Switched Supervisory Control

1
Switched Supervisory Control
Tutorial on Logic-based Control

• João P. Hespanha
• University of Californiaat Santa Barbara

2
Summary

• Supervisory control overview
• Estimator-based linear supervisory control
• Estimator-based nonlinear supervisory control
• Examples

3
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
exogenous disturbance/ noise
s
switching signal
measured output
w
controller 1
s
y
bank of candidate controllers
process
u
controller n
control signal

• Key ideas
• Build a bank of alternative controllers
• Switch among them online based on measurements

For simplicity we assume a stabilization problem,
otherwise controllers should have a reference
input r
4
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
exogenous disturbance/ noise
s
switching signal
measured output
w
controller 1
s
y
bank of candidate controllers
process
u
controller n
control signal

• Supervisor
• places in the feedback loop the controller that
  seems more promising based on the available
  measurements
• typically logic-based/hybrid system

5
Multi-controller
s
switching signal
controller 1
s
bank of candidate controllers
y
u
measured output
control signal
controller n
Conceptual diagram not efficient for many
controllers not possible for unstable
controllers
6
Multi-controller
s
switching signal
controller 1
s
bank of candidate controllers
y
u
measured output
control signal
controller n
Given a family of (n-dimensional) candidate
controllers
s
switching signal
y
u
measured output
control signal
7
Multi-controller
switching times
s(t)
switching signal
s 1
s 1
s 3
s 2
t
Given a family of (n-dimensional) candidate
controllers
s
switching signal
y
u
measured output
control signal
8
Supervisor
y
measured output
supervisor
u
control signal
s
switching signal
y
u
Typically an hybrid system ? continuous
state d discrete state
continuous vector field
discrete transition function
output function
9
Supervisory vs. Adaptive control
hybrid supervisor
continuous adaptive tuner
y
y
u
u
s
s
u
y
u
y

• Rapid adaptation s need not vary continuously
• Flexibility modularity can use off-the-shelf
  candidate controllers, estimators, and several
  alternative switching logics (allows reuse of
  existing, nonadaptive theory)
• Between switching times one recovers the
  nonadaptive behavior

10
Types of supervision
Pre-routed supervision
Performance-based supervision (direct)
s 2

• 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

s 1
s 3

• try one controllers after another in a
  pre-defined sequence
• stop when the performance seems acceptable

Estimator-based supervision (indirect)

• estimate process model from observed data
• select controller based on current estimate
  Certainty Equivalence

not effective when the number of controllers is
large
11
Summary

• Supervisory control overview
• Estimator-based linear supervisory control
• Estimator-based nonlinear supervisory control
• Examples

12
Estimator-based supervisions setup Example 1
process
control signal
u
y
measured output
Process is assumed to be
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 gt
.1 b 1 controller C3 u 1.1y to be
used when a gt .1 b 1
process parameter p is equal to p?(a,b) 2 P
use controller Cq with
controller selection function
13
Estimator-based supervisions setup
w
exogenous disturbance/noise
process
control signal
u
y
measured output
unmodeled dynamics
Process is assumed to be in a family
Mp small family of systems around a nominal
process model Np
parametric uncertainty
for each process in a family Mp, at least one
candidate controller Cq, q 2 Q provides adequate
performance.
controller selection function
controller Cq with q c(p)provides adequate
performance
process in Mp, p2 P
14
Estimator-based supervisor Example 1
y
measured output

y
multi-estimator


u
control signal

Process is assumed to be
unknown parameters
p(a, b) 2 P ? 1,1 1,1
Multi-estimator
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)
15
Estimator-based supervisor Example 1
y
measured output

y
switching signal
decisionlogic
multi-estimator

s

u
control signal

Process is assumed to be
unknown parameters
p(a, b) 2 P ? 1,1 1,1
Multi-estimator
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
processlikely in Mp
should useCq, q c(p)
set s c(p)
ep small
Certainty equivalence inspired
16
Estimator-based supervisor
y
measured output

y
switching signal
decisionlogic
multi-estimator

s

u
control signal

Process is assumed to be in family
process inMp, 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
processlikely in Mp
should useCq, q c(p)
set s c(p)
ep small
Certainty equivalence inspired
17
Estimator-based supervisor
y
measured output

y
switching signal
decisionlogic
multi-estimator

s

u
control signal

A stability argument cannot be based on this
because typically process in Mp ) ep small but
not the converse
Process is assumed to be in family
process inMp, 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
processlikely in Mp
should useCq, q c(p)
set s c(p)
ep small
Certainty equivalence inspired
18
Estimator-based supervisor
y
measured output

y
switching signal
decisionlogic
multi-estimator

s

u
control signal

Process is assumed to be in family
process inMp, 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
overall system is detectable through ep
overall state is small
sets c(p)
ep small
Certainty equivalence inspired, but formally
justified by detectability
19
Performance-based supervision
measured output
y
switching signal
decisionlogic
performancemonitor
s
u
control 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
20
Abstract supervision
Estimator and performance-based architectures
share the same common architecture
multi-est. or perf. monitor
decision logic
switching signal
w
s
multi- controller
u
process
y
measured output
control signal
In this talk we will focus mostly on an
estimator-based supervisor
21
Abstract supervision
Estimator and performance-based architectures
share the same common architecture
multi-estimator
decision logic
switching signal
w
s
multi- controller
u
process
measured output
y
control signal
switched system
22
The four basic properties (1-2) Example 1
Matching property At least one of the ep is
small Why?
decision logic
ep issmall
essentially a requirement on the multi-estimator
switching signal
detectable means small ep ) small state
s
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
23
Detectability
a system is detectable if for every pair
eigenvalue/eigenvector (li,vi) of A ltli 0 )
C vi ¹ 0
for short pair (A,C) is detectable
From solution to linear ODEs
¹ 0
(assuming A diagonalizable, otherwise terms in tk
eli t appear)
y(t) bounded ) ai C vi 0 (for ltli 0) ) ai
0 (for ltli 0) ) y(t) bounded
Lemma For any detectable system y(t) bounded )
x(t) bounded y(t) ! 0 ) x(t) ! 0
24
Detectability
system is detectable if for every pair
eigenvalue/eigenvector (li,vi) of A ltli 0 )
C vi ¹ 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
25
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 lt 0
Thus, ep small ) yp small ) y yp ep small ) u
small etc.

• Questions
• Where we just lucky in getting .6 lt 0 ? NO
  (why?)
• Does detectability hold if u 1.1y does not
  stabilize the process (e.g., a .5, b-1)?
  YES (why?)

26
The four basic properties (1-2)
Matching property At least one of the ep is
small Why?
decision logic
9 p2 P processin Mp
ep issmall
essentially a requirement on the multi-estimator
switching signal
s
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
27
The four basic properties (3-4)
Small error property There is a parameter
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)
r(t) can be viewed as current parameter estimate
decision logic
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) Typically requires some
form of slow switching
Both are essentially (conflicting) properties of
the decision logic
28
Decision logic

• For boundedness one wants er small for some
  parameter estimate r consistent with s (i.e., s
  c (r))

decision logic
small error

• To recover the static detectability of the
  time-varying switched system one wants slow
  switching.

non-destabilization
switching signal
s

• These are conflicting requirements
• r should follow smallest ep
• s c(r) should not vary

29
Dwell-time switching
monitoring signals
start
p 2 P
measure of the size of ep over a window of
length 1/l
forgetting factor
wait tD seconds
Non-destabilizing propertyThe minimum interval
between consecutive discontinuities of s is tD gt
0. (by construction)
30
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
• Switching will stop in finite time T at some p 2
  P

C
lt 1
31
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
• After some finite time T switching will occur
  only among elements of a subset P of P, each
  appearing in r infinitely many times

lt 1
C
32
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
33
Implementation issues
monitoring signals
start
How to efficiently compute a large number of
monitoring signals?
Example 1 Multi-estimator
input is linear comb. of y and u
wait tD seconds
ß
by linearity
ß
state-sharing
we can generate as many errors as we want with a
2-dim. systems
34
Implementation issues
monitoring signals
start
How to efficiently compute a large number of
monitoring signals?
Example 1 Multi-estimator
state-sharing
wait tD seconds
ß
1. we can generate as many monitoring signals as
we want with a (26)-dim. system
2. finding r is really an optimization
35
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 2. model is linearly parameterized
on p (leads to quadratic optimization) 3. there
are closed form solutions (e.g., mp polynomial
on p) 4. mp is convex on p
wait tD seconds
usual requirement in adaptive control
results still hold if there exists a
computational delay tC in performing the
optimization, i.e.
36
The four basic properties
r(t) can be viewed as current parameter estimate
Small error property There is a parameter
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)
decision logic
controller consistent with parameter estimate
Non-destabilization property Detectability is
preserved for the time-varying switched system
(not just for constant s)
switching signal
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
37
Analysis outline (linear case, w 0)
1st by the Matching property 9 p2 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
decision logic
switching signal
s
injected system
asymptotically stable by non-destabilization
property
small by 2nd step
) x is small (and converges to zero if, e.g., er2
L2)
38
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
IH
T
axiss inertia (.023)
PDE (small bending)
Boundary conditions
beams length(113 cm)
transversalslices inertia
beams elasticity
beamsmass density(.68Kg total mass)
39
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
IH
T
axiss inertia (.023)
Series expansion and truncation
eigenfunctions of the beam
40
Example 2 One-link flexible manipulator
mass at the tip
x
mt
y(x, t)
q
torque applied at the base
Assumed not known a priori mt 2 0, .1Kg xsg 2
40cm, 60cm
IH
T
axiss inertia (.023)
Control measurements
base angle
base angular velocity
tip position
bending at position xsg (measured by a strain
gauge attached to the beam at position xsg)
41
Example 2 One-link flexible manipulator
u
torque
transfer functions as mt ranges over 0, .1Kg
and xsg ranges over 40cm, 60cm
42
Example 2 One-link flexible manipulator
u
torque
Class of admissible processes
Mp family around a nominal transfer function
corresponding to parameters p ? (mt, xsg)
parameter set grid of 18 points in 0, .140,
60
unknown parameter p ? (mt, xsg)
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
43
Example 2 One-link flexible manipulator
44
Example 2 One-link flexible manipulator
(open-loop)
45
Example 2 One-link flexible manipulator
(closed-loop with fixed controller)
46
Example 2 One-link flexible manipulator
(closed-loop with supervisory control)
47
Example 3 Uncertain gain
C
1 k lt 4
The maximum gain margin achievable by a single
linear time-invariant controller is 4
Doyle, Francis, Tannenbaum, Feedback Control
Theory, 1992
48
Example 3 Uncertain gain
multi- controller
1 k lt 40
49
Example 3 Uncertain gain
50
Example 3 2-dim SISO linear process
Class of admissible processes
nominal transfer function nonlinear parameterized
on p
Any re-parameterization that makes the
coefficients of the transfer function lie in a
convex set will introduce an unstable zero-pole
cancellation
unstable zero-pole cancellations
But the multi-estimator is still separable and
state-sharing can be used
51
Example 3 2-dim SISO linear process
(without noise)
52
Example 3 2-dim SISO linear process
(with noise)
53
Example 3 Disturbance Rejection
(unknown frequency)
(rejection of one sinusoid)
54
Example 3 Disturbance Rejection
(unknown frequency)
(rejection of two sinusoids)
55
Example 3 Disturbance Rejection
(unknown frequency)
(rejection of a square wave)
56
Summary

• Supervisory control overview
• Estimator-based linear supervisory control
• Estimator-based nonlinear supervisory control
• Examples

57
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
exogenous disturbance/ noise
s
switching signal
measured output
w
controller 1
s
y
bank of candidate controllers
process
u
controller n
control signal

• Key ideas
• Build a bank of alternative controllers
• Switch among them online based on measurements

For simplicity we assume a stabilization problem,
otherwise controllers should have a reference
input r
58
Estimator-based nonlinear supervisory control
multi-estimator
decision logic
NON LINEAR
switching signal
w
s
multi- controller
u
process
measured output
y
control signal
59
Class of admissible processes Example 4
process
control signal
u
y
measured output
state accessible
(a, b) unknown parameters
Process is assumed to be in the family
p(a, b) 2 P ? 1,1 1,1
60
Class of admissible processes
w
exogenous disturbance/noise
process
control signal
u
y
measured output
unmodeled dynamics
Process is assumed to be in a family
Mp small family of systems around a nominal
process model Np
parametric uncertainty
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)

61
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 stabilizes the system
after the coordinate transformation the new state
is no longer accessible
Candidate controllers
62
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)
s
Multi-controller
switching signal
y
u
measured output
control signal
63
Multi-estimator
measured output

y
multi-estimator


u
control signal

How to design a multi-estimator?
we want Matching property there exist some p2
P such that ep is small Typically obtained by
9 p2 P processin Mp
ep issmall
when process matches Mp the corresponding
error must be small
64
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
)
)
Matching property
g for p p
65
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 p2 P, c0, l gt0 ep(t) c0 e-l t t
0
66
Designing multi-estimators - II
(output-injection away from stable linear system)
Suppose nominal models Np, p 2 P are of the form
asymptotically stable Ap
nonlinear output injection
(generalization of case I)
Multi-estimator
When process matches the nominal model Np
?
Matching property Assume M Np p 2 P 9
p2 P, c0, cw, l gt0 ep(t) c0 e-l t
cw t 0 with cw 0 in case w(t) 0, 8 t 0
State-sharing is possible when all Ap are equal
an Hp( y, u ) is separable
67
Designing multi-estimators - III
(output-inj. and coord. transf. away from stable
linear system)
Suppose nominal models Np, p 2 P are of the form
asymptotically stable Ap
(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
p2 P, c0, cw, l gt0 ep(t) c0 e-l t
cw t 0 with cw 0 in case w(t) 0, 8 t 0
68
Switched system
detectability property?
multi-estimator
decision logic
w
s
multi- controller
u
process
y
switched system
The switched system can be seen as
theinterconnection of the process with the
injected system
essentially the multi-controller
multi-estimator but now quite
69
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
yr

multi- controller
multi-estimator
y
70
Switched system process injected system
w
Q How to get detectability on the switched
system ? A Stability of the injected system
y
process
u

r
injectedsystem
v



s
r
71
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 a0,1) ! 0,1) is class K
continuous, strictly increasing, a(0) 0 is
class K1 class K and unbounded b0,1)0,1) !
0,1) is class KL b(,t) 2 K for fixed t
limt!1 b(s,t) 0 (monotonically) for fixed s
72
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
• for ISS systems u ! 0 ) solution exist globally
  x ! 0
• for iISS systems s01 g(u) lt 1 ) solution
  exist globally x ! 0

73
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
strictly weaker
Integral detectable (iIOSS) if 9 a2K1, b2KL,
gu, gy2K

• One can show
• for IOSS systems u, y ! 0 ) x ! 0
• for iIOSS systems s01 gu(u), s01 gy(y) lt
  1 ) x ! 0

74
Certainty Equivalence Stabilization Theorem
y
process
switched system
u

r
injectedsystem

v


s
r
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)
75
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 1. injected
system ISS ? switched system
detectable. 2. injected system integral ISS ?
switched system integral detectable
s
injected system
v
r p 2 Ps q 2 Q
u
yr

multi- controller
multi-estimator
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)

76
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)
77
Decision logic
For nonlinear systems dwell-time logics do not
work because of finite escape
decision logic
switching signal
s
switchedsystem
estimation errors
78
Scale-independent hysteresis switching
class K function from detectability property
start
monitoring signals
p 2 P
measure of the size of ep over a window of
length 1/l
hysteresis constant
forgetting factor
y
n
All the mp can be generated by a system with
small dimension if gp(ep) is separable. i.e.,
wait until current monitoring signal becomes
significantly larger than some other one
79
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)
80
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
81
Analysis
(w 0, no unmodeled dynamics)
1st by the Matching property 9 p2 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.
82
Summary

• Supervisory control overview
• Estimator-based linear supervisory control
• Estimator-based nonlinear supervisory control
• Examples

83
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
84
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
exponentially
)
)
Matching property
Candidate controller q c(p)
makes injected system ISS
?
Detectability property
85
Example 4 System in strict-feedback form
b
a
u
areference
a
86
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
nonlinearity is cancelled (even when a lt 0 andit
introduces damping)
?
drives g ! 0
One could instead make
still leads to exponential decrease of a (without
canceling nonlinearity when a lt 0 )
pointwise min-norm design
87
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
88
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
exponentially
?
?
Matching property
Candidate controller q c(p)
?
Detectability property
89
Example 4 System in strict-feedback form
b
a
u
areference
a
90
Example 4 System in strict-feedback form
pointwise min-norm design
feedback linearization design
91
Example 5 Unstable-zero dynamics
(stabilization)
92
Example 5 Unstable-zero dynamics
(stabilization with noise)
93
Example 5 Unstable-zero dynamics
(set-point with noise)
94
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
95
Example 6 Kinematic unicycle robot
96
Example 7 Induction motor in current-fed mode
l 2 Ñ2 rotor flux u 2 Ñ2 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