1. João P. Hespanha University of California at Santa Barbara
Hybrid Control and Switched Systems
Lecture #8 Stability and convergence of hybrid systems (topological view)
João P. Hespanha
University of California at Santa Barbara

2. Summary
Lyapunov stability of hybrid systems

3. Properties of hybrid systems
Xsig ´ set of all piecewise continuous signals x:[0,T) ! Rn, T2(0,1]
Qsig ´ set of all piecewise constant signals q:[0,T)! Q, T2(0,1]
Sequence property ´ p : Qsig £ Xsig ! {false,true}
E.g.,
A pair of signals (q, x) 2 Qsig £ Xsig satisfies p if p(q, x) = true
A hybrid automaton H satisfies p ( write H ² p ) if
p(q, x) = true, for every solution (q, x) of H
“ensemble properties” ´ property of the whole family of solutions (cannot be checked just by looking at isolated solutions) e.g., continuity with respect to initial conditions…

4. Lyapunov stability of ODEs (recall)
Xsig ´ set of all piecewise continuous signals taking values in Rn
Given a signal x2Xsig, ||x||sig › supt¸0 ||x(t)||
signal norm
ODE can be seen as an operator
T : Rn ! Xsig
that maps x0 2 Rn into the solution that starts at x(0) = x0
Definition (continuity definition):
A solution x* is (Lyapunov) stable if T is continuous at x*0› x*(0), i.e.,
8 e > 0 9 d >0 : ||x0 – x*0|| · d ) ||T(x0) – T(x*0)||sig · e
supt¸0 ||x(t) – x*(t)|| · e d e
x(t)
x*(t)
pend.m

5. Lyapunov stability of hybrid systems
x › F2(q1,x–)
F1(q1,x–) = q2 ?
mode q2
mode q1
Xsig ´ set of all piecewise continuous signals x:[0,T) ! Rn, T2(0,1]
Qsig ´ set of all piecewise constant signals q:[0,T)! Q, T2(0,1]
Hybrid automaton can be seen as an operator
T : Q £ Rn ! Qsig £ Xsig
that maps (q0, x0) 2 Q £ Rn into the solution that starts at q(0) = q0, x(0) = x0
Definition (continuity definition):
A solution (q*,x*) is (Lyapunov) stable if T is continuous at (q*(0), x*(0)).
To make sense of continuity we need ways to measure “distances” in
Q £ Rn and Qsig £ Xsig

6. Lyapunov stability of hybrid systems
A few possible “metrics”
one cares very much about the discrete states matching
one does not care at all about the discrete states matching
Definition (continuity definition):
A solution (q*,x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e.,
8 e > 0 9 d >0 : d( (q*(0), x*(0)) , (q(0), x(0)) ) · d ß
supt¸ 0 d ( (q*(t), x*(t)) , (q(t), x(t)) ) · e
not really metrics because no addition is defined on \scr{Q}\times\R^n
If the solution starts close to (q*,x*) it will remain close to it forever
e can be made arbitrarily small by choosing d sufficiently small
Note: may actually not be metrics on Q £ Rn because one may want “zero-distance” between points. However, still define a topology on Q £ Rn, which is what is really needed to make sense of continuity…

7. A point is only “arbitarily close” to itself (Hausdorff space)
Topological spaces
Given a set X and a collection TX of subsets of X
(X, TX) is a topological space if
;, X 2 TX
A, B 2 TX ) AÅ B 2 TX
A1, A2, …, An2 TX ) [i=1n Ai 2 TX (1· n · 1)
T is called a topology and the sets in T are called open and their complements are called closed
Intuitively: two elements of X are “arbitrarily close” if for every open set one belongs to, the other also belongs to
Examples:
Q › { q1, q2, …, qn } (finite)
TQ › {; , Q } (trivial topology – all points are close to each other)
TQ › {;} [ {all subsets of Q } (discrete topology – no two distinct points are close to each other)
TQ › {;, {1}, {1,2} } of {1,2}
X › Rn
TX › { (possibly infinite) union of all open balls } (norm-induced topology)
open ball ´ { x 2 Rn : ||x – x0|| < e }
A point is only “arbitarily close” to itself (Hausdorff space)
How to prove that 2. holds? (Hint: [ and Å are distributive & intersection of two open balls is in T)

8. Continuity in Topological spaces
Given a set X and a collection TX of subsets of X
(X, TX) is a topological space if
;, X 2 TX
A, B 2 TX ) AÅ B 2 TX
A1, A2, …, An2 TX ) [i=1n Ai 2 TX (1· n · 1)
T is called a topology and the sets in T are called open and their complements are called closed
Given a function f: X ! Y with X, Y topological space
f is continuous at a point x0 in X if for every neighborhood (i.e., set containing an open set) V of f(x0) there is a neighborhood U of x0 such that f(U) ½ V.
8 V U
f(U)
Intuitively: “arbitrarily close” points in X are transformed into “arbitrarily close” points in Y
f(x0) f x0
For norm-induced topologies we need only consider balls V › { y : || y – f(x0) || < e } and U › { x : || x – x0 || < d }

9. Continuity in Topological spaces
Given a function f: X ! Y with X, Y topological space
f is continuous at a point x0 in X if for every neighborhood (i.e., set containing an open set) V of f(x0) there is a neighborhood U of x0 such that f(U) ½ V.
Examples: X › Rn, Y › Rm
TX, TY › { (possibly infinite) union of all open balls } (norm-induced top.)
open ball ´ { x 2 Rn : ||x – x0|| < e }
leads to the usual definition of continuity in Rn: f continuous at x0 if
8 e > 0 9 d > 0 : ||x – x0|| < d ) || f(x) – f(x0)|| < e
Could be restated as: for every ball
V › { y : || y – f(x0) || < e }
there is a ball
U › { x : || x – x0 || < d }
such that
x 2 U ) f(x) 2 V
or equivalently
f(U) ½ V
8 V U
f(U)
f(x0) f x0

10. Continuity in Topological spaces
Given a function f: X ! Y with X, Y topological space
f is continuous at a point x0 in X if for every neighborhood (i.e., set containing an open set) V of f(x0) there is a neighborhood U of x0 such that f(U) ½ V.
Examples: Q › { q1, q2, …, qn } (finite)
1. TQ › {; , Q } (trivial topology – all points are close to each other)
2. TQ › {;} [ {all subsets of Q } (discrete topology – no two distinct points are close to each other)
3. TQ › {;, {1}, {1,2} } of {1,2}
Is any of these functions f : R ! Q continuous? (usual norm-topology in R) x
f(x) x
f(x) x
f(x) x
f(x)

11. Continuity in Topological spaces
Given a function f: X ! Y with X, Y topological space
f is continuous at a point x0 in X if for every neighborhood (i.e., set containing an open set) V of f(x0) there is a neighborhood U of x0 such that f(U) ½ V.
Examples: Q › { q1, q2, …, qn } (finite)
1. TQ › {; , Q } (trivial topology – all points are close to each other)
2. TQ › {;} [ {all subsets of Q } (discrete topology – no two points are close to each other)
3. TQ › {;, {1}, {1,2} } of {1,2} (2 is ?close? to 1 but 1 is not ?close? to 2)
Is any of these functions f : R ! Q continuous? (usual norm-topology in R) x
f(x) x
f(x) x
f(x) x
f(x)
continuous for 1., 2., 3.
continuous for 1., 3.
continuous only for 1.
continuous only for 1.

12. (for those that don’t want to leave anything to the imagination…)
Given a sets Q, X with topologies TQ and TX
One can construct a topology TQ£X on Q£X:
TQ£X › { A £ B : A 2 Q, B 2X }
Example: Q › {1, 2}, TQ › {;, {1}, {1,2} } X › R, TR ´ norm-induced topology
some open sets: ; , { (1,x): x2(1,2) }, { (q,x): q=1,2, x2(1,1) }
not open sets: { (1,x): x2(1,2] }, { (2,x): x2(1,2) }
One can construct a topology TQsig on the set Qsig of signals q:[0,T)!Q, T2(0,1]:
TQsig › sets of the form A › { q2 Qsig : q(t) 2 A(t) 8 t}
where the A(t) are a collection of open sets
Example: Q › {1, 2}, TQ › {;, {1}, {1,2} } some open sets: { q : q(t) = 1 8 t· 1 }, { q : q(t) = 1 8 t 2 Q }
not open sets: {q : q(t) = 2 8 t· 1 }
X › R, TR ´ norm-induced topology
some open sets: { x : x(t) < 0 8 t }, { x : |x(t)| < 1 8 t }
non open sets: { x : s01 x(t) dt <1 }

13. Back to hybrid systems…
Hybrid automaton can be seen as an operator
T : Q £ Rn ! Qsig £ Xsig
that maps (q0, x0) 2 Q £ Rn into the solution that starts at q(0) = q0, x(0) = x0
Definition (continuity definition):
A solution (q*, x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e., for every neighborhood V of T(q0*, x0*) there is a neighborhood U of (q0*, x0*) such that T(U) ½ V
Case 1: domain of T: TQ ´ trivial topology (all points close to each other)
T Rn ´ usual topology induced from Euclidean norm
co-domain of T: TQsig ´ trivial topology (all signals close to each other)
TXsig ´ usual topology induced from sup-norm
8 e > 0 9 d >0 : 8 q(0), || x*(0) – x(0))|| < d ) 8 t || x*(t) – x(t)|| < e U V
one does not care at all about the discrete states matching

14. Back to hybrid systems…
Hybrid automaton can be seen as an operator
T : Q £ Rn ! Qsig £ Xsig
that maps (q0, x0) 2 Q £ Rn into the solution that starts at q(0) = q0, x(0) = x0
Definition (continuity definition):
A solution (q*, x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e., for every neighborhood V of T(q0*, x0*) there is a neighborhood U of (q0*, x0*) such that T(U) ½ V
Case 2: domain of T: TQ ´ discrete topology (all points far from each other)
T Rn ´ usual topology induced from Euclidean norm
co-domain of T: TQsig ´ discrete topology (all signals far from each other)
TXsig ´ usual topology induced from sup-norm
8 e > 0 9 d >0 : q*(0) = q(0), || x*(0) – x(0))|| < d ) 8 t q*(t) = q(t), || x*(t) – x(t)|| < e U V
one cares very much about the discrete states matching

15. Back to hybrid systems…
Hybrid automaton can be seen as an operator
T : Q £ Rn ! Qsig £ Xsig
that maps (q0, x0) 2 Q £ Rn into the solution that starts at q(0) = q0, x(0) = x0
Definition (continuity definition):
A solution (q*, x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e., for every neighborhood V of T(q0*, x0*) there is a neighborhood U of (q0*, x0*) such that T(U) ½ V
Case 3: domain of T: TQ ´ discrete topology (all points far from each other)
T Rn ´ usual topology induced from Euclidean norm
co-domain of T: TQsig ´ trivial topology (all signals close to each other)
TXsig ´ usual topology induced from sup-norm
8 e > 0 9 d >0 : q*(0) = q(0), || x*(0) – x(0))|| < d ) 8 t || x*(t) – x(t)|| < e U V
one cares very much about the discrete states matching

16. Back to hybrid systems…
Hybrid automaton can be seen as an operator
T : Q £ Rn ! Qsig £ Xsig
that maps (q0, x0) 2 Q £ Rn into the solution that starts at q(0) = q0, x(0) = x0
Definition (continuity definition):
A solution (q*, x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e., for every neighborhood V of T(q0*, x0*) there is a neighborhood U of (q0*, x0*) such that T(U) ½ V
Case 4: domain of T: TQ ´ {;, {1}, {1,2} }, Q › {1,2}
T Rn ´ usual topology induced from Euclidean norm
co-domain of T: TQsig ´ trivial topology (all signals close to each other)
TXsig ´ usual topology induced from sup-norm
for q*(0) = 1: 8 e > 0 9 d >0 : q(0) = 1, || x*(0) – x(0))|| < d ) 8 t || x*(t) – x(t)|| < e
for q*(0) = 2: 8 e > 0 9 d >0 : 8 q(0), || x*(0) – x(0))|| < d ) 8 t || x*(t) – x(t)|| < e
small perturbation in x (but no perturbation in q) leads to small change in x
small perturbation in x leads to small change in x, regardless of q(0)

17. Back to hybrid systems…
Hybrid automaton can be seen as an operator
T : Q £ Rn ! Qsig £ Xsig
that maps (q0, x0) 2 Q £ Rn into the solution that starts at q(0) = q0, x(0) = x0
Definition (continuity definition):
A solution (q*, x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e., for every neighborhood V of T(q0*, x0*) there is a neighborhood U of (q0*, x0*) such that T(U) ½ V
Case 4: domain of T: TQ ´ discrete topology (all points far from each other)
T Rn ´ usual topology induced from Euclidean norm
co-domain of T: TQ ´ {;, {1}, {1,2} }, Q › {1,2} (signal version…)
TXsig ´ usual topology induced from sup-norm
8 e > 0 9 d >0 : q*(0) = q(0), || x*(0) – x(0))|| < d ) 8 t q*(t) = 2 or q*(t) = q(t), || x*(t) – x(t)|| < e
small perturbation in x (but no perturbation in q) leads to small change in x, q* and q may differ only when q* = 2

18. Example #2: Thermostat x ´ mean temperature room heater
q = 1
q = 2
room
heater
x ¸ 77 ?
turn heater off 77 x 73
turn heater on 1 1 1
q = 2 2 2 2
Why? t
not stable because an infinitesimal perturbation in the initial condition will make the qs not match for short (but nonzero) periods of time
some trajectories are stable others unstable
no trajectory is stable
all trajectories are stable
for discrete topology on Q for the domain and the trivial topology for the codomain
for discrete topology on Q
(all points far from each other)
for trivial topology on Q
(all points close to each other)

19. Example #5: Tank system pump
goal ´ prevent the tank from emptying or filling up
pump-on inflow ´ l = 3
d = .5 ´ delay between command is sent to pump and the time it is executed y
constant outflow ´ m = 1
t ¸ .5 ?
pump off
(q = 1)
wait to off
(q = 4)
y · 1 ?
t › 0
wait to on
(q = 2)
pump on
(q = 3)
t › 0
y ¸ 2 ?
t ¸ .5 ?
this topology only distinguishes between modes based on the state of the pump
A possible topology for Q: TQ › {;, {1,2}, {3,4}, {1,2,3,4} }

20. Example #4: Inverted pendulum swing-upq
Hybrid controller:
1st pump/remove energy into/from the system by applying maximum force, until E ¼ 0
(energy control)
2nd wait until pendulum is close to the upright position
3th next to upright position use feedback linearization controller
u 2 [-1,1]
remove energy
E 2 [-e,e] ?
The d is choose so that the pendulum is close enough to the upright position so that the feedback linearization controller does not require |u| larger than 1
The epsilon is chosen sufficiently small so that the trajectory will pass through the region defined by |w| + |q| · d
Answer: if one starts close to this configuration just make u=+1 for some period of time (extra state from which one transitions to pump-energy)
wait
stabilize
E < – e
E > e
pump energy
|w| + |q| · d ?
E 2 [-e,e] ?

21. Example #4: Inverted pendulum swing-upq
u 2 [-1,1]
A possible topology for Q › {r,p,w,s} :
TQ › {;, {s}, {r,p,w,s} }
1. for solutions (q*, x*) that start in “s,” we only consider perturbations that also start in “s”
2. for solutions (q*, x*) that start outside “s,” the perturbations can start in any state
remove energy
E 2 [-e,e] ?
wait
stabilize
The d is choose so that the pendulum is close enough to the upright position so that the feedback linearization controller does not require |u| larger than 1
The epsilon is chosen sufficiently small so that the trajectory will pass through the region defined by |w| + |q| · d
Answer: if one starts close to this configuration just make u=+1 for some period of time (extra state from which one transitions to pump-energy)
E < – e
E > e
pump energy
|w| + |q| · d ?
E 2 [-e,e] ?

22. Asymptotic stability for hybrid systems
Hybrid automaton can be seen as an operator
T : Q £ Rn ! Qsig £ Xsig
that maps (q0, x0) 2 Q £ Rn into the solution that starts at q(0) = q0, x(0) = x0
Definition (continuity definition):
A solution (q*, x*) is (Lyapunov) stable if T is continuous at (q0*, x0*)›(q*(0), x*(0)), i.e., for every neighborhood V of T(q0*, x0*) there is a neighborhood U of (q0*, x0*) such that T(U) ½ V
Definition:
A solution (q*, x*) is asymptotically stable if it is stable, every solution (q,x) exists globally, and q ! q*, x ! x* as t!1
in the sense of the topology on Qsig

23. Example #7: Server system with congestion control
q ¸ qmax ?
incoming rate r
r › m r –
qmax q
server B
every solution exists globally and converges to a periodic solution that is stable.
Do we have asymptotic stability?
rate of service
(bandwidth)

24. Example #7: Server system with congestion control
q ¸ qmax ?
incoming rate r
r › m r –
qmax q
server
Note that the synchronization even destroys stability because of the discontinuities in r
Unless we use the topology induced by the L_2 norm for the signal space
every solution exists globally and converges to a periodic solution that is stable.
Do we have asymptotic stability? B
rate of service
(bandwidth)
No, its difficult to have asymptotic stability for non-constant solutions due to the “synchronization” requirement.
(not even stability… Always?)

25. Example #2: Thermostat x ´ mean temperature room heater
q = 1
q = 2
room
heater
x ¸ 77 ?
turn heater off 77 x 73
turn heater on 1 1 1
q = 2 2 2 2 t
all trajectories are stable but not asymptotically
no trajectory is stable
for discrete topology on Q
(all points far from each other)
for trivial topology on Q
(all points close to each other)
Why?

26. Stability of sets Hybrid automaton can be seen as an operator
T : Q £ Rn ! Qsig £ Xsig
that maps (q0, x0) 2 Q £ Rn into the solution that starts at q(0) = q0, x(0) = x0
Poincaré distance between (q, x), (q*,x*) 2 Qsig £ Xsig after t0
distance at the point t where the (q(t), x(t)) is the furthest apart from (q*, x*)
can also be viewed as the distance from the trajectory (q, x) to the set
{(q*(t), x*(t)) :t¸ t0}
(q*(t), x*(t))
(q(t), x(t))
To prove is a topology, the only nontrivial is that the intersection of open balls is open:
to prove this one needs the triangular inequality
show that if a point belongs to the intersection of the two balls, there is a ball around it that is contained on both balls (why, point belongs to open ball of radius epsilon => belongs to open ball of radius epsilon-\delta (delta>0) => open ball of radius delta around that point still belongs to original ball==do this construction for both balls and pick smallest delta)
2. the intersection of the two balls is the union of the balls constructed above for every point in the intersection.
For constant trajectories (q*,x*) its just the sup-norm:

27. Stability of sets Hybrid automaton can be seen as an operator
T : Q £ Rn ! Qsig £ Xsig
that maps (q0, x0) 2 Q £ Rn into the solution that starts at q(0) = q0, x(0) = x0
Poincaré distance between (q, x), (q*,x*) 2 Qsig £ Xsig after t0
Definition: A solution (q*, x*) is Poincaré stable if T is continuous at (q0*, x0*) › (q*(0), x*(0)) for the topology on Xsig induced by the Poincaré distance,
8 e > 0 9 d >0 : d0((q(0), x(0)), (q*(0), x*(0))) · d ß
dP((q*,x*), (q,x); 0) = supt¸ 0 inf¸ 0 dT ((q(t), x(t)), (q*(), x*())) · e
in more modern terminology one would say that the following set is stable
{ (q*(t), x*(t)) : t ¸ 0 } ½ Q£X
To prove is a topology, the only nontrivial is that the intersection of open balls is open:
to prove this one needs the triangular inequality
show that if a point belongs to the intersection of the two balls, there is a ball around it that is contained on both balls (why, point belongs to open ball of radius epsilon => belongs to open ball of radius epsilon-\delta (delta>0) => open ball of radius delta around that point still belongs to original ball==do this construction for both balls and pick smallest delta)
2. the intersection of the two balls is the union of the balls constructed above for every point in the intersection.
(open sets are unions of open Poincaré balls { x 2 Xsig : dP(x – x0) < e }. Show this is a topology…)

28. Stability of sets Hybrid automaton can be seen as an operator
T : Q £ Rn ! Qsig £ Xsig
that maps (q0, x0) 2 Q £ Rn into the solution that starts at q(0) = q0, x(0) = x0
Poincaré distance between (q, x), (q*,x*) 2 Qsig £ Xsig after t0
Definition: A solution (q*, x*) is Poincaré asymptotically stable if it is Poincaré stable, every solution (q, x) exists globally, and dP((q,x), (q*,x*); t )!0 as t!1
in more modern terminology one would say that the following set is asymptotically stable: { (q*(t), x*(t)) : t ¸ 0 } ½ Q£X
To prove is a topology, the only nontrivial is that the intersection of open balls is open:
to prove this one needs the triangular inequality
show that if a point belongs to the intersection of the two balls, there is a ball around it that is contained on both balls (why, point belongs to open ball of radius epsilon => belongs to open ball of radius epsilon-\delta (delta>0) => open ball of radius delta around that point still belongs to original ball==do this construction for both balls and pick smallest delta)
2. the intersection of the two balls is the union of the balls constructed above for every point in the intersection.

29. Example #7: Server system with congestion control
q ¸ qmax ?
incoming rate r
r › m r –
qmax q
server
If one starts close enough to a trajectory, one remains close for all times r B
all trajectories are Poincaré asymptotically stable
rate of service
(bandwidth)
qmax q

30. To think about …
With hybrid systems there are many possible notions of stability. (especially due to the topology imposed on the discrete state) WHICH ONE IS THE BEST?
(engineering question, not a mathematical one)
What type of perturbations do you want to consider on the initial conditions?
(this will define the topology on the initial conditions)
What type of changes are you willing to accept in the solution?
(this will define the topology on the signals)
Even with ODEs there are several alternatives: e.g.,
8 e > 0 9 d >0 : ||x0 – xeq|| · d ) supt¸ 0||x(t) – xeq|| · e or
8 e > 0 9 d >0 : ||x0 – xeq|| · d ) s01 ||x(t) – xeq|| dt · e or
8 e > 0 9 d >0 : ||x0 – x0*|| · d ) dP( x, x*; 0 ) · e
(even for linear systems these definitions may differ: Why?)
Lyapunov
First is lyapunov stability
for linear systems, second is asymptotic stability (what about for nonlinear?)
integral
Poincaré

31. Next lecture… Analysis tools for hybrid systems Impact maps
Fixed-point theorem
Stability of periodic solutions
Decoupling
Switched systems
Supervisors