Hybrid Control and Switched Systems
Lecture #13
Stability under
slow switching &
state-dependent switching
João P. Hespanha
University of California
at Santa Barbara
Summary
Stability under slow switching
• Dwell-time switching
• Average dwell-time
• Stability under brief instabilities
Stability under state-dependent switching
• State-dependent common Lyapunov function
• Stabilization through switching
• Multiple Lyapunov functions
• LaSalle’s invariance principle
Switched system
parameterized family of vector fields ´ fp : Rn ! Rn
switching signal ´ piecewise constant signal s : [0,1) ! Q
p2Q
parameter set
S ´ set of admissible pairs (s, x) with s a switching signal and x a signal in Rn
switching times
s=1
s=2
s=3
s=1
t
A solution to the switched system is a pair (s, x) 2 S for which
1. on every open interval on which s is constant, x is a solution to
2.
at every switching time t, x(t) = r(s(t), s–(t), x–(t) )
time-varying ODE
Three notions of stability
a is independent
of x(t0) and s
||x(t) – xeq|| · a(||x(t0) – xeq||) 8 t¸ t0¸ 0, ||x(t0) – xeq||· c
along any solution (s, x) 2 S to the switched system
Definition (class K function definition):
The equilibrium point xeq is stable if 9 a 2 K:
Definition:
The equilibrium point xeq 2 Rn is asymptotically stable if
it is Lyapunov stable and for every solution that exists on [0,1)
x(t) ! xeq as t!1.
Definition (class KL function definition):
The equilibrium point xeq 2 Rn is uniformly asymptotically stable if 9 b2KL:
||x(t) – xeq|| · b(||x(t0) – xeq||,t – t0) 8 t¸ t0¸ 0
b is independent
along any solution (s, x) 2 S to the switched system
of x(t0) and s
exponential stability when b(s,t) = c e-l t s with c,l > 0
Stability under slow switching
So far …
Sall ´ set of all pairs (s, x) with s
piecewise constant and x piecewise
any switching signal is
continuous
r(p, q, x) = x 8 p,q 2 Q, x 2 Rn
no resets
admissible
Now…
switched linear systems
Slow switching:
Sdwell[tD] ´ switching signals with “dwell-time” tD > 0, i.e., interval
between consecutive discontinuities larger or equal to tD
Slow switching on the average:
Save[tD, N0] ´ switching signals with “average dwell-time” tD > 0 and
“chatter-bound” N0 > 0, i.e.,
# of discontinuities of s in
the open interval (t,t)
S[tD] = Save[tD,1] Why?
Stability under slow switching
switched linear systems
S[tD] ´ switching signals with “dwell-time” tD > 0, i.e., interval
between consecutive discontinuities larger or equal to tD
Theorem: (Q finite)
If all Aq, q 2 Q are asymptotically stable, there exists a dwell-time tD such that the
switched system is uniformly (exponentially) asymptotically stable over Sdwell[tD]
Why?
1st For a switched linear system
state-transition matrix (s-dependent)
t1, t2, t3, …, tk ´ switching times of s in the interval [t,t )
Stability under slow switching
switched linear systems
S[tD] ´ switching signals with “dwell-time” tD > 0, i.e., interval
between consecutive discontinuities larger or equal to tD
Theorem: (Q finite)
If all Aq, q 2 Q are asymptotically stable, there exists a dwell-time tD such that the
switched system is uniformly (exponentially) asymptotically stable over Sdwell[tD]
Why?
2st Since all the Aq, q 2 Q are asymptotically stable: 9 c,l0 > 0 ||eAq t|| · c e–l0 t
3nd Taking norms of the state-transition matrix…
Stability under slow switching
switched linear systems
S[tD] ´ switching signals with “dwell-time” tD > 0, i.e., interval
between consecutive discontinuities larger or equal to tD
Theorem: (Q finite)
If all Aq, q 2 Q are asymptotically stable, there exists a dwell-time tD such that the
switched system is uniformly (exponentially) asymptotically stable over Sdwell[tD]
Why?
3nd
4th Pick tD > 0, l 2 (0,l0) such that
Always possible? yes:
can pick
Stability under slow switching
switched linear systems
S[tD] ´ switching signals with “dwell-time” tD > 0, i.e., interval
between consecutive discontinuities larger or equal to tD
Theorem: (Q finite)
If all Aq, q 2 Q are asymptotically stable, there exists a dwell-time tD such that the
switched system is uniformly (exponentially) asymptotically stable over Sdwell[tD]
Why?
3nd
4th
5th Then
exponential convergence to zero
(with rate independent of s)
Stability under slow switching
switched linear systems
S[tD] ´ switching signals with “dwell-time” tD > 0, i.e., interval
between consecutive discontinuities larger or equal to tD
Theorem: (Q infinite)
Assuming the sets {Aq : q 2 Q } & { Rp,q : p, q2 Q } are compact.
If all Aq, q 2 Q are asymptotically stable, there exists a dwell-time tD such that the
switched system is uniformly (exponentially) asymptotically stable over Sdwell[tD]
Stability under slow switching on the average
switched linear systems
Save[tD, N0] ´ switching signals with “average dwell-time” tD > 0 and
“chatter-bound” N0 > 0, i.e.,
# of switchings in (t,t)
Theorem: (Q finite)
If all the Aq, q 2 Q are asymptotically stable, there exists an average dwell-time
tD such that for every chatter-bound N0 the switched system is uniformly
(exponentially) asymptotically stable over Save[tD, N0]
Why?
1st As before …
(w.l.g we assume r c > 1)
2nd But k is the number of switchings in [t,t ) so
exponential decrease as long as
Stability under slow switching on the average
switched linear systems
Save[tD, N0] ´ switching signals with “average dwell-time” tD > 0 and
“chatter-bound” N0 > 0, i.e.,
# of switchings in (t,t)
Theorem: (Q infinite)
Assuming the sets {Aq : q 2 Q } & { Rp,q : p, q2 Q } are compact.
If all the Aq, q 2 Q are asymptotically stable, there exists an average dwell-time
tD such that for every chatter-bound N0 the switched system is uniformly
(exponentially) asymptotically stable over Save[tD, N0]
1. Same results would hold for any subset of Save[tD, N0]
2. Some versions of these results also exist for nonlinear systems
3. One may still have stability if some of the Aq are unstable,
provided that s does not “dwell” on these values for a long time
(switching under brief instabilities)
So far… state-independent switching
no resets
Arbitrary switching:
Sall ´ set of all pairs (s, x) with s piecewise constant and x piecewise
continuous
switched linear systems
Slow switching:
S[tD] ´ switching signals with “dwell-time” tD > 0, i.e., interval
between consecutive discontinuities larger or equal to tD
Slow switching on the average:
Save[tD, N0] ´ switching signals with “average dwell-time” tD > 0 and
“chatter-bound” N0 > 0, i.e.,
# of discontinuities of s in the
open interval (t,t)
Current-state dependent switching
no resets
c {cq 2 Rn: q 2 Q} ´ (not necessarily disjoint) covering of Rn, i.e., [q2Q cq = Rn
Current-state dependent switching
S[c] ´ set of all pairs (s, x) with s piecewise constant and x piecewise
continuous such that 8 t, s(t) = q is allowed only if x(t) 2 cq
c2
c1
s = 1 or 2
s=1
s=2
Thus (s, x) 2 S[c] if and only if x(t) 2 cs(t) 8 t
Common Lyapunov function for arbitrary switching
Theorem:
Suppose there exists a continuously differentiable, positive definite, radially
unbounded function V: Rn ! R such that
Then for arbitrary switching Sall
1. the equilibrium point xeq is Lyapunov stable
2. if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable.
Why? (for simplicity consider xeq = 0)
1st Take an arbitrary solution (s, x) and define v(t) V( x(t) ) 8 t ¸ 0
2nd Therefore
V( x(t) ) is always bounded…
Common Lyapunov function for current-state dep. switching
Theorem:
Suppose there exists a continuously differentiable, positive definite, radially
unbounded function V: Rn ! R such that
Then for current-state dependent switching S[c]
1. the equilibrium point xeq is Lyapunov stable
2. if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable.
Why? (for simplicity consider xeq = 0)
1st Take an arbitrary solution (s, x) and define v(t) V( x(t) ) 8 t ¸ 0
still holds because
x(t) 2 cs(t)
2nd Therefore
Same conclusions as before …
Common Lyapunov function for current-state dep. switching
Theorem:
Suppose there exists a continuously differentiable, positive definite, radially
unbounded function V: Rn ! R such that
Then for current-state dependent switching S[c]
1. the equilibrium point xeq is Lyapunov stable
2. if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable.
Note that:
• Same conclusion would hold for any subset of S[c]
• Some (or all) the unswitched systems may not be stable
•
This theorem does not guarantee existence of solutions (as
opposed to the usual Lyapunov Theorem and the ones for state
independent switching)…
Common Lyapunov function for current-state dep. switching
Theorem:
Suppose there exists a continuously differentiable, positive definite, radially
unbounded function V: Rn ! R such that
Then for current-state dependent switching S[c]
1. the equilibrium point xeq is Lyapunov stable
2. if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable.
E.g., Q {–1, +1}, c–1 [0,1), c+1 (–1,0)
no solutions
exists
For xeq = 0 is an equilibrium point and for V(z) z2
Stabilization through switching
Given a family of unstable vector fields fq, q 2 Q
Is there a covering c for which the current-state dependent
set of switching signals S[c] results in stability?
Theorem:
If there exists a set of constants lq ¸ 0, q 2 Q such that q lq =1 and xeq is an
(asymptotically) stable equilibrium point of the ODE
convex combination of the fq
then there is a current-state dependent set of switching signals S[c] for which xeq
is an (asymptotically) stable equilibrium point of the switched system.
Why?
1st Since the convex combination is asymptotically stable, it has a Lyapunov
function V:
since all the lq¸0, for
every z, at least one of
the terms must be · 0
Stabilization through switching
Given a family of unstable vector fields fq, q 2 Q
Is there a covering c for which the current-state dependent
set of switching signals S[c] results in stability?
Theorem:
If there exists a set of constants lq ¸ 0, q 2 Q such that q lq =1 and xeq is an
(asymptotically) stable equilibrium point of the ODE
convex combination of the fq
then there is a current-state dependent set of switching signals S[c] for which xeq
is an (asymptotically) stable equilibrium point of the switched system.
Why?
2nd Define
V is a common Lyapunov
function for current-state
dep. switching
1. every point in Rn belongs to one of the cq
) c { cq : q 2 Q } form a covering
2.
Stabilization through switching
Given a family of unstable vector fields fq, q 2 Q
Is there a covering c for which the current-state dependent
set of switching signals S[c] results in stability?
Theorem:
If there exists a set of constants lq ¸ 0, q 2 Q such that q lq =1 and xeq is an
(asymptotically) stable equilibrium point of the ODE
convex combination of the fq
then there is a current-state dependent set of switching signals S[c] for which xeq
is an (asymptotically) stable equilibrium point of the switched system.
But these covers may lead to non-existence of solution (Zeno)
Example
s=2
s=1
s=2
s=1
The two regions actually
intersect. One can use this to
prevent Zeno
(e.g., through hysteresis)…
Multiple Lyapunov functions
Vq : Rn ! R, q 2 Q ´ family of Lyapunov functions (cont. dif., pos. def., rad. unb.)
Given a solution (s, x) and defining v(t) Vs(t)( x(t) ) 8 t ¸ 0
1. On an interval [t, t) where s = q (constant)
2. But at a switching time t, where s–(t) = p  s(t) = q,
v may be discontinuous
(even without reset)
v decreases
Multiple Lyapunov functions
Vq : Rn ! R, q 2 Q ´ family of Lyapunov functions (cont. dif., pos. def., rad. unb.)
Given a solution (s, x) and defining v(t) Vs(t)( x(t) ) 8 t ¸ 0
1. On an interval [t, t) where s = q (constant)
v decreases
2. But at a switching time t, where s–(t) = p  s(t) = q,
we would be okay if v
would not increase at
switching times
v=V1(x)
v=V1(x)
v=V1(x)
v=V1(x)
v=V2(x)
s=1
s=2
v=V2(x)
s=1
t
s=1
s=2
s=1
t
Multiple Lyapunov functions
Theorem: (Q finite)
Suppose there exists a family of continuously differentiable, positive definite,
radially unbounded functions Vq: Rn ! R, q 2 Q such that
and at any z 2 Rn where a switching signal in S can jump from p to q
Then
1. the equilibrium point xeq is Lyapunov stable
2. if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable.
Why? (for simplicity consider xeq = 0)
1st Take an arbitrary solution (s, x) and define v(t) Vs( x(t) ) 8 t ¸ 0
while s is constant:
and, at points of discontinuity of s: v–(t) ¸ v(t) does not increase
from now on same as before …
Multiple Lyapunov functions
Theorem: (Q finite)
Suppose there exists a family of continuously differentiable, positive definite,
radially unbounded functions Vq: Rn ! R, q 2 Q such that
and at any z 2 Rn where a switching signal in S can jump from p to q
Then
1. the equilibrium point xeq is Lyapunov stable
2. if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable.
Why? (for simplicity consider xeq = 0)
2nd Since 9 a1,a22K1:
a1(||x||) · Vq(x) · a2(||x||)
3rd If 9 a3: W(x) · –a3(||x||)
class KL function
independent of s
Multiple Lyapunov functions
Theorem: (Q finite)
Suppose there exists a family of continuously differentiable, positive definite,
radially unbounded functions Vq: Rn ! R, q 2 Q such that
and at any z 2 Rn where a switching signal in S can jump from p to q
Then
1. the equilibrium point xeq is Lyapunov stable
2. if W(z) = 0 only for z = xeq then xeq is (glob) uniformly asymptotically stable.
The Vq’s need not be positive definite and radially unbounded “everywhere”
It is enough that 9 a1,a22K1: a1(||z||) · Vq(z) · a2(||z||)
8 q 2 Q, z 2 cq
LaSalle’s Invariance Principle (ODE)
M 2 Rn is an invariant set ´ x(t0) 2 M ) x(t) 2 M 8 t¸ t0
Theorem (LaSalle Invariance Principle):
Suppose there exists a continuously differentiable, positive definite, radially
unbounded function V: Rn ! R such that
Then xeq is a Lyapunov stable equilibrium and the solution always exists globally.
Moreover, x(t) converges to the largest invariant set M contained in
E { z 2 Rn : W(z) = 0 }
Note that:
1. When W(z) = 0 only for z = xeq then E = {xeq }.
Since M ½ E, M = {xeq } and therefore x(t) ! xeq ) asympt. stability
2. Even when E is larger then {xeq } we often have M = {xeq } and can
conclude asymptotic stability.
LaSalle’s Invariance Principle (linear system)
M 2 Rn is an invariant set if x(0) 2 M ) x(t) 2 M 8 t ¸ 0
Theorem (LaSalle Invariance Principle–linear system, quadratic V):
Suppose there exists a positive definite matrix P
A’ P + P A · – Q · 0
Then the system is stable.
Moreover, x(t) converges to the largest invariant set M contained in
E { z 2 Rn : Q z = 0 }
Note that:
1. Since Q ¸ 0 we can always write Q = C’ C …
LaSalle’s Invariance Principle (linear system)
M 2 Rn is an invariant set if x(0) 2 M ) x(t) 2 M 8 t ¸ 0
Theorem (LaSalle Invariance Principle–linear system, quadratic V):
Suppose there exists a positive definite matrix P
A’ P + P A · – C’C · 0
Then the system is stable.
Moreover, x(t) converges to the largest invariant set M contained in
E { z 2 Rn : C z = 0 }
Why? show that C’Cz = 0 ) Cz = 0
Note that:
2. When Q > 0 then E = {0}.
Since M ½ E, M = {0} and therefore x(t) ! 0 ) asympt. stability
3. Even when E is larger then {0} we often have M = {0} and can conclude
asymptotic stability.
When does this happen ?
Asymptotic stability from LaSalle’s IP
M 2 Rn is an invariant set if x(0) 2 M ) x(t) 2 M 8 t ¸ 0
M ´ largest invariant set contained in E { z 2 Rn : C z = 0 }
x0 2 M if and only if x(t) eA t x0 2 M ½ E 8 t ¸ 0
m (Why?)
Asymptotic stability from LaSalle’s IP
M 2 Rn is an invariant set if x(0) 2 M ) x(t) 2 M 8 t ¸ 0
M ´ largest invariant set contained in E { z 2 Rn : C z = 0 }
x0 2 M if and only if x(t) eA t x0 2 M ½ E 8 t ¸ 0
(check that this is indeed an invariant set …)
LaSalle’s Invariance Principle (linear system)
M 2 Rn is an invariant set if x(0) 2 M ) x(t) 2 M 8 t ¸ 0
Theorem (LaSalle Invariance Principle–linear system, quadratic V):
Suppose there exists a positive definite matrix P
observability matrix
A’ P + P A · – C’C · 0
of the pair (C,A)
Then the system is stable. Moreover, x(t) converges to
When O is nonsingular, we have asymptotic stability
(pair (C,A) is said to be observable)
Back to switched linear systems…
Theorem: (Q finite)
Suppose there exist positive definite matrices Pq2 Rn£ n, q 2 Q such that
Aq’ Pq + Pq Aq · – Cq’Cq · 0 8 q 2 Q
and at any z 2 Rn where a switching signal in S[c] can jump from p to q
z’ Pp z ¸ z’ R’q p PqRq p z
from general theorem
Then the switched system is stable.
Moreover, if every pair (Cq,Aq), q 2 Q is observable then
1. if S ½ Sweak-dwell then it is asymptotically stable
2. if S ½ Sp-dwell[tD,T] then it is uniformly asymptotically stable.
Sets of switching signals
Sdwell[tD]
´ switching signals with “dwell-time” tD > 0, i.e., interval
between consecutive discontinuities larger or equal to tD
Save[tD, N0] ´ switching signals with “average dwell-time” tD > 0 and
“chatter-bound” N0 > 0, i.e.,
Sp-dwell[tD,T] ´ switching signals with “persistent dwell-time” tD > 0 and
“period of persistency” T > 0, i.e., 9 infinitely many intervals of
length ¸ tD on which sigma is constant & consecutive intervals
with this property are separated by no more than T
¸tD
·T
¸tD
·T
¸tD
Sweak-dwell [tD > 0 Sp-dwell[tD,+1] ´ each s has persistent dwell-time > 0
Sdwell[tD] ½ Save[tD, N0] ½ Sp-dwell[g tD,T] ½ Sweak-dwell ½ Sall
LaSalle’s IP for switched systems
Theorem: (Q finite)
Suppose there exist positive definite matrices Pq2 Rn£ n, q 2 Q such that
Aq’ Pq + Pq Aq · – Cq’Cq · 0 8 q 2 Q
and at any z 2 Rn where a switching signal in S[c] can jump from p to q
Vp(z) ¸ Vq(Rq p z)
from general theorem
Then the switched system is stable.
Moreover, if every pair (Cq,Aq), q 2 Q is observable then
1. if S ½ Sweak-dwell then it is asymptotically stable
2. if S ½ Sp-dwell[tD,T] then it is uniformly asymptotically stable.
Sp-dwell[tD,T] ´ switching signals with “persistent dwell-time” tD > 0 and “period
of persistency” T > 0, i.e., 9 infinitely many intervals of length ¸
tD on which sigma is constant & consecutive intervals with this
property are separated by no more than T
Sweak-dwell [tD > 0 Sp-dwell[tD,+1] ´ each s has persistent dwell-time > 0
Sdwell[tD] ½ Save[tD, N0] ½ Sp-dwell[g tD,T] ½ Sweak-dwell ½ Sall
Example
Choosing P1 = P2 = I
common Lyapunov function
nonsingular (observable)
1.
2.
One can find s  Sweak-dwell for which we do not have asymptotic stability
Stability is not uniform on Sweak-dwell, because one can find s 2 Sweak-dwell for
which convergence is “arbitrarily slow”
(problems, e.g., close to the x2=0 axis)
LaSalle’s IP for switched systems
Theorem: (Q finite)
Suppose there exist positive definite matrices Pq2 Rn£ n, q 2 Q such that
Aq’ Pq + Pq Aq · – Cq’Cq · 0 8 q 2 Q
and at any z 2 Rn where a switching signal in S[c] can jump from p to q
Vp(z) ¸ Vq(Rq p z)
from general theorem
Then the switched system is stable.
Moreover, if every pair (Cq,Aq), q 2 Q is observable then
1. if S ½ Sweak-dwell then it is asymptotically stable
2. if S ½ Sp-dwell[tD,T] then it is uniformly asymptotically stable.
a) Finiteness of Q could be replaced by compactness
b) In some cases it is sufficient for all pairs (Cq,Aq), q 2 Q to be detectable
(e.g., when Aq = A + B Fq)
c) When the pairs (Cq,Aq), q 2 Q are not observable x converges to the
smallest subspace M that is invariant for all unswitched system and
contains the kernels of all Oq
d) There are nonlinear versions of this result (no uniformity?)
Next lecture…
• Computational methods to construct multiple Lyapunov
functions—Linear Matrix Inequalities (LMIs)
• Applications (vision-based control)