Application of Genetic Algorithms in Optimization and Neural Network Training for Nonlinear Dynamic Plants August 2005

University of the Ryukyus Graduate School of Engineering and Science Mechanics and Control Systems Engineering MUHANDO, Billy Endusa (038374B)

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Abstract This thesis is based on research in two areas of genetic algorithm (GA) application. One part of the study deals exclusively with binary-coded GAs in search and optimization. The other is research on intelligent control design for nonlinear dynamic plants, with aid of real-coded GAs for neural network (NN) training. The clean world of quadratic objective functions with ideal constraints and ever present derivatives has fast become unsuitable for all but a very limited problem domain. The real world of search is fraught with discontinuities and vast multi-modal, noisy search spaces of less calculus-friendly functions. Chapter 2 introduces the Recessive Gene Model (RGM) as a tool in numerical function optimization with binary coded GAs. Unlike most traditional schemes that work well in a narrow problem domain, RGM is robust and works well across a broad spectrum of problems. Use of the dual gene system ensures there is an ample pool of better-fit individuals for successful evolution and rapid convergence. Its efficacy was tested on both unimodal and multimodal functions in the form of Easom’s, Grienwangk’s and Schaffer’s F6 functions. Results of simulations confirm the effectiveness of the scheme. Next two chapters give a synopsis of my research on both off-line and on-line design of control systems for nonlinear dynamic plants. In Chapter 3 a hybrid control scheme is proposed for the stabilization of backward movement along simple paths for a vehicle composed of a truck and six trailers. The hybrid comprises a linear quadratic regulator (LQR) and a neurocontroller (NC) that is trained by a realcoded GA. The evaluation function of the NC in the hybrid design has been modified from the conventional type to incorporate the running steps errors to enhance faster training of the NC. Training of the NC is performed off-line until the best design is evolved, then the plant is run. Achieving good backward movement is difficult because of the restraints of physical angular limitations; the system is impossible to globally stabilize since some initial states necessarily lead to jack-knife locks. Results from trials show the hybrid can be used effectively for the control of nonlinear dynamical systems. Chapter 4 presents a continuation of the above research, with the main focus being on real-time design of the control system for a plant comprising a truck with seven trailers. Generally, difficulty in control intensifies with increase in the number of connected trailers. Firstly, an emulator of the plant is designed and a trajectory formulated. The plant is allowed to back up on the trajectory to a desired destination. Due to the possibility of model uncertainty, the plant may deviate from this constrained path, thus the emulator undergoes on-line training and re-design to formulate another trajectory hence. Simulations with a kinematic model of the trailer-truck verify that the adaptive design can guide the actual plant to desired destinations in an errorless condition.

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Acknowledgments I am greatly indebted to my Supervisor Professor Tetsuhiko Yamamoto and my Advisor, Associate Professor Hiroshi Kinjo, who have given freely of their time to my academic growth and social adjustment since I came to the University of the Ryukyus, Okinawa, Japan on October 3, 2003. They have been invaluable in my life for their guidance, advice and encouragement. I am truly grateful for their commitment to ensuring I made successful presentations at international Conferences, both locally and abroad, on the ongoing research on genetic algorithm optimization and the neurocontrol of complex nonlinear dynamical systems. Special thanks to Dr Koji Kurata, Dr Eiho Uezato, Dr Naoki Oshiro and Dr Kunihiko Nakazono of the Mechanical Systems Engineering Department, University of the Ryukyus, who have assisted me either individually or collectively, in my studies. Throughout my MSc course my fellow students in the Mechanics and Control laboratory were particularly keen on ensuring there was a conducive research atmosphere and accorded me ample encouragement and moral support in respect of both academic and social spheres. The much I learned from you is indelibly imprinted in my psyche and I thank you bountifully. I express my deepest gratitude and respect to the Ministry of Education, Culture, Sports, Science and Technology, Japan (MONBUKAGAKUSHO) for supporting my studies throughout the course. I dedicate this work to my son Glen Muhando: as young as you may be, the evolutionary world of engineering awaits your genius.

Contents 1 Introduction 1.1 Biological Basis for Genetic Algorithms . . . . . . . . . . . . . . . . . 1.2 Artificial Neural Networks in Nonlinear Dynamical Control . . . . . . 2 Recessive Gene Model in Genetic Algorithm Optimization 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Recessive Gene Model . . . . . . . . . . . . . . . . . . . . . . 2.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Easom Function . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Grienwangk Function . . . . . . . . . . . . . . . . . . . 2.3.3 Schaffer’s F6 Function . . . . . . . . . . . . . . . . . . 2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 GA Parameters . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Search Criteria . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Effect of Recessive Gene P on Searching Performance . 2.4.4 Effect of Population Size N on Searching Performance 2.4.5 Effect of Generations G on Searching Performance . . . 2.5 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Hybrid Controller for a Nonlinear Dynamical Plant 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.2 Modeling and Problem Formulation . . . . . . . . . . 3.2.1 Trailer-truck Model . . . . . . . . . . . . . . . 3.2.2 Statement of the Problem . . . . . . . . . . . 3.3 Control System . . . . . . . . . . . . . . . . . . . . . 3.3.1 Hybrid Controller . . . . . . . . . . . . . . . . 3.3.2 LQR Controller . . . . . . . . . . . . . . . . . 3.3.3 Construction of the NC . . . . . . . . . . . . . 3.3.4 Training of the NNs with Real-coded GA . . . 3.3.5 Evaluation Function of the NC . . . . . . . . 3.4 Simulations and Results . . . . . . . . . . . . . . . . 3.4.1 Parameters . . . . . . . . . . . . . . . . . . . 3

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CONTENTS

3.5 3.6

3.4.2 Design Performance . . . . . . . . . . . . . . . 3.4.3 Control Performance . . . . . . . . . . . . . . 3.4.4 Control Performance for the Initial Conditions Analysis of Results . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . .

4 Real-time Control Design for a 4.1 Introduction . . . . . . . . . . 4.2 Modeling . . . . . . . . . . . . 4.3 Control System . . . . . . . . 4.4 On-line Control Sequence . . . 4.5 Results of Simulations . . . . 4.5.1 Parameters . . . . . . 4.5.2 Design Performance . . 4.5.3 Control Performance . 4.6 Highlights . . . . . . . . . . . 4.7 Conclusion . . . . . . . . . . .

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Multi-trailer System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Concluding Remarks

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A List of Publications

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Chapter 1 Introduction 1.1

Biological Basis for Genetic Algorithms

Essentially, GAs are a method of ‘breeding’ computer programs and solutions to optimization or search problems by simulating biological evolution. GAs are based on the Darwinian model of survival of the fittest: by this theory, an individual with the most favorable genetic characteristics is more likely to survive and produce offspring [1,2]. Processes loosely based on natural selection, crossover, and mutation are repeatedly applied to a population of binary strings which represent potential solutions. Over time, the number of above-average individuals increases and better fit individuals are created, until a good solution to the problem at hand is found. Main aspects of using GAs are: 1. definition of the objective function, 2. definition and implementation of the genetic representation, and 3. definition and implementation of the genetic operators. Once these have been defined, the generic GA should work fairly well. The solutions to the problem are encoded in a structure that can be stored in the computer in chromosome form. A chromosome comprises a collection of genes, which are simply the parameters to be optimized (Figure 1.1).

Figure 1.1: Elemental coding representation The GA creates an initial population, evaluates this population for fitness, then evolves the population through multiple generations using randomized genetic operators of selection, crossover, and mutation, in the search for a good solution for 5

6

CHAPTER 1. INTRODUCTION

the problem at hand (Fig. 1.2). It uses various selection criteria so that the best individuals are chosen for subsequent mating. The objective function determines how ‘good’ each individual is. GAs search for the optimal solution until a specified termination criterion is met. Initial parents (randomized) Population

NC ...

NC ...

Gene (16-bit variable

...

Evaluation

Selection Death Survival NC ...

NC ...

...

Crossover NC

NC

Mutation NC

Parents ...

...

...

Offspring

Figure 1.2: GA implementation, with 2-point crossover GAs have been successfully employed in applications related to artificial intelligence, particularly machine learning and game theory, and also in image processing, manufacturing, economics, political science etc. The most widely reported application, however, is in the area of search and optimization related to complex engineering problems, for which the conventional optimization methods are inadequate. They have proved to be robust search methods for a wide range of optimization problems.

1.2. ARTIFICIAL NEURAL NETWORKS IN NONLINEAR DYNAMICAL CONTROL7

1.2

Artificial Neural Networks in Nonlinear Dynamical Control

The idea of constructing an artificial brain or neural network (NN) has been proposed many times (McCulloch and Pitts, 1943; Rosenblatt, 1957; and others). The brain is an immensely complex network of neurons, synapses, axons, dendrites and so forth (Figure 1.3(a)). Through detailed modeling of these elements, a simulated network that is capable of diverse behaviours may be constructed. Earlier works by Hebb (1949) indicated that neural networks could learn to recognize patterns by weakening and strengthening the synaptic connections between the neurons. The NN would be sterile in the absence of an algorithm to usefully adjust the connections of the architecture. It is now widely accepted that multiple layered NNs with variable connection strengths, bias terms, and nonlinear Sigmoid functions can approximate arbitrary measurable mapping functions (Cybenk, 1989; Hornik et al., 1989; Barron, 1993). APICAL DENDRITES

SOMA BIAS

bk

INPUT SIGNALS

BASAL DENDRITES

x1

wk1

x2

wk2

AXON

. . . xm

SYNAPTIC TERMINALS

(a) Biological NN

ACTIVATION FUNCTION

vk

. . .

(.)

yk OUTPUT

SUMMING JUNCTION

wkm SYNAPTIC WEIGHTS

(b) Stochastic model, ANN

Figure 1.3: Basic neuron structure

8

CHAPTER 1. INTRODUCTION

An Articicial neural network (ANN) is synonymous with a plastic brain: plasticity permits the system to adapt to its sorrounding environment, and is essential to the functioning of the processing units of the NN. A neural network derives its computing power through its massively parallel distributed structure and its ability to learn and therefore generalize. The use of ANNs offers the following properties and capabilities: 1 Nonlinearity - may aptly be applied to the analysis of input signals generated by nonlinear physical mechanisms; 2 Learning - modification of the free parameters (synaptic weights) from generic training samples to minimize difference between desired and actual response of input signal in accord with an appropriate statistical criterion; 3 Adaptivity - the NN buit-in capability to adapt their synaptic weights to changes in their sorrounding environment. System modeling and control applications may use soft computing techniques, particularly NNs and GAs, in formulating designs that offer overall system stability. A dynamic system is one whose state varies with time and may be cast in the form of a system of first-order differentals, viz: d x(t) = F (x(t)) dx

(1.1)

where the nonlinear function F is vector valued, each element of which operates on a corresponding element of the state vector: x(t) = [x1 (t), x2 (t)....., xN (t)]T

(1.2)

The state space may be Euclidean or some other differentiable manifold, such as a circle, sphere, torus etc. It is important because it provides a conceptual tool for analyzing the dynamics of the nonlinear system described in Eq. (1.1). It does so by focusing attention on the global characteristics of the motion rather than the detailed aspects of analytic or numeric solutions of the equation. Use of NNs is crucial in achieving stability of the nonlinear system. The equilibrium state x¯ is said to be asymptotically stable if it is stable and all trajectories of the system converge to x¯ as time t approaches infinity. Design of a successful controller for a nonlinear plant requires that global asymptotic stability is attained, whereby the system ultimately settles down to a steady state for any choice of initial conditions. This is the basis of the controller design for multi-trailer systems.

Chapter 2 Recessive Gene Model in Genetic Algorithm Optimization 2.1

Introduction

GAs aptly perform multi-directional global searches and have successfully been widely applied in many optimization problems [3,4]. However, their main setback is loss of diversity, leading to either evolutionary stagnation or premature convergence. RGM is a dual gene system that offers diversity and exploits local continuities thereby avoiding the mechanism of evolutionary stagnation [A.1(1)]. I used the basic information on Mendelian genetics to illustrate that a recessive characteristic might significantly affect a closed population [1,5]. In observing living organisms, characteristics of the offspring do not always resemble those of parents. A dual gene system exists whereby some alleles are dominant hence always expressed, while some are recessive, that is, only expressed under certain conditions. However, the individual preserves the recessive gene, which is sent to the next generation thus maintaining the diversity of the characteristics of the living organism. RGM is an effective evolutionary tool in numerical function optimization with binary coded GAs.

2.2

Recessive Gene Model

Figure 2.1(a) shows the structure of the double gene scheme. In usual GA systems only the dominant genes appear, hence in the first generation of two individual parents F1 and F2 , only the dominant characteristics A and B appear, respectively. For the purely dominant genes case, only one offspring, S1 , appears in the second generation. By introducing the dual gene, there is an additional three offspring in the second filial generation: S2 , S3 and S4 . Further, each of the individual offspring S1 , S2 , S3 and S4 with 1 dominant and 1 recessive gene, will simultaneously have their individual complements S10 , S20 , S30 and S40 appearing, respectively. 9

10CHAPTER 2. RECESSIVE GENE MODEL IN GENETIC ALGORITHM OPTIMIZATION Dominant Chromosome

F1

F2

A

B D

C

Recessive Chromosome First Generation

S1 A B

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B C G H

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(a) Structure

D

A

(b) Possible offspring of F1 and F2 in the 2nd filial generation.

Figure 2.1: Schematic of RGM.

As an illustration, Fig. 2.1(b) shows parents F1 and F2 from the 1st generation will produce 2 offspring, S3 and its complement S30 , in the 2nd generation. Offspring S3 is comprised the dominant gene BC and recessive AD, while its complement S30 has CB and DA as dominant and recessive, respectively. Let P denote the probability that a recessive gene is selected to be a dominant gene in the next generation. With the functional rise in P , there is a large pool of better-fit recessive individuals transforming to dominant status for generational mating. Two special cases may occur: when P = 0%, the evolutionary strategy is the same as using only the dominant gene and the offspring is S1 , while when P = 100% the offspring is S4 , the absence of any dominant genes from the parents. The appeal of the dual gene concept lies in potential to yield diversity in offspring.

2.3

Problem Formulation

To work in a context, RGM was applied on a set of three benchmark numerical functions of which global optima are known [6,7,8]. The test environment comprised Easom’s, the Grienwangk and Schaffer’s F6 functions.

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2.3. PROBLEM FORMULATION 2.3.1

Easom Function f(x,y) 0 -0.2 -0.4 -0.6 -0.8 -1 -10

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0 x

5

10 -10

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5 y

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Figure 2.2: Easom function The Easom function is unimodal, described by Eq. (2.1) and takes the form of Figure 2.2. The function has been inverted for minimization. f (x, y) = − cos(x) cos(y)e−((x−π)

2 +(y−π)2 )

(2.1)

The analytical global minimum of the Easom function has a small area relative to the search space, and has numerical value f (x, y)min = −1.0, 2.3.2

when (x, y) = (π, π).

(2.2)

Grienwangk Function

f(x,y) 5 4 3 2 1 0 -20

12 10

-10

0 x

10

(a) Form

20 -20

-10

0

10 y

20

f(x,y)

8 6 4 2 0 -200 -150 -100 -50

0 y

50

100 150 200

(b) Effect of cosine modulation on objective minimum Figure 2.3: Grienwangk function.

This is a typical nonlinear multimodal function, with a complexity O(n ln(n)), where n is the number of the function’s parameters. It has the general form ! ! n n X Y x2i xi f (xi ) = 1 + − cos( √ ) (2.3) i i=1 4000 i=1

12CHAPTER 2. RECESSIVE GENE MODEL IN GENETIC ALGORITHM OPTIMIZATION The summation terms produce a parabola while the product term creates strongly codependent subpopulations. Dimensions of the search range increase on the basis of the product term, resulting in the decrease of the local minimums, that is, the more the search range is increased, the flatter the function. The global minimum is: f (x, y)min = 0, (x(i), y(i)) = (0, 0), i = [1, n].

(2.4)

In this study, n=2 and the descriptive equation for the function becomes x2 + y 2 f (x, y) = 1 + 4000

!

x y − cos( √ )cos( √ ) 1 2

(2.5)

Fig. 2.3(a) shows the Grienwangk function when n=2. It is symmetrical about the global minimum on both axes. Fig. 2.3(b) illustrates that the function is highly multimodal due to the addition of the cosine modulation that produces many widespread local minima whose location is regularly distributed. The local optima are above the parabola level that is produced by the summation term. 2.3.3

Schaffer’s F6 Function

1 0.8 0.6 0.4 0.2 0 -15 -10 -5

0 x

5

(a) Form

-5 -10 10 15 -15

0

5

15 10 y

f(x,y)

f(x,y) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -15

-10

-5

0 y

5

10

15

(b) Sectional view on the axis x=0 Figure 2.4: Schaffer’s F6 function

This parametric optimization problem is multimodal, with the descriptive form: √ sin2 ( x2 + y 2 ) − 0.5 f (x, y) = 0.5 + . (2.6) 1 + 0.001(x2 + y 2 )2 The function is a two-parameter “ripple”, like the waves in a pond caused by a pebble and is symmetrical on x and y axes (Fig. 2.4(a)). For (x, y) ∈ [−15, 15] there are several local optima but the centermost two peaks represent the circular global optimum (Fig. 2.4(b)). Table 2.1 gives the numerical values for the various √ 2 maxima. Each set of values at the respective optima is defined on a locus r = x + y 2 .

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2.4. SIMULATION RESULTS Table 2.1: Numerical solutions Optima r Global π/2 1st local 3π/2 2nd local 5π/2 3rd local 7π/2

2.4

for Schaffer’s F6 f (x, y) 0.996989 0.838081 0.606185 0.532483

Simulation Results

In this section, results of simulations for the search performance utilizing RGM for Easom’s, the Grienwangk and Schaffer’s F6 functions are presented. 2.4.1

GA Parameters

A random generation of (x, y) values in Euclidean space was used in the GA search for the three functions. Table 2.2 shows the set of constant parameters utilized in the GA search. The criterion for performance was the rate of successful evolution S% against the variables: percentage of recessive gene P , mutation probability M , generations G, and population size N . The optimal population size was 50 for most analyses, but I also investigated the search performance for N ∈ [20, 100]. Table 2.2: Constant GA parameters Parameter Value/Method Binary bit length, B 16 Selection pressure, parents Pp 0.5 Selection pressure, children Pc 0.6 Selection scheme Roulette wheel Crossover 2-point 2.4.2

Search Criteria

Success in evolution was established against the conditions set in Table 2.3 for the respective functions. The global search space was utilized as the initial sampling region for both Easom and Grienwangk’s functions. However, spot distribution was used as initial sampling space for Schaffer’s F6 to confirm robustness of the recessive gene scheme in multimodal optimization. Table 2.3: Search space and performance criterion Function Search space Initial sampling space Success criterion Easom (x, y) = [−10, 10] (x, y) = [−10, 10] f (x, y) < 0.999 Grienwangk (x, y) = [−20, 20] (x, y) = [−20, 20] f (x, y) < 0.1 Schaffer’s F6 (x, y) = [−15, 15] (x, y) = [11, 13] f (x, y) > 0.99

14CHAPTER 2. RECESSIVE GENE MODEL IN GENETIC ALGORITHM OPTIMIZATION 2.4.3

Effect of Recessive Gene P on Searching Performance

80

80

60

60 S [%]

100

S [%]

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M = 0% M = 10% M = 20% M = 30% 0

20

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(a) Easom

(b) Grienwangk

100

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20 M = 0% M = 10% M = 20% M = 30%

40

20

0

M = 0% M = 10% M = 20% M = 30% 0

20

40

60

80

100

P [%]

(c) Schaffer’s F6 Figure 2.5: Variation of search performance with P as the primary variable.

Figure 2.5 shows variation of search perfomance S% with recessive gene P % for the three functions as mutation is consecutively increased from 0% to 30%. It is generally observed that for all the functions search is more successful when utilizing the recessive gene, (P 6= 0%), relative to the dominant gene, P =0%. For the Easom function (Fig. 2.5(a)), a significant observation made is that in the absence of mutation, the search improves only with increase in P . A similar pattern emerges in the Grienwangk search (Fig. 2.5(b)). However, when P is too large, search performance is degraded in either case. In Fig. 2.5(c) it is seen that mutation is essential in this multimodal function, as there is practically no search when M = 0%.

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2.4. SIMULATION RESULTS 2.4.4

Effect of Population Size N on Searching Performance

80

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S [%]

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30

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(b) Grienwangk

100

80

60 S [%]

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20 P = 0% P = 10% P = 20% P = 30%

40

20

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P = 0% P = 10% P = 20% P = 30% 20

30

40

50

60 N

70

80

90

100

(c) Schaffer’s F6 Figure 2.6: Effect of population size N on search performance. (M = 20%).

From Figure 2.6 it is observed that in all the three cases, firstly, success in search improves with population size, N . Secondly, performance is a notch higher when P 6= 0% than when P = 0%.

Figures 2.6(a) and (b) show the implementation of the scheme for Easom’s and Grienwangk’s functions respectively. More importantly, search performance is still high at low population sizes when P 6= 0%. It is seen from Fig. 2.6(c) for Schaffer’s F6 that when P = 0% the evolution of solutions generally stagnates at N =70; by the nature of this function, the algorithm converges to some local optima when P = 0%.

16CHAPTER 2. RECESSIVE GENE MODEL IN GENETIC ALGORITHM OPTIMIZATION 2.4.5

Effect of Generations G on Searching Performance

0

4.5

P = 0% P = 10% P = 20% P = 30%

-0.1

P = 0% P = 10% P = 20% P = 30%

4

-0.2

3.5

-0.3

3

-0.4 Fm

2 -0.6 1.5

-0.7

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1 0.95 0.9 0.85 0.8 Fm

Fm

2.5 -0.5

0.75 0.7 0.65 0.6 P = 0% P = 10% P = 20% P = 30%

0.55 0.5

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50 G

60

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(c) Schaffer’s F6 Figure 2.7: Effect of recessive gene on rate of convergence. (M = 20%).

Figure 2.7 shows variation of the mean of the best values of the respective functions on 100 trials per generation Fm , with generations, G, as the recessive gene in the population is gradually increased. It is noted that both rate and accuracy of convergence to the optimal solution is enhanced by the presence of the recessive gene than when P =0%. Figures 2.7(a) and (b) give a strong indication of the advantage of the recessive gene in these global minima searches, where the attainment of the best mean value, Fm , of the functions is more rapid when P 6= 0%. Figures 2.7(a) and (c) indicate that for P = 0% there is premature convergence and stagnation at some local optima.

2.5. ANALYSIS OF RESULTS

2.5

17

Analysis of Results

Using RGM, I compared the performance of a GA driven by a gradually increasing recessive probability to the dominant genes case. The main performance metric is the efficiency of the GA. Realizing that a significant population size is necessary for GA performance so as to avoid premature convergence and stagnation, this invariably has a computational penalty. Introducing the recessive gene ensures rapidity and precision in attaining convergence by virtue of the diversity in selectable individuals. This ensures effective optimization in few generations thereby reducing CPU time costs. The benchmark functions utilized were selected for their associated complexity. The unimodal Easom function’s global minimum has a small area relative to the search space. The Grienwangk function is difficult for most GAs because the product term causes the variables to be highly interdependent. Schaffer’s F6 function is usually very difficult for most hill-climbing heuristics due to its circular local maxima; it is all too easy for a hill-climbing algorithm to wander around in circles rather than abandoning whatever ring it has found for an inner one. Figures 2.5(a), (b) and (c) are the basis of the research. For all the test functions, the success rate in search performance is lower when using only dominant genes but increases with P . For the Easom and Grienwangk functions it is noted that when M = 0% and P 6= 0%, search performance is still quite high relative to the search with dominant genes. Since mutation is integral to the search, it may be concluded that the recessive gene performs the function of mutation. The recessive gene introduces a certain amount of randomness thus helps the search find solutions that crossover alone might not encounter in the absence of the mutation operator. Figures 2.6(a), (b) and (c) show that population size N , greatly influences the search. Recognizing that a small N provides an insufficient sample for effective GA performance while a large N undoubtedly raises the probability of the algorithm performing an informed and directed search, the latter is computationally expensive. Generally, convergence is expected after N log(N ) functional evaluations, since computational cost, Cc , in terms of time and memory, is in direct proportion to N [6,9,10]. As an example, theoretically, Cc for N = 100, P = 0% should be equal to the Cc for N = 50, P 6= 0%. It can be seen from the respective Figs. 2.6(a), (b) and (c) that performance is superior for the case N = 50, P 6= 0%, than for N = 100, P = 0%. Figure 2.7 shows that for all the functions the search performance increases with generations, but there is inadequate search with purely dominant genes due to stagnation. More specifically, P greatly influences rate of convergence. There is premature convergence to solution for Easom’s function when P = 0%. For the Grienwangk function rate is slowest when P = 0% while there is premature convergence to some local optima for Schaffer’s F6 function when P = 0%. Maintaining population diversity in the GA search ensures maintainability of several sub-optima yet avoids quick convergence and stagnation to local sub-optima.

18CHAPTER 2. RECESSIVE GENE MODEL IN GENETIC ALGORITHM OPTIMIZATION RGM ensures accurate convergence at low N and this is desirable for memory storage during computations. For Schaffer’s F6, spot distribution was used for the initial sample space so as to test the efficacy of RGM. The results, though slightly lower, show the same characteristic as for global search. RGM requires no prior knowledge of the modality of the fitness landscape and works based on local convergence which is quick and assumes that the global optimum can be obtained by the interaction of locally optimized individuals. The search is comprehensive even when those optima have small basins of attraction and also when N is not large enough to form the species at each optima. The major setback with RGM is that it is computationally expensive.

2.6

Conclusions

In this chapter, the author has proposed a new, evolutionary technique for evaluating continuous, multi-dimensional and multimodal functions by optimization. Genetics provided a very good conceptional framework for optimization inspired by nature to show that RGM as a GA tool can be efficiently applied to complex functional optimization. Simulation results have shown that search performance with recessive genes is superior to the purely dominant genes case. RGM avoids stagnation by offering diversity, works very well at low sampling population sizes and that the recessive gene ocassionally performs the function of mutation. RGM has the capability of performing consistently well on a broad range of problem types, unimodal and multi-modal, thus very robust. Perspectively, the author believes the management of computational time could be further improved.

Chapter 3 Hybrid Controller for a Nonlinear Dynamical Plant 3.1

Introduction

The truck-backing-up exercise is a kinematic inverse modeling problem. Difficulty in control intensifies with increase in the number of trailers, as the appropriate instantaneous response is changing and dependent on the angles between the trailers, and the angle between truck and trailer. Complexity arises, firstly, due to the system being a single-input, multiple-output problem, whereby the plant’s outputs should closely follow the independent reference signals. Secondly, achieving good backward movement is difficult because of the restraints of physical angular limitations. Due to these constraints the system is impossible to globally stabilize with standard smooth control techniques, since some initial states lead to jack-knife locks. In the recent past, neural networks have been trained to control reversing trucktrailers and many control systems for the problem have been proposed [11,12]. With regard to the neurocontrol system, the seminal work on the subject is by Nguyen & Widrow [13]. Others include Widrow et al [14] and Hougen et al [15]. Jenkins and Yuhas [16] have presented a small-sized NC based on back-propagation. In previous studies, Kinjo et al proposed various control methods for a single trailertruck combination [17], for a truck connected to two trailers [18] and for a five trailer-truck combination [19], using NCs evolved by GA. Though successful, the methods are computationally expensive. In this Chapter, a hybrid control scheme is proposed for the stabilization of backward movement along simple paths for a vehicle composed of a truck and six trailers. The hybrid comprises the combination of a linear quadratic regulator (LQR) and a neurocontroller (NC) that is trained by a GA. The evaluation function of the NC in the hybrid has been modified from the conventional type to incorporate both the squared errors and the running steps errors [A.1(2), A.2(1)]. 19

20CHAPTER 3. HYBRID CONTROLLER FOR A NONLINEAR DYNAMICAL PLANT

3.2

Modeling and Problem Formulation

3.2.1

Trailer-truck Model

u(t) x0(t)

L

L

L

L

L

l

L

x1(t) x2(t)

x3(t) x4(t)

x5(t)

x7(t) x9(t)

x6(t) x8(t)

x11(t) x10(t) x12(t) x14(t)

Figure 3.1: Schematic model of the trailer-truck

Table 3.1: Parameters of the trailer-truck system Quantity l, L ∆t v u(t) x0 , x2 , x4 , x6 , x8 , x10 and x12 x1 x3 x5 x7 x9 x11 x13 x14

Description Length (truck, trailer) Sampling time Speed of system Steering angle Angles of truck, 1st, 2nd, 3rd, 4th, 5th and 6th trailers Angular Differences, between: Truck and 1st trailer 1st and 2nd trailer 2nd and 3rd trailer 3rd and 4th trailer 4th and 5th trailer 5th and 6th trailer Vertical position of 6th trailer Horizontal position of 6th trailer

x13(t)

0

3.2. MODELING AND PROBLEM FORMULATION

21

Figure 3.1 details the geometry of the control object: the six trailer-truck model with an actuated front steering, and its orientation in a coordinate base. The steering angle u(t), is the input to the plant and is determined by the state variables x1 −x13 . Table 3.1 explains the terminology used for the trailer-truck system parameters. The kinematics of the system are described by the set of equations (3.1) - (3.15). v∆t tan[u(t)] l x1 (t) = x0 (t) − x2 (t) v∆t sin[x1 (t)] x2 (t + 1) = x2 (t) + L x3 (t) = x2 (t) − x4 (t) v∆t x4 (t + 1) = x4 (t) + sin[x3 (t)] L x5 (t) = x4 (t) − x6 (t) v∆t x6 (t + 1) = x6 (t) + sin[x5 (t)] L x7 (t) = x6 (t) − x8 (t) v∆t x8 (t + 1) = x8 (t) + sin[x7 (t)] L x9 (t) = x8 (t) − x10 (t) v∆t x10 (t + 1) = x10 (t) + sin[x9 (t)] L x11 (t) = x10 (t) − x12 (t) v∆t x12 (t + 1) = x12 (t) + sin[x11 (t)] L x0 (t + 1) = x0 +

(3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) (3.13)

x12 (t + 1) + x12 (t) ] (3.14) 2 x12 (t + 1) + x12 (t) ] (3.15) x14 (t + 1) = x14 (t) + v∆t × cos[x11 (t)] × cos[ 2 The expressions (3.1) - (3.15) have the form of time-dependent state equations and by setting the initial conditions, ongoing nominal values of the position and orientation of the vehicle are produced. This restricts the controlled statement to only the truck steering angle u(t). The equations describe a system that has nonlinear characteristics hence linear system theory may not be applied. x13 (t + 1) = x13 (t) + v∆t × cos[x11 (t)] × sin[

3.2.2

Statement of the Problem

The steering angle u(t) is to be controlled such that the system is asymptotically stabilized along a desired trajectory, in this case, along the straight line x13 (t)=0. This requires that the angular differences, angle of last trailer and the vertical position are kept to a minimum, thus X(t) = [x1 (t), x3 (t), x5 (t), x7 (t), x9 (t), x11 (t), x12 (t), x13 (t)]T −→ 0.

(3.16)

22CHAPTER 3. HYBRID CONTROLLER FOR A NONLINEAR DYNAMICAL PLANT

3.3

Control System

3.3.1

Hybrid Controller

Figure 3.2 shows the Hybrid control system. The plant to be controlled is the six trailer-truck system. The stabilizing controller of the hybrid is based on both the LQR, which covers the linear part, and the NC that covers the nonlinear part of the system. X ref is the reference for the state variables while GA denotes the genetic algorithm procedure. Hybrid controller Xref +

uL +

LQR

_

+

u

Trailer-truck

X

system

GA Neuro controller

uN

Figure 3.2: Hybrid control system The LQR and NC receive the error of angles x1 , x3 , x5 , x7 , x9 , x11 , x12 and position x13 as inputs and output the steering angles uL (t) and uN (t), respectively. The input to the plant is the hybrid steering angle u(t), comprising the outputs from the LQR and the NC respectively, u(t) = uL (t) + uN (t)

(3.17)

The system outputs the state vector X. Considering that this is a regulator problem, necessarily X ref = 0. 3.3.2

LQR Controller

I used the linear quadratic regulator from optimal control theory [20,21] to solve the linear part of the design problem in which the state is accessible. The stochastic formulation for the LQR design in this system is linearized and described by X(t + 1) = AX(t) + BuL (t)

(3.18)

where A and B are linearized parameters of the nonlinear system described by Eqs.(3.1) - (3.15). A and B are obtained from the assumption that the angular differences and angle of last trailer have magnitudes, respectively [x1 , x3 , x5 , x7 , x9 , x11 ]  1,

x12  1

(3.19)

23

3.3. CONTROL SYSTEM The obtained parameters are as follows [11,12,17]: 

A=

                      

v∆t 0 0 1− L v∆t v∆t 1− 0 L L v∆t v∆t 0 1− L L

0

···

0

···

0

···



0  0

0

··· 0

···

0

0

···

0

v∆t , 0 , ··· , 0 B= l 

T

.

v∆t 1 0 L 2 (v∆t) v∆t 1 2L

          ,           

(3.20)

(3.21)

The LQR cost function is the sum of the steady-state mean-square weighted state X, and the steady-state mean-square weighted actuator signal uL (t): J=

∞ X t=0

{X(t)W X(t) + wuL (t)2 }

(3.22)

where W is a positive semidefinite weight matrix and w is a positive scalar quantity; the first term penalizes deviations of X(t) from zero, and the second represents the cost of using the actuator signal. One of the methods of LQR design is by use of the system gain to control the system errors. The gain values of the linear combiner are calculated through some state-feedback technique, in this case, by use of the Riccati equation. Thus, linear output of the discrete system is: uL (t) = −GX(t).

(3.23)

where G is the system gain. The gain values are predetermined and set prior to running the actual plant. 3.3.3

Construction of the NC

The NC covers the nonlinear part of the system. I used a 3-layered, 8-5-1 configuration neural network with nonlinear activity functions for both the hidden and output layers. For the hidden layer I utilized the Sigmoid function: 1 f (x) = (3.24) 1 + e−x while the activity function of the output layer is a cubic, f (x) = ax3 .

(3.25)

With this function, when the magnitude of the state variable X(t) is small, the control object tends to be linear and it is easy to get the desired output, uN (t)=0.

24CHAPTER 3. HYBRID CONTROLLER FOR A NONLINEAR DYNAMICAL PLANT 3.3.4

Training of the NNs with Real-coded GA

Table 3.2: Initial starting configurations Pattern x0 , x2 , x4 , x6 , x8 No. x10 and x12 [rad] 1 0 2 π/4 3 π/2 4 0 5 π/4 6 π/2 7 0 8 π/4 9 π/2

x13 [m] 0.0

3.0

6.0

Training the NN involved setting the position and orientation of the system, initializing the backward control and evaluating the errors. Table 3.2 gives the initial configurations of the starting positions. The number of patterns used, P =9. The methodology of training the NCs is as follows. 1 Connecting weights of the NN are randomly set. 2 The trailer-truck is set to an initial configuration, such as pattern no. 1. 3 The truck backs up using the NC, undergoing several individual cycles of backing up until it stops on attaining a preset number of running steps, or gets out of control. 4 The final error of the trailer-truck system is recorded. This error is a function of the state variables and the running steps. 5 Next, the trailer-truck is placed in another initial configuration, say, pattern no. 2 and let to back up until it stops. 6 On completion of control trials from all the nine configurations P = 9, the control performance of the NC is evaluated, as explained in subsection 3.3.5. All the NCs in the population are evaluated in the same fashion. A real-coded GA is used to evaluate and adjust the NN connecting weights appropriately and obtain the best individual from among the evolved NCs. The GA procedure relies on the Blend crossover (BLX) method. The BLX tool utilizes interval schemata and has been shown to have good training characteristics for NNs by real-coded GAs [22,23]. A real-coded GA is employed for two reasons: the range of the connecting weights’ values is unbounded, and quantization errors normally associated with bit-string GAs are non-existent.

3.3. CONTROL SYSTEM 3.3.5

25

Evaluation Function of the NC

In the control simulation, the NCs are awarded a huge penalty if they cause the plant to run out of control, Epenalty = 1.0 × 106 . Else, the NC is gauged with the summation of weighted error of the evaluation function. The evaluation function E for the NC training by the GA consists of the squared errors ES , and the cumulative running steps error ET when system is out of control, computed from the relation: E = αES + βET

(3.26)

where α and β are weights of the squared errors and running steps errors respectively. The first term ES gives the squared errors accumulated for all the patterns, P ES =

P X

Ep

(3.27)

p=1

and the squared error per pattern Ep , is evaluated by the expression ref ref 2 end 2 end 2 Ep = q1 (xref − xend 1 1p ) + q3 (x3 − x3p ) + q5 (x5 − x5p )

ref ref 2 end 2 end 2 + q7 (xref − xend 7 7p ) + q9 (x9 − x9p ) + q11 (x11 − x11p ) ref end 2 end 2 + q12 (xref 12 − x12p ) + q13 (x13 − x13p )

(3.28)

where xref is the reference variable and xend is the final value of the state variable p which starts from any initial configuration of p. The factor q is the weight of the squared error function and adjusts the importance of the control variables. The second term ET , in Eq. (3.26) refers to the penalty suffered by the system when out of control ET =

P X

p=1

(tmax − tp )

(3.29)

where tmax is the maximum steps set for the design and tp refers to the advanced steps from initial configuration per pattern p, prior to the out of control state. Ideally, ET = 0 when no pattern exhibits the out of control state. Generally, an excellent NC is one that is highly evolved, with a small value of E. In the selection process for the parent individuals, the Roulette wheel technique is utilized. Denoting P as the probability that an NC with error E will serve as parents, then f (3.30) F where f is the fitness function of the NC and F is the total fitness for all the NCs, P =

1 E The selected offspring are then used in the crossing operation. f=

(3.31)

26CHAPTER 3. HYBRID CONTROLLER FOR A NONLINEAR DYNAMICAL PLANT

3.4

Simulations and Results

3.4.1

Parameters

(a) The 6 Trailer-truck Model The modeling of the six trailer-truck is based on the quantities in Table 3.3.

Table 3.3: Parameter values for the six-trailer-truck system Parameter Value Trailer length, l 0.3m Trailer length, L 1.0m Velocity of system, v -0.2m/s Sampling time, ∆t 0.25s

For the state and input constraints of the system the following limits were set, respectively, for the steering angle and the relative angles: |u| < π/2

and

| x1 , x3 , x5 , x7 , x9 , x11 | < π/2

(3.32)

A consequence of the latter constraints in Eq. (3.32) is the appearance of the jackknife configurations, corresponding to at least one of the relative angles reaching its saturation value, π/2 rad. (b) Design of LQR and NC In solving the Riccati equation, the diagonal of the matrix W in the Cost function (Eq. (3.22)) may be set variously, to achieve the optimum gain. In these simulations W is weighted such that W =diag[1,1,1,1,1,1,1,100], and weighting factor w=100. With uL (t) as given in Eq. (3.23), the following values of the gain G were utilized: G = [ − 4.01, 22.87, − 71.72, 132.04, − 138.49, 68.86, − 3.76, 0.72].

(3.33)

It is appreciated that some of the G values may be large thus by Eq. (3.23) the steering angle command uL (t), may exceed the physical limitations of the system leading to the trailer-truck getting out of control. For the construction of the NCs, trial and error was used in determining and setting a = 0.1 for the output neuron activity function, thus f (x) = 0.1x3 in Eq. (3.25). The idea being to linearize the output state variables, this cubic output neuron activity function had the best performance as its effective range is near zero in the region of interest.

27

3.4. SIMULATIONS AND RESULTS (c) GA Parameters and Evaluation function

The range factor for the BLX interval schemata is 0.8. The other GA properties are shown in Table 3.4. Table 3.4: Constant parameters of the GA Parameter Value/method Population 50 No. of offspring 30 Selection scheme Roulette wheel Crossover BLX

For the evaluation function, the values in Table 3.5 were determined and assigned accordingly: Table 3.5: Parameters of the Evaluation function Parameter Value Remark α, β 1.0 Eq. (3.26) q1 , q3 , q5 , q7 , q9 , q11 , q12 1.0 Eq. (3.28) q13 0.1 Eq. (3.28) tmax (steps) 600 Eq. (3.29)

3.4.2

Design Performance

Table 3.6: Rate of evolutionary success of Hybrid and NC controller. (α = 1.0). Success Rate, [%] Hybrid controller Only NC No. of trailers 4 5 6

β = 1.0 90 81 20

β = 0.0 69 31 0

β = 1.0 β = 0.0 78 0 6 0 0 0

Table 3.6 shows the results of investigation on trucks with 4, 5 and 6 trailers and the success rate of the evolution of both hybrid and NC controllers. Criterion for successful evolution is percentage of NCs in 3000 generations with error E ≤ 0.001. β = 0.0 means the running steps error is not considered in the evaluation function.

28CHAPTER 3. HYBRID CONTROLLER FOR A NONLINEAR DYNAMICAL PLANT

E, ES

Firstly, it is observed that optimal performance is degraded in either controller as the number of connected trailers increases. Secondly, controller design with the NC only is not successful with β = 0.0; though the NC improves with β = 1.0, it does not yield a suitable controller for the six trailer case. The results show that modifying the evaluation function to include the running steps error improves either controller design. It is clear that the hybrid controller with β = 1.0 is the most suitable design for the six trailer-truck configuration.

104 103 102 10 1 −1 10 10−2 10−3 10−4 10−5

E ES

0

500 1000 1500 2000 2500 3000 Generation

Figure 3.3: Effect of generational training of the NCs on system errors ES and E. (α = 1.0, β = 1.0).

Figure 3.3 shows one of the successful evolutions of the hybrid controller when α=1.0 and β=1.0. The system is designed for 3000 generations. It is observed that the squared error ES is relatively small throughout the evolution. At 300 generations the running steps error ET = 0 and beyond this only ES is minimized. Thus the hybrid is able to control all the state variables of the six trailer-truck system for the maximum running steps from all the patterns. 3.4.3

Control Performance

Figure 3.4 is an example of the controlled results, for pattern no. 9, where the initial orientation and starting vertical position are set to π/2 rad and 6.0m respectively. It is observed that the hybrid controller is able to guide the trailertruck along x13 successfully, within the training region on x14 ∈ [-5.0, 10.0]m.

29

3.4. SIMULATIONS AND RESULTS

15 10 Start

x13 [m]

5 End

0 -5 -10 -15 -15

-10

-5

0

5

x14 [m]

10

15

Figure 3.4: Trajectory for pattern no. 9. Display interval: 140 time steps.

6

1

x11 x9

0

x7 x5 x3 x1

-1 -2

4

x12 x13 [m]

Angles [rad]

2

0

100

2 0 -2 -4

200

300

400

Step

(a) Angular differences

500

600

-6

0

100

200

300

400

500

600

Step

(b) Vertical position of trailer 6

Figure 3.5: Variation of the state variables X with running steps Figure 3.5 (a) shows variation of the relative angles (x1 , x3 , x5 , x7 , x9 , x11 ) and angle of last trailer x12 with running steps. It is seen that the angles are controlled successfully by the hybrid controller to within the design range [−π/2, π/2] rad. Figure 3.5(b) shows that the vertical position x13 of the last trailer smoothly falls to the desired trajectory, x13 =0. The hybrid successfully controls all the state variables. At 300 steps all the state variables are very small and the system eventually realigns and stabilizes at the desired path x13 =0.

2

2

1

1 uN [rad]

uL [rad]

30CHAPTER 3. HYBRID CONTROLLER FOR A NONLINEAR DYNAMICAL PLANT

0 -1 -2

0 -1

0

100

200

300

400

500

-2

600

0

100

200

Step

300

400

500

600

Step

(a) LQR

(b) NC

2

u [rad]

1 0 -1 -2

0

100

200

300

400

500

600

Step

(c) Hybrid Figure 3.6: Controller outputs Figures 3.6(a), (b) and (c) show, respectively, the variation of the LQR, NC and hybrid controller outputs uL (t), uN (t) and u(t), with running steps. uL (t) is initially out of range but stabilizes after a while. At about 200 running steps uN (t) is practically zero and only the LQR is effective. The initial high values uL (t) are systematically regulated by the NC, thereby ensuring that u(t) is within the desired range, [−π/2, π/2]rad. 3.4.4

Control Performance for the Initial Conditions

Further investigation was on the workable ranges for both the orientation angle and the starting vertical position for the hybrid controller. The 3 vertical lines in Figures 3.7(a) and (b) denote, respectively, the trained initial angles and positions while the squared error value, 104 , refers to the penalty when plant is out of control. In Fig. 3.7(a), starting vertical position is 3.0m and training values of the angles are 0, π/4, and π/2 rad. It is observed that there is a wider range of untrained starting angles [−3π/16, π/2]. For second case (Fig. 3.7(b)), orientation is π/4 rad and training values are 0.0, 3.0 and 6.0m. It is seen that the hybrid offers a wider operating range [-3.0, 9.0]m, of the initial vertical position.

31

104

104

102

102

Squared error

Squared error

3.5. ANALYSIS OF RESULTS

1 10−2 10−4 10−6 -π/2

0 π/2 Initial angle [rad]

(a) Initial angles

π

1 10−2 10−4 10−6 -10

-5 0 5 10 Initial vertical position x13 [m]

(b) Initial vertical positions

Figure 3.7: Range of effective hybrid control performance.

3.5

Analysis of Results

Table 3.6 forms the basis of this research. Difficulty in designing a controller intensifies as the number of trailers increases. Notably, using only the NC fails to control the 6 trailer-truck configuration. The proposed hybrid controller effectively controls the backing up motion of the multi-trailer system. To shorten the NC training times, firstly, the control problem is distributed such that the LQR handles the linear part while the NC is purely for the nonlinear part. Secondly, the evaluation function is modified to include the running steps error. For the hybrid system, the LQR and NC controllers are designed separately. The best gain factor generated by use of the Riccati equation for the LQR is chosen for the LQR, while for the NC the design criterion is based on evaluating both ES and ET . Trial and error is used to get the best combination of the weight matrix W in Eq. (3.22) for the LQR and the weight q for the NC in Eq. (3.28). The respective values of weight, 100 for the position magnitude in W and q13 = 0.1 gave the best results in the simulations. Figure 3.3 shows the sequence of evolution of the hybrid controller when α = 1.0 and β = 1.0. First, ET is successfully minimized, then E is evaluated by minimizing the squared errors. However, in case α and β are changed then the evaluation strategy changes. Figure 3.4 is representative of the effectiveness of the hybrid controller in steering the trailer-truck along the desired path from the various starting configurations while backing up. It is noted from Figures 3.5 and 3.6 that at 200 steps, all the state variables are relatively small and the NC output gradually falls to zero. From this point onward only the LQR is active, thus u(t) = uL (t). The problems of out of control steering and jack-knife locks are eliminated since all the relative angles are kept to within [−π/2, π/2]rad as per Eq. (3.32). Superiority of the hybrid controller is noted in Figure 3.6(c), where the output steering angle u(t) oscillates within the

15

32CHAPTER 3. HYBRID CONTROLLER FOR A NONLINEAR DYNAMICAL PLANT desired range, [−π/2, π/2]rad. The neural network property of adaptive learning is significant in Figures 3.7(a) and (b), where extrapolated ranges of operation for the hybrid controller are obtained from the three-point training values. By the NN ability to generalize, a wider range of patterns is generated from untrained starting orientations and positions, from which the system is able to successfully realign on the trajectory to the desired destination.

3.6

Conclusion

A hybrid control scheme comprising a LQR and a NC has been proposed to stabilize the backward motion of a six trailer-truck configuration. Motivation for the study stems from the inability of conventional neurocontrollers to optimally control the system. Difficult in control is rendered by the nonlinear dynamics of the system and the physical limitations such as jack-knife locks. The system is nonlinear hence the linear system methods are not suitable. The core objective has been to design a high performance controller utilizing neural networks coupled with a linear method. In total the trailer system is able to successfully execute the manoeuvre over a given trajectory from generic initial conditions while in the backward mode. The hybrid has shown very good performance in controlling the six trailer-truck configuration. In perspective, the method may be applied to other nonlinear dynamical control problems.

Chapter 4 Real-time Control Design for a Multi-trailer System 4.1

Introduction

Previous research, as reported in Chapter 2, centered on off-line control of a multi-trailer system, where performance evaluation of the optimization method depended only on the best fitness level achieved after a given number of trials, irrespective of the fitness of the intermediate samples. Realizing that in contemporary times industry requires real-time control, I have attempted to simulate the control of a similar plant by on-line design. Having increased the number of trailers to seven, the smooth backing up movement of the vehicle is highly unstable due to angular limitations. To effectively execute tracking of the designed trajectory, the dynamic optimization algorithm uses the plant model as emulator for the training and design of the NC. First an emulator is designed for the control problem and the trajectory determined. The plant is then allowed to back up along the trajectory, from different starting configurations. If there exists model uncertainty from any starting pattern then the scheme adaptively modifies the design and determines a new trajectory. Simultaneously, the plant is allowed to continue with the reverse movement along this new trajectory.

4.2

Modeling

Figure 4.1 depicts the truck with seven trailers. Nomenclature in Table 3.1 applies, the only difference being: x13 = Angular difference between 6th and 7th trailers x14 = Angle of 7th trailer x15 = Vertical position of 7th trailer x16 = Horizontal position of 7th trailer 33

34CHAPTER 4. REAL-TIME CONTROL DESIGN FOR A MULTI-TRAILER SYSTEM

L

L

L

L

L

L

l u(t)

L

x0(t) x1(t) x2(t)

x3(t) x4(t)

x5(t)

x7(t) x9(t) x13(t)

x6(t) x8(t)

x11(t) x10(t) x14(t)

x15(t)

x12(t)

x16(t)

0

Figure 4.1: Schematic model of the trailer-truck Similarly, the kinematics of the system are described by the set of expressions as given in Eqns. (3.1) - (3.13). The following additional equations apply: x13 (t) = x12 (t) − x14 (t) v∆t x14 (t + 1) = x14 (t) + sin[x13 (t)] L x14 (t + 1) + x14 (t) ] 2 x14 (t + 1) + x14 (t) x16 (t + 1) = x16 (t) + v∆t × cos[x13 (t)] × cos[ ] 2 x15 (t + 1) = x15 (t) + v∆t × cos[x13 (t)] × sin[

(4.1) (4.2) (4.3) (4.4)

For this case, the steering angle u(t) is to be controlled such that the system is asymptotically stabilized along the straight line x15 (t)=0. X(t) = [x1 (t), x3 (t), x5 (t), x7 (t), x9 (t), x11 (t), x13 (t), x14 (t), x15 (t)]T −→ 0.

(4.5)

35

4.3. CONTROL SYSTEM

4.3

Control System

Figure 4.2 shows the control system. The plant to be controlled is the trailertruck system; it receives the steering command u(t) and outputs the state variables X(t). The error in state variables (X ref − X(t)) is used to adjust the design for optimality.

Xref + _

uL +

LQR

+

u

Trailer-truck

X

system

GA Neuro controller

uN

Figure 4.2: Control system. The LQR has same characteristics as those described for the regulator in subsection 3.3.2. The NC in this adaptive design is also similar to that of the hybrid, described in subsection 3.3.3. However, the NN has a 9-5-1 configuration. The design is simulation-based; an emulator of the plant is used to design the NC and determine the trajectory. The evaluation function E comprises both the squared and running steps errors, and has the form of Eq. (3.26). A variation in the squared error term ES is due to the additional trailer: ref ref end 2 2 2 − xend − xend ES = q1 (xref 1 1 ) + q3 (x3 3 ) + q5 (x5 − x5 ) ref ref 2 end 2 2 − xend + q7 (xref − xend 7 9 ) + q11 (x11 − x11 ) 7 ) + q9 (x9 ref ref end 2 end 2 end 2 + q13 (xref 13 − x13 ) + q14 (x14 − x14 ) + q15 (x15 − x15 )

(4.6)

where xref is the reference variable and xend is the final value of the state variable that starts from the initial configuration. The factor q adjusts the importance of the control variables. The running steps error ET is computed thus, ET = tmax − t

(4.7)

where tmax is the maximum allowable steps and t refers to the running steps from the initial position along the desired trajectory. As for the Hybrid controller, the NCs are awarded a huge penalty if they cause the plant to run out of control, Epenalty = 1.0 × 106 . Else, the NC is evaluated on the basis of the summation of the weighted errors ES and ET , as in Eq. (3.26).

36CHAPTER 4. REAL-TIME CONTROL DESIGN FOR A MULTI-TRAILER SYSTEM

4.4

On-line Control Sequence

The emulator is designed for the desired trajectory based on the dynamic characteristics of the trailer-truck and the initial starting configuration. The idea is to have the actual plant track this trajectory; if there is deviation between the trajectories of the emulator and actual plant then re-design is necessary. Figure 4.3(a) shows the sequence of adaptive training and re-design. The emulator is designed such that trajectory no. 1 from the initial position is the optimum. The emulator is then used to control the actual dynamic system. If the plant tracks a different trajectory, the emulator is re-designed say, at position no. 2 so as to formulate trajectory no. 2. Subsequently, in case trajectory no. 2 is not conformed to, the system is instantaneously trained at position no. 3 and a new trajectory, no. 3, designed for. The learning continues as the controller improves by tracking the physical process. All these training and re-design are performed on-line so that the plant is controlled by minimizing the associated errors. START

SET INITIAL POSITION DETERMINE TRAJECTORY

Initial position

RUN THE PLANT Desired trajectory #1 Position #2

YES

Desired trajectory #2

D > DT NO

Desired trajectory #3

NO

STOP YES

Position #3

END

(a) Control sequence

(b) Training and design algorithm

Figure 4.3: Adaptive design. The flow chart in Figure 4.3(b) depicts the sequence: 1 Set the vehicle at the initial configuration. Randomly set the connecting weights of the NN. 2 The emulator is then trained for a particular starting pattern and the desired trajectory designed for.

4.5. RESULTS OF SIMULATIONS

37

3 Run the actual plant and emulator simultaneously while observing index D, D = (XM − X)T (XM − X) ≤ DT

(4.8)

where XM and X are emulator and actual plant state variables, respectively. DT is the threshold value of index D. 6 If plant does not trace the desired trajectory (i.e. D>DT ), goto process no. 2. 5 If the actual plant does not reach the desired destination, goto process no. 3. 6 End the control.

4.5

Results of Simulations

4.5.1

Parameters

(a) The 7 trailer-truck model Table 4.1: Parameter values for the trailer-truck system Truck length, l 0.3m Trailer length for design, LM 1.0m Trailer length for actual plant, L 0.7m & 1.2m Velocity of system, v -0.2m/s Sampling time, ∆t 0.25s

Table 4.1 gives the parameters of the system. These limits were applied: |u| < π/2 and |x1 , x3 , x5 , x7 , x9 , x11 , x13 | < π/2.

(4.9)

(b) Design of LQR & NC, Evaluation function and GA parameters For the LQR the following values of the gain G were determined: G = [ − 4.58, 30.49, − 115.07, 267.69, − 387.73, 329.84, − 133.47, 4.14, − 0.68].

(4.10)

For the construction of the NCs, the output neuron activity function was set as for the hybrid controller: f (x) = 0.1x3 . The evaluation function has same form as given in Eq. (3.26), where α = 1.0 and β = 1.0. As previously set for the hybrid controller, {q1 , q3 , q5 , q7 , q9 , q11 , q13 , q14 }=1.0 and q15 =0.1 in Eq. (4.6). The running steps limit in Eq. (4.7), tmax = 600 steps while in Eq. (4.8) the threshold value for the index D is set as: DT = 0.05. A real-coded GA was employed in the NN training, with similar properties as shown in Table 3.4.

38CHAPTER 4. REAL-TIME CONTROL DESIGN FOR A MULTI-TRAILER SYSTEM 4.5.2

Design Performance

Figure 4.4(a) shows movement along the desired trajectory when the lengths of the emulator and actual plant are equivalent (L=LM ). The initial configuration for the design is set at vertical position x15 =8.0m and orientation of π/2 rad. After confirming the effectiveness of the design, I simulated for actual plants with different trailer lengths and starting configurations. Results of trials follow in subsection 4.5.3. 20

20

Start

10

x15 [m]

x15 [m]

10

End

0

-10

0

Start

End

-10

-20 -20

-10

0 x16 [m]

10

20

(a) Emulator design, L = LM =1.0m. Display interval: 150 time steps

-20 -20

-10

0 x16 [m]

10

20

(b) Trajectory, actual plant, L=1.2m. Display interval: 180 time steps

Figure 4.4: Performance, actual plant trajectory tracking. Lengths L=LM and L=1.2m. 4.5.3

Control Performance

Figure 4.4(b) is an example of the controlled results, for the real plant when L=1.2m, where the initial orientation and starting vertical position are set to π/2 rad and 8.0m respectively. The emulator has been designed for LM =1.0m. It is observed that the plant successfully executes the desired trajectory, backing up along the straight line x15 =0m. Fig. 4.5 shows that the proposed adaptive controller optimally controls the state variables. Fig. 4.5(a) shows variation of the relative angles (x1 , x3 , x5 , x7 , x9 , x11 , x13 ) and angle of last trailer x14 with running steps. It is seen that the angles are controlled successfully by the adaptive controller to within the set range [−π/2, π/2] rad. Fig. 4.5(b) gives the vertical position x15 of the last trailer. The trailer smoothly reaches the desired trajectory with no zig zag. Generally, at 200 steps all the state variables are very small and the system eventually realigns and stabilizes at the desired path, x15 =0m.

39

4.5. RESULTS OF SIMULATIONS x x31 x5 x7 x x119 x13 x14

1 0

x15 [m]

Angles [rad]

2

-1 -2

0

100

200

300

400

500

8 6 4 2 0 -2 -4 -6 -8

600

0

100

200

300

400

Step

Step

(a) Angles

(b) Vertical position

500

600

Figure 4.5: State variables, L=1.2m 0.15

100 Generation

0.1 0.05

60 40 20

0

0

100

200

300

400

500

0

600

0

100

200

300

Step

(b) Generations

Figure 4.6: Adaptive training results, L=1.2m 2

u uN uL

1 0 -1 -2

0

400

Step

(a) Index D

u [rad]

Index D

80

100

200

300 Step

400

500

Figure 4.7: Control Inputs, L=1.2m

600

500

600

40CHAPTER 4. REAL-TIME CONTROL DESIGN FOR A MULTI-TRAILER SYSTEM Figure 4.6 shows adaptive training and re-design. From Fig. 4.6(a) it is seen that initially there are several instances when D>DT hence there is need to retrain the NN on-line and determine new trajectories. After 453 steps the best NC is realized. It is noted in Fig. 4.6(b) that on-line re-design takes very few generations. Figure 4.7 shows the variation of the LQR, NC and combined controller outputs, uL (t), uN (t) and u(t) respectively, with running steps. It is observed that the LQR output is initially high but stabilizes after a while. At about 100 running steps uN (t) is practically zero and only the LQR is effective. The output u(t) does not exceed the set range [−π/2, π/2] rad of the steering angle since the high value of uL (t) is suppressed by the NC. Next, simulations were performed for a real plant, trailer length L=0.7m and Figures 4.8 - 4.11 are the graphical results of the simulations. It is more difficult to evolve the controller from this starting pattern because the orientation is 0 rad and the starting vertical position of 5.0m is very near the desired asymptote x15 = 0.

20

10

x15 [m]

Start 0

End

-10

-20 -20

-10

0

x16 [m]

10

20

Figure 4.8: Trajectory, L=0.7m. Display interval: 120 time steps

As seen from Figure 4.8, the plant is successfully stabilized along the line x15 =0m. Fig. 4.9 confirms that the state variables are controlled to within required limits in minimal steps. Figure 4.10 shows that there are several trainings and re-designs prior to D falling below DT , and that to evolve the initial design takes many generations. However, it takes very few generations to evolve the NCs for the re-designs and after about 280 steps there is no more training as the system is fully controlled along the trajectory. Figure 4.11 confirms that the steering command u(t) is controlled successfully to within the limit [−π/2, π/2]rad and eventually stabilizes at 0 rad.

41

4.5. RESULTS OF SIMULATIONS x x31 x5 x7 x x119 x13 x14

1 0

x15 [m]

Angles [rad]

2

-1 -2

0

100

200

300

400

500

8 6 4 2 0 -2 -4 -6 -8

600

0

100

200

Step

300

400

500

600

500

600

Step

(a) Angles

(b) Position

Figure 4.9: State variables, L=0.7m 0.15

175 Generation

0.1 0.05

125 100 75 50 25

0

0

100

200

300

400

500

0

600

0

100

200

Step

300

(b) Generations

Figure 4.10: Training results, L=0.7m 2

u uN uL

1 0 -1 -2

0

400

Step

(a) Index D

u [rad]

Index D

150

100

200

300 Step

400

500

Figure 4.11: Control Inputs, L=0.7m

600

42CHAPTER 4. REAL-TIME CONTROL DESIGN FOR A MULTI-TRAILER SYSTEM

4.6

Highlights

In this chapter the problem of backing the truck with seven trailers to a target location by on-line design of the controller is examined. The proposed scheme uses real-time acquired data to re-train the NCs and generate a suitable trajectory. The emulator is trained and designed for only one starting configuration; if there exists model uncertainty resulting in deviation from desired trajectory, then the scheme adaptively trains the NC to evolve another suitable trajectory. Having successfully designed the emulator, the actual plant with similar parameters is run. Figure 3.4(a) shows that the trailer-truck system successfully tracks the trajectory. However, when the trailer length L changes or even a small variation in D exists, significant offsets from the given destinations in the case of backing-up motion result. In running the actual plant, the D-factor determines whether the controller is able to guide the plant along the desired trajectory. Once the threshold value DT is exceeded, the plant deviates from the trajectory and gets out of control. Thus the NC is adaptively redesigned by an instantaneous update of its weights and another trajectory formulated from this point. Generally, a sudden step change in u(t) results in backing along a different trajectory after the transient takes place and dies out. The input u(t)=0, causes the vehicle to back along a straight line in steady state. Figures 3.4(b) and 3.8 show the effectiveness of adaptive design in steering the actual plant, lengths L=1.2m and L=0.7m, respectively, along the desired path while backing up. Figures 3.5 and 3.9 show that the state variables are successfully controlled in minimal steps to conform to the reference variables. The problems of out of control steering and jack-knife locks are eliminated since all the relative angles are kept to within [−π/2, π/2]rad as per Eq. (3.9). Figs. 3.6 and 3.10 show that to evolve the best controller successfully, several on-line trainings are necessary, else the system gets out of control. However, it takes very few generations to subsequently evolve the emulator after the initial design, hence very few computations. From Figs. 3.7 and 3.11, uL (t), uN (t) and u(t) diminish significantly with running steps along the trajectory thereby stabilizing the vehicle on the desired path.

4.7

Conclusion

A paradigm for successfully controlling the backward movement of a multi-trailer system on-line has been described. Simulation results to support the proposed approach have been presented. Guaranteeing that the vehicle never leaves a predefined corridor constitutes a valuable property for the entire process of motion control and trajectory planning. Complexity has been rendered by the increased number of trailers and the fact that training is from a singular starting pattern. Due to its ability for generalization, the NN acquires many ranges thus one starting pattern is representative enough for the evolution of the design. The system is able to adaptively learn very quickly, enable the design of the system based on on-line acquired data, and effectively control the real plant along the trajectory without any annoying zig zag or jack-knife locks. The paradigm is rich with possibilities for further study.

Chapter 5 Concluding Remarks In Chapter 1 the author has proposed RGM as a new, evolutionary technique for evaluating continuous, multi-dimensional and multimodal functions by optimization. Borrowing heavily from the elemental concepts of genetics, it is seen that a recessive trait can manifest significantly in subsequent offspring across generations. This characteristic has been exploited in the formulation of RGM, where search is enhanced by optimization of the dual gene system. Simulation results have shown that search performance with recessive genes is superior to the purely dominant genes. RGM avoids stagnation by offering diversity, works very well at low sampling population sizes and that the recessive gene occasionally performs the function of mutation. RGM has the capability of performing consistently well on a broad range of problem types, unimodal and multi-modal, thus very robust. Perspectively, the author suggests the management of computational time could be further improved. The essential aspect of safety is to limit the space for the vehicle in motion. Guaranteeing that the vehicle never leaves a predefined corridor constitutes a valuable property for the entire process of motion control and trajectory planning. The author has described two variations of controller design for guiding multi-trailer systems along designed trajectories to desired destinations while backing up. Simulation results of the research have been presented to support the appeal of the proposed paradigms. In Chapter 2 the main contribution is a hybrid control scheme, comprising a LQR and a NC to stabilize the backward motion of a six trailer-truck configuration vehicle. Motivation for the research stems from the inability of conventional neurocontrollers to optimally control the system. Difficulty in control is rendered by the nonlinear dynamics of the system and the physical limitations such as jack-knife locks. The core objective was to design a high performance controller; this was achieved by utilizing NNs, coupled with a linear regulator to effect shorter training times. In total the six trailer-truck configuration is able to successfully execute the manoeuvre to a given trajectory from generic initial conditions while in the backward mode. I strongly believe this method may be applied to other nonlinear dynamical control problems. 43

44

CHAPTER 5. CONCLUDING REMARKS

In Chapter 3 the author has presented a control system able to adaptively learn very quickly, enable the design of the emulator based on real-time acquired data, and effectively control the real plant along a predetermined trajectory without any annoying zig zag or jack-knife locks. Real-time design is utilized firstly, because the number of trailers have been increased to seven, implying more inputs to the system. Secondly, off-line GA training is usually computationally expensive as the time of convergence may be too lengthy for real-time adaptive schemes. Simulations with a kinematic model of a truck with seven trailers verify that the on-line design can guide the system to desired destinations in an errorless condition. The paradigm is rich with possibilities for further study.

Appendix A List of Publications The publications referred to in the thesis are based on papers submitted for presentations at the following Conferences: 1 Muhando Endusa, Kinjo Hiroshi and Yamamoto Tetsuhiko: Enhanced Performance for Multivariable Optimization Problems by Use of GAs with Recessive Gene Structure, International Symposium on Artificial Life and Robotics (AROB 10th Conference), Oita, Japan. February 2005. 2 Muhando Endusa, Kinjo Hiroshi, Uezato Eiho and Yamamoto Tetsuhiko: Hybrid Controller of Neural Network and Linear Regulator for Multi-trailer Systems Optimized by Genetic Algorithms, International Conference on Control, Automation and Systems (ICCAS2005), KINTEX, Seoul, S. Korea. June 2005.

45

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APPENDIX A. LIST OF PUBLICATIONS THIS PAGE INTENTIONALLY LEFT BLANK

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