Motivations and Perspectives
1. Classification of Control Theory
General control system:
input u
Process
output y
Controller
(1)
Linear model
Continuous-time
(2)
Nonlinear model
Process
(3)
Linear model
Discrete-time
(4)
Nonlinear model
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(1.1)
Model known
(1.2)
Model unknown
(2.1)
Model known
(2.2)
Model unknown
(3.1)
Model known
(3.2)
Model unknown
(4.1)
Model known
(4.2)
Model unknown
Case (1)
Process model:
y(n) (t) = a 1y(n-1) (t) + … + a n y(t) + b1u(t) + … + bm u (m-1) (t)
(1.1) The model parameters a1 ,..., an , b1 ,..., bm are known:
Linear control theory (ELEC271 and ELEC560): transfer
functions, stability test, pole-zero analysis, pole placement
controller design, etc..
(1.2) The model parameters a1 ,..., an , b1 ,..., bm are unknown:
Linear adaptive control theory (Lyapunov synthesis
approach): propose a controller based on the unknown
parameters a1 ,..., an , b1 ,..., bm , then develop an algorithm to
estimate the parameters a1 ,..., an , b1 ,..., bm by minimizing a
Lyapunov function candidate.
input u
Process
(linear with unknown parameters)
output y
Controller
(with free parameters)
Parameter estimation
Key issues:
 How to choose the controller structure
 How to choose a Lyapunov function candidate
 How to design the parameter estimation algorithm
 Convergence of the parameter estimation algorithm
(whether the estimated parameters converge to the true
values, how fast, …)
 Stability of the overall closed-loop system
2
Case (2)
Process model:
y (n) (t) = g[y (n -1) (t), …, y(t), u(t), …, u (m-1) (t)]
(2.1) The nonlinear model g [*] is known:
Nonlinear control theory (ELEC562): Lyapunov stability
theory, linearization around equilibrium point, differential
geometric methods, input-output linearization, etc..
(2.2) The nonlinear model g [*] is unknown:
 PID control
u(t )  ae(t )  b  e(t )dt  c
de(t )
,
dt
e( t )  y ( t )  y d ( t )
 Fuzzy control
Rule-based controller, using knowledge from human
experts.
 Nonlinear adaptive control
o Neural network based control
o Adaptive fuzzy control (ELEC541)
 Indirect adaptive fuzzy control:
Develop a fuzzy system model with free
parameters for the unknown g [*] first, and
then design a controller based on the fuzzy
system model. The free parameters of the
fuzzy system model are estimated on-line
based on an adaptation law.
 Direct adaptive fuzzy control:
Use a fuzzy system with free parameters
directly as the controller, and then develop
an algorithm to estimate the parameters of
the controller.
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Indirect adaptive fuzzy control
input u
Process
(nonlinear and unknown)
output y
Controller
(based on fuzzy model of process)
fuzzy system model of the process
(with free parameters estimated by
an adaptation law)
Key issues:
 What is the structure of the fuzzy system model
 What are the properties of the fuzzy system model
 Why use the fuzzy system models
 How to construct the fuzzy system model
 How to design the controller based on the fuzzy model
 What is the performance of the closed-loop control system
(stability, convergence, robustness, optimality, etc..)
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Direct adaptive fuzzy control
input u
Process
(nonlinear and unknown)
output y
Fuzzy controller
(fuzzy system with free parameters)
Adaptation law
(provide fuzzy controller parameters)
Key issues:
 How to design the adaptation law
 How to guarantee the stability of the closed-loop control
system
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Case (3)
Process model:
y(t  1) = a 1y(t) + …+ a n y(t - n  1) + b1u(t) + …+ bm u(t - m  1)
(3.1) The model parameters a1 ,..., an , b1 ,..., bm are known:
Digital control theory (ELEC377): z-transform models,
sampling continuous-time systems, stability test, pole-zero
analysis, pole placement controller design, etc..
(3.2) The model parameters a1 ,..., an , b1 ,..., bm are unknown:
Linear adaptive control theory (self-tuning controllers):
design a controller as if the parameters a1 ,..., an , b1 ,..., bm are
known, then develop an algorithm to estimate the parameters
a1 ,..., an , b1 ,..., bm .
input u
Process
(linear with unknown parameters)
output y
Controller
Parameter estimation
Key issues:
 Parameter estimation algorithms (least squares, gradient
descent, projection based, …)
 Convergence of the parameter estimation algorithms
(whether the estimated parameters converge to the true
values, how fast, …)
 Controller design (pole placement, optimal control, …)
 Stability of the overall closed-loop system
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Case (4)
Process model:
y(t + 1) = g[y(t), …, y(t - n + 1), u(t), …, u(t - m + 1)]
(4.1) The nonlinear model g [*] is known:
Nonlinear discrete-time control theory: Extensions of the
concepts and theory for continuous-time systems
(4.2) The nonlinear model g [*] is unknown:
 PID control
u(t )  ae(t )  b e(t )  ce(t ),
e(t )  y(t )  yd (t )
 Fuzzy control
Same as the continuous-time case
 Nonlinear adaptive control
o Neural network based control
o Adaptive fuzzy control (ELEC541)
 Indirect adaptive fuzzy control
 Direct adaptive fuzzy control
Key issues: Similar to the continuous-time case
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2. The gaps between control theory and control
engineering practice
The evolution of control theory
 1940-1950s: the early days, classical control theory, more
heuristic based approach.
 1960-1970s: modern control theory, more mathematical
approach.
 1980-1990s: in-depth study versus playing mathematics.
The control engineering practice
 PID is still the most commonly-used controller
Why:
o No need to know the model of the process under control
o Good-enough performance
o Easy to tune
 Systems are getting more and more complex with the
extensive use of computers, networks, smart sensors and
other new technologies, to:
o Increase productivity
o Reduce cost
 Control is just a small part of the whole system
The gaps
 Unrealistic assumptions: linear, known models, etc., which
make the theory unusable for practical systems.
 The real hard problems (nonlinear) are still unsolved.
Nonlinear adaptive control: targetting the real hard problem with
weakest assumptions.
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3. Key issues in nonlinear adaptive control
<1> How to find a good nonlinear structure f [*] to model the
unknown nonlinear function g[*] in
y (n) (t) = g[y (n -1) (t), …, y(t), u(t), …, u (m-1) (t)]
or
y(t + 1) = g[y(t), …, y(t - n + 1), u(t), …, u(t - m + 1)]
The requirements:
 f [*] should be general enough to approximate a wide variety
of nonlinear functions, that is, Can the f [*] achieves
sup x | f [ x ]  g[ x ] | 
for any small  ?
 f [*] should be efficient to represent general nonlinear
functions.
 The structure and parameters of f[*] should have clear
meanings so that humans can easily understand it.
The main choices:
 Polynomials
f [x] = constant term + first-order terms + secondorder terms + …
Advantages:
o Direct extension of linear functions
o Mathematically well-understood
Disadvantages:
o Efficiency is low, too many terms are needed to
approximate nonlinear functions
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 Neural networks

( 2) 
(1)
f [ x ]   wi    wij x j 
i
 j

where w are weights and  is a  -function
Network representation of a neural network
f [x ]
x1
x2
…
xp
Advantages:
o Support from human brain research
o Good mathematical properties
Disadvantages:
o Black-box, difficult to understand what is going on
inside
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 Fuzzy system
Advantages:
o Rule-based systems, easy to understand for humans
o Good mathematical properties
o Capable of incorporating human knowledge
o A variety of structures to choose
Disadvantages:
o Mathematically not well-studied
<2> How to determine the structures of the choosing nonlinear
models and develop adaptation laws to identify the parameters
<3> How to develop controllers based on the identified nonlinear
models
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4. Different types of fuzzy systems
<1> Standard fuzzy systems
fuzzy rule base
x
f[x]
fuzzifier
defuzzifier
fuzzy inference engine
fuzzy set
fuzzy set
<2> Takagi-Sugeno fuzzy systems
f[x] =

(THEN-part of rule i) * (strength of rule i)
i
strength
IF-part of rule 1
THEN-part of rule 1
strength
IF-part of rule 2
THEN-part of rule 2
x
f[x]
+
…
strength
IF-part of rule M
THEN-part of rule M
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<3> Hierarchical fuzzy systems
f[x]
fuzzy sub-system
fuzzy sub-system
fuzzy sub-systems
fuzzy sub-system
x1
xp
x2
<4> Fuzzy adaptive rule systems
<5> Projection pursuit fuzzy systems
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