1
Crisp Control Is Always Better Than
Fuzzy Feedback Control
MICHAEL ATHANS
Professor of Electrical Engineering (Emeritus)
MIT, Cambridge, Mass., USA
and
Visiting Scientist, Instituto de Sistemas e Robotica
Instituto Superior Tecnico, Lisbon, PORTUGAL
mathans@mit.edu
or
athans@isr.ist.utl.pt
EUFIT
´
99 DEBATE WITH PROF. L.A. ZADEH
Aachen, Germany, September 1999
2
Debating Points
•
I like Fuzzy Logic as an alternative to probability theory, especially in
applications involving man

machine interactions
•
Fuzzy feedback control methods represent
inferior
engineering
practice, often by people that never bothered to learn control theory
and design
•
Fuzzy feedback control is a
vacuous technology
for the design of high

performance control systems
•
Fuzzy control methods are “
parasitic
;” they simply implement trivial
interpolations of control strategies obtained by other means
•
Theological arguments about “fuzzification”, “defuzzification”,
nonlinear control, and inherent robustness are simply nonsense
•
Fuzzy feedback control has
failed to capture and utilize
alternative
means in dealing with uncertainty using Fuzzy Sets and Fuzzy Logic
•
Prof. Zadeh should communicate to his disciples the sorry state of
affairs in fuzzy feedback control and tell them to “shape

up”
3
Crisp Vs Fuzzy Feedback Control
•
Crisp control:
Normative

prescriptive
•
Quantitative models of plant dynamics and disturbances
•
Precise definition of performance specifications
•
Modeling and environmental uncertainty accounted for
•
Rigorous optimization

based design
•
Fuzzy control:
Empirical

descriptive
•
1st generation (Mamdani). Ad

hoc interpolation of “expert” control
rule

based system
•
Vast majority of “fuzzy applications” use this method
•
2nd generation (Takagi

Sugeno). Ad

hoc interpolation of control
strategies derived from crisp feedback control methodologies
•
Fuzzy control has failed the noble goal of “fuzzy logic”
in providing alternatives in dealing with uncertainty
4
The Joy of Feedback
•
Measure system response, including effects of disturbances, using
(noisy) sensors
•
Compare actual system response to desired system response at
each time
•
“Error” signal(s) = (Desired response)

(Actual response)
•
Use “error” signals to drive compensator (controller) so as to
generate real

time control corrections so as to keep “errors” small
for all time
•
FEEDBACK ESSENTIAL TO GUARANTEE GOOD
PERFORMANCE IN THE PRESENCE OF UNCERTAINTY
5
Why Feedback?
•
Automatic feedback control systems have been used since the 1930
´
s
to provide
superior performance and higher fidelity
than manual
control systems requiring human operators
•
The
SCIENCE
of Feedback Control was developed to allow
engineering designs that deliver this superior performance,
NOT
to
duplicate poor human control performance
•
The performance payoffs are even more dramatic in the case of
coupled
multivariable systems
, i.e. systems with many sensors and
control inputs
•
crisp control theory
exploits
the tight dynamic coupling
•
humans are
notorious in lacking ability
to develop control rules for
such multivariable systems
•
Increased cost of feedback (sensors, actuators, processors,...) is
justified by increased performance capabilities
•
sensor/actuator hardware costs greatly exceed
processing
costs
6
Fixed Structure Feedback
•
Compensator structure does not change (no learning)
•
No change in digital processor algorithms that approximate the
solution of compensator differential equations and gains
•
Design methodologies available for general multivariable case using
(crisp) robust

control theories and algorithms
DISTURBANCES
DYNAMIC SYSTEM
(PLANT)
CONTROLLER
(COMPENSATOR)
COMMANDS
CONTROLS
OUTPUTS
7
Adaptive Feedback Control
•
Uncertain plant parameters identified in real

time and compensator
parameters are adjusted also in real

time
DISTURBANCES
DYNAMIC SYSTEM
(PLANT)
CONTROLLER
(COMPENSATOR)
COMMANDS
CONTROLS
OUTPUTS
IDENTIFICATION
PARAMETER
ADJUSTMENT
LOGIC
REALTIME
8
Fault

Tolerant Feedback
•
“Supervisory” level monitors for failures
•
Failure isolated and identified
•
Compensator structure and algorithms modified
DISTURBANCES
DYNAMIC SYSTEM
(PLANT)
CONTROLLER
(COMPENSATOR)
COMMANDS
CONTROLS
OUTPUTS
REALTIME
FAILURE
DETECTION
ISOLATION
CONTROL
RECONFIGURATION
LOGIC
9
Crisp Mathematical Control
•
Based upon analytical description of plant dynamics, model errors,
environment, constraints, and performance objectives
•
Optimal Control Theory
•
Used to generate “open

loop” preprogrammed control and state
variable trajectories as a function of time
•
Feedback Control Theory
•
Used to ensure precise command

following and disturbance

rejection performance, in the presence of uncertainty, using
feedback of sensed variables
•
stability guarantees are
essential
•
performance guarantees (in the presence of uncertain models)
are desirable
10
Closed

Loop Stability
•
“Models have limitations, stupidity does not!”
•
Feedback control can result in
superior performance
•
Careless feedback strategies can cause
instabilities
•
Closed

loop stability must be guaranteed for
family of plants
(stability

robustness)
•
stability guarantees for “nominal” plant and nominal plant
simulations are
not
enough
•
control engineers
must be paranoid
about closed

loop stability
11
Crisp Feedback Theory Status
•
Start with
global nonlinear dynamic model
of plant (nonlinear
differential or difference equations)
•
Using “linearization” establish a collection of linear models in vicinity of
operating conditions
•
Generate linear multivariable dynamic compensator with
guaranteed
stability

robustness and performance

robustness properties for each
linear model
•
Use “gain

scheduling” of the parameters of the linear compensator
collection to derive a
single global nonlinear dynamic compensator
for
the global nonlinear plant
12
Linearization, Gain

Scheduling
LM:1
LM:2
LM:3
LM:4
LM:k
LC
:1
LC:2
LC:3
LC:4
LC:k
Global
Nonlinear
Plant
Global
Nonlinear
Dynamic
Compensator
Family of linear dynamic models
Family of linear dynamic
compensators
13
Robust Feedback Control Design
•
Start with nominal state

space model of linear MIMO dynamic system
•
Define bounds on model errors (class of “legal” errors)
•
parametric uncertainty
; upper and lower bounds for key
coefficients
•
unstructured uncertainty
; worst size of dynamic errors as a function
of frequency (bending modes, torsional modes, actuator/sensor
errors, ....)
•
Model exogenous signals (a key requirement for superior
performance)
•
power spectral densities of commands, disturbances and sensor
noise
•
Quantify robust

performance specifications in the frequency domain
Design is meaningless unless performance specs are quantified
14
Robust MIMO Feedback Design
•
LQG or H2 method
•
performance goal: minimize RMS errors of stochastic performance
variables
•
H
• method
•
performance goal: minimize maximum errors assuming worst

case
exogenous disturbances
•
Robust feedback design is done via
mixed

(structured singular
value) synthesis. Iterative generation of H
• dynamic compensators (of
increasing complexity) to guarantee
stability

robustness and
performance

robustness
•
Generate linear multivariable dynamic compensator with guaranteed
stability

robustness and performance

robustness properties for each
linear model (model errors are explicitly accounted for)
•
Use
“gain scheduling”
of the parameters of the linear compensator
collection to derive a single global nonlinear dynamic compensator for
the global nonlinear plant
•
controller involves real

time solution of coupled nonlinear
differential equations
15
Fuzzy Feedback Systems (Mamdani)
•
1st generation fuzzy feedback control systems
•
start with set of “expert” discrete

valued control rules (if

then...),
often obtained from human operators
•
interpolate between discrete control rules using “membership
functions” from fuzzy set theory
•
No
explicit quantitative statement of performance specifications
•
No
quantitative modeling of plant dynamics, disturbance and sensor
noise characteristics
•
No
stability

robustness or performance

robustness guarantees
•
Lots of “theology”, hand

waiving and scientifically unfounded claims
•
Simulation based results (
where does model used for simulation come
from?
)
16
Fuzzy Control (Mamdani)
d
e
\
e
N
L
N
S
Z
E
P
S
P
L
N
L
P
L
P
L
P
S
Z
E
N
S
N
S
P
L
P
S
P
S
Z
E
N
S
Z
E
P
L
P
S
Z
E
N
S
N
L
P
S
P
S
Z
E
N
S
N
S
N
L
P
L
Z
E
N
S
N
S
N
L
N
L
17
Weakness of Mamdani

Type Fuzzy
Control Philosophy
•
Attempt to emulate or duplicate human control behavior
•
Basic problem
•
premise:
Human is good controller
•
fallacy:
Human is very poor controller for complex, multivariable,
marginally stable dynamic plants
•
Fuzzy feedback controllers “work” for very simple SISO dynamic
systems where high precision is not required
•
mostly PI controllers (a few PID with a crisp channel)
•
no guarantees of closed

loop stability, stability

robustness and of
performance in presence of uncertainty
•
hard to extrapolate designs to new applications
99% of fuzzy feedback control applications deal with essentially
1st or 2nd

order, overdamped, SISO systems
18
Michio Sugeno Says....
•
“Stability has been one of the central issues since Mamdani
´
s
pioneering work.
Most of the critical comments to fuzzy control are
due to the lack of a general method for its stability analysis
. We are
still seeking an appropriate tool for the stability analysis of fuzzy
control systems, though this situation is now improved......The success
of fuzzy control, however, does not imply that we do not need a
stability theory for it.
Perhaps the main drawback of the lack of stability
analysis would be that we cannot take a model

based approach to
fuzzy controller design.
”
•
Reference:
M. Sugeno, “On Stability of Fuzzy Systems Expressed by
Fuzzy Rules with Singleton Consequences,”
IEEE Trans. on Fuzzy
Systems,
Vol. 7, April 1999
19
From Jenkins and Passino...
•
Reference:
D.F. Jenkins and K.M. Pasino, “An Introduction to Nonlinear
Analysis of Fuzzy Control Systems,”
J. Intelligent and Fuzzy Systems,
Vol. 7,
1999
•
“The fuzzy controller design methodology primarily involves distilling human
expert knowledge about how to control a system into a set of rules. While a
significant amount of attention has been given to the advantages of the
heuristic fuzzy control design methodology .... relatively little attention has
been given to its potential disadvantages. For example, the following
questions are cause for concern
•
will the behaviors observed by a human expert include all possible
unforseen situations
that can occur due to disturbances, noise, or
plant
parameter variations
?
•
can the human expert realisticaly and reliably foresee problems that could
arise from
closed

loop
system instabilities or limit cycles
•
will the expert really know how to
incorporate stability criteria and
performance objectives
into a rule

base to ensure that reliable operation
can be obtained?
•
Authors advocate the use of Tagaki

Sugeno models with crisp stability criteria
20
Shortcomings of Fuzzy Controller
Methodology
e
1
e
2
e
n
u
1
u
2
u
m
MIMO
Fuzzy
Controller
u
1
h
1
(
e
1
,
e
2
,
...,
e
n
)
u
2
h
2
(
e
1
,
e
2
,
...,
e
n
)
.............
u
m
h
m
(
e
1
,
e
2
,
...,
e
n
)
•
Fuzzy rules just generate nonlinear static functions
•
Impossible to generate multidimensional “if

then” rule tables
•
Cannot generate “differential equation” controller rules
•
It is not easy to differentiate noisy sensor signals by finite differencing, as it is
almost always done in fuzzy applications
•
no utilization of dynamic (e.g. Kalman) filtering of sensor noise
•
I have never seen a multiple

input multiple

output (MIMO) fuzzy control
application using Mamdani

type methods
•
combinatorial complexity for high

order and multivariable applications
21
Challenge to Fuzzy Control Experts
•
Observe only noisy position x(t)
•
with broadband sensor noise
•
Find force f(t) to relocate cart
•
not just balance stick
•
No static fuzzy rule

based system
can solve this problem
•
human cannot stabilize system
with knowledge only of x(t)
•
To change cart position and for
inverted pendulum stabilization, the
controller must be dynamic, i.e. it
must implement “differential
equations” from x(t) to f(t)
f(t)
x
o
x
1
x(t)
M
m
M
~
m
22
Why is Fuzzy Control Popular with the
Masses
LEARNING FUZZY CONTROL
•
Working pragmatic knowledge of
fuzzy sets and membership
functions .....
1 week
•
Working pragmatic knowledge of
Mamdani method .....
1 week
LEARNING CRISP CONTROL
•
Differential equations ...
8 weeks
•
Linear algebra ...
10 weeks
•
SISO servos ....
14 weeks
•
State space methods/stability
theory ...
14 weeks
•
Optimal control ....
8 weeks
•
Multivariable robust control ...
14
weeks
23
Takagi

Sugeno Fuzzy Control
•
Approach developed to overcome criticism regarding closed

loop
stability guarantees
•
Approximate global nonlinear dynamics by “interpolating” linear state

space models with membership functions
•
Design full

state feedback controllers for each linear model
(using
crisp control methods, e.g. LQR, H
2
, H
•,
etc.)
and “interpolate” using
membership functions
•
technique is inferior to that of “gain

scheduling”
•
It is possible to use quadratic Lyapunov functions to obtain sufficient
conditions for nominal stability
•
results are disappointing; at best applicable to low performance
systems
•
Current methodology does not address stability

robustness and
performance

robustness issues
•
Current methodology does not address output feedback requiring
dynamic compensator designs
24
Recent References on Fuzzy Stability
•
M. Sugeno, “On Stability of Fuzzy Systems Expressed by Fuzzy Rules
with Singleton Consequences,”
IEEE Trans. on Fuzzy Systems,
Vol.
7, April 1999
•
S.H. Zak, “Stabilizing Fuzzy System Models Using Linear Controllers,”
IEEE Trans. on Fuzzy Systems,
Vol. 7, April 1999
•
M. Margaliot and G. Langholz, “Fuzzy Lyapunov

based Approach to
the Design of Fuzzy Controllers,”
Fuzzy Sets and Systems,
Vol. 106,
August 1999
•
D.F. Jenkins and K.M. Pasino, “An Introduction to Nonlinear Analysis
of Fuzzy Control Systems,”
J. Intelligent and Fuzzy Systems,
Vol. 7,
1999
•
A. Kandel, Y. Luo,and Y.Q. Zhang, “Stability Analysis of Fuzzy Control
Systems,”
Fuzzy Sets and Systems,
Vol. 105, July 1999
•
Y. Tang, N. Zhang and Y. Li, “Stable Fuzzy Adaptive Control for a
Class of Nonlinear Systems,”
Fuzzy Sets and Systems,
Vol. 104, June
1999
25
Trends in Fuzzy Stability Studies
•
Must have a (linear, nonlinear, multi

model,...) state

space model
•
Classical crisp stability theory results are applied
•
Popov criterion
•
Circle criterion
•
Lyapunov stability theory
•
Linear Matrix Inequalities (LMI)
•
Bounded

input bounded

output (L2) stability theory
B
I
G
Q
U
E
ST
I O
N
I
f
a s
t
a
t
e s
p
ace
mo
d
e
l i s av a
i
l abl
e
w
h
y
no
t
u
se
s
up
er i
o
r c
r
i s
p
de
s
i g
n t
ec
h
n
i
q
u
es
th
at
g
u
ar
a
n
t
ee
s
t
a
b
i l i t
y
, s
t
abi
l
i t y

r
ob
u
s
t
n
ess,
an
d p
er
fo
r
m
a
n
ce
r
ob
u
s
t
n
ess
?
26
Takagi

Sugeno Models
• Start with
R
linear state

space models,
each valid in a specific region
S
k
of
R
n
Ý
x
(
t
)
A
k
x
(
t
)
B
k
u
(
t
);
k
1
,
2
,
...,
R
;
x
(
t
)
S
k
• Define
R
scalar valued membership functions,
k
(
x
(
t
)),
0
k
(
x
(
t
))
1
,
such that
k
(
x
(
t
))
1
if
x
(
t
)
S
k
0
if
x
(
t
)
S
j
for
k
j
linear
int
erpolations
otherwise
;
let
(
x
(
t
))
1
R
1
k
(
x
(
t
))
1
R
1
(
x
(
t
))
R
(
x
(
t
))
Global nonlinear model
Ý
x
(
t
)
A
k
k
(
x
(
t
))
k
1
R
x
(
t
)
B
k
k
(
x
(
t
))
k
1
R
u
(
t
)
Almost impossible to define the
membership functions
k
(
x
(
t
))
for
high

dimensional problems
27
Takagi

Sugeno Feedback Law
• For each linear plant,
design full

state feedback gain matrices,
typically by crisp
feedback methods (eigenstructure

assignment,
LQR,
H
,
etc.) of the form
K
k
x
(
t
),
k
1
,
2
,
...,
R
.
• Generate global nonlinear feedback by interpolating with the same membership
functions
u
(
t
)
K
j
j
(
x
(
t
))
j
1
R
x
(
t
)
• Global closed

loop system
Ý
x
(
t
)
A
k
k
(
x
(
t
))
B
k
k
(
x
(
t
))
k
1
R
K
j
j
(
x
(
t
))
j
1
R
k
1
R
x
(
t
)
• Quadratic Lyapunov functions provide sufficient conditions for stability.
Find
P
0
so
P
(
A
k
B
k
K
j
)
(
A
k
B
k
K
j
)
T
P
0
for
all
j
,
k
1
,
2
,
...,
R
i.e. all mismatched linear plant/linear gain combinations must be stable!
!
!
This
seldom happens in high

performance designs.
28
Set

Point Vs. Task

Based Control
•
Prof. Zadeh asserts
•
crisp control theory only deals with set

point control; it cannot
handle task

based control
•
Fact
•
hybrid control systems do provide the methodology for integrating
task

based and set

point control
29
Hybrid Control
DYNAMIC SYSTEM
(PLANT)
CONTROLLER
(COMPENSATOR)
Discretestate system
Continuoustime system
•
Architectures involving interactions between a finite

state event

driven
system and a continuous

state continuous

time system
•
Discrete level can establish different modes of operation (tasks) for feedback
system
30
Car Parking
•
Prof. Zadeh asserts that control theory cannot solve parallel parking problem
•
Fact: Time

optimal solution using simplified dynamics is shown
•
optimal control theory using more complex nonholonomic car dynamic
model can also be used using arbitrary initial car location and orientation
•
automated crisp solution can be implemented if customer is willing to pay
the price
31
Highway Driving
•
Prof. Zadeh asserts that it will never be possible to construct an
automated automobile driving system using conventional control
theory
•
FACT: Such a prototype system has been already been
demonstrated by PATH on the I

5 freeway in San Diego including
•
longitudinal control with minimal inter

car spacing to triple freeway
lane capacity
•
lateral control (lane changing and lane

centerline following)
•
automated merging and demerging capabilities
•
using hybrid control methodologies
•
by some of Prof. Zadeh
´
s colleagues (Varayia, Sastry, Hedrick, ...)
at UC

Berkeley, among others
•
Most certainly the fatality rate of such automated highway systems will
be far less that those involving human drivers
•
Similar efforts are ongoing by Daimler

Benz in Europe
32
Barriers to “Computing With Words”
•
Prof. Zadeh advocates computing with words using fuzzy logic
concepts
•
noble task; provides a foundation for a computational theory of
perceptions
•
What is not usually stressed is that such computations require the
solution of
exceedingly complex equations in real

time
•
in June 1997 talk at the Portuguese Academy of Sciences, Prof.
Zadeh showed an example which illustrated that
even simple
“word computations” require solution of systems of complex
nonlinear integro

differential equations
•
such real

time computations are beyond capabilities of current and
projected computers
•
must wait for completely new computers with novel architectures
and software
33
Fuzzy Dynamical Systems
•
Appropriate framework for capturing system uncertainty
•
References
•
P.E. Kloeden, “Fuzzy Dynamical Systems,”
Fuzzy Sets and
Systems,
Vol. 7, 1982
•
Y. Friedman and U. Sandler, “Evolution of Systems under Fuzzy
Dynamics Laws,”
Fuzzy Sets and Systems,
Vol. 84, 1996
•
Y. Friedman and U. Sandler, “Fuzzy Dynamics as Alternative to
Statistical Mechanics,”
Fuzzy Sets and Systems,
Vol. 106, 1999
•
Must propagate the
Possibility Density Function
using Chapman

Kolmogorov integral equations
•
to solve these requires enormous computational power
•
feedback control system design using such Chapman

Kolmogorov
equations is extremely complex
and its real

time computational
requirements are astronomical
34
Linear

Quadratic

Fuzzy (LQF) Optimal
Control
• Formulation of standard LQ problem using
fuzzy membership functions for process
and measurement noise
x
(
t
1
)
Ax
(
t
)
Bu
(
t
)
Lw
(
t
)
y
(
t
1
)
Cx
(
t
1
)
v
(
t
1
)
J
lim
T
1
2
T
x
T
(
t
)
Qx
(
t
)
u
T
(
t
)
Ru
(
t
)
k
T
T
• Technical difficulties
(1). The conditional state membership function,
given past observations,
involves the solution of nonlinear partial differential equations
(2). Min/max fuzzy arithmetic further complicates life
(3). Common membership functions are nondifferentiable
35
The Numbers Game: So What?
•
Prof. Zadeh claims that from 1981 to 1996 there are 15,631 INSPEC
and 5,660 Math Reviews citations with “fuzzy”, and 2,997 INSPEC
citations with “fuzzy control”
•
There are at least 250,000 citations on Kalman filtering alone, and
there must be
several million
citations on other aspects of “crisp”
modern control theory
•
Note that Modern Control Theory started in about 1959 and Zadeh
´
s
seminal paper on Fuzzy sets was written in 1965
36
The Numbers Game: Comparisons
•
Prof. Zadeh credits Japanese with innovative insight to popularize
fuzzy control applications and bring “fuzzy” commercial products into
the marketplace
•
oriental vs western philosophy
•
Numerical facts
•
in December 1989 the Nikkei 225 was at
39,000
•
in December 1989 the Dow Jones was at
2,700
•
on August 18, 1999 the Nikkei 225 was at
17,879
•
on August 18, 1999 the Dow Jones was at
10,991
37
Fuzzy Applications
•
Lot
´
s of “hoopla” about commercial applications (air

conditioners,
washing machines, camcorders, ...)
•
The innovation is adding special sensors/actuators and feedback to
previously open

loop systems
•
even better performance would be obtained for the same
sensor/actuator architectures if engineers used crisp control
methods
•
Example: Phillips design for Mercedes CD player rejecting fuzzy
control design in favor of H
•

based one
It is time we moved from "
voodoo engineering"
into solid and respectable
science and technology,
and
Prof. Zadeh should take a leadership role in this transition
38
Crisp and Fuzzy Control Complement?
•
Prof. Zadeh
´
s asserts:
Fuzzy controls do not replace crisp
controls, but they can complement each other
•
Basic engineering problem:
How does an engineer integrate a
crisp and a fuzzy control design (and why???)
Disturbance
y(t)
r(t)
e(t)
Fuzzy
controller
Dynamic
system
u(t)
Command
Output
Error

Control
d(t)
39
My Dillema
•
Without stability guarantees, Mamdani fuzzy controllers cannot be
used for 3rd or higher order systems
•
To obtain stability guarantees, even fuzzy control afficionados admit
that they
must use some nominal state space model
for system
dynamics for fuzzy control designs (Sugeno
et al
)
•
plus,
lots of crisp tools
(Lyapunov theory, circle criterion, Popov
criterion, linear quadratic regulators, pole placement, ...)
•
they still have to worry about unmodeled dynamics and uncertain
parameters
•
Given that a state space model is necessary,
why bother
to introduce
fuzzy ideas when conventional crisp control methods can deal with the
design problem directly???
•
and, at the same time, address
explicitly and directly
disturbances,
sensor noise, model errors, performance specifications, nominal
stability, robust stablity, and performance

robustness
40
Optimal Control
•
Used for determining best way of adjusting controls, as functions of
time, such that system response is “optimal” (in well

defined sense)
from any initial state
State
Dynamics
(continuous

time)
:
Ý
x
(
t
)
f
x
(
t
),
u
(
t
)
;
x
(
t
o
)
x
o
Cost
Functional
:
J
(
u
)
K
(
x
(
t
f
))
L
x
(
t
),
u
(
t
)
t
o
t
f
dt
State
Dynamics
(discrete

time)
:
x
(
t
1
)
f
x
(
t
),
u
(
t
)
;
x
(
0
)
x
o
Cost
Function
:
J
(
u
)
K
x
(
T
)
L
x
(
t
1
),
u
(
t
)
t
0
T
1
41
An Example
•
Oldfashioned F4 aircraft
•
Objective: Reach operational altitude in minimum time
•
Shown is expected flight path
Range
Altitude
60,000
ft
Conventional
T=720s
42
43
Optimal Control Theory
•
Pontryagin
maximum principle
(1957) main theoretical tool for
analyzing and solving optimal control problems
•
Extension of Kuhn

Tucker conditions in Nonlinear Programming
problems to dynamic case
•
Maximum Principle leads to numerical solution of
Two

Point

Boundary

Value
(TPBV) problem to calculate
•
optimal controls vs. time
•
resulting optimal dynamic state trajectories and responses
•
Several algorithms exist for solving TPBV problems
44
Linearization, Gain

Scheduling
LM:1
LM:2
LM:3
LM:4
LM:k
LC
:1
LC:2
LC:3
LC:4
LC:k
Global
Nonlinear
Plant
Global
Nonlinear
Dynamic
Compensator
Family of linear dynamic models
Family of linear dynamic
compensators
45
MIMO Linear Feedback
•
Must design MIMO compensator to ensure stability and satisfaction of
performance specifications
•
Digital approximation of MIMO compensator solves in real

time high

order LTI differential equations
46
Concluding Remarks
•
Crisp control theory offers a powerful methodology for designing SISO
and MIMO optimal and high

performance feedback control systems
•
extensive knowledge of theoretical developments required
•
quantitative modeling of plant, disturbances, specs. is essential
•
systematic prescriptive/normative approach to control design
•
leads to
high

performance
(high

gain, high

bandwidth) designs
•
Fuzzy feedback control methods (Mamdani) are suitable for
trivial
control problems requiring low accuracy (minimal performance)
•
no training in control theory necessary
•
no models, no specifications, no guarantees
•
impossible to guarantee stability
•
empirical ad

hoc approach to design
•
leads to
low

performance
(low

gain, low

bandwidth) designs
Fuzzy control is a “parasitic” technology
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