Institute of Intelligent Power Electronics Publications

Espoo, May 1999

Publication 2

SOFT COMPUTING METHODS FOR CONTROL

AND INSTRUMENTATION

Thesis for the degree of Doctor of Science in Technology

Xiao-Zhi Gao

Institute of Intelligent Power Electronics Publications

Espoo, May 1999

Publication 2

SOFT COMPUTING METHODS FOR CONTROL

AND INSTRUMENTATION

Xiao-Zhi Gao

Dissertation for the degree of Doctor of Science in Technology to be presented with

due permission for public examination and debate in Auditorium S4 at Helsinki

University of Technology (Espoo, Finland) on the 21st of June, 1999, at 12 o’clock

noon.

Helsinki University of Technology

Department of Electrical and Communications Engineering

Institute of Intelligent Power Electronics

Distribution:

Helsinki University of Technology

Institute of Intelligent Power Electronics

P. O. Box 3000

FIN-02015 HUT

Tel. +358-9-451 2434

Fax. +358-9-460 224

Xiao-Zhi Gao and Helsinki University of Technology

ISBN 951-22-4529-9

ISSN 1456-0445

Libella Painopalvelu Oy

Espoo 1999

i

Abstract

The development of soft computing methods has attracted considerable research interest over the

past decade. They are applied to important fields such as control, signal processing, and system

modeling. Although soft computing methods have shown great potential in these areas, they share

some common shortcomings that hinder them from being used more widely. For example, neural

networks, a component of soft computing, often suffer from a slow learning rate. This drawback

renders neural networks less than suitable for time critical applications. Therefore, the objective of

this thesis is to explore and investigate the soft computing theory so that new and enhanced methods can be put forward. The applications of soft computing in control and instrumentation are also

studied to solve demanding real-world problems.

In this work, the existing soft computing techniques have been enhanced, and applied to control

and instrumentation areas. First, new soft computing methods are proposed. A Modified Elman

Neural Network (MENN) is introduced to provide fast convergence speed. Based on Muller’s

method, we propose a new reinforcement learning method, which can converge faster than the

original algorithm. As a fusion of fuzzy logic and neural networks, a new fuzzy filter using the selforganizing map to fine tune the membership functions is studied. The new soft computing schemes

presented in this thesis improve the performance of those earlier methods.

Second, we study the MENN-based identification and control problems. A dynamical system

identification scheme as well as a trajectory tracking configuration using the MENNs are discussed,

respectively. Our MENN-based identification structure belongs to the ‘black box’ identification

catalogue. It has the advantageous feature of not knowing the exact order of the system. The inverted pendulum is utilized here as a testbed for the MENN-based trajectory control scheme. It is

shown that neural networks are very efficient in dealing with nonlinear system identification and

control. In addition, they need little prior information of the plant to be identified or controlled.

However, the existence of local minima, under-fitting, and over-fitting may reduce the identification and control accuracy.

Third, the applications of soft computing methods in velocity and acceleration acquisition in

motion control systems are discussed. The aforementioned fuzzy filter is applied to filter out the

velocity noise in the feedback loop without introducing any harmful delay. This could lead to a

better servo control performance. Moreover, we construct a neural network-based acceleration acquisition scheme to obtain clean and delayless acceleration signals. Our method has the advantage

ii

of implicit adaptation. It can be used for any slowly altering velocity signal, which overcomes the

drawback of polynomial predictor-based approaches.

Finally, the power prediction and regulation in mobile communications systems are studied with

these soft computing methods. An optimal neural predictor is selected by applying the Predictive

Minimum Description Length (PMDL) principle. A Temporal Difference (TD)-based multi-step

ahead prediction scheme is also considered for the fading signals. Simulations demonstrate that the

neural predictors offer better results than conventional filters. Their weakness is the accompanying

high computational complexity. We introduce an MENN-based power controller at the base station,

which takes advantage of the inverse radio channel model. On the other hand, the efficiency of the

power controller employed is clearly limited by the single-bit command transmission mode. Meanwhile, an Embedded Fuzzy Unit (EFU) is proposed to provide versatile alternatives. The EFU can

effectively tackle the bottle-neck of incremental command transmission mode, and thus achieve

better power regulation. Additionally, a comparison between the conventional and soft computingbased power control methods is made. This research gives us useful guidance in determining appropriate power regulation configurations in terms of effectiveness and complexity.

In conclusion, the theory and applications of soft computing methods are studied in this thesis.

Nevertheless, our research is not aiming at implementation of the proposed schemes. Therefore, the

validation is verified only by numerical simulations. All the simulations are based on simplified application models without consideration to practical details.

iii

Preface

It was a great pleasure for me to work on this thesis. I hope, too, readers will find it useful and comfortable to read.

First and foremost, I am truly indebted to my advisor, Professor Seppo J. Ovaska, who guided

me throughout the whole of my research work. Professor Ovaska has proposed countless suggestions during this thesis writing procedure as well as through my course study. I have learnt quite a

lot from his comments, which are always inspiring and fruitful. I also thank him for providing me

with the invaluable opportunity enabling me to come to Finland and study for my D.Sc. (Tech.) degree at the Helsinki University of Technology. Here, I can only extend my best wishes to him in his

future career. Thank you very much indeed, Seppo.

And, I want to thank all the personnel at the laboratory of Electric Drives and Power Electronics. Professor Jorma Kyyr is especially thanked for his warm-hearted help. Secretary Leena

V is nen deserves my gratitude for helping me cope with practical matters including all the difficulties that a student may encounter in a foreign country. Laboratory manager Ismo Vainiom ki and

laboratory technician Ilkka Hanhivaara are thanked for their efficient efforts in creating such a fresh

and pleasant research environment. I really enjoy working in this laboratory.

I am very grateful to my colleagues, Jarno Tanskanen, Sami V liviita, Vlad Grigore, and Adrian

Dumitrescu, with whom I had so many helpful discussions. Special thanks go to Jarno Tanskanen

and Sami V liviita for their cooperative work in our joint publications.

I am also very much obliged to the secretary of the Graduate School of Electronics, Telecommunications, and Automation (GETA), Marja Lepp harju, for her kind assistance in my graduate

study at GETA.

Joe O’Reilly is thanked for checking the language used in this thesis. All the errors possibly

remaining in the text have been introduced by me alone at the final stages of revision.

Last but not the least, I want to say a heart-felt ‘thank you’ to my parents and brother. They play

an important role in my life, study and work, both morally and financially.

The author wishes to express his deep thanks to the Center for International Mobility (CIMO)

and GETA for their financial support.

Otaniemi, May 1999

Xiao-Zhi Gao

iv

Table of Contents

1. Introduction……………………………………………………………………………………... 1

1.1 Definition of Soft Computing………………………………………………………………. 2

1.2 Intelligent Control…………………………………………………………………………... 6

1.3 Aim of This Dissertation…………………………………………………………………… 8

2. Introduction to Soft Computing Methods……………………………………………………….12

2.1 Neural Networks……………………………………………………………………………. 12

2.1.1 Back-Propagation Neural Network…………………………………………………...13

2.1.2 Elman Neural Network………………………………………………………………. 16

2.1.3 Self-organizing Map…………………………………………………………………. 19

2.2 Fuzzy Logic………………………………………………………………………………… 22

2.2.1 Basic Theory of Fuzzy Logic Systems……………………………………………… 22

2.2.2 Fuzzy Logic-based Control………………………………………………………….. 26

2.2.3 Fuzzy Neural Network……………………………………………………………… 30

2.3 Reinforcement Learning……………………………………………………………………. 33

2.3.1 Single-step-ahead Predictor-based Critic-Actor Algorithm………………………….. 35

2.3.2 Temporal Difference Method-based Prediction…………………………………….. 38

3. Related Research………………………………………………………………………………… 42

3.1 Predictive Filtering Methods and Their Applications……………………………………... 42

3.1.1 Predictive Filtering Methods………………………………………………………… 42

3.1.2 Power Prediction in Mobile Communications Systems……………………………... 46

3.1.3 Acceleration Acquisition in Motor Control Systems………………………………... 49

3.2 Neural Network-based Dynamical System Identification…………………………………. 53

3.2.1 Neural Network-based Forward Model Identification………………………………. 54

3.2.2 Neural Network-based Inverse Model Identification………………………………... 57

3.3 Neural Network-based Control Applications………………………………………………60

3.3.1 Neural Network-based Control Schemes……………………………………………. 60

3.3.2 Power Regulation in Mobile Communications Systems……………………………...66

3.3.3 Inverted Pendulum Control…………………………………………………………...71

4. Summary of Publications………………………………………………………………………... 78

4.1 Neural Network-based System Identification Techniques…………………………………..78

4.1.1 Publication [P1]……………………………………………………………………… 78

v

4.2 Neural Network-based Control Methods…………………………………………………... 83

4.2.1 Publication [P2]……………………………………………………………………….83

4.3 Soft Computing Methods in Instrumentation……………………………………………….87

4.3.1 Publication [P3]……………………………………………………………………….87

4.3.2 Publication [P4].……………………………………………………………………... 90

4.3.3 Publication [P5]..…………………………………………………………………….. 93

4.3.4 Conclusions of Publications [P3]—[P5]………………………………………………96

4.4 Power Prediction in Mobile Communications Systems……………………………………..96

4.4.1 Publication [P6]……………………………………………………………………….96

4.4.2 Publication [P7]…………………………………………………………………….. 101

4.4.3 Conclusions of Publications [P6]—[P7]…………………………………………….103

4.5 Power Control in Mobile Communications Systems……………………………………... 103

4.5.1 Publication [P8]…………………………………………………………………….. 103

4.5.2 Publication [P9]…………………………………………………………………….. 107

4.5.3 Publication [P10]…………………………………………………………………… 111

4.5.4 Conclusions of Publications [P8]—[P10]………………………………………….. 115

4.6 Contribution of the Author……………………………………………………………….. 116

5. Conclusions and Discussion…………………………………………………………………….119

5.1 Main Results……………………………………………………………………………….119

5.2 Scientific Importance of the Author’s Work……………………………………………… 120

5.3 Topics for Future Research………………………………………………………….….… 122

6. References………………………………………………………………………………………124

Appendix A: Publications [P1]—[P10]

Appendix B: Errata

vi

List of Publications

This thesis consists of an introduction and the following ten publications which are referred to by

[P1], [P2], …, [P10] in the text:

[P1]

X. Z. Gao, X. M. Gao, and S. J. Ovaska, “A modified Elman neural network model with

application to dynamical systems identification,” in Proceedings of the 1996 IEEE

International Conference on Systems, Man, and Cybernetics, Beijing, P. R. China, October

1996, pp. 1376-1381.

[P2]

X. Z. Gao, X. M. Gao, and S. J. Ovaska, “Trajectory control based on a modified Elman

neural network,” in Proceedings of the 1997 IEEE International Conference on Systems,

Man, and Cybernetics, Orlando, FL, October 1997, pp. 2505-2510.

[P3]

X. M. Gao, X. Z. Gao, and S. J. Ovaska, “Power command enhancement in mobile

communication systems using an embedded fuzzy unit,” in Proceedings of the 1997 IEEE

International Conference on Systems, Man, and Cybernetics, Orlando, FL, October 1997,

pp. 4364-4369.

[P4]

“Neural networks-based approach for

the acquisition of acceleration from noisy velocity signal,” in Proceedings of the 1998 IEEE

Instrumentation and Measurement Technology Conference, St. Paul, MN, May 1998, pp.

935-940.

[P5]

X. Z. Gao and S. J. Ovaska, “A new fuzzy filter with application in motion control

systems,” in Proceedings of the 1999 IEEE International Conference on Systems, Man, and

Cybernetics, Tokyo, Japan, October 1999, IN PRESS.

[P6]

X. Z. Gao, “A temporal difference method-based prediction scheme applied to fading power

signals,” in Proceedings of the 1998 IEEE International Joint Conference on Neural

Networks, Anchorage, AK, May 1998, pp. 1954-1959.

vii

[P7]

X. M. Gao, X. Z. Gao, J. M. A. Tanskanen, and S. J. Ovaska, “Power prediction in mobile

communication systems using an optimal neural-network structure,” IEEE Transactions on

Neural Networks, vol. 8, no. 6, pp. 1446-1455, November 1997.

[P8]

X. M. Gao, X. Z. Gao, J. M. A. Tanskanen, and S. J. Ovaska, “Power control for mobile

DS/CDMA systems using a modified Elman neural network controller,” in Proceedings of

the 47th IEEE Vehicular Technology Conference, Phoenix, AZ, May 1997, pp. 750-754.

[P9]

X. Z. Gao, X. M. Gao, and S. J. Ovaska, “Fast reinforcement learning algorithm for power

control in cellular communication systems,” in Proceedings of the 1997 IEEE International

Conference on Systems, Man, and Cybernetics, Orlando, FL, October 1997, pp. 3883-3888.

[P10] X. Z. Gao and S. J. Ovaska, “Comparison of conventional and soft computing-based control

methods in a power regulation application,” in Proceedings of the 1998 IEEE International

Conference on Systems, Man, and Cybernetics, San Diego, CA, October 1998, pp. 20752082.

viii

List of Abbreviations

ACE

Adaptive Critic Element

Adaline

Adaptive Linear Element

ADC

Analog to Digital Converter

AI

Artificial Intelligence

ANFIS

Adaptive Neural Fuzzy Inference System or

Adaptive Network-based Fuzzy Inference System

ASE

Associative Search Element

BER

Bit Error Rate

BP

Back-Propagation

CMAC

Cerebellar Model Articulation Controller

COA

Center Of Area

DC

Direct Current

DCL

Differential Competitive Learning

DS/CDMA

Direct Sequence Code Division Multiple Access

DSP

Digital Signal Processing

EFU

Embedded Fuzzy Unit

ENN

Elman Neural Network

FIR

Finite Impulse Response

H-N

Heinonen-Neuvo

IIR

Infinite Impulse Response

IMC

Internal Model Control

LN

Large Negative

LP

Large Positive

LS

Least Squares

LSN

Linear Smoothed Newton

MIMO

Multiple-Input and Multiple-Output

MDL

Minimum Description Length

MENN

Modified Elman Neural Network

MLP

Multilayer Perceptron

MN

Medium Negative

MP

Medium Positive

ix

MRAC

Model Reference Adaptive Control

MOM

Mean Of Maximum

NB

Negative Big

NM

Negative Medium

NS

Negative Small

PB

Positive Big

PI

Proportional-Integral

PID

Proportional-Integral-Derivative

PM

Positive Medium

PMDL

Predictive Minimum Description Length

PS

Positive Small

RLSN

Recursive Linear Smoothed Newton

RMS

Root Mean Square

SN

Small Negative

SNR

Signal to Noise Ratio

SOM

Self-Organizing Map

SP

Small Positive

SSE

Sum Squared Error

TD

Temporal Difference

TDL

Tapped Delay Line

ZE

Zero

x

List of Symbols

a

Node activation function in the neural network

a′

First order derivative of a

A

Af

Fuzzy set

Fuzzy predictate

A′f

Fuzzy predictate

Aji

Linguistic value

Bf

Fuzzy predictate

Bj

Linguistic value

B′f

Fuzzy predictate

c

Consequent parameters in the ANFIS

C

Fuzzy set or

Multiple controllers or

Oscillator coefficients

d

Vector of desired output of the neural network

di

Desired output of the neural network or

Euclidean distance in the self-organizing map

e

Feedback error in control systems or

Identification error

∆e

Feedback error change

E

Cost function of the neural network

e˙

First order derivative of the feedback error

e˙˙

Second order derivative of the feedback error

f(⋅)

Function relationship between reinforcement signal and control actions

F

Fuzzy set or

Force applied to the cart

g

Gravity acceleration

G

Syntactic rule

h(⋅)

Function of fuzzy rule base

H LN ( z )

Transfer function of the Newton predictor for polynomial degree L and the number

of prediction steps N

i

Index variable

I MENN

Input set of the MENN

xi

j

Index variable

J

Trajectory control performance index of the Elman neural network-based controller

or

Instant power regulation index

J*

Cumulative trajectory control index of the Elman neural network-based controller or

Cumulative control criterion for power regulation

k

Index variable or

Iteration step index in neural network training or

Scaling constant in reinforcement learning

K

Length of the Newton predictor

l

Number of nodes in the hidden layer in neural network or

lp

Index variable for predictor

Half length of the pole in the inverted pendulum

L

Polynomial degree assumed for the input signal

m

Number of nodes in the input layer in neural network or

Length of a sequence or

Index variable for predictor or

Order of the plant output

mc

Mass of the cart in the inverted pendulum

mp

Mass of the pole in the inverted pendulum

M

Carrier modular or

Order of the acceleration estimator or

Multiple identification models or

Semantic rule

M

Node function in the ANFIS

ms

Number of support values

n

Number of nodes in the output layer in neural network or

Order of the input of the plant

nf

Number of fuzzy inputs

nr

Number of fuzzy rules

N

Number of prediction steps ahead or

Order of the neuro predictive filter

NG

White noise gain of the Newton predictor

xii

n˜

Noise in the input signal of the predictor

Nc

Neighborhood in the self-organizing map

net q

Input of node q in the hidden layer

net _ ci

Input of context node i in the Elman neural network

net _ hi

Input of node i in the hidden layer in the Elman neural network

Oil

Output of node i in layer l in the ANFIS

OContext

Output set of the context nodes in the MENN

O MENN

Output set of the MENN

o _ ci

Output of context node i in the Elman neural network

o _ hi

Output of node i in the hidden layer in the Elman neural network

p

Index variable or

Number of inputs in the ANFIS or

Prediction output

P

Consequent parameters in the ANFIS or

Transmitting signal power

∆p

Discrete power step

P Ref

Desired power level for the received power at the base station

q

Index variable or

Number of membership functions for each input in the ANFIS

q

q

Number of quantization levels

r

Reinforcement signal

R

General variable form of the reinforcement learning signal

Ref

Reference signal for closed loop control systems

Ri

Fuzzy reasoning rule i

S

Oscillator coefficients

t

Discrete sampling point

T

Length of training sequence or

Term set

Ts

Sampling period

u

Fuzzy set element or

Control output or

Identification excitation input

U

Universe of discourse or

xiii

General variable form of control action in reinforcement learning

u nn

Identification output of the neural network-based inverse model

u∗

Optimal output for the action network to learn in reinforcement learning

u˜

Actual control applied to the plant in reinforcement learning or

Output of the fuzzy power command enhancement unit

u

Approximation of u ∗ by secant method

∆u

Output of the stochastic unit

∆u FLC

Incremental output of the fuzzy power controller

v

vqj

Fuzzy set element

Weight connecting node j in the input layer with node q in the hidden layer in the BP

neural network

v˜

Measured motion velocity

vˆ

v˜˙

Velocity measurement

vˆ˙

Acceleration estimate

w

W

General presentation of weights in neural networks

Both the weights vqj and wiq

wi

Weights in the self-organizing map

w new

General presentation of updated weights in neural networks

w old

wiq

General presentation of previous weights in neural networks

Weight connecting node q in the hidden layer with node i in the output layer in the

Noisy acceleration

BP neural network

∆w

∆wiq

Update change of w

Update change of wiq

w1i, j

Weight connecting node i in the input layer to node j in the hidden layer in the

w2 i , j

Elman neural network

Weight connecting node i in the hidden layer to node j in the output layer in the

w3i, j

Elman neural network

Weight connecting context node i to node j in the hidden layer in the Elman neural

w4 i, j

network

Weight connecting context node i to node j in the output layer in the Elman neural

W1

network

Weight set of w1i, j

xiv

W2

Weight set of w2 i, j

W3

Weight set of w3i, j

W4

Weight set of w4 i, j

x

Fading signal or

General variable or

Position of the cart in the inverted pendulum or

Sequence samples

x˙

Velocity of the cart in the inverted pendulum

x˙˙

Acceleration of the cart in the inverted pendulum

x

Input pattern for the neural network

X

Time series

xf

Linguistic input of the fuzzy rule base

x0

Crisp input

xi( k )

Input of node i at iteration k in the Elman neural network

xif

Linguistic input variables

y

Clean primary signal or

General variable or

Trajectory output of the plant

y

Output of the predictor

yˆ

Identification output of the Elman neural network or

Noisy input signal of the predictor

yf

Linguistic output of the fuzzy rule base

ym

Output of the reference model in MRAC

yn

Prediction output of the neural network model in IMC

y nn

Neural network identification output

yp

Output response of the plant

yd

Reference trajectory

yi

Output of node i in the output layer in the neural network

y

(k )

j

Output of node j in the output layer in the Elman neural network at iteration k

Yi ( k )

Desired output for node i in the Elman neural network at iteration k

z

zq

Final outcome of a sequence signal

Output of node q in the hidden layer

z −1

Unit delay

xv

z oj

Amount of output at quantization level j

z sj

Support value at which the membership function reaches the maximum value

*

zCOA

Crisp output of the COA defuzzification method

z*MOM

Crisp output of the MOM defuzzification method

θ

θ˙

Angle of the pole in the inverted pendulum

θ˙˙

Angular acceleration of the pole in the inverted pendulum

λ

Scaling constant in reinforcement learning or

Angular velocity of the pole in the inverted pendulum

Forgetting factor in TD learning

δ oi

General back-propagation error term for node i in the output layer

δ hi

General back-propagation error term for node i in the hidden layer

η

Learning rate of the neural network

φ

Gaussian density function or

Inverse function of ϕ

ϕ

Function relationship realized by the neuro predictor or

General function of a system

ϕ −1

Inverse function of ϕ

ϕˆ

Approximated mapping of the system realized by the neural network

ϕˆ −1

Approximated inverse mapping of the system realized by the neural network

ϑ

Mapping realized by the fuzzy command enhancement unit

ξ

Index variable

σ

Variance of the output of the stochastic unit

µA

Fuzzy membership function A

µc

Coefficient of friction of cart on track

µC

Fuzzy membership function C

µF

Fuzzy membership function F

µp

Coefficient of friction of pole on cart

1

1. Introduction

Soft computing, as pointed out by Dr. Zadeh, is not a single methodology [Zad94]. Instead, it is a

fusion of several methodologies, i.e., neural networks, fuzzy logic, and genetic algorithms. Different

from conventional (hard) computing, soft computing takes advantage of intuition, which implies that

the human mind-based intuitive and subjective thinking is realized in soft computing. The motivation of applying the human intuition is that a large number of real-world problems cannot be solved

by hard computing methods due to the fact that either they are too complex to handle or they cannot be described or catalogued by analytical and exact models. However, in some cases, human experts are marvellously successful in dealing with these problems, e.g., face recognition in a noisy

background. Dr. Zadeh emphasises that precise measurement and control approaches are not always

effective in coping with such difficult problems, but perception can often help [Zad98]. Therefore,

the goal of soft computing is to exploit the imprecision and uncertainty in human decision making

procedure, and achieve simple, reliable, and low-cost solutions.

Currently, the techniques in control and instrumentation fields are facing difficulties in meeting

the growing needs of modern industry. For example, numerous nonlinear and time-variant plants

cannot be efficiently stabilized or regulated using classical control methods. Additionally, as for the

acceleration acquisition in motion control systems, the existing signal processing techniques fail to

provide clean and delayless acceleration estimates because of the inherent noise in the measured velocity. For instance, conventional Finite Impulse Response (FIR) and Infinite Impulse Response

(IIR) filters always introduce some delay in the filtered output. It is advantageous to apply soft

computing methods in these cases, since soft computing offers us a totally new perspective by providing a set of techniques to solve practical problems in which ambiguity and uncertainty prevail

[Jan97].

Fuzzy logic, a branch of soft computing, has been an active partner in the process control for at

least the last two decades [Dri93]. Besides control engineering, fuzzy logic in instrumentation is

drawing great research interest [Rus96]. Neural networks are also investigated in control and instrumentation such as on-line tuning of controller parameters [Kaw98], friction compensation in servo

motion control systems [Gao99], and Analog to Digital Converter (ADC) resolution enhancement

[Gao97a]. The application of soft computing, in fact, covers a variety of application areas. Besides

control and instrumentation, other important aspects include speech recognition [Kom92], signal

processing [Cic93], telecommunications [Yuh94], power electronics systems [Dot98, pp. 143-185],

and system diagnosis [Cho92]. Recent advances of soft computing methods and their applications

2

in engineering design and manufacturing can be found in [Roy98]. With the rapid development of

hardware platforms, e.g., Digital Signal Processing (DSP) and neural networks chips, it is becoming

more feasible to apply soft computing methods into practice.

1.1 Definition of Soft Computing

Soft computing is a collection of methods to construct computationally intelligent systems. By ‘intelligent systems’, we here mean those systems that are capable of imitating the human reasoning

process as well as handling quantitative and qualitative knowledge. It is well known that the intelligent systems, which can provide human like expertise such as domain knowledge, uncertain reasoning, and adaptation to a noisy and time-varying environment, are important in tackling practical

computing problems. Specifically, in the modern control system design and analysis, there is a

promising trend going on to employ some heuristic methods that can benefit from human experts,

because the currently existing complex plants cannot be accurately described by rigorous mathematical models, and are, therefore, difficult to control using conventional model-based methods.

Meanwhile, in practice, experienced operators are often able to obtain fairly satisfactory control

quality. Soft computing is an appropriate candidate for creating these knowledge-based intelligent

systems. It has attracted the growing interest of researchers from various scientific and engineering

communities during recent years [Lin96].

Basically, soft computing is considered as an emerging approach to computing which parallels

the remarkable ability of the human mind to reason and learn in a circumstance of uncertainty and

imprecision. The pioneer of fuzzy logic, Dr. Zadeh, has pointed out that ‘the guiding principle of

soft computing is to exploit the tolerance for imprecision, uncertainty, and partial truth to achieve

tractability, robustness, low solution cost, better rapport with reality’ [Zad92]. In contrast with

hard computing methods, which only deal with precision, certainty, and rigor, soft computing is effective in acquiring imprecise or sub-optimal but economical and competitive solutions. In short,

because of its unique feature in coping with real-world problems, e.g., intelligent control, decision

making support, nonlinear programming and optimization, soft computing has drawn increasing research attention from people of different backgrounds [Jan97].

In general, soft computing methods consist of three essential paradigms: neural networks

[Hay98], fuzzy logic [Wan97], and genetic algorithms (evolutionary programming) [Gol89]. Nevertheless, soft computing is an open instead of conservative concept. That is, it is evolving those relevant techniques together with the important advances in other new computing methods such as arti-

3

ficial immune systems [Das98]. To condense the following presentation, we concentrate only on the

above three principal methods. In the triumvirate of soft computing, neural networks are concerned

with adaptive learning, nonlinear function approximation, and universal generalization; fuzzy logic

with imprecision and approximate reasoning; and genetic algorithms with uncertainty and propagation of belief. Table 1.1 lists these three methodologies together with their advantages.

Table 1.1. Soft computing constituents.

Methodology

Advantage

Neural Networks

Learning and Approximation

Fuzzy Logic

Approximate Reasoning

Genetic Algorithms

Systematic Random Search

However, in soft computing, they are complementary rather than competitive. More precisely, it

is advantageous to employ neural networks, fuzzy logic, and genetic algorithms in combination instead of exclusively. A typical example to support this argument is the popular fuzzy neural network model, which takes advantage of the capabilities of both fuzzy logic and neural networks

[Buc94]. The fuzzy neural network is constructed to merge fuzzy inference mechanism and neural

networks into an integrated system so that their individual weaknesses are overcome. Normally, the

neuro-fuzzy technique can have the same topology with the feedforward neural network, i.e., nodes

and layers. On the other hand, the node functions inside are replaced with fuzzy membership functions such as Gaussian functions. And, the commonly used back-propagation learning algorithm is

applied to adjust the parameters of these fuzzy membership functions. In this way, the fuzzy neural network retains the approximate inference characteristics and imprecise information processing

capability of fuzzy logic, meanwhile it also has the strength of adaptation and generalization by introducing the learning algorithm from neural networks. It has found many application prospects,

e.g., in image processing [Gho93] and speech recognition [Qi93]. A growing number of consumer

products, for example, washing machines and air conditioners, are released with the embedded

neuro-fuzzy technique. The same principal idea applies to other kinds of hybrid schemes among

neural networks, fuzzy logic, and genetic algorithms, which are in fact already available [Tak97].

Refer to Figure 1.1 for other alternatives.

4

Fuzzy Logic

Neuro-Fuzzy

Genetic-Fuzzy

Neural Network

Genetic Algorithm

Genetic-Neuro

Figure 1.1. Framework of soft computing.

We only present two representative examples, genetic-neuro and genetic-fuzzy techniques here.

In general, a genetic algorithm is a derivative-free and stochastic optimization method [Man96]. Its

orientation comes from ideas borrowed from natural selection as well as the evolutionary process.

As a general purpose solution to demanding problems, it has the distinguishing feature of parallel

search and global optimization. In addition, genetic algorithm needs less prior information about the

problems to be solved than the conventional optimization schemes, such as the steepest descent

method, which often require the derivative of the objective functions [Jan97, pp. 129-168]. Thus, it

is attractive to employ a genetic algorithm to optimize the parameters and structures of neural networks and fuzzy logic systems instead of using the back-propagation learning algorithm alone. For

instance, to optimize the weights of the neural network for a specified problem, we can first encode

each weight of the neural network into a binary bit string. Next, we concatenate these sub-strings

into a complete string, which is usually called a chromosome. A fixed number of the chromosome

candidates forms one generation. The value of the cost function to be optimized of every chromosome is then calculated and assigned to the chromosome as its ‘fitness’. The cost function is always

problem dependent. Based on individual fitness value, genetic algorithm uses the operators such as

reproduction, crossover and mutation to get the next generation that may contain chromosomes

providing better fitnesses. This evolutionary process evolves from generation to generation until a

preset cost function criterion has been reached. Due to the unique advantages of genetic algorithms,

i.e., derivative free, stochastic, and global search [Tan96a], this kind of genetic-neuro technique is

5

suitable for some special situations, where no derivative or other auxiliary knowledge of the cost

function is available for the neural network back-propagation learning algorithm. Additionally, since

the genetic algorithms are all based on probabilistic rather than deterministic search, the geneticneuro technique overcomes the severe shortcoming of applying pure back-propagation learning algorithm to train the neural networks. The steepest descent-based method is easily trapped into the

local minima in the nonlinear search space of weights.

For the genetic-fuzzy scheme, the genetic algorithms are applied to get optimal fuzzy inference

parameters [Far98]. We point out that not only the parameters of the fuzzy inference structure,

such as the antecedent and consequent coefficients, can be optimized by the genetic algorithms, but

also the tailoring of the fuzzy reasoning rules, e.g., adding new rules and deleting invalid ones, is appropriate for the utilization of the genetic algorithms. In a word, genetic algorithms play an important role in the parameter and structure learning in both the genetic-neuro and genetic-fuzzy techniques. However, it is to be emphasized that the combination of soft computing methods is not

only limited on the algorithm level as discussed previously. There are also potential fusion alternatives on the system level. For example, in a hierarchical system, a fuzzy logic controller and a neural

network predictor can act independently and co-operatively with different tasks attributed by the

top level process supervisor. We will demonstrate the effectiveness of such a hybrid scheme by the

illustrative example of power regulation in a mobile communications system in the following chapters.

Among the attempts to mimic the human brain intelligence, soft computing is not the first trial.

In fact, soft computing has some similarities with the conventional expert systems in Artificial Intelligence (AI) [Rus95]. Their common goal is to explore and realize machine intelligence. Basically,

expert systems target to imitate intelligence in the form of language expressions or symbolic rules.

On the other hand, there are no symbolic manipulations in soft computing. This feature makes it

free from the handicap inherent in the expert systems-oriented classical symbolicism, which can result in the ‘dimension curse’. Instead, the knowledge acquisition procedure in soft computing is

based on learning from practical data samples as well as operator experience. Neural networks are

often trained by the application data from measurement or provided by the supervisor. The antecedent and consequent of the mostly used IF-THEN rules in the fuzzy inference are interpreted by

the linguistic variables arising from the knowledge of the operation experts. Random but global

search using genetic algorithms is preferred in obtaining the optimal solution. Therefore, soft computing can also be viewed as a kind of data driven intelligent technique.

6

1.2 Intelligent Control

Over the past four decades, we have observed numerous successful applications and implementations of conventional control, ranging from autonomous robotics to chemical process and air traffic

control

The classic control approaches are based on analytical methods derived from the

physical models, such as Laplace transformation and linear state space. Bode diagrams, Nyquist and

Lyapunov stability criteria, and root-locus methods are still widely utilized in practical control engineering. However, control of a wider class of modern complex systems, e.g., time-varying, nonlinear, multi-variable, and multi-loop industrial systems, is difficult with these conventional techniques. An interesting example is the famous control benchmark truck backer-upper problem. To

back a trailer truck when applying the classical control scheme, we must first write out the fulldefined mathematical model of the truck including the dynamical kinematics between the displacement of the truck with the angels of the truck and the trailer. The finalized model is instinctually

nonlinear. Once this step is finished, we have to solve the equations with some numerical methods,

because normally there are no straightforward analytical methods for nonlinear systems. Nevertheless, even using the numerical methods, the acquisition of the approximated solutions is still unpractically time consuming. The underlying reason is due to the existing high dimensional input variables

and strong nonlinearity of the model. On the other hand, for the skilled truck drivers, this work is

reasonably easy. They just look at the distance between the truck and the dock, as well as the angle

of the truck and trailer, and based on this vague information together with a few reasoning rules,

they make their judgements to turn the truck a little bit in the right direction, and repeat the above

process until the truck is backed into the dock. Thus, it is clear that intelligent, for example, soft

computing-based, methods are advantageous for such kind of problems. Nguyen has applied a twolayer feedforward network to control the above simulated truck [Ngu90]. Kong proposes a fuzzy

system approach based on the Differential Competitive Learning (DCL) to get an even improved

result [Kon92]. In conclusion, the demand for higher performance criteria and economical solutions

necessities the invention of intelligent control systems.

There is no formal or single definition of an intelligent control system. Generally, an intelligent

system should satisfy the famous Turing test, which can be concisely expressed as follows: if a man

and a machine (or a program) perform the same task, then if one cannot distinguish between the machine and the human by examining only the nature of their performances, the machine is said to be

intelligent, otherwise not [Tur50]. However, intelligent control systems can be broadly described as

the use of artificial intelligence-related methods to design and implement automatic control systems.

7

Dr. Fu first introduced the term ‘intelligent control’, and initiated studies in this area as early as the

1950s [Fu71]. At the initial stage of the development of intelligent control, there were tight ties between artificial intelligence and automatic control. The advances in computer science, especially operations research, also contributed to intelligent control during the years that followed. The consequence is the study of cybernetics, robotics, and learning machines such as Michie’s broom balancer

‘BOXES’ [Mic68]. Figure 1.2 gives an illustrative diagram of the relationship between the intelligent

control and these four aforementioned fields.

dynamics

feedback

optimization

Automatic Control

management

Operations Research

planning

coordination

communications

Intelligent Control

object oriented

design

Computer Science

heuristics

memory

learning

distributed processing

Artificial Intelligence

Figure 1.2. Techniques employed in intelligent control [Bro94, pp. 5].

Today, several AI methods have been well established. Expert systems are applied in many application areas including dynamic decision making and medical diagnosis [Rus95]. Soft computing is

the most prominent AI technique in the control field. Modern control system design and implementation face three main difficulties [Nar91]: first, the heavy computational complexity, which the

new control methods require. The second is the presence of nonlinear systems with a large number

Espoo, May 1999

Publication 2

SOFT COMPUTING METHODS FOR CONTROL

AND INSTRUMENTATION

Thesis for the degree of Doctor of Science in Technology

Xiao-Zhi Gao

Institute of Intelligent Power Electronics Publications

Espoo, May 1999

Publication 2

SOFT COMPUTING METHODS FOR CONTROL

AND INSTRUMENTATION

Xiao-Zhi Gao

Dissertation for the degree of Doctor of Science in Technology to be presented with

due permission for public examination and debate in Auditorium S4 at Helsinki

University of Technology (Espoo, Finland) on the 21st of June, 1999, at 12 o’clock

noon.

Helsinki University of Technology

Department of Electrical and Communications Engineering

Institute of Intelligent Power Electronics

Distribution:

Helsinki University of Technology

Institute of Intelligent Power Electronics

P. O. Box 3000

FIN-02015 HUT

Tel. +358-9-451 2434

Fax. +358-9-460 224

Xiao-Zhi Gao and Helsinki University of Technology

ISBN 951-22-4529-9

ISSN 1456-0445

Libella Painopalvelu Oy

Espoo 1999

i

Abstract

The development of soft computing methods has attracted considerable research interest over the

past decade. They are applied to important fields such as control, signal processing, and system

modeling. Although soft computing methods have shown great potential in these areas, they share

some common shortcomings that hinder them from being used more widely. For example, neural

networks, a component of soft computing, often suffer from a slow learning rate. This drawback

renders neural networks less than suitable for time critical applications. Therefore, the objective of

this thesis is to explore and investigate the soft computing theory so that new and enhanced methods can be put forward. The applications of soft computing in control and instrumentation are also

studied to solve demanding real-world problems.

In this work, the existing soft computing techniques have been enhanced, and applied to control

and instrumentation areas. First, new soft computing methods are proposed. A Modified Elman

Neural Network (MENN) is introduced to provide fast convergence speed. Based on Muller’s

method, we propose a new reinforcement learning method, which can converge faster than the

original algorithm. As a fusion of fuzzy logic and neural networks, a new fuzzy filter using the selforganizing map to fine tune the membership functions is studied. The new soft computing schemes

presented in this thesis improve the performance of those earlier methods.

Second, we study the MENN-based identification and control problems. A dynamical system

identification scheme as well as a trajectory tracking configuration using the MENNs are discussed,

respectively. Our MENN-based identification structure belongs to the ‘black box’ identification

catalogue. It has the advantageous feature of not knowing the exact order of the system. The inverted pendulum is utilized here as a testbed for the MENN-based trajectory control scheme. It is

shown that neural networks are very efficient in dealing with nonlinear system identification and

control. In addition, they need little prior information of the plant to be identified or controlled.

However, the existence of local minima, under-fitting, and over-fitting may reduce the identification and control accuracy.

Third, the applications of soft computing methods in velocity and acceleration acquisition in

motion control systems are discussed. The aforementioned fuzzy filter is applied to filter out the

velocity noise in the feedback loop without introducing any harmful delay. This could lead to a

better servo control performance. Moreover, we construct a neural network-based acceleration acquisition scheme to obtain clean and delayless acceleration signals. Our method has the advantage

ii

of implicit adaptation. It can be used for any slowly altering velocity signal, which overcomes the

drawback of polynomial predictor-based approaches.

Finally, the power prediction and regulation in mobile communications systems are studied with

these soft computing methods. An optimal neural predictor is selected by applying the Predictive

Minimum Description Length (PMDL) principle. A Temporal Difference (TD)-based multi-step

ahead prediction scheme is also considered for the fading signals. Simulations demonstrate that the

neural predictors offer better results than conventional filters. Their weakness is the accompanying

high computational complexity. We introduce an MENN-based power controller at the base station,

which takes advantage of the inverse radio channel model. On the other hand, the efficiency of the

power controller employed is clearly limited by the single-bit command transmission mode. Meanwhile, an Embedded Fuzzy Unit (EFU) is proposed to provide versatile alternatives. The EFU can

effectively tackle the bottle-neck of incremental command transmission mode, and thus achieve

better power regulation. Additionally, a comparison between the conventional and soft computingbased power control methods is made. This research gives us useful guidance in determining appropriate power regulation configurations in terms of effectiveness and complexity.

In conclusion, the theory and applications of soft computing methods are studied in this thesis.

Nevertheless, our research is not aiming at implementation of the proposed schemes. Therefore, the

validation is verified only by numerical simulations. All the simulations are based on simplified application models without consideration to practical details.

iii

Preface

It was a great pleasure for me to work on this thesis. I hope, too, readers will find it useful and comfortable to read.

First and foremost, I am truly indebted to my advisor, Professor Seppo J. Ovaska, who guided

me throughout the whole of my research work. Professor Ovaska has proposed countless suggestions during this thesis writing procedure as well as through my course study. I have learnt quite a

lot from his comments, which are always inspiring and fruitful. I also thank him for providing me

with the invaluable opportunity enabling me to come to Finland and study for my D.Sc. (Tech.) degree at the Helsinki University of Technology. Here, I can only extend my best wishes to him in his

future career. Thank you very much indeed, Seppo.

And, I want to thank all the personnel at the laboratory of Electric Drives and Power Electronics. Professor Jorma Kyyr is especially thanked for his warm-hearted help. Secretary Leena

V is nen deserves my gratitude for helping me cope with practical matters including all the difficulties that a student may encounter in a foreign country. Laboratory manager Ismo Vainiom ki and

laboratory technician Ilkka Hanhivaara are thanked for their efficient efforts in creating such a fresh

and pleasant research environment. I really enjoy working in this laboratory.

I am very grateful to my colleagues, Jarno Tanskanen, Sami V liviita, Vlad Grigore, and Adrian

Dumitrescu, with whom I had so many helpful discussions. Special thanks go to Jarno Tanskanen

and Sami V liviita for their cooperative work in our joint publications.

I am also very much obliged to the secretary of the Graduate School of Electronics, Telecommunications, and Automation (GETA), Marja Lepp harju, for her kind assistance in my graduate

study at GETA.

Joe O’Reilly is thanked for checking the language used in this thesis. All the errors possibly

remaining in the text have been introduced by me alone at the final stages of revision.

Last but not the least, I want to say a heart-felt ‘thank you’ to my parents and brother. They play

an important role in my life, study and work, both morally and financially.

The author wishes to express his deep thanks to the Center for International Mobility (CIMO)

and GETA for their financial support.

Otaniemi, May 1999

Xiao-Zhi Gao

iv

Table of Contents

1. Introduction……………………………………………………………………………………... 1

1.1 Definition of Soft Computing………………………………………………………………. 2

1.2 Intelligent Control…………………………………………………………………………... 6

1.3 Aim of This Dissertation…………………………………………………………………… 8

2. Introduction to Soft Computing Methods……………………………………………………….12

2.1 Neural Networks……………………………………………………………………………. 12

2.1.1 Back-Propagation Neural Network…………………………………………………...13

2.1.2 Elman Neural Network………………………………………………………………. 16

2.1.3 Self-organizing Map…………………………………………………………………. 19

2.2 Fuzzy Logic………………………………………………………………………………… 22

2.2.1 Basic Theory of Fuzzy Logic Systems……………………………………………… 22

2.2.2 Fuzzy Logic-based Control………………………………………………………….. 26

2.2.3 Fuzzy Neural Network……………………………………………………………… 30

2.3 Reinforcement Learning……………………………………………………………………. 33

2.3.1 Single-step-ahead Predictor-based Critic-Actor Algorithm………………………….. 35

2.3.2 Temporal Difference Method-based Prediction…………………………………….. 38

3. Related Research………………………………………………………………………………… 42

3.1 Predictive Filtering Methods and Their Applications……………………………………... 42

3.1.1 Predictive Filtering Methods………………………………………………………… 42

3.1.2 Power Prediction in Mobile Communications Systems……………………………... 46

3.1.3 Acceleration Acquisition in Motor Control Systems………………………………... 49

3.2 Neural Network-based Dynamical System Identification…………………………………. 53

3.2.1 Neural Network-based Forward Model Identification………………………………. 54

3.2.2 Neural Network-based Inverse Model Identification………………………………... 57

3.3 Neural Network-based Control Applications………………………………………………60

3.3.1 Neural Network-based Control Schemes……………………………………………. 60

3.3.2 Power Regulation in Mobile Communications Systems……………………………...66

3.3.3 Inverted Pendulum Control…………………………………………………………...71

4. Summary of Publications………………………………………………………………………... 78

4.1 Neural Network-based System Identification Techniques…………………………………..78

4.1.1 Publication [P1]……………………………………………………………………… 78

v

4.2 Neural Network-based Control Methods…………………………………………………... 83

4.2.1 Publication [P2]……………………………………………………………………….83

4.3 Soft Computing Methods in Instrumentation……………………………………………….87

4.3.1 Publication [P3]……………………………………………………………………….87

4.3.2 Publication [P4].……………………………………………………………………... 90

4.3.3 Publication [P5]..…………………………………………………………………….. 93

4.3.4 Conclusions of Publications [P3]—[P5]………………………………………………96

4.4 Power Prediction in Mobile Communications Systems……………………………………..96

4.4.1 Publication [P6]……………………………………………………………………….96

4.4.2 Publication [P7]…………………………………………………………………….. 101

4.4.3 Conclusions of Publications [P6]—[P7]…………………………………………….103

4.5 Power Control in Mobile Communications Systems……………………………………... 103

4.5.1 Publication [P8]…………………………………………………………………….. 103

4.5.2 Publication [P9]…………………………………………………………………….. 107

4.5.3 Publication [P10]…………………………………………………………………… 111

4.5.4 Conclusions of Publications [P8]—[P10]………………………………………….. 115

4.6 Contribution of the Author……………………………………………………………….. 116

5. Conclusions and Discussion…………………………………………………………………….119

5.1 Main Results……………………………………………………………………………….119

5.2 Scientific Importance of the Author’s Work……………………………………………… 120

5.3 Topics for Future Research………………………………………………………….….… 122

6. References………………………………………………………………………………………124

Appendix A: Publications [P1]—[P10]

Appendix B: Errata

vi

List of Publications

This thesis consists of an introduction and the following ten publications which are referred to by

[P1], [P2], …, [P10] in the text:

[P1]

X. Z. Gao, X. M. Gao, and S. J. Ovaska, “A modified Elman neural network model with

application to dynamical systems identification,” in Proceedings of the 1996 IEEE

International Conference on Systems, Man, and Cybernetics, Beijing, P. R. China, October

1996, pp. 1376-1381.

[P2]

X. Z. Gao, X. M. Gao, and S. J. Ovaska, “Trajectory control based on a modified Elman

neural network,” in Proceedings of the 1997 IEEE International Conference on Systems,

Man, and Cybernetics, Orlando, FL, October 1997, pp. 2505-2510.

[P3]

X. M. Gao, X. Z. Gao, and S. J. Ovaska, “Power command enhancement in mobile

communication systems using an embedded fuzzy unit,” in Proceedings of the 1997 IEEE

International Conference on Systems, Man, and Cybernetics, Orlando, FL, October 1997,

pp. 4364-4369.

[P4]

“Neural networks-based approach for

the acquisition of acceleration from noisy velocity signal,” in Proceedings of the 1998 IEEE

Instrumentation and Measurement Technology Conference, St. Paul, MN, May 1998, pp.

935-940.

[P5]

X. Z. Gao and S. J. Ovaska, “A new fuzzy filter with application in motion control

systems,” in Proceedings of the 1999 IEEE International Conference on Systems, Man, and

Cybernetics, Tokyo, Japan, October 1999, IN PRESS.

[P6]

X. Z. Gao, “A temporal difference method-based prediction scheme applied to fading power

signals,” in Proceedings of the 1998 IEEE International Joint Conference on Neural

Networks, Anchorage, AK, May 1998, pp. 1954-1959.

vii

[P7]

X. M. Gao, X. Z. Gao, J. M. A. Tanskanen, and S. J. Ovaska, “Power prediction in mobile

communication systems using an optimal neural-network structure,” IEEE Transactions on

Neural Networks, vol. 8, no. 6, pp. 1446-1455, November 1997.

[P8]

X. M. Gao, X. Z. Gao, J. M. A. Tanskanen, and S. J. Ovaska, “Power control for mobile

DS/CDMA systems using a modified Elman neural network controller,” in Proceedings of

the 47th IEEE Vehicular Technology Conference, Phoenix, AZ, May 1997, pp. 750-754.

[P9]

X. Z. Gao, X. M. Gao, and S. J. Ovaska, “Fast reinforcement learning algorithm for power

control in cellular communication systems,” in Proceedings of the 1997 IEEE International

Conference on Systems, Man, and Cybernetics, Orlando, FL, October 1997, pp. 3883-3888.

[P10] X. Z. Gao and S. J. Ovaska, “Comparison of conventional and soft computing-based control

methods in a power regulation application,” in Proceedings of the 1998 IEEE International

Conference on Systems, Man, and Cybernetics, San Diego, CA, October 1998, pp. 20752082.

viii

List of Abbreviations

ACE

Adaptive Critic Element

Adaline

Adaptive Linear Element

ADC

Analog to Digital Converter

AI

Artificial Intelligence

ANFIS

Adaptive Neural Fuzzy Inference System or

Adaptive Network-based Fuzzy Inference System

ASE

Associative Search Element

BER

Bit Error Rate

BP

Back-Propagation

CMAC

Cerebellar Model Articulation Controller

COA

Center Of Area

DC

Direct Current

DCL

Differential Competitive Learning

DS/CDMA

Direct Sequence Code Division Multiple Access

DSP

Digital Signal Processing

EFU

Embedded Fuzzy Unit

ENN

Elman Neural Network

FIR

Finite Impulse Response

H-N

Heinonen-Neuvo

IIR

Infinite Impulse Response

IMC

Internal Model Control

LN

Large Negative

LP

Large Positive

LS

Least Squares

LSN

Linear Smoothed Newton

MIMO

Multiple-Input and Multiple-Output

MDL

Minimum Description Length

MENN

Modified Elman Neural Network

MLP

Multilayer Perceptron

MN

Medium Negative

MP

Medium Positive

ix

MRAC

Model Reference Adaptive Control

MOM

Mean Of Maximum

NB

Negative Big

NM

Negative Medium

NS

Negative Small

PB

Positive Big

PI

Proportional-Integral

PID

Proportional-Integral-Derivative

PM

Positive Medium

PMDL

Predictive Minimum Description Length

PS

Positive Small

RLSN

Recursive Linear Smoothed Newton

RMS

Root Mean Square

SN

Small Negative

SNR

Signal to Noise Ratio

SOM

Self-Organizing Map

SP

Small Positive

SSE

Sum Squared Error

TD

Temporal Difference

TDL

Tapped Delay Line

ZE

Zero

x

List of Symbols

a

Node activation function in the neural network

a′

First order derivative of a

A

Af

Fuzzy set

Fuzzy predictate

A′f

Fuzzy predictate

Aji

Linguistic value

Bf

Fuzzy predictate

Bj

Linguistic value

B′f

Fuzzy predictate

c

Consequent parameters in the ANFIS

C

Fuzzy set or

Multiple controllers or

Oscillator coefficients

d

Vector of desired output of the neural network

di

Desired output of the neural network or

Euclidean distance in the self-organizing map

e

Feedback error in control systems or

Identification error

∆e

Feedback error change

E

Cost function of the neural network

e˙

First order derivative of the feedback error

e˙˙

Second order derivative of the feedback error

f(⋅)

Function relationship between reinforcement signal and control actions

F

Fuzzy set or

Force applied to the cart

g

Gravity acceleration

G

Syntactic rule

h(⋅)

Function of fuzzy rule base

H LN ( z )

Transfer function of the Newton predictor for polynomial degree L and the number

of prediction steps N

i

Index variable

I MENN

Input set of the MENN

xi

j

Index variable

J

Trajectory control performance index of the Elman neural network-based controller

or

Instant power regulation index

J*

Cumulative trajectory control index of the Elman neural network-based controller or

Cumulative control criterion for power regulation

k

Index variable or

Iteration step index in neural network training or

Scaling constant in reinforcement learning

K

Length of the Newton predictor

l

Number of nodes in the hidden layer in neural network or

lp

Index variable for predictor

Half length of the pole in the inverted pendulum

L

Polynomial degree assumed for the input signal

m

Number of nodes in the input layer in neural network or

Length of a sequence or

Index variable for predictor or

Order of the plant output

mc

Mass of the cart in the inverted pendulum

mp

Mass of the pole in the inverted pendulum

M

Carrier modular or

Order of the acceleration estimator or

Multiple identification models or

Semantic rule

M

Node function in the ANFIS

ms

Number of support values

n

Number of nodes in the output layer in neural network or

Order of the input of the plant

nf

Number of fuzzy inputs

nr

Number of fuzzy rules

N

Number of prediction steps ahead or

Order of the neuro predictive filter

NG

White noise gain of the Newton predictor

xii

n˜

Noise in the input signal of the predictor

Nc

Neighborhood in the self-organizing map

net q

Input of node q in the hidden layer

net _ ci

Input of context node i in the Elman neural network

net _ hi

Input of node i in the hidden layer in the Elman neural network

Oil

Output of node i in layer l in the ANFIS

OContext

Output set of the context nodes in the MENN

O MENN

Output set of the MENN

o _ ci

Output of context node i in the Elman neural network

o _ hi

Output of node i in the hidden layer in the Elman neural network

p

Index variable or

Number of inputs in the ANFIS or

Prediction output

P

Consequent parameters in the ANFIS or

Transmitting signal power

∆p

Discrete power step

P Ref

Desired power level for the received power at the base station

q

Index variable or

Number of membership functions for each input in the ANFIS

q

q

Number of quantization levels

r

Reinforcement signal

R

General variable form of the reinforcement learning signal

Ref

Reference signal for closed loop control systems

Ri

Fuzzy reasoning rule i

S

Oscillator coefficients

t

Discrete sampling point

T

Length of training sequence or

Term set

Ts

Sampling period

u

Fuzzy set element or

Control output or

Identification excitation input

U

Universe of discourse or

xiii

General variable form of control action in reinforcement learning

u nn

Identification output of the neural network-based inverse model

u∗

Optimal output for the action network to learn in reinforcement learning

u˜

Actual control applied to the plant in reinforcement learning or

Output of the fuzzy power command enhancement unit

u

Approximation of u ∗ by secant method

∆u

Output of the stochastic unit

∆u FLC

Incremental output of the fuzzy power controller

v

vqj

Fuzzy set element

Weight connecting node j in the input layer with node q in the hidden layer in the BP

neural network

v˜

Measured motion velocity

vˆ

v˜˙

Velocity measurement

vˆ˙

Acceleration estimate

w

W

General presentation of weights in neural networks

Both the weights vqj and wiq

wi

Weights in the self-organizing map

w new

General presentation of updated weights in neural networks

w old

wiq

General presentation of previous weights in neural networks

Weight connecting node q in the hidden layer with node i in the output layer in the

Noisy acceleration

BP neural network

∆w

∆wiq

Update change of w

Update change of wiq

w1i, j

Weight connecting node i in the input layer to node j in the hidden layer in the

w2 i , j

Elman neural network

Weight connecting node i in the hidden layer to node j in the output layer in the

w3i, j

Elman neural network

Weight connecting context node i to node j in the hidden layer in the Elman neural

w4 i, j

network

Weight connecting context node i to node j in the output layer in the Elman neural

W1

network

Weight set of w1i, j

xiv

W2

Weight set of w2 i, j

W3

Weight set of w3i, j

W4

Weight set of w4 i, j

x

Fading signal or

General variable or

Position of the cart in the inverted pendulum or

Sequence samples

x˙

Velocity of the cart in the inverted pendulum

x˙˙

Acceleration of the cart in the inverted pendulum

x

Input pattern for the neural network

X

Time series

xf

Linguistic input of the fuzzy rule base

x0

Crisp input

xi( k )

Input of node i at iteration k in the Elman neural network

xif

Linguistic input variables

y

Clean primary signal or

General variable or

Trajectory output of the plant

y

Output of the predictor

yˆ

Identification output of the Elman neural network or

Noisy input signal of the predictor

yf

Linguistic output of the fuzzy rule base

ym

Output of the reference model in MRAC

yn

Prediction output of the neural network model in IMC

y nn

Neural network identification output

yp

Output response of the plant

yd

Reference trajectory

yi

Output of node i in the output layer in the neural network

y

(k )

j

Output of node j in the output layer in the Elman neural network at iteration k

Yi ( k )

Desired output for node i in the Elman neural network at iteration k

z

zq

Final outcome of a sequence signal

Output of node q in the hidden layer

z −1

Unit delay

xv

z oj

Amount of output at quantization level j

z sj

Support value at which the membership function reaches the maximum value

*

zCOA

Crisp output of the COA defuzzification method

z*MOM

Crisp output of the MOM defuzzification method

θ

θ˙

Angle of the pole in the inverted pendulum

θ˙˙

Angular acceleration of the pole in the inverted pendulum

λ

Scaling constant in reinforcement learning or

Angular velocity of the pole in the inverted pendulum

Forgetting factor in TD learning

δ oi

General back-propagation error term for node i in the output layer

δ hi

General back-propagation error term for node i in the hidden layer

η

Learning rate of the neural network

φ

Gaussian density function or

Inverse function of ϕ

ϕ

Function relationship realized by the neuro predictor or

General function of a system

ϕ −1

Inverse function of ϕ

ϕˆ

Approximated mapping of the system realized by the neural network

ϕˆ −1

Approximated inverse mapping of the system realized by the neural network

ϑ

Mapping realized by the fuzzy command enhancement unit

ξ

Index variable

σ

Variance of the output of the stochastic unit

µA

Fuzzy membership function A

µc

Coefficient of friction of cart on track

µC

Fuzzy membership function C

µF

Fuzzy membership function F

µp

Coefficient of friction of pole on cart

1

1. Introduction

Soft computing, as pointed out by Dr. Zadeh, is not a single methodology [Zad94]. Instead, it is a

fusion of several methodologies, i.e., neural networks, fuzzy logic, and genetic algorithms. Different

from conventional (hard) computing, soft computing takes advantage of intuition, which implies that

the human mind-based intuitive and subjective thinking is realized in soft computing. The motivation of applying the human intuition is that a large number of real-world problems cannot be solved

by hard computing methods due to the fact that either they are too complex to handle or they cannot be described or catalogued by analytical and exact models. However, in some cases, human experts are marvellously successful in dealing with these problems, e.g., face recognition in a noisy

background. Dr. Zadeh emphasises that precise measurement and control approaches are not always

effective in coping with such difficult problems, but perception can often help [Zad98]. Therefore,

the goal of soft computing is to exploit the imprecision and uncertainty in human decision making

procedure, and achieve simple, reliable, and low-cost solutions.

Currently, the techniques in control and instrumentation fields are facing difficulties in meeting

the growing needs of modern industry. For example, numerous nonlinear and time-variant plants

cannot be efficiently stabilized or regulated using classical control methods. Additionally, as for the

acceleration acquisition in motion control systems, the existing signal processing techniques fail to

provide clean and delayless acceleration estimates because of the inherent noise in the measured velocity. For instance, conventional Finite Impulse Response (FIR) and Infinite Impulse Response

(IIR) filters always introduce some delay in the filtered output. It is advantageous to apply soft

computing methods in these cases, since soft computing offers us a totally new perspective by providing a set of techniques to solve practical problems in which ambiguity and uncertainty prevail

[Jan97].

Fuzzy logic, a branch of soft computing, has been an active partner in the process control for at

least the last two decades [Dri93]. Besides control engineering, fuzzy logic in instrumentation is

drawing great research interest [Rus96]. Neural networks are also investigated in control and instrumentation such as on-line tuning of controller parameters [Kaw98], friction compensation in servo

motion control systems [Gao99], and Analog to Digital Converter (ADC) resolution enhancement

[Gao97a]. The application of soft computing, in fact, covers a variety of application areas. Besides

control and instrumentation, other important aspects include speech recognition [Kom92], signal

processing [Cic93], telecommunications [Yuh94], power electronics systems [Dot98, pp. 143-185],

and system diagnosis [Cho92]. Recent advances of soft computing methods and their applications

2

in engineering design and manufacturing can be found in [Roy98]. With the rapid development of

hardware platforms, e.g., Digital Signal Processing (DSP) and neural networks chips, it is becoming

more feasible to apply soft computing methods into practice.

1.1 Definition of Soft Computing

Soft computing is a collection of methods to construct computationally intelligent systems. By ‘intelligent systems’, we here mean those systems that are capable of imitating the human reasoning

process as well as handling quantitative and qualitative knowledge. It is well known that the intelligent systems, which can provide human like expertise such as domain knowledge, uncertain reasoning, and adaptation to a noisy and time-varying environment, are important in tackling practical

computing problems. Specifically, in the modern control system design and analysis, there is a

promising trend going on to employ some heuristic methods that can benefit from human experts,

because the currently existing complex plants cannot be accurately described by rigorous mathematical models, and are, therefore, difficult to control using conventional model-based methods.

Meanwhile, in practice, experienced operators are often able to obtain fairly satisfactory control

quality. Soft computing is an appropriate candidate for creating these knowledge-based intelligent

systems. It has attracted the growing interest of researchers from various scientific and engineering

communities during recent years [Lin96].

Basically, soft computing is considered as an emerging approach to computing which parallels

the remarkable ability of the human mind to reason and learn in a circumstance of uncertainty and

imprecision. The pioneer of fuzzy logic, Dr. Zadeh, has pointed out that ‘the guiding principle of

soft computing is to exploit the tolerance for imprecision, uncertainty, and partial truth to achieve

tractability, robustness, low solution cost, better rapport with reality’ [Zad92]. In contrast with

hard computing methods, which only deal with precision, certainty, and rigor, soft computing is effective in acquiring imprecise or sub-optimal but economical and competitive solutions. In short,

because of its unique feature in coping with real-world problems, e.g., intelligent control, decision

making support, nonlinear programming and optimization, soft computing has drawn increasing research attention from people of different backgrounds [Jan97].

In general, soft computing methods consist of three essential paradigms: neural networks

[Hay98], fuzzy logic [Wan97], and genetic algorithms (evolutionary programming) [Gol89]. Nevertheless, soft computing is an open instead of conservative concept. That is, it is evolving those relevant techniques together with the important advances in other new computing methods such as arti-

3

ficial immune systems [Das98]. To condense the following presentation, we concentrate only on the

above three principal methods. In the triumvirate of soft computing, neural networks are concerned

with adaptive learning, nonlinear function approximation, and universal generalization; fuzzy logic

with imprecision and approximate reasoning; and genetic algorithms with uncertainty and propagation of belief. Table 1.1 lists these three methodologies together with their advantages.

Table 1.1. Soft computing constituents.

Methodology

Advantage

Neural Networks

Learning and Approximation

Fuzzy Logic

Approximate Reasoning

Genetic Algorithms

Systematic Random Search

However, in soft computing, they are complementary rather than competitive. More precisely, it

is advantageous to employ neural networks, fuzzy logic, and genetic algorithms in combination instead of exclusively. A typical example to support this argument is the popular fuzzy neural network model, which takes advantage of the capabilities of both fuzzy logic and neural networks

[Buc94]. The fuzzy neural network is constructed to merge fuzzy inference mechanism and neural

networks into an integrated system so that their individual weaknesses are overcome. Normally, the

neuro-fuzzy technique can have the same topology with the feedforward neural network, i.e., nodes

and layers. On the other hand, the node functions inside are replaced with fuzzy membership functions such as Gaussian functions. And, the commonly used back-propagation learning algorithm is

applied to adjust the parameters of these fuzzy membership functions. In this way, the fuzzy neural network retains the approximate inference characteristics and imprecise information processing

capability of fuzzy logic, meanwhile it also has the strength of adaptation and generalization by introducing the learning algorithm from neural networks. It has found many application prospects,

e.g., in image processing [Gho93] and speech recognition [Qi93]. A growing number of consumer

products, for example, washing machines and air conditioners, are released with the embedded

neuro-fuzzy technique. The same principal idea applies to other kinds of hybrid schemes among

neural networks, fuzzy logic, and genetic algorithms, which are in fact already available [Tak97].

Refer to Figure 1.1 for other alternatives.

4

Fuzzy Logic

Neuro-Fuzzy

Genetic-Fuzzy

Neural Network

Genetic Algorithm

Genetic-Neuro

Figure 1.1. Framework of soft computing.

We only present two representative examples, genetic-neuro and genetic-fuzzy techniques here.

In general, a genetic algorithm is a derivative-free and stochastic optimization method [Man96]. Its

orientation comes from ideas borrowed from natural selection as well as the evolutionary process.

As a general purpose solution to demanding problems, it has the distinguishing feature of parallel

search and global optimization. In addition, genetic algorithm needs less prior information about the

problems to be solved than the conventional optimization schemes, such as the steepest descent

method, which often require the derivative of the objective functions [Jan97, pp. 129-168]. Thus, it

is attractive to employ a genetic algorithm to optimize the parameters and structures of neural networks and fuzzy logic systems instead of using the back-propagation learning algorithm alone. For

instance, to optimize the weights of the neural network for a specified problem, we can first encode

each weight of the neural network into a binary bit string. Next, we concatenate these sub-strings

into a complete string, which is usually called a chromosome. A fixed number of the chromosome

candidates forms one generation. The value of the cost function to be optimized of every chromosome is then calculated and assigned to the chromosome as its ‘fitness’. The cost function is always

problem dependent. Based on individual fitness value, genetic algorithm uses the operators such as

reproduction, crossover and mutation to get the next generation that may contain chromosomes

providing better fitnesses. This evolutionary process evolves from generation to generation until a

preset cost function criterion has been reached. Due to the unique advantages of genetic algorithms,

i.e., derivative free, stochastic, and global search [Tan96a], this kind of genetic-neuro technique is

5

suitable for some special situations, where no derivative or other auxiliary knowledge of the cost

function is available for the neural network back-propagation learning algorithm. Additionally, since

the genetic algorithms are all based on probabilistic rather than deterministic search, the geneticneuro technique overcomes the severe shortcoming of applying pure back-propagation learning algorithm to train the neural networks. The steepest descent-based method is easily trapped into the

local minima in the nonlinear search space of weights.

For the genetic-fuzzy scheme, the genetic algorithms are applied to get optimal fuzzy inference

parameters [Far98]. We point out that not only the parameters of the fuzzy inference structure,

such as the antecedent and consequent coefficients, can be optimized by the genetic algorithms, but

also the tailoring of the fuzzy reasoning rules, e.g., adding new rules and deleting invalid ones, is appropriate for the utilization of the genetic algorithms. In a word, genetic algorithms play an important role in the parameter and structure learning in both the genetic-neuro and genetic-fuzzy techniques. However, it is to be emphasized that the combination of soft computing methods is not

only limited on the algorithm level as discussed previously. There are also potential fusion alternatives on the system level. For example, in a hierarchical system, a fuzzy logic controller and a neural

network predictor can act independently and co-operatively with different tasks attributed by the

top level process supervisor. We will demonstrate the effectiveness of such a hybrid scheme by the

illustrative example of power regulation in a mobile communications system in the following chapters.

Among the attempts to mimic the human brain intelligence, soft computing is not the first trial.

In fact, soft computing has some similarities with the conventional expert systems in Artificial Intelligence (AI) [Rus95]. Their common goal is to explore and realize machine intelligence. Basically,

expert systems target to imitate intelligence in the form of language expressions or symbolic rules.

On the other hand, there are no symbolic manipulations in soft computing. This feature makes it

free from the handicap inherent in the expert systems-oriented classical symbolicism, which can result in the ‘dimension curse’. Instead, the knowledge acquisition procedure in soft computing is

based on learning from practical data samples as well as operator experience. Neural networks are

often trained by the application data from measurement or provided by the supervisor. The antecedent and consequent of the mostly used IF-THEN rules in the fuzzy inference are interpreted by

the linguistic variables arising from the knowledge of the operation experts. Random but global

search using genetic algorithms is preferred in obtaining the optimal solution. Therefore, soft computing can also be viewed as a kind of data driven intelligent technique.

6

1.2 Intelligent Control

Over the past four decades, we have observed numerous successful applications and implementations of conventional control, ranging from autonomous robotics to chemical process and air traffic

control

The classic control approaches are based on analytical methods derived from the

physical models, such as Laplace transformation and linear state space. Bode diagrams, Nyquist and

Lyapunov stability criteria, and root-locus methods are still widely utilized in practical control engineering. However, control of a wider class of modern complex systems, e.g., time-varying, nonlinear, multi-variable, and multi-loop industrial systems, is difficult with these conventional techniques. An interesting example is the famous control benchmark truck backer-upper problem. To

back a trailer truck when applying the classical control scheme, we must first write out the fulldefined mathematical model of the truck including the dynamical kinematics between the displacement of the truck with the angels of the truck and the trailer. The finalized model is instinctually

nonlinear. Once this step is finished, we have to solve the equations with some numerical methods,

because normally there are no straightforward analytical methods for nonlinear systems. Nevertheless, even using the numerical methods, the acquisition of the approximated solutions is still unpractically time consuming. The underlying reason is due to the existing high dimensional input variables

and strong nonlinearity of the model. On the other hand, for the skilled truck drivers, this work is

reasonably easy. They just look at the distance between the truck and the dock, as well as the angle

of the truck and trailer, and based on this vague information together with a few reasoning rules,

they make their judgements to turn the truck a little bit in the right direction, and repeat the above

process until the truck is backed into the dock. Thus, it is clear that intelligent, for example, soft

computing-based, methods are advantageous for such kind of problems. Nguyen has applied a twolayer feedforward network to control the above simulated truck [Ngu90]. Kong proposes a fuzzy

system approach based on the Differential Competitive Learning (DCL) to get an even improved

result [Kon92]. In conclusion, the demand for higher performance criteria and economical solutions

necessities the invention of intelligent control systems.

There is no formal or single definition of an intelligent control system. Generally, an intelligent

system should satisfy the famous Turing test, which can be concisely expressed as follows: if a man

and a machine (or a program) perform the same task, then if one cannot distinguish between the machine and the human by examining only the nature of their performances, the machine is said to be

intelligent, otherwise not [Tur50]. However, intelligent control systems can be broadly described as

the use of artificial intelligence-related methods to design and implement automatic control systems.

7

Dr. Fu first introduced the term ‘intelligent control’, and initiated studies in this area as early as the

1950s [Fu71]. At the initial stage of the development of intelligent control, there were tight ties between artificial intelligence and automatic control. The advances in computer science, especially operations research, also contributed to intelligent control during the years that followed. The consequence is the study of cybernetics, robotics, and learning machines such as Michie’s broom balancer

‘BOXES’ [Mic68]. Figure 1.2 gives an illustrative diagram of the relationship between the intelligent

control and these four aforementioned fields.

dynamics

feedback

optimization

Automatic Control

management

Operations Research

planning

coordination

communications

Intelligent Control

object oriented

design

Computer Science

heuristics

memory

learning

distributed processing

Artificial Intelligence

Figure 1.2. Techniques employed in intelligent control [Bro94, pp. 5].

Today, several AI methods have been well established. Expert systems are applied in many application areas including dynamic decision making and medical diagnosis [Rus95]. Soft computing is

the most prominent AI technique in the control field. Modern control system design and implementation face three main difficulties [Nar91]: first, the heavy computational complexity, which the

new control methods require. The second is the presence of nonlinear systems with a large number

## CS 205 Mathematical Methods for Robotics and Vision docx

## Mathematical methods for physics and engineering riley, hobson

## Mathematical Methods for Physical and Analytical Chemistry pot

## MATERIALS AND METHODS (for definitions and additional details, see the technical appendix at end of chap- ter): Sources of data pdf

## intro to methods of appl. math - adv math methods for scientists and engineers - s. mauch

## numerical methods for engineers and scientists

## applied quantitative methods for trading and investment- dunis 2003

## formal methods for components and objects 9th international symposium, fmco 2010, graz, austria, november 29-december 1, 2010 revised papers

## METHODS FOR DETERMINATION AND APPROXIMATION OF THE DOMAIN OF ATTRACTION IN THE CASE OF AUTONOMOUS pdf

## studies in surface science and catalysis METHODS FOR MONITORING AND DIAGNOSING THE doc

Tài liệu liên quan