1. Lecture #10 Switched systems
Hybrid Control and Switched Systems
Lecture #10 Switched systems
João P. Hespanha
University of California at Santa Barbara

2. Summary Switched systems Linear switched systems
Lyapunov stability, asymptotic stability, exponential stability
Using switched systems to analyze complex hybrid systems

3. Switched system
parameterized family of vector fields ´ fp : Rn ! Rn p 2 Q
parameter set
switching signal ´ piecewise constant signal s : [0,1) ! Q
S ´ set of admissible switching signals
E.g., S › { s : Ns(t, t)· 1 + (t – t), 8 t > t¸ 0 }
# of discontinuities of s in the interval (t, t)
switching times (discontinuities of s)
s = 1
s = 1
s = 3
s = 2
To study REACHABILITY we saw that it is convenient to abstract hybrid systems as TRANSITION SYSTEMS (essentially ignore timing information)
To study STABILITY it will be convenient to abstract hybrid systems as SWITCHED SYSTEMS (essentially ignore the discrete dynamics) t
A solution to the switched system is any pair (s, x) with s 2 S and x a solution to
time-varying ODE

4. Switched system with state-dependent switching
parameterized family of vector fields ´ fp : Rn ! Rn p 2 Q
parameter set
switching signal ´ piecewise constant signal s : [0,1) ! Q
S ´ set of admissible pairs (s, x) with s a switching signal and x a signal in Rn
E.g., S › {(s, x) : Ns(t, t)· 1+ sups2(t,t) ||x(s)|| (t – t), 8 t > t ¸ 0 }
for each x only some s may be admissible
switching times
s = 1
s = 1
s = 3
s = 2 t
A solution to the switched system is a pair (s, x) 2 S for which x is a solution to
time-varying ODE

5. Switched system with resets
parameterized family of vector fields ´ fp : Rn ! Rn p 2 Q
parameter set
switching signal ´ piecewise constant signal s : [0,1) ! Q
S ´ set of admissible pairs (s, x) with s a switching signal and x a signal in Rn
switching times
s = 2
s = 1
s = 1
s = 3 t
A solution to the switched system is a pair (s, x) 2 S for which
on every open interval on which s is constant, x is a solution to
at every switching time t, x(t) = r(s(t), s–(t), x–(t) )
time-varying ODE

6. Time-varying systems vs. Hybrid systems vs. Switched systems
Time-varying system ´ for each initial condition x(0) there is only one solution
(all fp locally Lipschitz)
Hybrid system ´ for each initial condition q(0), x(0) there is only one solution
Switched system ´ for each x(0) there may be several solutions, one for each admissible s
the notions of stability, convergence, etc. must address “uniformity” over all solutions

7. Stability of ODEs equilibrium point ´ xeq 2 Rn for which f(xeq) = 0
class K ´ set of functions a : [0,1)![0,1) that are 1. continuous 2. strictly increasing 3. a(0)=0
a(s) s
Definition (class K function definition): The equilibrium point xeq is (Lyapunov) stable if 9 a 2 K:
||x(t) – xeq|| · a(||x(t0) – xeq||) 8 t¸ t0¸ 0, ||x(t0) – xeq||· c
x(t)
a(||x(t0) – xeq||)
||x(t0) – xeq||
xeq t

8. Stability of switched systems
equilibrium point ´ xeq 2 Rn for which fq(xeq) = 0 8 q 2 Q
class K ´ set of functions a : [0,1)![0,1) that are 1. continuous 2. strictly increasing 3. a(0)=0
a(s) s
Definition (class K function definition):
The equilibrium point xeq is (Lyapunov) stable if 9 a 2 K:
||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
a is independent of x(t0) and s
x(t)
a(||x(t0) – xeq||)
||x(t0) – xeq||
xeq t
in switched systems one is only concerned about boundedness or convergence of the continuous state

9. Asymptotic stability of ODEs
equilibrium point ´ xeq 2 Rn for which f(xeq) = 0
class K ´ set of functions a : [0,1)![0,1) that are 1. continuous 2. strictly increasing 3. a(0)=0
a(s) s
Definition:
The equilibrium point xeq is (globally) asymptotically stable if
it is Lyapunov stable and for every initial state the solution exists on [0,1) and
x(t) ! xeq as t!1.
x(t)
a(||x(t0) – xeq||)
||x(t0) – xeq||
xeq t

10. Asymptotic stability of switched systems
equilibrium point ´ xeq 2 Rn for which fq(xeq) = 0 8 q 2 Q
class K ´ set of functions a : [0,1)![0,1) that are 1. continuous 2. strictly increasing 3. a(0)=0
a(s) s
Definition:
The equilibrium point xeq is (globally) asymptotically stable if
it is Lyapunov stable and for every solution that exists on [0,1)
x(t) ! xeq as t!1.
x(t)
a(||x(t0) – xeq||)
||x(t0) – xeq||
xeq t

11. Asymptotic stability of ODEs
b(s,t)
(for each fixed t)
equilibrium point ´ xeq 2 Rn for which f(xeq) = 0
class KL ´ set of functions b : [0,1)£[0,1)![0,1) s.t. 1. for each fixed t, b(¢,t) 2 K 2. for each fixed s, b(s,¢) is monotone decreasing and b(s,t) ! 0 as t!1 s
b(s,t)
(for each fixed s) t
Definition (class KL function definition):
The equilibrium point xeq is (globally) asymptotically stable if 9 b2KL:
||x(t) – xeq|| · b(||x(t0) – xeq||,t – t0) 8 t¸ t0¸ 0
b(||x(t0) – xeq||,t)
b(||x(t0) – xeq||,0)
x(t)
||x(t0) – xeq||
We have exponential stability when
b(s,t) = c e-l t s
with c,l > 0
xeq t

12. Uniform asymptotic stability of switched systems
b(s,t)
(for each fixed t)
equilibrium point ´ xeq 2 Rn for which f(xeq) = 0
class KL ´ set of functions b : [0,1)£[0,1)![0,1) s.t. 1. for each fixed t, b(¢,t) 2 K 2. for each fixed s, b(s,¢) is monotone decreasing and b(s,t) ! 0 as t!1 s
b(s,t)
(for each fixed s) t
Definition (class KL function definition):
The equilibrium point xeq is uniformly asymptotically stable if 9 b2KL:
||x(t) – xeq|| · b(||x(t0) – xeq||,t – t0) 8 t¸ t0¸ 0
along any solution (s, x) 2 S to the switched system
b is independent of x(t0) and s
b(||x(t0) – xeq||,t)
b(||x(t0) – xeq||,0)
We have exponential stability when
b(s,t) = c e-l t s
with c,l > 0
x(t)
||x(t0) – xeq||
xeq t

13. Three notions of stability
Definition (class K function definition):
The equilibrium point xeq is stable if 9 a 2 K:
||x(t) – xeq|| · a(||x(t0) – xeq||) 8 t¸ t0¸ 0, ||x(t0) – xeq||· c
along any solution (x, s) 2 S to the switched system
a is independent of x(t0) and s
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
along any solution (s, x) 2 S to the switched system
b is independent of x(t0) and s
exponential stability when b(s,t) = c e-l t s with c,l > 0

14. Example
S ´ set of piecewise constant switching signals taking values in Q › {–1, +1}
S ´ set of piecewise constant switching signals taking values in Q › {–1, 0}
S ´ set of piecewise constant switching signals taking values in Q › {–1, 0} with infinitely many switches
S ´ set of piecewise constant switching signals taking values in Q › {–1, 0} with infinitely many switches and interval between consecutive discontinuities bounded below by 1
S ´ set of piecewise constant switching signals taking values in Q › {–1, 0} with infinitely many switches and interval between consecutive discontinuities below by 1 and above by 2

15. Example
S ´ set of piecewise constant switching signals taking values in Q › {–1, +1}
unstable
S ´ set of piecewise constant switching signals taking values in Q › {–1, 0}
stable but not asympt.
S ´ set of piecewise constant switching signals taking values in Q › {–1, 0} with infinitely many switches
stable but not asympt.
S ´ set of piecewise constant switching signals taking values in Q › {–1, 0} with infinitely many switches and interval between consecutive discontinuities bounded below by 1
asympt. stable
S ´ set of piecewise constant switching signals taking values in Q › {–1, 0} with infinitely many switches and interval between consecutive discontinuities below by 1 and above by 2
uniformly asympt. stable

16. Linear switched systems
Aq, Rq,q’ 2 Rn£ n q,q’2 Q
vector fields and reset maps linear on x
s = 2
s = 1
s = 1
s = 3 t0 t1 t2 t3 t

17. Linear switched systems
Aq, Rq,q’ 2 Rn£ n q,q’2 Q
vector fields and reset maps linear on x
s = 2
s = 1
s = 1
s = 3 t0 t1 t2 t3 t
state-transition matrix for the switched system (s-dependent)
t1, t2, t3, …, tk ´ switching times of s in the interval [t,t)

18. Linear switched systems
Aq, Rq,q’ 2 Rn£ n q,q’2 Q
state-transition matrix (s-dependent)
t1, t2, t3, …, tk ´ switching times of s in the interval [t,t)
Analogous to what happens for (unswitched) linear systems:
Fs(t,t) = I 8 t
Fs(t,s) Fs(s,t) = Fs (t,t) 8 t ¸ s ¸ t (semi-group property)
if t is not a switching time, Fs (t,t) is differentiable at t and
if t is a switching time,
variation of constants formula holds for systems with inputs
for a given s, Fs is a “solution” to the switched system with resets
but now Fs may not be nonsingular (will be singular if one of the Rq q’ are)

19. Uniform vs. exponential stability
Aq, Rq,q’ 2 Rn£ n q,q’2 Q
state-independent switching ´ S is such that
(s, x) 2 S ) (s, z) 2 S
for any other piecewise continuous z
only s determines whether or not (s, x) is admissible
Theorem:
For switched linear systems with state-independent switching, uniform asymptotic stability implies exponential stability (two notions are equivalent)
Outline…
1st By uniform asymptotic stability 9 b 2 KL: ||x(t)|| · b(||x(t0)||,t – t0) 8 t¸ t0¸ 0
2nd Choose T sufficiently large so that b(1,T) = g = e–l < 1 ( l > 0)
3rd Pick arbitrary solution (s, x ) 2 S
4th Consider another solution (s, x*) starting at x*(t1) = z › x(t1)/||x(t1)||. Then x(t2) = Fs(t2,t1) x(t1) = ||x(t1)|| Fs(t2,t1) z = ||x(t1)|| x*(t2) ||x*(t2)|| · b(||z||,t2 – t1) = b(1, t2 – t1)
) || x(t2) || · b(1, t2 – t1) ||x(t1)||
exponential decrease of gk any interval of length ¸ k T

20. Uniform vs. exponential stability
Aq, Rq,q’ 2 Rn£ n q,q’2 Q
state-independent switching ´ S is such that
(s, x) 2 S ) (s, z) 2 S
for any other piecewise continuous z
only s determines whether or not (s, x) is admissible
Theorem:
For switched linear systems with state-independent switching, uniform asymptotic stability implies exponential stability (two notions are equivalent)
Outline…
4th … || x(t2) || = b(1, t2 – t1) ||x(t1)||
5th Given an arbitrary interval [t0,t], break it into k› floor((t – t0)/T) intervals of length T plus one interval of length smaller than T …

21. Example #2: Thermostat y ´ mean temperature room heater
off on
room
heater
turn heater off y
turn heater on t
The state of the system remains bounded as t ! 1:

22. Example #2: Thermostat y ´ mean temperature room heater
off on
room
heater
turn heater off y
turn heater on t
A0, A1 asymptotically stable (all eigenvalues with negative real part)
if system would stay in off mode forever then eq. state xeq = A0–1 b0 is asymptotically stable & y ! yoff › c0 A0–1 b0 · y*-h
if system would stay in on mode forever then eq. state xeq = A1–1 b1 is asymptotically stable & y ! yon › c1 A1–1 b1 ¸ y*
We are assuming here that y remains in the bounded band (to check one would compute \dot y)
With switching, does the overall state x of the system remains bounded as t ! 1?

23. a (sequence) property of the discrete-component of the state
Example #2: Thermostat
y ´ mean temperature
off on
room
heater
One option to prove that the state remains bounded:
1st Establish a bound of how fast switching can occur: on an interval (t, t) the maximum number of switchings N(t, t) is bounded by
Why? maximum derivative of y is proportional to ||x|| and between two consecutive switchings y must have a variation of h
a (sequence) property of the discrete-component of the state

24. a (sequence) property of the discrete-component of the state
Example #2: Thermostat
y ´ mean temperature
off on
room
heater
One option to prove that the state remains bounded:
1st Establish a bound of how fast switching can occur: on an interval (t, t) the maximum number of switchings N(t, t) is bounded by
x is a solution to the following (state-dependent) switching system:
Why? maximum derivative of y is proportional to ||x|| and between two consecutive switchings y must have a variation of h
with
a (sequence) property of the discrete-component of the state
(tough to analyze directly…)

25. a (sequence) property of the continuous-dynamics
Example #2: Thermostat
y ´ mean temperature
off on
room
heater
One option to prove the state remains bounded:
2nd Estimate how large x can be from y: For the following (state independent) switching systems
a (sequence) property of the continuous-dynamics
· y*
there exist constants a ¸ 1, b, g > 0 such that
constants , ,  depend on N0 & tD
to prove this one needs the system to be observable from y

26. Example #2: Thermostat
1st On an interval (t,t) the maximum number of switchings N(t,t) is bounded by
2nd Assuming that the max. number of switchings N(t,t) on (t,t) is bounded by
Then there exist constants a ¸ 1, b, g > 0 such that
3rd For any choice of tD and h such that
x must be bounded for any solution compatible with 1 & 2 above.
Hint: prove by contradiction that

27. Proof… We will show that (*) 1nd For s = 0, (*) holds because …
2nd By contradiction suppose that (*) holds strictly for t 2 [0,t*) and with equality at t = t*. Then Therefore, we conclude that
(*)
a ¸ 1
||x(t*)|| can never reach h / (c D) !

28. Discrete/continuous decoupling
1st x is a solution to the following (state-dependent) switching system:
2nd For the following (state-independent) switching system:
There exist constants a, b, g such that
property of the discrete evolution +
property of a (state-independent) switching systems =
property of the interconnection

29. Next lecture… Stability under arbitrary switching
Instability caused by switching
Common Lyapunov function
Converse results
Algebraic conditions