Tutorial on Logic-based Control
Switched Supervisory Control
João P. Hespanha
University of California
at Santa Barbara
Summary
•
•
•
•
Supervisory control overview
Estimator-based linear supervisory control
Estimator-based nonlinear supervisory control
Examples
Supervisory control
Motivation: in the control of complex and highly uncertain systems, traditional
methodologies based on a single controller do not provide satisfactory performance.
supervisor
s
controller 1
bank of candidate
controllers
controller n
exogenous
disturbance/
noise
switching signal
measured
output
w
s
u
process
y
control
signal
Key ideas:
1. Build a bank of alternative controllers
2. Switch among them online based on measurements
For simplicity we assume a stabilization problem, otherwise controllers should have a reference input r
Supervisory control
Motivation: in the control of complex and highly uncertain systems, traditional
methodologies based on a single controller do not provide satisfactory performance.
supervisor
s
controller 1
bank of candidate
controllers
controller n
exogenous
disturbance/
noise
switching signal
measured
output
w
s
u
process
y
control
signal
Supervisor:
• places in the feedback loop the controller that seems more
promising based on the available measurements
• typically logic-based/hybrid system
Multi-controller
s
controller 1
y
measured
output
bank of candidate
controllers
controller n
switching signal
s
u
control
signal
Conceptual diagram: not
efficient for many
controllers & not possible
for unstable controllers
Multi-controller
s
switching signal
s
controller 1
bank of candidate
controllers
y
measured
output
u
control
signal
controller n
Given a family of (n-dimensional) candidate controllers
s
y
measured
output
switching signal
u
control
signal
Multi-controller
switching times
s(t)
switching
signal
s=1
s=3
s=2
s=1
t
Given a family of (n-dimensional) candidate controllers
s
y
measured
output
switching signal
u
control
signal
Supervisor
supervisor
s
y
measured output
u
control signal
switching signal
y
u
Typically an hybrid system:  ´ continuous state
d ´ discrete state
continuous
vector field
discrete
transition
function
output
function
Types of supervision
Pre-routed supervision
s=2
s=1
s=3
• try one controllers after
another in a pre-defined
sequence
• stop when the performance
seems acceptable
not effective when the
number of controllers is
large
Performance-based supervision
(direct)
• keep controller while observed
performance is acceptable
• when performance of current
controller becomes unacceptable,
switch to controller that leads to
best expected performance based
on available data
Estimator-based supervision
(indirect)
• estimate process model from
observed data
• select controller based on current
estimate – Certainty Equivalence
Summary
•
•
•
•
Supervisory control overview
Estimator-based linear supervisory control
Estimator-based nonlinear supervisory control
Examples
Estimator-based supervision’s setup: Example #1
process
control
signal
u
Process is assumed to be
y
measured
output
unknown parameters
p*=(a*, b*) 2 P [–1,1] £ {–1,1}
we consider three candidate controllers:
controller C1: u = 0
to be used when a*· –.1
controller C2: u = 1.1y to be used when a* > –.1 & b* = –1
controller C3: u = – 1.1y to be used when a* > –.1 & b* = +1
process parameter
p* is equal to
p(a,b) 2 P
use controller Cq with
controller selection function
Estimator-based supervision’s setup
w
control
signal
u
process
Process is assumed to be in a family
parametric uncertainty
exogenous
disturbance/
noise
y
measured
output
unmodeled
dynamics
Mp ´ small family of systems around a
“nominal” process model Np
for each process in a family Mp, at least one candidate controller Cq, q 2 Q
provides adequate performance.
controller selection function
process in Mp, p2 P
controller Cq with q =c(p)
provides adequate performance
Estimator-based supervisor: Example #1
measured
output y
control
signal
u
y
–
multiestimator
Process is assumed to be
Multi-estimator
+
–
+
unknown parameters
p*=(a*, b*) 2 P [–1,1] £ {–1,1}
8 p=(a, b) 2 P [–1,1] £ {–1,1}
yp ´ estimate of the output y that would be correct if the parameter was p=(a,b)
ep ´ output estimation error that would be small if the parameter was p=(a,b)
consider the yp corresponding to p = p*
Since P has infinitely many elements, this multi-estimator would have
to be infinite dimensional !?
(not a very good multi-estimator)
Estimator-based supervisor: Example #1
measured
output y
control
signal
u
y
–
multiestimator
+
decision
logic
–
s
switching
signal
+
Process is assumed to be
Multi-estimator
unknown parameters
p*=(a*, b*) 2 P [–1,1] £ {–1,1}
8 p=(a, b) 2 P [–1,1] £ {–1,1}
yp ´ estimate of the output y that would be correct if the parameter was p=(a,b)
ep ´ output estimation error that would be small if the parameter was p=(a,b)
Decision logic:
ep small
process
likely in Mp
should use
Cq, q = c(p)
Certainty equivalence inspired
set s = c(p)
Estimator-based supervisor
measured
output y
control
signal
u
y
–
+
multiestimator
decision
logic
–
s
switching
signal
+
Process is assumed to be in family
process in
Mp, p2 P
controller Cq, q =c(p)
provides adequate performance
Multi-estimator
yp ´ estimate of the process output y that would be correct if the process was Np
ep ´ output estimation error that would be small if the process was Np
Decision logic:
ep small
process
likely in Mp
should use
Cq, q = c(p)
Certainty equivalence inspired
set s = c(p)
Estimator-based supervisor
measured
output y
control
signal
u
y
–
multiestimator
+
decision
logic
–
s
switching
signal
+
A stability argument cannot be
based on this because typically
Process is assumed to be in family
process in Mp ) ep small
process in
controller Cq, q =c(p)
but
not
the
converse
Mp, p2 P
provides adequate performance
Multi-estimator
yp ´ estimate of the process output y that would be correct if the process was Np
ep ´ output estimation error that would be small if the process was Np
Decision logic:
ep small
process
likely in Mp
should use
Cq, q = c(p)
Certainty equivalence inspired
set s = c(p)
Estimator-based supervisor
measured
output y
control
signal
u
y
–
+
multiestimator
–
decision
logic
s
switching
signal
+
Process is assumed to be in family
process in
Mp, p2 P
controller Cq, q =c(p)
provides adequate performance
Multi-estimator
yp ´ estimate of the process output y that would be correct if the process was Np
ep ´ output estimation error that would be small if the process was Np
detectable means
“small ep ) small state”
Decision logic:
ep small
set
s = c(p)
overall system
is detectable
through ep
overall state
is small
Certainty equivalence inspired, but formally justified by detectability
Performance-based supervision
measured
output y
control
signal
u
performance
monitor
decision
logic
s
switching
signal
Candidate controllers:
Performance monitor:
pq ´ measure of the expected performance of controller Cq inferred from past data
Decision logic:
ps is acceptable
keep current controller
ps is unacceptable
switch to controller Cq corresponding to best pq
Abstract supervision
Estimator and performance-based architectures share the same common architecture
multi-est.
or
perf. monitor
decision
logic
switching
s signal
multicontroller
w
u
control
signal
process
In this talk we will focus mostly on an estimator-based supervisor…
y
measured
output
Abstract supervision
Estimator and performance-based architectures share the same common architecture
multiestimator
decision
logic
switching
s signal
multicontroller
switched system
w
u
control
signal
process
y
measured
output
The four basic properties (1-2) : Example #1
Matching property:
At least one of the ep is “small”
decision
logic
switching
signal s
Why?
Process is
Multi-estimator is
ep* is
“small”
essentially a requirement on the multi-estimator
detectable means
“small ep ) small state”
Detectability property:
For each p2 P, the switched system is detectable
through ep when s = c(p)
index of controller
that stabilizes
processes in Mp
Detectability
a system is detectable if for every pair eigenvalue/eigenvector (li,vi) of A
<[li] ¸ 0 ) C vi  0
for short: pair (A,C) is detectable
From solution to linear ODEs…
(assuming A diagonalizable,
otherwise terms in tk eli t appear)
0
y(t) bounded ) ai C vi = 0 (for <[li] ¸ 0) ) ai = 0 (for <[li] ¸ 0) ) y(t) bounded
Lemma: For any detectable system: y(t) bounded ) x(t) bounded
& y(t) ! 0
) x(t) ! 0
Detectability
system is detectable if for every pair eigenvalue/eigenvector (li,vi) of A
<[li] ¸ 0 ) C vi  0
Lemma: For any detectable system: y(t) bounded ) x(t) bounded
& y(t) ! 0
) x(t) ! 0
Lemma: For any detectable system, there exists a matrix K such that
is asymptotically stable (all eigenvalues of A – KC with negative real part)
Can re-write system as
(output injection)
asympt. stable
confirms that: y is bounded /!0 ) x is bounded /!0
The four basic properties (1-2) : Example #1
detectable means
“small ep ) small state”
Detectability property:
For each p2 P, the switched system is detectable through ep when s = c(p)
index of controller that
stabilizes processes in Mp
Why? consider, e.g., p=(a,b)= (.5,1) ) use controller C3: u=– 1.1y (3 = s = c(p))
.5 – 1.1 = –.6 < 0
Thus, ep small ) yp small ) y = yp– ep small ) u small etc.
Questions:
• Where we just lucky in getting –.6 < 0 ? NO (why?)
• Does detectability hold if u=– 1.1y does not stabilize the process
(e.g., a* = .5, b*=-1)? YES (why?)
The four basic properties (1-2)
Matching property:
At least one of the ep is “small”
decision
logic
switching
signal s
Why?
process in
9 p*2 P:
process
in Mp*
ep* is
“small”
essentially a requirement on the multi-estimator
Detectability property:
For each p2 P, the switched system is detectable
through ep when s = c(p)
index of controller
that stabilizes
processes in Mp
essentially a requirement on the candidate controllers
This property justifies using the candidate controller
that corresponds to a small estimation error.
Why? Certainty equivalence stabilization theorem…
The four basic properties (3-4)
Small error property:
There is a parameter “estimate”
decision
logic
r : [0,1) ! P
as current parameter
“estimate”
for which er is “small” compared to any fixed ep and
that is consistent with s, i.e.,
s = c(r)
switching
signal s
r(t) can be viewed
controller consistent with
parameter “estimate”
Non-destabilization property:
Detectability is preserved for the time-varying
switched system (not just for constant s)
Typically requires some form of “slow switching”
Both are essentially (conflicting) properties of the decision logic
Decision logic
decision
logic
switching
signal s
1. For boundedness one wants er small for
some parameter estimate r consistent with s
(i.e., s = c (r))
“small error”
2. To recover the “static” detectability of the
time-varying switched system one wants slow
switching.
“non-destabilization”
These are conflicting requirements:
1. r should follow smallest ep
2. s = c(r) should not vary
Dwell-time switching
start
monitoring signals
p2P
measure of the size of ep over a
“window” of length 1/l
forgetting factor
wait tD seconds
Non-destabilizing property:
The minimum interval between consecutive discontinuities of s is tD > 0.
(by construction)
Small error property
(e.g., L2 noise and no
unmodeled dynamics)
Assume P finite and 9 p* 2 P :


when we select r = p at time t we must have

Two possible cases:
1. Switching will stop in finite time T at some p 2 P:
<1
· C*
Small error property
(e.g., L2 noise and no
unmodeled dynamics)
Assume P finite and 9 p* 2 P :


when we select r = p at time t we must have

Two possible cases:
2. After some finite time T switching will occur only among elements of a
subset P* of P, each appearing in r infinitely many times:
<1
· C*
Small error property
(e.g., L2 noise and no
unmodeled dynamics)
Assume P finite and 9 p* 2 P :


when we select r = p at time t we must have

Small error property:
(L2 case)
Assume that P is a finite set. If 9 p* 2 P for which
then
at least one error L2
“switched” error will be L2
Implementation issues
monitoring signals
start
How to efficiently compute a large number
of monitoring signals?
Example #1: Multi-estimator
wait tD seconds
input is linear
comb. of y and u


by linearity
state-sharing
we can generate as many
errors as we want with a 2dim. systems
Implementation issues
monitoring signals
start
How to efficiently compute a large number
of monitoring signals?
Example #1: Multi-estimator
wait tD seconds
1. we can generate as many
monitoring signals as we want with
a (2+6)-dim. system
state-sharing

2. finding r is really an optimization
Implementation issues
monitoring signals
start
When P is a continuum (or very large), it may be
issues with respect to the optimization for r.
Things are easy, e.g.,
1. P has a small number of elements
wait tD seconds
2. model is linearly parameterized on p
(leads to quadratic optimization)
3. there are closed form solutions
usual requirement
(e.g., mp polynomial on p)
in adaptive control
4. mp is convex on p
results still hold if there exists a computational
delay tC in performing the optimization, i.e.
The four basic properties
decision
logic
r(t) can be viewed
Small error property:
as current parameter
There is a parameter “estimate”
“estimate”
r : [0,1) ! P
for which er is “small” compared to any fixed ep and
that is consistent with s, i.e.,
s = c(r) controller consistent with
parameter “estimate”
switching
signal s
Non-destabilization property:
Detectability is preserved for the time-varying
switched system (not just for constant s)
Matching property:
At least one of the ep is “small”
Detectability property:
For each p2 P, the switched system is detectable
through ep when s = c(p)
index of controller
that stabilizes
processes in Mp
Analysis outline (linear case, w = 0)
decision
logic
switching
signal s
1st by the Matching property:
9 p*2 P such that ep* is “small”
2nd by the Small error property:
9 r such that s = c(r) and er is “small”
(when compared with ep*)
3rd by the Detectability property:
there exist matrices Kp such that the matrices
Aq– Kp Cp, q = c(p)
are asymptotically stable
4th the switched system can be written as
asymptotically stable by
non-destabilization property
injected
system
“small” by
2nd step
) x is small (and converges to zero if, e.g., er2 L2)
Example #2: One-link flexible manipulator
mass at the tip
x
mt
y(x, t)
deviation with
respect to rigid body
q
torque applied
at the base
T
PDE (small bending):
beam’s
elasticity
transversal
slice’s inertia
beam’s
mass density
(.68Kg total mass)
IH
axis’s inertia (.023)
Boundary conditions:
beam’s
length
(113 cm)
Example #2: One-link flexible manipulator
mass at the tip
x
mt
y(x, t)
deviation with
respect to rigid body
q
torque applied
at the base
T
IH
axis’s inertia (.023)
Series expansion and truncation:
eigenfunctions of the beam
Example #2: One-link flexible manipulator
mass at the tip
x
mt
y(x, t)
q
torque applied
at the base
T
IH
axis’s inertia (.023)
Control measurements:
Assumed not known a priori:
mt 2 [0, .1Kg]
xsg 2 [40cm, 60cm]
´ base angle
´ base angular velocity
´ tip position
´ bending at position xsg (measured by a strain gauge
attached to the beam at position xsg)
Example #2: One-link flexible manipulator
torque
u
Bode Diagrams
Bode Diagrams
From: torque
0
0
-100
1500
1000
500
0
-500
-1
10
0
10
1
10
2
10
-50
-100
1500
To: base angular velocity
Phase (deg); Magnitude (dB)
50
-50
To: base angle
Phase (deg); Magnitude (dB)
From: torque
50
1000
500
0
-500
-1000
-1
10
3
10
0
10
Frequency (rad/sec)
1
10
2
10
3
10
Frequency (rad/sec)
transfer functions as mt ranges over [0, .1Kg] and xsg ranges over [40cm, 60cm]
Bode Diagrams
Bode Diagrams
From: torque
From: torque
50
40
20
Phase (deg); Magnitude (dB)
-100
2000
1500
1000
500
-20
-40
1000
500
0
-500
0
-500
-1
10
0
To: bending
-50
To: tip position
Phase (deg); Magnitude (dB)
0
0
10
1
10
Frequency (rad/sec)
2
10
3
10
-1000
-1
10
0
10
1
10
Frequency (rad/sec)
2
10
3
10
Example #2: One-link flexible manipulator
torque
u
Class of admissible processes:
Mp ´ family around a nominal transfer function
corresponding to parameters p (mt, xsg)
unknown parameter
p (mt, xsg)
parameter set:
grid of 18 points in
[0, .1]£[40, 60]
For this problem it is not possible to write the coefficients of the nominal
transfer functions as a function of the parameters because these coefficients are
the solutions to transcendental equations that must be computed numerically.
Family of candidate controllers:
18 controllers designed using LQR/LQE, one for each nominal model
Example #2: One-link flexible manipulator
3
0.7
0.6
0.5
2
0.4
mt
xsg
0.3
0.2
0.1
1
0
0
5
10
15
20
25
t
0
-1
18
16
14
12
10
-2
-3
0
tip position
torque
set point
8
6
4
0
0
5
s
2
10
15
t
20
25
5
10
15
t
20
25
30
Example #2: One-link flexible manipulator
(open-loop)
Example #2: One-link flexible manipulator
(closed-loop with fixed controller)
Example #2: One-link flexible manipulator
(closed-loop with supervisory control)
Example #3: Uncertain gain
+
C
-
1·k<4
The maximum gain margin achievable by a
single linear time-invariant controller is 4
Doyle, Francis, Tannenbaum, Feedback Control Theory, 1992
Example #3: Uncertain gain
+
-
multicontroller
1 · k < 40
Example #3: Uncertain gain
output
reference
true parameter value
monitoring signals
Example #3: 2-dim SISO linear process
Class of admissible processes:
nominal transfer function
nonlinear parameterized on p
–1
unstable zero-pole
cancellations
+1
Any re-parameterization that makes
the coefficients of the transfer function
lie in a convex set will introduce an
unstable zero-pole cancellation
But the multi-estimator is still separable and state-sharing can be used …
Example #3: 2-dim SISO linear process
output
reference
true parameter value
(without noise)
Example #3: 2-dim SISO linear process
output
reference
true parameter value
(with noise)
Example #3: Disturbance Rejection
(unknown frequency)
disturbance
frequency estimate
output
(rejection of one sinusoid)
Example #3: Disturbance Rejection
(unknown frequency)
disturbance
frequency estimate
output
(rejection of two sinusoids)
Example #3: Disturbance Rejection
(unknown frequency)
disturbance
frequency estimate
output
(rejection of a square wave)
Summary
•
•
•
•
Supervisory control overview
Estimator-based linear supervisory control
Estimator-based nonlinear supervisory control
Examples
Supervisory control
Motivation: in the control of complex and highly uncertain systems, traditional
methodologies based on a single controller do not provide satisfactory performance.
supervisor
s
controller 1
bank of candidate
controllers
controller n
exogenous
disturbance/
noise
switching signal
measured
output
w
s
u
process
y
control
signal
Key ideas:
1. Build a bank of alternative controllers
2. Switch among them online based on measurements
For simplicity we assume a stabilization problem, otherwise controllers should have a reference input r
Estimator-based nonlinear supervisory control
multiestimator
decision
logic
NON LINEAR
switching
signal
s
multicontroller
w
u
control
signal
process
y
measured
output
Class of admissible processes: Example #4
control
signal
u
process
(a*,
b*)
y
measured
output
´ unknown parameters
Process is assumed to be in the family
p=(a, b) 2 P [–1,1] £ {–1,1}
state accessible
Class of admissible processes
w
control
signal
u
process
Process is assumed to be in a family
parametric uncertainty
exogenous
disturbance/
noise
y
measured
output
unmodeled
dynamics
Mp ´ small family of systems around a
nominal process model Np
Typically
metric on set of
state-space model (?)
Most results presented here:
• independent of metric d (e.g., detectability)
• or restricted to case ep = 0 (e.g., matching)
Candidate controllers: Example #4
Process is assumed to be in the family
p=(a, b) 2 P [–1,1] £ {–1,1}
state accessible
To facilitate the controller design, one can first “back-step” the system to
simplify its stabilization:
virtual input
now the control law
after the coordinate transformation the
new state is no longer accessible
Candidate controllers
stabilizes the system
Candidate controllers
Class of admissible processes
Mp ´ small family of systems around a
nominal process model Np
Assume given a family of candidate controllers
(without loss of generality all with same dimension)
Multi-controller:
y
measured
output
s
switching signal
u
control
signal
Multi-estimator
measured
output y
control
signal
u
–
multiestimator
+
–
+
How to design a multi-estimator?
we want:
Matching property: there exist some p*2 P such that ep* is “small”
Typically obtained by:
process in
9 p*2 P:
process
in Mp*
ep* is
“small”
when process “matches” Mp* the
corresponding error must be “small”
Candidate controllers: Example #4
Process is assumed to be
state not accessible
Multi-estimator:
p=(a, b) 2 P [–1,1] £ {–1,1}
we can do state-sharing to
generate all the errors with a
finite-dimensional system
for p = p* …
exponentially
)
g for p = p*
)
Matching property
Designing multi-estimators - I
(state accessible)
Suppose nominal models Np, p 2 P are of the form
no exogenous
input w
state
accessible
Multi-estimator:
asymptotically
stable A
When process matches the nominal model Np*
exponentially


Matching property: Assume M = { Np : p 2 P }
9 p*2 P, c0, l* >0 :
|| ep*(t) || · c0 e-l* t
t¸0
Designing multi-estimators - II
Suppose nominal models Np, p 2 P are of the form
asymptotically
stable Ap
nonlinear output
injection
(output-injection away
from stable linear system)
(generalization of case I)
Multi-estimator:
When process matches the nominal model Np*

Matching property: Assume M = { Np : p 2 P }
9 p*2 P, c0, cw, l* >0 :
with cw = 0 in case w(t) = 0, 8 t ¸ 0
|| ep*(t) || · c0 e-l* t + cw
t¸0
State-sharing is possible when all Ap are equal an Hp( y, u ) is separable:
Designing multi-estimators - III
Suppose nominal models Np, p 2 P are of the form
asymptotically
stable Ap
(output-inj. and coord.
transf. away from stable
linear system)
(generalization of case I & II)
´ cont. diff. coordinate transformation with continuous
inverse xp-1 (may depend on unknown parameter p)
zp  xp‘ ± xp-1
The Matching property is an input/output property so the same multi-estimator
can be used:
Matching property: Assume M = { Np : p 2 P }
9 p*2 P, c0, cw, l* >0 :
with cw = 0 in case w(t) = 0, 8 t ¸ 0
|| ep*(t) || · c0 e-l* t + cw
t¸0
Switched system
detectability property?
multiestimator
decision
logic
s
multicontroller
w
u
process
y
switched system
The switched system can be seen as the
interconnection of the process with the “injected system”
essentially the multi-controller &
multi-estimator but now quite…
Constructing the injected system
1st Take a parameter estimate signal r : [0,1)! P.
2nd Define the signal v  er = yr - y
3rd Replace y in the equations of the multi-estimator and multi-controller
by yr - v.
s
v
u
multicontroller
multiestimator
yr
–
y
Switched system = process + injected system
w
Q: How to get “detectability” on the
switched system ?
y
process
A: “Stability” of the injected system
u
–
v
injected
system
+
–
+
r
s
r
Stability & detectability of nonlinear systems
Stability:
input u “small”  state x “small”
Input-to-state stable (ISS) if 9 b2KL, g2K
Integral input-to-state stable (iISS) if 9 a2K1, b2KL, g2K
strictly
weaker
Notation:
a:[0,1) ! [0,1) is class K
´ continuous, strictly increasing, a(0) = 0
is class K1 ´ class K and unbounded
b:[0,1)£[0,1) ! [0,1) is class KL ´ b(¢,t) 2 K for fixed t &
limt!1 b(s,t) = 0 (monotonically) for fixed
Stability & detectability of nonlinear systems
Stability:
input u “small”  state x “small”
Input-to-state stable (ISS) if 9 b2KL, g2K
Integral input-to-state stable (iISS) if 9 a2K1, b2KL, g2K
strictly
weaker
One can show:
1.
for ISS systems:
2.
for iISS systems:
u!0
) solution exist globally & x ! 0
s01 g(||u||) < 1 ) solution exist globally & x ! 0
Stability & detectability of nonlinear systems
Detectability:
input u & output y “small”  state x “small”
Detectability (or input/output-to-state stability IOSS) if 9 b2KL, gu, gy2K
Integral detectable (iIOSS) if 9 a2K1, b2KL, gu, gy2K
One can show:
1.
2.
for IOSS systems: u, y ! 0
)x!0
for iIOSS systems: s01 gu(||u||), s01 gy(||y||) < 1 ) x ! 0
strictly
weaker
Certainty Equivalence Stabilization Theorem
process
y
switched system
u
v
–
injected
system
r
+
–
+
r
s
Theorem: (Certainty Equivalence Stabilization Theorem)
Suppose the process is detectable and take fixed r = p 2 P and s = q 2 Q
1. injected system ISS
 switched system detectable.
2. injected system integral ISS  switched system integral detectable
Stability of the injected system is not the only mechanism to achieve detectability:
e.g., injected system i/o stable + process “min. phase” ) detectability of switched system
(Nonlinear Certainty Equivalence Output Stabilization Theorem)
Achieving ISS for the injected system
Theorem: (Certainty Equivalence Stabilization Theorem)
Suppose the process is detectable and take fixed r = p 2 P and s = q 2 Q
 switched system detectable.
1. injected system ISS
2. injected system integral ISS  switched system integral detectable
s
injected system
u
multicontroller
multiestimator
v
yr
–
r=p2P
s=q2Q
y
We want to design candidate controllers that make the injected system
(at least) integral ISS with respect to the “disturbance” input v
Nonlinear robust control design problem, but…
• “disturbance” input v can be measured (v = er = yr – y)
• the whole state of the injected system is measurable (xC, xE)
Designing candidate controllers: Example #4
Multi-estimator:
p=(a, b) 2 P [–1,1] £ {–1,1}
ep
To obtain the injected system, we use
Candidate controller q = c( p ):
Detectablity property
injected system is exponentially
stable linear system (ISS)
Decision logic
For nonlinear systems dwell-time logics
do not work because of finite escape
decision
logic
switching
signal s
switched
system
estimation
errors
process
injected
system
Scale-independent hysteresis switching
monitoring signals
start
class K function from
detectability property
measure of the size of ep over a
“window” of length 1/l
hysteresis
constant
p2P
forgetting factor
y
n
wait until current monitoring
signal becomes significantly
larger than some other one
All the mp can be generated by a system with small
dimension if gp(||ep||) is separable. i.e.,
Scale-independent hysteresis switching
Theorem: Let P be finite with m elements. For every p 2 P
number of switchings in [t, t )
and
Assume P is finite, the gp are locally Lipschitz and
maximum interval of
existence of solution


uniformly bounded on [0, Tmax)
uniformly bounded on [0, Tmax)
Scale-independent hysteresis switching
Theorem: Let P be finite with m elements. For every p 2 P
number of switchings in [t, t )
and
Assume P is finite, the gp are locally Lipschitz and
maximum interval of
existence of solution
Non-destabilizing property: Switching will stop at some finite time T* 2 [0, Tmax)
Small error property:
Analysis
(w = 0, no unmodeled dynamics)
1st by the Matching property: 9 p*2 P such that || ep*(t) || · c0 e-l* t
t¸0
2nd by the Non-destabilization property: switching stops at a finite time
T* 2 [0, Tmax) ) r(t) = p & s(t) = c(p) 8 t 2 [T*,Tmax)
3rd by the Small error property:
4th by the Detectability property:
the state x of the switched system is bounded on [T*,Tmax)

solution exists globally Tmax = 1 & x ! 0 as t ! 1
Theorem: Assume that P is finite and all the gp are locally Lipschitz.
The state of the process, multi-estimator, multi-controller, and all other signals
converge to zero as t ! 1.
Summary
•
•
•
•
Supervisory control overview
Estimator-based linear supervisory control
Estimator-based nonlinear supervisory control
Examples
Example #4: System in strict-feedback form
Suppose nominal models Np, p 2 P are of the form
state accessible
To facilitate the controller design, one can first “back-step” the system to
simplify its stabilization:
now the control law
stabilizes the system
Example #4: System in strict-feedback form
Suppose nominal models Np, p 2 P are of the form
Multi-estimator:
it is separable so we
can do state-sharing
When process matches the nominal model Np*
)
Candidate controller q = c(p):
makes injected
system ISS
exponentially
)
Matching property
 Detectability property
Example #4: System in strict-feedback form
b
a
u
areference
a
Example #4: System in strict-feedback form
Suppose nominal models Np, p 2 P are of the form
state accessible
In the previous back-stepping procedure:
the controller
drives g ! 0

One could instead make
pointwise min-norm design
nonlinearity is cancelled
(even when a < 0 and
it introduces damping)
still leads to exponential
decrease of a
(without canceling
nonlinearity when a < 0 )
Example #4: System in strict-feedback form
Suppose nominal models Np, p 2 P are of the form
state accessible
A different recursive procedure:
In this case
 g ! 0 exponentially


pointwise min-norm
recursive design
a ! 0 exponentially
Example #4: System in strict-feedback form
Suppose nominal models Np, p 2 P are of the form
Multi-estimator ( option III ):
When process matches the nominal model Np*

Candidate controller q = c(p):
exponentially

Matching property
 Detectability property
Example #4: System in strict-feedback form
b
a
u
areference
a
Example #4: System in strict-feedback form
b
b
a
a
u
a
pointwise min-norm design
u
a
feedback linearization design
Example #5: Unstable-zero dynamics
unknown parameter
estimate
output y
(stabilization)
Example #5: Unstable-zero dynamics
unknown parameter
estimate
output y
(stabilization with noise)
Example #5: Unstable-zero dynamics
unknown parameter
reference
output y
(set-point with noise)
Example #6: Kinematic unicycle robot
x2
q
u1 ´ forward velocity
u2 ´ angular velocity
x1
p1, p2 ´ unknown parameters determined by the radius of the driving wheel
and the distance between them
This system cannot be stabilized by a continuous time-invariant controller.
The candidate controllers were themselves hybrid
Example #6: Kinematic unicycle robot
Example #7: Induction motor in current-fed mode
l 2 2 ´ rotor flux
u 2 2 ´ stator currents
w ´ rotor angular velocity
t ´ torque generated
w is the only measurable output
Unknown parameters:
tL 2 [tmin, tmax] ´ load torque
R 2 [Rmin, Rmax] ´ rotor resistance
“Off-the-shelf” field-oriented candidate controllers: