Crisp Control Is Always Better Than

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16 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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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
REAL-TIME
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
REAL-TIME
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)
Discrete-state system
Continuous-time 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

Old-fashioned F-4 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