Hybrid Control and Switched Systems
Lecture #12
Controller realizations for stable switching
João P. Hespanha
University of California
at Santa Barbara
Summary
Controller realization for stable switching
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 (x, s) 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 (x, s) 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 (x, s) 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 arbitrary switching
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
Can we change the switching system to make it stable?
Example #11: Roll-angle control
q
roll-angle
q is uniquely determined by u
and the initial conditions
input-output model
q
u
process
state-space realization
AP
cP
bP
Example #11: Roll-angle control
roll-angle
q
set-point control ´ drive the roll angle q to a desired value qreference
qreference
etrack
+
–
set-point
controller
measurement
noise
n
q
u
process
+
+
Example #11: Roll-angle control
set-point control ´ drive the roll angle q to a desired value qreference
qreference
etrack
+
–
set-point
controller
controller 1
slow but
not very
sensitive
to noise
(low-gain)
measurement
noise
n
q
u
process
+
+
controller 2
fast but
very
sensitive to
noise
(high-gain)
Switching controller
measurement
noise
switching signal taking
s values in Q{1,2}
qreference
etrack
+
–
switching
controller
n
q
u
process
+
+
How to build the
switching controller
to avoid instability ?
s=2
s=1
s=2
Realization theory (SISO)
nth order input-output model
u, y 2 R
for short a(y) = b(u)
state-space model
x 2 Rm ´ state
Definition:
(A, b, c) is called a realization of the input-output model if the two models have
the same solution y for every given u and zero initial conditions.
Theorem:
1. (A, b, c) is a realization of the IO model if and only if
2. Any nth order IO model has a realization with x 2 Rn
3. If all roots of a(s) have negative real part, A can be chosen asymptotically stable
4. For any nonsingular matrix T 2 Rm£m, if (A, b, c) is a realization of an IO model
then (TAT–1, Tb, cT–1) is also a realization of the same model
Realization theory (SISO)
u, y 2 R
nth order input-output model
state-space realization of the IO model
for short a(y) = b(u)
x 2 Rm ´ state
Suppose A is asymptotically stable: 9 P > 0, P A + A’ P = – I
(P1/2 AP–1/2, P1/2b, cP–1/2) is also a realization of the IO model
Theorem:
Given any nth order input-output model for which all roots of a(s) have negative
real parts, it is always possible to find a realization for it, for which
A + A’ = Q < 0
Switching between input-output models
M { aq(y) = bq(u) : q 2 Q } ´ finite family of nth order input-output models,
with all roots of all aq(s) with negative real parts
Theorem:
There exists a family of realizations for M
R { (Aq, bq, cq) : q 2 Q }
such that the switched system
is exponentially stable for arbitrary switching
Why?
1st Choose realizations such that Aq + Aq’ = – Qq < 0 8 q 2 Q
2nd The function V(x) = x’ x is a common Lyapunov function for the switched
system: continuously differentiable, positive definite, radially unbounded,
system is uniformly asymptotically stable ) exponentially stable
Back to switching controllers…
controller 1
controller 2
realization:
realization:
measurement
noise
s
n
qreference
etrack
+
–
u
q
+
+
Back to switching controllers…
switching signal taking
s values in Q{1,2}
qreference
etrack
+
–
u
measurement
noise
n
q
+
+
overall system:
Assuming each controller was properly designed, each Aq is asymptotically
stable but the overall switched systems could still be unstable
This can be avoided by proper choice of the controller realizations
Youla parameterization (non-switched systems)
Assume process is asymptotically stable
Q
(asympt. stable)
u
real process
v
–
+
process copy
1. If the real process and its copy have the same initial conditions ) v = 0 8 t
otherwise v converges to zero exponentially fast
2. Since the Q system is asymptotically stable, u converges to zero exponentially fast
No matter what we choose for Q, as long as it is asymptotically
stable, the overall system is asymptotically stable
Youla parameterization (non-switched systems)
Assume process is asymptotically stable
Q
(asympt. stable)
u
real process
v
–
+
process copy
stabilizing controller
1. If the real process and its copy have the same initial conditions ) v = 0 8 t
otherwise v converges to zero exponentially fast
2. Since the Q system is asymptotically stable, u converges to zero exponentially fast
No matter what we choose for Q, as long as it is asymptotically
stable, the overall system is asymptotically stable
Youla parameterization (non-switched)
Assume process is asymptotically stable
v
–
+
Q
(asympt. stable)
u
real process
process copy
Theorem [Youla-Bongiorno]:
1. For any asymptotically stable Q, this controller asymptotically stabilizes the
overall system
2. Any controller that asymptotically stabilizes the overall system is of this form,
for an asymptotically stable Q with the same IO model as:
a similar parameterization also exists when
the process is not asymptotically stable…
–
controller
process
Why?
“Youla” realizations
e
v
+
Q
(asympt. stable)
u
process copy
realization for Q
realization for the
process copy
realization for the controller
In general these realizations are not minimal
and back again to multiple controllers…
e
controller 1
v
u
–
Q1
u
process copy
e
controller 2
v
u
–
Q2
u
process copy
realization for Qq
Switching controller
s
e
v
u
switched Q
+
process copy
Switched Q
switched controller
realization for the
process copy
Switched closed-loop
u
real process
v
–
+
process copy
1. If the real process and its copy have the same initial conditions ) v = 0 8 t
otherwise v converges to zero exponentially fast
2. If the switched Q system is asymptotically stable, u converges to zero
exponentially fast and the overall system is asymptotically stable
Always possible by appropriate choice of realizations for each Qq
(e.g., by choosing realizations so that V(z) = z’ z is a common Lyapunov function)
Switched closed-loop
v
–
u
real process
+
the construction in this
slide is only valid for
stable processes
Theorem:
For every family of input-output controller models, there always exist a family a
controller realizations such that the switched closed-loop systems is
exponentially stable for arbitrary switching.
One can actually show that there exists a common
quadratic Lyapunov function for the closed-loop.
In general the realizations are not minimal
Non-asymptotically stable processes
1st Pick one stabilizing “nominal” controller
u
–
real process
controller 0
asymptotically stable
Non-asymptotically stable processes
2nd repeat previous construction
u
Q
(asympt. stable)
–
v
real process
–
+
controller 0
closed-loop
copy
Theorem [Youla-Bongiorno]:
1. For any asymptotically stable Q, this controller asymptotically stabilizes the
overall system
2. Any controller that asymptotically stabilizes the overall system is this form,
for an appropriately chosen Q:
desired controller
–
– controller 0
closed-loop
copy
Why?
Non-asymptotically stable processes
2nd repeat previous construction
u
Q
(asympt. stable)
–
v
real process
–
+
controller 0
1. Q will be stable as long as the controller 0 is stable
closed-loop
2. other more complicated constructions
exist when
copy
one cannot find a stable controller
0
Theorem [Youla-Bongiorno]:
1. For any asymptotically stable Q, this controller asymptotically stabilizes the
overall system
2. Any controller that asymptotically stabilizes the overall system is this form,
for an appropriately chosen Q:
desired controller
–
– controller 0
closed-loop
copy
Why?
Switching controller
measurement
noise
switching signal taking
s values in Q{1,2}
qreference
etrack
+
–
switching
controller
n
q
u
process
+
+
By proper choice of
the controllers
realization we can
have stability for
arbitrary switching.
s=2
s=1
s=2
Next lecture…
Stability under slow switching
• Dwell-time switching
• Average dwell-time
• Stability under brief instabilities