline:
Background
Input-to-state stability (ISS)
Main results
ISS under ADT switching
Invertibility of switched systems

19. INPUT-TO-STATE STABILITY (ISS)
class
Nonlinear gain functions:
ISS [Sontag ’89]:
class , e.g.
class
Equivalent Lyapunov characterization [Sontag–Wang]:
Think of u as disturbance, not control
Pause to interpret ISS in terms of BIBS and CICS
(means: pos. def., rad. unbdd.)
without loss of generality,
can replace by

20. ISS under ADT SWITCHING
Suppose functions .
class functions and constants
such that :
each
subsystem
is ISS
Explain the meaning of \alpha_{1,2}, that lambda is no loss of generality, and discuss \mu
Note independence of p (OK if P is finite or compact)
The proof will be different from the paper (it works better for ISS but not for iISS)
If has average dwell time
then switched system is ISS
[Vu–Chatterjee–L, Automatica, Apr 2007]

21. SKETCH of PROOF 1 2 1 3 Let be switching times on Consider
Recall ADT definition:
Norm of u is up to infinity or up to t, by causality doesn’t matter
Then there will be stage 4 and so on
Will now analyze each stage separately

22. SKETCH of PROOF 1 2 2 1 3 3 – ISS Special cases: GAS when
Common ISS-Lyapunov function
Last case: don’t exit the level set of V
Special cases:
GAS when
ISS without switching (single )
ISS under arbitrary switching if (common )

23. VARIANTS Output-to-state stability (OSS) [M. Müller]
Integral ISS:
finds application in switching adaptive control
Output-to-state stability (OSS) [M. Müller]
Stochastic versions of ISS for randomly switched
systems [D. Chatterjee]
First is in the same paper, also mention stability margins
Some subsystems not ISS [Müller, Chatterjee]

24. SWITCHED SYSTEMS with INPUTS and OUTPUTS
Outline:
Background
Input-to-state stability (ISS)
Main results
ISS under ADT switching
Invertibility of switched systems
[Vu–L, Automatica, Apr 2008; Tanwani–L, CDC 2008]

25. PROBLEM FORMULATION
Invertibility problem: recover uniquely from for given
If this is possible, we say that the system is invertible
Mention variants: strong invertibility around x_0, strong invertibility
Sundaram, Daafouz: know sigma , recover u, discrete time – and only for linear systems
Vidal, Babaali, De Santis: switched observability/mode identification, recover sigma and x (no u or known u). Also studied discernibility for control – again, only for linear
Desirable: fault detection (in power systems)
Undesirable: security (in multi-agent networked systems)
Related work: [Sundaram–Hadjicostis, Millerioux–Daafouz];
[Vidal et al., Babaali et al., De Santis et al.]

26. MOTIVATING EXAMPLE Guess: because
Will check this guess/conjecture later
Guess:

27. INVERTIBILITY of NON-SWITCHED SYSTEMS
Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham]
Background slide: invertibility problem has been well-studied

28. INVERTIBILITY of NON-SWITCHED SYSTEMS
Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham]
Nonlinear: [Hirschorn, Isidori–Moog, Nijmeijer, Respondek, Singh]

29. INVERTIBILITY of NON-SWITCHED SYSTEMS
Linear: [Brockett–Mesarovic, Silverman, Sain–Massey, Morse–Wonham]
Nonlinear: [Hirschorn, Isidori–Moog, Nijmeijer, Respondek, Singh]
SISO nonlinear system affine in control:
Suppose it has relative degree at :
Then we can solve for :
I’m dropping time arguments, stress that these are functions of time
Inversion works on some time interval
Important: the role of the inverse system is to give the input but also the state produced by it
Abuse of notation: in the paper, the state of the system is z. Emphasize that need to initialize with the same state.
Inverse system

30. BACK to the EXAMPLE We can check that each subsystem is invertible
– similar
Can mention reduced inverse
For MIMO systems, can use nonlinear structure algorithm

31. SWITCH-SINGULAR PAIRS
Consider two subsystems and
is a switch-singular pair if such that
|||
Invertibility of subsystems was the first ingredient for switched invertibility, this will be the second ingredient
Say “x_0 and y form a switch-singular pair”
Abuse of notation: p, q in the formula stand for sigma identically equal to p, q (in Linh’s prelim this is written via H_{p,x_0}, not sure if it’s better)

32. FUNCTIONAL REPRODUCIBILITY
SISO system affine in control with relative degree at :
For given and , that produces this output
if and only if
Slide transition: to check for switch-singular pairs, we need to characterize the set of ouputs that each subsystem can produce from a given initial condition (and then check their intersection)
Reason: (i) terms that we have no control over have to match, (ii) then choose u (state feedback)p to have y^{(r)}(t) \equiv y_d^{(r)}(t), and initial conditions match => integrate to get the result

33. CHECKING for SWITCH-SINGULAR PAIRS
is a switch-singular pair for SISO subsystems
with relative degrees if and only if
For linear systems, this can be characterized by a
matrix rank condition
MIMO systems – via nonlinear structure algorithm
Existence of switch-singular pairs is difficult to check in general

34. MAIN RESULT Theorem: Switched system is invertible at over output set
if and only if each subsystem is invertible at and
there are no switched-singular pairs
Idea of proof:
no switch-singular pairs can recover
Details: output set (differentiability), subintervals (finite escape), strong invertibility
subsystems are invertible can recover
The devil is in the details

35. BACK to the EXAMPLE Let us check for switched singular pairs:
Stop here because relative degree
For every , and with
form a switch-singular pair
Switched system is not invertible on the diagonal

36. OUTPUT GENERATION Given and , find (if exist) s. t.
may be unique for some but not all
Recall our example again:
A slightly different twist: switched system not necessarily invertible, want an algorithm to find all sigma, u
Tacitly assuming that subsystems are invertible, otherwise there will be infinitely many inputs
Will use the example one last time to illustrate how the algorithm works

37. OUTPUT GENERATION Given and , find (if exist) s. t.
may be unique for some but not all
Recall our example again:
Solution from :
switch-singular pair

38. OUTPUT GENERATION Given and , find (if exist) s. t.
may be unique for some but not all
Recall our example again:
Solution from :
For sigma=1 can continue up to \pi because we know there are no more singular pairs before \pi
For sigma=2 will have to check for singular pairs
switch-singular pair

39. OUTPUT GENERATION Given and , find (if exist) s. t.
may be unique for some but not all
Recall our example again:
Case 1: no switch at
Then up to
Must switch when y is discontinuous
At , must switch to 2
But then
won’t match the given output

40. OUTPUT GENERATION Given and , find (if exist) s. t.
may be unique for some but not all
Recall our example again:
Case 2: switch at
No more switch-singular pairs
Still need to investigate Case 2, because we don’t know if this output is generatable at all

41. OUTPUT GENERATION Given and , find (if exist) s. t.
may be unique for some but not all
Recall our example again:
Case 2: switch at
No more switch-singular pairs

42. OUTPUT GENERATION Given and , find (if exist) s. t.
may be unique for some but not all
Recall our example again:
Case 2: switch at
No more switch-singular pairs
Didn’t give the formulas for u just to save space, they are easy
We see how one switch can help
recover an earlier “hidden” switch
We also obtain from