Contents

Acknowledgements iv

List of Tables xiv

List of Figures xv

Abstract (English) xix

Abstract (Arabic) xx

1 INTRODUCTION 1

1.1 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Examples of Nonlinear Systems . . . . . . . . . . . . . . . . . 2

1.2 Nonlinear Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Multivariable Nonlinear Control . . . . . . . . . . . . . . . . . 5

1.3.2 Internal Model Control . . . . . . . . . . . . . . . . . . . . . . 11

vii

1.3.3 Modified Internal Model Control . . . . . . . . . . . . . . . . 17

1.3.4 Learning Feedforward Control . . . . . . . . . . . . . . . . . . 20

1.3.5 Hammerstein System Control . . . . . . . . . . . . . . . . . . 22

1.3.6 Control of Bilinear Systems . . . . . . . . . . . . . . . . . . . 23

1.4 Nonlinear Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . 25

1.5 Research Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.6 Dissertation Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.7 Dissertation Organization . . . . . . . . . . . . . . . . . . . . . . . . 32

2 THE MIMO U-MODEL AND THE NEWTON-RAPHSON BASED

CONTROLLER 33

2.1 SISO U-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2 U-model Equivalence for Nonlinear Models . . . . . . . . . . . . . . . 35

2.2.1 Bilinear Input-Output Model . . . . . . . . . . . . . . . . . . 35

2.2.2 Hammerstein Model . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.3 Lur’e Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.4 NARX Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.5 Nonlinear Finite Impulse Response Model . . . . . . . . . . . 39

2.2.6 Output Affine Model . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 Newton-Raphson Based Controller . . . . . . . . . . . . . . . . . . . 40

2.3.1 Advantages of U-Model Based Control . . . . . . . . . . . . . 42

viii

2.4 SISO to MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5 The Proposed MIMO U-model . . . . . . . . . . . . . . . . . . . . . . 45

2.5.1 Radial Basis Functions Neural Networks . . . . . . . . . . . . 46

2.5.2 The Adaptation Algorithm . . . . . . . . . . . . . . . . . . . . 48

2.6 The Newton-Raphson Based Controller for MIMO Systems . . . . . . 50

2.7 SISO U-model to MIMO U-model . . . . . . . . . . . . . . . . . . . . 52

2.8 U-model Based Multivariable Control Algorithms . . . . . . . . . . . 54

3 STABILITY AND CONVERGENCE ANALYSIS 55

3.1 Convergence Analysis for the U-model . . . . . . . . . . . . . . . . . 57

3.1.1 Tracking Robustness of U-model . . . . . . . . . . . . . . . . . 59

3.1.2 Optimal Learning Rate for Tracking Robustness of U-model . 60

3.1.3 The Feedback Structure for U-model Update Recursion . . . . 63

3.1.4 Optimal Learning Rate for U-model Via Small Gain Theorem 68

3.2 Convergence Analysis for the RBFNN . . . . . . . . . . . . . . . . . . 70

3.2.1 Tracking Robustness of RBFNN . . . . . . . . . . . . . . . . . 72

3.2.2 Optimal Learning Rate for Tracking Robustness of RBFNN . 72

3.2.3 Feedback Structure for RBFNN Update Recursion . . . . . . . 75

3.2.4 Optimal Learning Rate for RBFNN Via Small Gain Theorem 79

3.3 Effects on Overall Stability . . . . . . . . . . . . . . . . . . . . . . . . 81

4 Real-Time Implementation 82

ix

4.1 The Real-Time Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.1.1 Hardware Assembly . . . . . . . . . . . . . . . . . . . . . . . . 83

4.1.2 The Dynamic Model of the 2-Link Robot arm . . . . . . . . . 85

4.1.3 Kinematics and Inverse Kinematics . . . . . . . . . . . . . . . 87

4.1.4 Interfacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.1.5 Tracking with PID Controller . . . . . . . . . . . . . . . . . . 89

5 MIMO U-MODEL BASED INTERNAL MODEL CONTROL 91

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.1.1 Disturbance Rejection in IMC . . . . . . . . . . . . . . . . . . 93

5.1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 MIMO U-model Based IMC for Nonlinear Systems . . . . . . . . . . 95

5.2.1 Stability of the U-model based IMC Scheme . . . . . . . . . . 97

5.2.2 Real-Time Implementation . . . . . . . . . . . . . . . . . . . . 98

5.3 MIMO U-model Based Control of Hammerstein Systems . . . . . . . 101

5.3.1 Proposed U-model Based Hammerstein Control . . . . . . . . 103

5.3.2 Real-Time Implementation Results . . . . . . . . . . . . . . . 104

5.4 U-model Based IMC for MIMO Bilinear Systems . . . . . . . . . . . 106

5.4.1 Problem Statement for Bilinear Systems . . . . . . . . . . . . 106

5.4.2 Known Parameters Case . . . . . . . . . . . . . . . . . . . . . 107

5.4.3 Unknown Parameters Case . . . . . . . . . . . . . . . . . . . . 109

x

5.4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 111

6 MIMO U-MODEL BASED MODIFIED INTERNAL MODEL CON-

TROL 117

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.2 Modified IMC with Improved Disturbance Rejection . . . . . . . . . . 118

6.2.1 Closed-Loop Transfer Function . . . . . . . . . . . . . . . . . 119

6.2.2 Disturbance Transfer Function . . . . . . . . . . . . . . . . . . 121

6.2.3 Comparison of Original and Modified IMC . . . . . . . . . . . 123

6.2.4 U-model Based Modified IMC With Local Disturbance Rejector124

6.3 Modified IMC for Unstable Systems . . . . . . . . . . . . . . . . . . . 125

6.3.1 U-model Based Modified IMC for Unstable Systems . . . . . . 126

6.4 Stability of the U-model Based Modified IMC Scheme . . . . . . . . . 127

6.5 Real-time Experiment Results . . . . . . . . . . . . . . . . . . . . . . 127

6.5.1 Local Noise Rejector . . . . . . . . . . . . . . . . . . . . . . . 128

6.5.2 Unstable Systems . . . . . . . . . . . . . . . . . . . . . . . . . 130

7 MIMO U-MODEL BASED LEARNING FEEDFORWARD CON-

TROL 132

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.1.1 Closed Loop Transfer Function . . . . . . . . . . . . . . . . . 135

7.1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 136

xi

7.1.3 U-model Based LFFC for Nonlinear Systems . . . . . . . . . . 137

7.1.4 Stability of the U-model Based LFFC Scheme . . . . . . . . . 138

7.2 Real-Time Implementation Results . . . . . . . . . . . . . . . . . . . 139

8 PERFORMANCE COMPARISON 142

8.1 PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.1.1 Ziegler-Nichols Tuning Method . . . . . . . . . . . . . . . . . 143

8.1.2 Performance Using PID Controller with Ziegler-Nichols Tuning 144

8.2 Adaptive PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8.2.1 Performance Using Adaptive PD Controller . . . . . . . . . . 148

8.3 Nonlinear PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . 150

8.3.1 Performance Using Nonlinear PID Controller . . . . . . . . . . 151

8.4 Adaptive Inverse Control . . . . . . . . . . . . . . . . . . . . . . . . . 153

8.4.1 Performance Using AIC . . . . . . . . . . . . . . . . . . . . . 154

8.5 Comparison of MSE . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

9 CONCLUSIONS, SUMMARY AND FUTURE WORK 159

9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

9.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

9.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

REFERENCES 162

xii

VITA 175

xiii

DISSERTATION ABSTRACT

Name: SYED SAAD AZHAR ALI

Title: U-MODEL BASED MULTIVARIABLE NONLINEARADAPTIVE CONTROL

Degree: DOCTOR OF PHILOSOPHY

Major Field: ELECTRICAL ENGINEERING

Date of Degree: JANUARY 2007

Many of the real industrial systems are multivariable and nonlinear in nature.Establishing a nonlinear adaptive control algorithm has always been a major researcharea for control engineers. A generalized modelling framework with perfect approxi-mation capabilities is required which also assists in finding a simple control law. Inthis research a technique based on the control-oriented model called U-model is devel-oped for multi-input multi-output nonlinear adaptive control. The U-model inverseis developed based on Newton-Raphson method. The technique is applied to differ-ent control schemes such as Internal Model Control, Learning feed-forward Control,modified Internal Model Control and Nonlinear Hammerstein Control. Performancecomparison and stability analysis are also presented. The proposed U-model basedschemes are tested on a laboratory scale prototype 2-degree-of-freedom robot manip-ulator.

Keywords: U-MODEL, NONLINEAR SYSTEMS, ADAPTIVE CONTROL, RBFNN,ROBOTICS, MIMO.

KING FAHD UNIVERSITY OF PETROLEUM & MINERALS, DHAHRAN.

JANUARY 2007

xix

xx

خالصة االطروحه

على اظهرسيد سعد : االسم التحكم التوليفي لألنظمة اال خطية ومتعدد المتغيرات بإعتماد : لعنوان ا

"يو"نموذج ال

دكتوراه الفلسفه : الدرجة

ئيةالهندسه الكهربا : ميدان رئيسي

2007يناير : تاريخ الشهاده

تحكم ال خطية لل خوارزميه إنشاء. أنظمة غير خطية ومتعددة المتغيرانناعية هي كثر النظم الصأوتبرز الحاجة إلطار عام لنمذجة هذه االنظمة بهدف إيجاد . دائما بحثيا المجاوليفي هو الت

في سبيل " يو"، نطور طرقا تعتمد على نموذج ال في هذا البحث. خواريزميات سهلة للتحكميمكن إيجاده "يو" النموذج العكسي لنموذج . ير خطية متعددة المتغيراتالتحكم باألنظمة الغ

هذه الطرق سوف تطبق في التحكم بهذه األنظمة باستخدام . رابهسون-طريقة نيوتن باستخدام اطعام االمام ، وتعديل نموذج الرقابة ومثل المراقبة الداخلية نموذج التعلم خواريزيميات متعددة

مقارنة يشمل البحث أيضا . هامرستيينلألنظمة الغير اخطية المعروفة بإسم الداخلية والرقابة لكلكل الخواريميزات المطورة باستخدام عدة أنظمة للمحاكاة وكذالتحليل و األداء واالستقرار

. نموذج رجل آلي ذي ذراعين

وليفي، رجل آلي، أنظمة متعددة المتغيراتنظم التحكم ال، " يو" نموذج :الكلمات الرئيسية

جامعة الملك فهد للبترول والمعادن

2007يناير

Chapter 1

INTRODUCTION

1.1 Nonlinearity

In mathematics, nonlinear systems represent systems whose behaviors are not ex-

pressible as a sum of the behaviors of their descriptors. In particular, the behavior of

nonlinear systems is not subject to the principle of superposition, as linear systems

are.

Linearity of a system allows to make certain mathematical assumptions and

approximations, simplifying the computation of results. In nonlinear systems these

assumptions cannot be made. Nonlinear systems are often difficult or sometimes

impossible to model. The behavior with respect to a given variable, for example,

time, is extremely difficult to predict. Some properties of nonlinear systems are:

• The state of a nonlinear system can go to infinity in finite time.

1

2

• They do not follow the principle of superposition (linearity and homogeneity).

• They may have multiple isolated equilibrium points.

• They may exhibit properties such as limit-cycle, bifurcation, chaos.

• For a sinusoidal input, the output signal may contain many harmonics and

sub-harmonics with various amplitudes and phase differences (a linear system’s

output will only contain the sinusoid at the input).

1.1.1 Examples of Nonlinear Systems

Some examples of nonlinear systems are:

• Pendulum equation: The pendulum has multiple equilibrium points.

• Tunnel-diode circuit: The tunnel-diode has multiple equilibrium points

depending upon the source and other components.

• Mass-spring system: An example of periodic excitation for a periodic ex-

ternal force.

• Friction: produces discontinuity

• Control valve: Example of saturation

• Robot manipulators: high-order dynamic system

• Fluid dynamics: Viscous friction and high-order dynamics, etc.

3

1.2 Nonlinear Control

Many common control problems involve dynamic systems that exhibit nonlinear

behavior. The effect of nonlinearities is not severe if the nonlinearities are mild or

the operating conditions are constant and limited. A quite mature and developed

linear control theory is applicable to such nonlinear systems trading off with either

the performance or the operating region. However, conditions where performance

is nonnegotiable or operating region is not certain, linear control theory fails to

perform. In addition, the linear analysis tools are inapplicable to the nonlinear

systems, such as:

• Matrix and vector methods, transform methods, block-diagram algebra, fre-

quency response methods, poles and zeros and root loci are all inapplicable.

• Available methods of analysis are concerned almost entirely with providing

limited stability information.

• System design/synthesis methods barely exist.

• Numerical simulation of nonlinear systems may yield results that are mislead-

ing or at least difficult to interpret. This is because, in general, behavior of a

nonlinear system is structurally different in different regions of the state space.

Under these conditions, the conventional constant linear controller fails to maintain

the performance of the system at acceptable levels and does not respond well to

4

changes in system dynamics. Considering the problems and difficulties of using linear

control over the nonlinear systems, scientists have put efforts towards developing

nonlinear control techniques. These techniques can be classified roughly as:

• Graphical Controller Design

• Nonlinear State-Space Controller Design

• Feedback Linearization of Nonlinear Systems

• Nonlinear Output Regulation

• Lyapunov and Universal Lyapunov Design

• Back-stepping

• Sliding-Model Controller Design

• Nonlinear Adaptive Control

• Intelligent Control (neural, fuzzy control)

• Heuristic and Evolutionary techniques

A survey on the recent advancements in the area of multivariable nonlinear

control is reported in the following section.

5

1.3 Literature Review

This section presents a general review on the recent multivariable nonlinear con-

trol schemes. Literature reviews on internal model control (IMC), modified IMC,

learning feedforward control (LFFC) and Hammerstein control are also reported to

support the proposed algorithms in this section.

1.3.1 Multivariable Nonlinear Control

Multivariable nonlinear control has been an active area of research over the years.

Some remarkable surveys of historic progress and recent advances can be found in

[1, 2, 3].

As noted in [4], the extension of adaptive control methods for single-input single-

output (SISO) systems to multi-input multi-output (MIMO) systems is far from

trivial. This extension becomes even more challenging when nonlinear systems are

considered. Early results on adaptive control of MIMO linear systems can be found

in [4, 5]. With respect to nonlinear systems, adaptive control has been mostly tar-

geted at special cases of MIMO systems [6]. Most notably is the case of robot

manipulators e.g. in [7], which exploits several structural properties of system dy-

namics.

In [8], the output tracking problem is addressed for MIMO nonlinear non-minimum

phase systems. A feedback control law is designed such that outputs of the closed-

6

loop system track a commanded output function with internal stability. The ap-

proach is based on the decomposition of a system into two parts: a linear input-

output subsystem with trivial internal dynamics, the minimum phase part, and an

input-output subsystem with unstable internal dynamics, the non-minimum phase

part.

A stable inversion approach is used in [9] for the output tracking of a non-

minimum phase system. The non-minimum phase system is first stably inverted

offline to obtain desired (and stable) state and input trajectories that satisfy the

system dynamics equation and map exactly into a given desired output trajectory.

Then the desired input is used as a feed-forward signal and the state error deviating

from the desired state is used as a feedback signal to a stabilizing tracking controller.

A constrained multivariable control strategy is presented in [10]. A neural net-

work model-based non-linear long-range predictive control is reported, that provides

offset-free closed-loop behavior with a proper and consistent treatment of modelling

errors and other disturbances. The algorithm is applied to an efficient thermal power

plant.

In [11], a synthetic output is used that is equivalent to the plant output. The

synthetic outputs are selected using a systematic procedure and then used to con-

struct a model-state feedback controller. The controller is shown to induce zero

steady-state error to the original plant outputs.

In the predictive control approach, a multivariable continuous-time generalized

7

predictive controller (CGPC) is recast in a state-space form and shown to include

generalized minimum variance (GMV) [12]. Thus, proposed a new algorithm, pre-

dictive GMV (PGMV).

A design of a neural servocontroller for a nonlinear MIMO plant is reported

in [13]. The proposed control scheme uses the feedback error and the variables

representing the plant operating point. The authors used integrators in the control

loop to ensure low frequency setpoint following and disturbance rejection.

A nonlinear model of a cement mill is presented in [14], motivated by practical

observations about the linear quadratic Gaussian (LQG) controller on an industrial

milling circuit. Based on this nonlinear model a multivariable nonlinear receding

horizon (NRH) control strategy is applied to improve the performance and to enlarge

the stability region.

In [15], a fuzzy basis function vector (FBFV) approach for the adaptive control

of multivariable nonlinear systems is proposed. The linearized bias and uncertainties

as well as disturbances are assumed to be included in the linearized model structure.

The output of the FBFV is used as the parameters of the robust controller in the

sense that both the robustness and the asymptotic error convergence is obtained for

the multivariable nonlinear system.

A multivariable nonlinear control strategy [16] which accounts for unmeasured

state variables and input constraints is developed for the free-radical polymerization

of methyl methacrylate in a continuous stirred tank reactor. Input constraints are

8

handled explicitly by applying linear model predictive control to the constrained

linear system obtained after feedback linearization and constraint transformation.

A class of multi-input multi-output (MIMO) nonlinear systems having a triangu-

lar structure in control inputs is considered in [17]. By utilizing the system triangular

property, integral-type Lyapunov functions are introduced for deriving the control

structure and adaptive laws without the need of estimating the decoupling matrix

of the multivariable nonlinear system.

A decentralized robust adaptive output feedback control scheme for a class of

large-scale nonlinear systems of the output feedback canonical form with unmodelled

dynamics [18]. A modified dynamic signal is introduced for each subsystem to

dominate the unmodelled dynamics and an adaptive nonlinear damping is used to

counter the effects of the interconnections.

In [19], the authors investigate a fuzzy model reference adaptive controller for

continuous-time MIMO nonlinear systems. The adaptive scheme uses a TakagiSe-

guno fuzzy adaptive system, which allows for the inclusion of a priori information

in terms of qualitative knowledge about the plant operating points or analytical

regulators for those operating points.

A class of nonlinear generalized predictive controllers (NGPC) is derived for

MIMO nonlinear systems with offset or steady-state response error. The MIMO

composite controller consists of an optimal NGPC and a nonlinear disturbance ob-

server [20].

9

A novel approach is presented for the control of the entire photolithography track

using a combination of genetic programming and nonlinear model predictive control

[21].

Recurrent neural networks (RNN) are used for controlling the robot trajectory

in [22]. The control system comprises of RNN based controller and a fixed gain

feedback controller.

A practical design that combines a fuzzy adaptation technique with sliding mode

control to enhance robustness and sliding performance in a class of uncertain MIMO

nonlinear systems is presented in [23]. Using an online adaptation scheme, a fuzzy

sliding mode controller is used to approximate the equivalent control in the neigh-

borhood of the sliding manifold.

The problem of tracking control and maneuver regular control for a nonlinear

nonminimum phase control system with application to flight control is discussed

in [24]. A tracking controller, consisting of feedforward and static-state feedback,

is designed to guarantee uniform asymptotic trajectory tracking. The feedforward

is determined by solving a stable noncausal inversion problem. Constant feedback

gains are determined based on linear quadratic regulator optimization and assumed

satisfaction of a robustness inequality.

A genetic algorithm based multi-objective evolutionary algorithm for control

of the multivariable and nonlinear systems is presented in [25]. Problem design

considers time domain specifications such as overshoot, rising time, settling time

10

and stationary error as well as interaction effects.

Radial basis function neural networks (RBFNN) based IMC is presented in [26].

RBFNN is used for the model inversion.

In the paper [27], multivariable nonlinear control is applied to boiler-turbine unit.

A decentralized multivariable nonlinear controller system is obtained for a nonlinear

boiler-turbine model, and then evaluated under a wide range of operating conditions

and perturbations. The method can asymptotically estimate and compensate the

nonlinearity, coupling and disturbances owing to its observation ability.

The paper [28] adopts the theory of the nonlinear transformation and the feed-

back linearization of nonlinear system, establishes the nonlinear mathematic model

of the voltage source converter in d-q synchronous reference frame. The decoupled

control strategy with state variable feedback linearization, which is combined with

the optical control theory is presented.

An indirect adaptive decoupling controller is presented for a class of uncertain

nonlinear multivariable discrete time dynamical systems [29]. The indirect adaptive

decoupling controller is composed of a linear robust indirect adaptive decoupling

controller, a neural network nonlinear indirect adaptive decoupling controller and a

switching mechanism.

In this research, the adaptive tracking of multivariable nonlinear systems is per-

formed using the proposed MIMO U-model in the IMC, modified IMC and LFFC

schemes. These scheme are discussed in detail in subsequent chapters. For complete-

11

ness of the literature review, the advancements in these schemes are summarized

here.

1.3.2 Internal Model Control

Internal model control was first introduced in 1982 by Garcia and Morari [30] for

SISO, discrete time systems. The stability of IMC was discussed and concluded

about the IMC structure that it allows a rational design procedure where, in the

first step, the controller is selected to give perfect control. In the second step, a

filter is introduced which makes the system robust to model-plant mismatches.

The nonlinear version of the IMC was presented in [31], in 1986. The properties

of the nonlinear IMC were discussed in detail. A controller was developed based on

numerical methods and the overall stability of the system was looked into.

In [32], the authors proposed a novel technique of directly using artificial neural

networks for the adaptive control of nonlinear systems. The use of nonlinear function

inverses was investigated and IMC was used as the control structure. In a further

contribution [33], in 1993, they presented artificial neural network architectures for

the implementation of nonlinear IMC. This approach can be viewed as a nonlinear

analogue of adaptive inverse control; the network models used were equivalent to

nonlinear adaptive filters. They used two separate networks in the implementation

of nonlinear IMC; one network models the plant, and the second network models

the plant inverse.

12

In 1996, [34] the authors combined adaptation with an internal model control

structure to obtain an adaptive IMC scheme possessing theoretically provable guar-

antees of stability. The adaptive IMC scheme was designed for open-loop stable

plants using the traditional certainty equivalence approach of adaptive control and

it was shown that using a series-parallel identification model, for a stable plant,

one can adapt the internal model on-line and guarantee stability and asymptotic

performance in the ideal case.

In [35], in 1997, NMBC based on a Takagi-Sugeno fuzzy model is presented. The

controller proposed is a combination of a model predictive controller and an inverse

based controller (IMC). If no constraints are violated, an inverse based control al-

gorithm is used. When constraints are violated, the required optimization used is

a branch-and-bound algorithm. They illustrate their controller by applying it to a

laboratory scale air-conditioning system.

The paper [36] investigated in detail the possible application of neural networks

to the modelling and adaptive control of nonlinear systems. Nonlinear neural-

network-based plant modelling was discussed, based on the approximation capabil-

ities of the multi-layer perceptron. A novel nonlinear IMC strategy was suggested,

that utilizes a nonlinear neural model of the plant to generate parameter estimates

over the nonlinear operating region for an adaptive linear internal model, without

the problems associated with recursive parameter identification algorithms. Unlike

other neural IMC approaches the linear control law can then be readily designed.

13

The proposed schemes was applied to the tracking of the CSTR plant.

In [37], a robust adaptive controller based on the IMC structure for stable plants

was proposed. A stable high order model for the stable plants using the RLS algo-

rithm and its stable reduced order model is calculated using the ordered real Schur

form method. The stable adaptive IMC controller is designed for the reduced order

model and is augmented by the low-pass filter such that the closed loop stability for

the higher order model is ensured.

The paper [38] presented several theoretical results for the application of recur-

rent neural networks to the production of an IMC system for nonlinear plants. The

results include determination of the relative order of a recurrent neural network and

invertibility of such a network. A closed loop controller was produced without the

need to retrain the neural network plant model and the stability of the closed-loop

controller was demonstrated.

The 1999 tutorial [39], provides a quick overview of neural networks and explains

how they can be used in control systems. Included in the tutorial are several control

architectures, such as model reference adaptive control, model predictive control,

and internal model control.

In [40], a new nonlinear IMC using neural networks for control of processes with

delay is presented. The internal model used is a delay-deprived model cascaded with

fixed delay, while the inverse model is based on neural network models and consists

of the inverse of the delay-deprived model only.

14

In [41], a fuzzy adaptive internal model controller for open-loop stable plants is

presented. The control scheme consists of a dynamic model and a model-based fuzzy

controller. Fuzzy dynamic model which serves as the internal model is identified

online by using the input and output measurement of the plant. Based on the

identified fuzzy model, the fuzzy controller is designed.

The paper [42] extended the IMC scheme to nonlinear processes based on local

linear models where the properties of the linear design procedures can be exploited

directly. A local linear neuro-fuzzy model is used. The output of the model is

calculated as an interpolation of locally valid linear models. Local model architec-

tures allow the separate inversion of each local linear model. The control output is

calculated as a weighted sum of locally valid linear controllers yielding a globally

nonlinear controller which results in a gain-scheduled control approach.

In [43], the use of an artificial neural network in IMC both as process model

and as controller, for a class of nonlinear systems with separable non- linearity is

proposed. It is shown that an IMC with a neural network controller, in which the

linear part of the plant and its inverse are replaced by neural networks, cancels the

effects of nonlinear dynamics and measured disturbances.

In [44], the authors have developed and applied three control approaches to

adjust the speed of the drive system for an induction motor. The first control

design combines the variable structure theory with the fuzzy logic concept. In the

second approach neural networks are used in an internal model control structure.

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Finally, a fuzzy state feedback controller is developed based on the pole placement

technique. A simulation study of these methods is presented. The effectiveness

of these controllers is demonstrated for different operating conditions of the drive

system.

In [45], a nonlinear IMC for control of Switched Reluctance Motors (SRM) is

presented. It is shown that combining the simplicity of the feedback linearization

control and the robustness of IMC structure, gives a controller that exhibits excellent

dynamic and static performances for the torque and current control. Simulations

and experiments carried out on a 7.5 kW four-phase SRM, show that the ripple of

the output torque is very low in spite of model-plant mismatches.

In [46], the authors have proposed an adaptive IMC scheme based on adaptive

finite impulse response filters, which can be designed for both minimum and non-

minimum phase systems in the same fashion.

In [47], an IMC scheme based on locally-linear-model-tree modelling framework

is developed. The proposed control strategy is applied to the control of PH neu-

tralization process and the results are compared with those of IMC based on multi

layer perceptron neural networks. Simulation results demonstrate the superiority of

the new controller.

A neural network based multi-model IMC structure is presented in [48] for the

adaptive tracking of plants with strong nonlinear characteristics. Multi-model is an

effective method in parameters-varying and nonlinear process. The core idea is to

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represent a nonlinear dynamic system by a set of locally valid sub-models across the

operating range.

In [49], robust speed tracking of permanent magnet synchronous motor (PMSM)

using the IMC structure is achieved. It is shown that the IMC controller greatly

improves the performance of the current loop and simplifies the design procedure.

In [50], Passivity Theorem to develop a new IMC scheme for the adaptive track-

ing of multivariable nonlinear processes is used. The conventional IMC method

involves inversion of the process, which is often difficult or even impossible. In the

proposed method, the process is approximated using a passive system. The con-

troller is designed to effectively invert the passive approximation. The stability of

the closed-loop system is guaranteed by the passivity condition. The effectiveness

of the proposed method is illustrated by using a mixing tank control problem.

In [51], derivation of IMC controllers and tuning procedures for application to

second-order plus dead-time (SOPDT) processes for achieving set-point response

and disturbance rejection tradeoff is performed.

An IMC based controller for force control in a SUMI-Ink rubbing machine is

developed and implemented in [52]. It is shown that excellent force control and

disturbance rejection can be achieved by the use of the IMC structure.

In [53], an adaptive IMC design is presented for hard disk drive servo control

where low pass filters are used to handle high frequency mechanical modes and

disturbances. They apply the scheme to the real-time tracking of the drive head.

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In [54], the IMC structure for studying the effects of demand forecast error

on a tactical decision policy for a single node of a manufacturing supply chain is

used. The demand forecast is treated as an external measured disturbance in a

multi-degree-of-freedom IMC based inventory control system. The multi-degree- of-

freedom formulation allows the controller to be independently tuned for set-point

changes, forecasted demand changes, and unforecasted demand changes. A mathe-

matical framework for evaluating the effect of forecast revisions in an IMC controller

is developed and several useful results are achieved.

The paper [55] presented an adaptive internal model controller using neural net-

works for a tilt rotor aircraft platform. The controller includes an online learning

neural network of inverse model and an off-line trained neural network of forward

model. Lyapunov stability analysis is used to guarantee that tracking errors and

network parameters remain bounded. The performance of the controller is demon-

strated through real-time experiments.

In [56], the SISO U-model is used in the IMC structure for the adaptive track-

ing of nonlinear dynamic plants. This technique is similar to the pole placement

controller using the U-model [57].

1.3.3 Modified Internal Model Control

Ever since the IMC strategy was first introduced, several modifications have been

introduced to improve the performance or achieve specific desired objectives. Al-

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though IMC has inherent disturbance rejection capabilities, in practical applications,

even small plant model mismatch can lead to adequate tracking errors. Moreover,

the classical IMC structure cannot reject disturbances which enter the system from

the input of the plant. The problem of windup caused by the actuator saturation is

not considered in the classical IMC. It is also important to mention that the classical

IMC can only deal with situations where the plant is assumed stable. For unstable

plants IMC structure fails and the overall closed loop becomes unstable.

Therefore, these short comings are handled later by several researchers and dif-

ferent modified IMC schemes are introduced. Modified IMC is also referred to as

the 2-degree-of-freedom (2DoF) IMC.

A 2DoF controller for motor drives with first order model is proposed in [58].

The parameters of the proposed controller are designed using a systematic approach

to match the prescribed motor drive specification.

2DoF (IMC) is implemented [59] for the temperature control of a single tubular

heat exchanger system. The controller had two degree of freedom, the one tuning

filtering parameter of IMC controller effected the tracking of reference and the other

tuning parameter of IMC reduce disturbance.

A high performance input-output linearizing control law is claimed in [60]. The

authors presented a modified IMC scheme that can handle constraints to the plant

input as well. In the same perspective, windup scenarios were also considered and

applied to nonlinear control.

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The nonlinear MIMO neural network (NN) control based on a simple and straight-

forward modification of the IMC is presented in [61]. The nonlinear modified internal

model control (MIMC) structure is completely defined by the inverse process model

and guarantees the offset-free control. The inverse process model, defining the main

controller in Modified IMC, is obtained by the experimental NN inverse modelling

technique.

A modified IMC structure with simplified design is presented by [62]. The au-

thors presented the idea to avoid the modelling of the plant and in effect to avoid

the high modelling errors.

An anti-windup control strategy using the modified IMC is presented in [63] for

both stable and unstable systems.

A modified internal model control technique is presented for unstable systems in

[64]. The author presented a very simple feedback block that stabilizes the plant.

The overall stabilized loop was then controlled using the regular IMC for stable

systems. Properties of the system like stability and robust stability are also discussed

for the proposed scheme.

A disturbance observer-based control scheme for modified IMC is presented in

[65]. The authors have shown that the algorithm is very effective in controlling

integral processes with dead time.

A parametrization of all stabilizing 2DoF controllers for (possibly unstable) pro-

cesses with dead-time is presented in modified IMC structure by [66].