SLIDE 1 STABILITY OF SWITCHED SYSTEMS Daniel Liberzon
Coordinated Science Laboratory and
- Dept. of Electrical & Computer Eng.,
- Univ. of Illinois at Urbana-Champaign
U.S.A. DISC HS, June 2003
SLIDE 2 SWITCHED vs. HYBRID SYSTEMS
: stability Switching can be:
- State-dependent or time-dependent
- Autonomous or controlled
Details of discrete behavior are “abstracted away” Properties of the continuous state Switched system:
- is a family of systems
- is a switching signal
SLIDE 3
STABILITY ISSUE
Asymptotic stability of each subsystem is necessary for stability
SLIDE 4
STABILITY ISSUE
unstable
Asymptotic stability of each subsystem is (This only happens in dimensions 2 or higher) necessary but not sufficient for stability
SLIDE 5 TWO BASIC PROBLEMS
- Stability for arbitrary switching
- Stability for constrained switching
SLIDE 6 TWO BASIC PROBLEMS
- Stability for arbitrary switching
- Stability for constrained switching
SLIDE 7
GLOBAL UNIFORM ASYMPTOTIC STABILITY
GUAS is Lyapunov stability plus asymptotic convergence Reduces to standard GAS notion for non-switched systems
SLIDE 8 COMPARISON FUNCTIONS
class function is of class if
Example: GUAS: GUES
SLIDE 9
COMMON LYAPUNOV FUNCTION
Similarly: is GUAS iff s.t. where is positive definite Lyapunov theorem: is GAS iff pos def rad unbdd function s.t.
SLIDE 10
COMMON LYAPUNOV FUNCTION (continued)
Example: as Unless is compact and is continuous, is not enough if
SLIDE 11
CONVEX COMBINATIONS
Corollary: is GAS Proof: Define
SLIDE 12
SWITCHED LINEAR SYSTEMS
LAS for every σ GUES but not necessarily quadratic: common Lyapunov function
∃
(LMIs)
SLIDE 13
COMMUTING STABLE MATRICES => GUES
quadratic common Lyap fcn:
∃
) (
) 1 ... ( 1 ) 1 ... ( 2
→ =
+ + + +
x s s A e t t A e
k k
= ) (t x
) ( x
k
t A e
2 k
s A e
1 1 1s
A e
1 2t
A e …
I A P P AT − = +
1 1 1 1 1 2 2 2 2
P A P P AT − = +
1 2 2 1
}, 2 , 1 { A A A A P = =
t
1 = σ 1 = σ 2 = σ 2 = σ
1
s
2
s
1
t
2
t …
SLIDE 14
LIE ALGEBRAS and STABILITY
A L
} , { P p A g
p
∈ =
Lie algebra:
1 2 2 1 2 1
] , [ A A A A A A − =
Lie bracket:
,
1
g g =
k k k
g g g g ⊂ =
+
] , [
1
g is nilpotent if s.t.
=
k
g k ∃
,
) 1 (
g g =
) ( ) ( ) ( ) 1 (
] , [
k k k k
g g g g ⊂ =
+ ) ( = k
g
g is solvable if s.t.
k ∃
U
SLIDE 15
SOLVABLE LIE ALGEBRA => GUES
g
Lie’s Theorem: is solvable triangular form Example: quadratic common Lyap fcn
∃
diagonal exponentially fast exp fast
SLIDE 16 MORE GENERAL LIE ALGEBRAS
s r g ⊕ =
Levi decomposition:
radical (max solvable ideal)
s
- is compact => GUES, quadratic common Lyap fcn
s
- is not compact => not enough info in Lie algebra
SLIDE 17 NONLINEAR SYSTEMS
- Nothing is known beyond this
- Commuting systems
=> GUAS
- Linearization (Lyapunov’s indirect method)
SLIDE 18 REMARKS on LIE-ALGEBRAIC CRITERIA
- Checkable conditions
- In terms of the original data
- Independent of representation
- Not robust to small perturbations
SLIDE 19 SYSTEMS with SPECIAL STRUCTURE
- Triangular systems
- Feedback systems
- passivity conditions
- small-gain conditions
- 2-D systems
SLIDE 20
TRIANGULAR SYSTEMS
Recall: for linear systems, triangular => GUAS For nonlinear systems, not true in general Example: For stability need to know Not necessarily true
SLIDE 21 INPUT-TO-STATE STABILITY (ISS)
Linear systems: is AS is ISS:
- bounded bounded
- Nonlinear systems:
but bdd bdd, is input-to-state stable (ISS) if For switched systems, triangular + ISS => GUAS
SLIDE 22
FEEDBACK SYSTEMS: ABSOLUTE STABILITY
Circle criterion: quadratic common Lyapunov function is strictly positive real (SPR): For this reduces to SPR (passivity) Popov criterion not suitable: depends on Hurwitz
SLIDE 23
FEEDBACK SYSTEMS: SMALL-GAIN THEOREM
Hurwitz Small-gain theorem: quadratic common Lyapunov function
SLIDE 24
TWO-DIMENSIONAL SYSTEMS
Necessary and sufficient conditions for GUES known since 1970s quadratic common Lyap fcn <=>
∃
convex combinations of Hurwitz worst-case switching
SLIDE 25 WEAK LYAPUNOV FUNCTION
Barbashin-Krasovskii-LaSalle theorem: is GAS if pos def rad unbdd function s.t.
- (weak Lyapunov function)
- is not identically zero along any nonzero solution
(observability with respect to )
=> GAS
Example:
SLIDE 26 COMMON WEAK LYAPUNOV FUNCTION
Extends to nonlinear switched systems and nonquadratic common weak Lyapunov functions using a suitable nonlinear observability notion Theorem: is GAS if
- s.t. there are infinitely many
switching intervals of length
SLIDE 27 TWO BASIC PROBLEMS
- Stability for arbitrary switching
- Stability for constrained switching
SLIDE 28 MULTIPLE LYAPUNOV FUNCTIONS
− = = ) ( , ) (
2 1
x f x x f x & &
GAS
−
2 1 , V
V
respective Lyapunov functions t
1 = σ 1 = σ 2 = σ 2 = σ
) (
) (
t V
t σ
) (x f x
σ
= &
is GAS Very useful for analysis of state-dependent switching
SLIDE 29 MULTIPLE LYAPUNOV FUNCTIONS
t
1 = σ 1 = σ 2 = σ 2 = σ
) (
) (
t V
t σ
) (x f x
σ
= &
is GAS decreasing sequence decreasing sequence
SLIDE 30
DWELL TIME
The switching times satisfy
... , ,
2 1 t
t
D i i
t t τ ≥ −
+1
dwell time
− = = ) ( , ) (
2 1
x f x x f x & &
GES
−
2 1 , V
V
respective Lyapunov functions
SLIDE 31
DWELL TIME
The switching times satisfy
... , ,
2 1 t
t
D i i
t t τ ≥ −
+1
− = = ) ( , ) (
2 1
x f x x f x & &
GES , | | ) ( | |
2 1 1 2 1
x b x V x a ≤ ≤
) ( ) (
1 1 1 1
x V x f
x V
λ − ≤
∂ ∂
, | | ) ( | |
2 2 2 2 2
x b x V x a ≤ ≤
) ( ) (
2 2 2 2
x V x f
x V
λ − ≤
∂ ∂
t
1 = σ 1 = σ 2 = σ
1
t
2
t t Need: ) ( ) (
1 2 1
t V t V <
SLIDE 32 DWELL TIME
The switching times satisfy
... , ,
2 1 t
t
D i i
t t τ ≥ −
+1
− = = ) ( , ) (
2 1
x f x x f x & &
GES , | | ) ( | |
2 1 1 2 1
x b x V x a ≤ ≤
) ( ) (
1 1 1 1
x V x f
x V
λ − ≤
∂ ∂
, | | ) ( | |
2 2 2 2 2
x b x V x a ≤ ≤
) ( ) (
2 2 2 2
x V x f
x V
λ − ≤
∂ ∂
) ( 2
1 t
V
) ( 2
2 2 1
t V a b ≤ ) ( 1
2 2 2 1
t V e a b
D
τ
λ −
≤ ) ( 1
1 2 1 2 2 1
t V e a b a b
D
τ
λ −
≤
) ( 0
1 ) 2 ( 1 2 2 1
1
t V e a b a b
D
τ
λ λ + −
≤
must be
1 <
Need: ) ( ) (
1 2 1
t V t V <
SLIDE 33
AVERAGE DWELL TIME
AD
t T N t T N
τ
σ
− + ≤ ) , ( # of switches on ) , ( T t average dwell time
− =1 N
dwell time: cannot switch twice if
AD
t T τ < −
− = 0 N
no switching: cannot switch if
AD
t T τ < −
) (x f x
σ
= &
SLIDE 34 AVERAGE DWELL TIME
AD
t T N t T N
τ
σ
− + ≤ ) , ( # of switches on ) , ( T t average dwell time
) (x f x
σ
= &
=>
is GAS if
λ µ
τ
log >
AD
) | (| ) ( ) | (|
2 1
x x V x
p
α α ≤ ≤ ) ( ) ( x V x f
p p p
x V
λ − ≤
∂ ∂
P q p x V x V
q p
∈ ≤ , ), ( ) ( µ
) (x f x
σ
= &
SLIDE 35 SWITCHED LINEAR SYSTEMS
x A x
σ
= &
with large enough
- Finite induced norms for
- The case when some subsystems are unstable
σ
AD
τ u B x A x
σ σ
+ = & x C y
σ
=
SLIDE 36
STATE-DEPENDENT SWITCHING
But switched system is stable for (many) other Switched system unstable for some no common switch on the axes is a Lyapunov function
SLIDE 37
STATE-DEPENDENT SWITCHING
But switched system is stable for (many) other level sets of level sets of Switched system unstable for some no common Switch on y-axis t
1 = σ 1 = σ 2 = σ
=> GAS
SLIDE 38 MULTIPLE WEAK LYAPUNOV FUNCTIONS
Theorem: is GAS if
- s.t. there are infinitely many
switching intervals of length
- bservable for each
- For every pair of switching times
s.t. have (each is a weak Lyapunov function)
SLIDE 39
STABILIZATION by SWITCHING
− = = x A x x A x
2 1 , &
&
both unstable Assume: stable for some
2 1
) 1 ( A A A α α − + = ) 1 , ( ∈ α
< + PA P AT
SLIDE 40
STABILIZATION by SWITCHING
− = = x A x x A x
2 1 , &
&
both unstable Assume: stable for some
2 1
) 1 ( A A A α α − + = ) 1 , ( ∈ α
) )( 1 ( ) (
2 2 1 1
< + − + + PA P A PA P A
T T
α α
So for each : either or
) (
1 1
< + x PA P A x
T T
) (
2 2
< + x PA P A x
T T
≠ x
SLIDE 41
UNSTABLE CONVEX COMBINATIONS
Can also use multiple Lyapunov functions LMIs
SLIDE 42
REFERENCES
Branicky, DeCarlo, Hespanha
