Order Number 9421027

On-line damage identification and motion control of adaptive structures: An intelligent control approach

Tsou, Po-Yu, Ph.D.

The Ohio State University, 1994

Copyright ©1994 by Tsou, Po-Yu. All rights reserved.

UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 On-line Damage Identification and Motion Control of Adaptive Structures: An Intelligent Control Approach

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

Po-Yu Tsou, B.S., M.S.

$ $ $ + $

The Ohio State University

1994

Dissertation Committee: Approved by

M.-H. Herman Shen

R. K. Yedavalli Adviser H. A. Oz Department of Aeronautical Y. F. Zheng and Astronautical Engineering © Copyright by

Po-Yu Tsou

1994 This is dedicated to my parents and my wife.

ii Acknowledgements

I wish to thank Professor Herman Shen’s continuous guidance, encouragement and support throughout my disseration work. Because of him, I have such a wonderful opportunity and enjoy working on this subject. With his tireless caring and extensive advices, the benefits from this study are not only academic but also spiritual. I would also like to thank the committee members, Professor Rama K. Yedavalli, Professor

Hayrani A. Oz of the Aeronautical and Astronautical Engineering Department and

Professor Yuan F. Zheng of the Electrical Engineering Department, for their academic instructions and valuable comments and suggestions on this work.

During the graduate study period, the friendships and assistances from my col­ leagues are truely gratiful. Thanks to Dr. Yong Liu, Yiping Yao, C.R. Ashokkumar,

Keiji Yano, Yangyao Niu, and Anjali Diwekar.

I want to express my affective appreciation to my parents and my dear wife, Limei.

Without their devoted love and faithful support, this disseration cannot be completed.

This research work is partially supported by the 1991 Research Initiation Award of the Engineering Foundation, the Wright-Patterson Air Force Base, the Office of the Vice President for Research at the Ohio State University and the Department of

Aeronautical and Astronautical Engineering. Vita

July 3, 1964 ...... Born - Taichung, Taiwan, R.O.C.

June 1986 ...... B.S. Aeronautical and Astronautical Engineering, National Cheng Kung University, Taiwan June 1988 ...... M.S. Aeronautical and Astronautical Engineering, National Cheng Kung University, Taiwan 1990-1991 ...... Graduate Student, The Ohio State University 1992-August 1993 ...... Graduate Research Assistant, The Ohio State University 1994-present ...... Graduate Teaching Assistant, The Ohio State University

Publications Poyu Tsou, and M.-H. Herman Shen, “Structural Damage Detection and Identifica­ tion Using Neural Networks,” AIAA Journal, 32(1):176-183, Jan. 1994.

Poyu Tsou, and M.-H. Herman Shen, “Design of Fuzzy Controller for Trajectory Tracking of Adaptive Truss Structures,” Smart Materials and Structures.

Poyu Tsou, and M.-H. Herman Shen, “Structural Damage Detection and Identifica­ tion Using Neural Networks,” Proceedings of 34th Structures, Structural Dynamics and Material Conference , AIAA/ASME/ACSE/AHS/ACS, La Jolla, CA, 3515-3519, Apr. 1993.

Poyu Tsou, and M.-H. Herman Shen, “Fuzzy Control for Pursuing Problem of an Adaptive Planar Structure,” 1994 ACM Symposium of Applied Computing, Mar. 1994. Poyu Tsou, and M.-H. Herman Shen, “A Neural Net Based Damage Identifica­ tion Method for Discrete Structural Systems,” Pacific International Conference on Aerospace Science and Technology, Taiwan, 1367-1374, Dec. 1993.

Fields of Study

Major Field: Aeronautical and Astronautical Engineering

Studies in: Structures Prof. M.-H. H. Shen Controls Prof. R. K. Yedavalli Dynamics Prof. H. A. Oz Table of Contents

DEDICATION ...... ii

ACKNOWLEDGEMENTS ...... iii

VITA ...... iv

LIST OF TABLES ...... viii

LIST OF FIGURES ...... ix

CHAPTER PAGE

I Introduction ...... 1

1.1 Background Information of Adaptive S tructure ...... 1 1.2 Review of Intelligent C ontrol ...... 4 1.3 Research Objectives ...... 9 1.4 Dissertation O utline ...... 10

II Overview of an Intelligent Control S; ..tern ...... 13

2.1 Architecture of the intelligent control system ...... 14 2.2 Detailed descriptions of functional modules ...... 17 2.3 Information flow p a t h ...... 23

III Structural Damage Detection and Identification ...... 26

3.1 Review of Neural N e tw o rk s ...... 29 3.2 Three DOF System ...... 33 3.2.1 Single Damage C a s e ...... 35 3.2.2 Multiple Damage Case ...... 37 3.3 Kabe’s M o d e l ...... 43 3.3.1 Dyn&mic Residual M ethod ...... 44 3.3.2 Simulation Result ...... 46 3.4 Truss S tru c tu re ...... 52

IV Fuzzy Logic Control D esign ...... 58

4.1 Fuzzy Logic Control T h e o ry ...... 61 4.2 Kinematics of Adaptive Structures ...... 63 4.3 Fuzzy Controller D esign ...... 66 4.4 Illustrative Examples ...... 71 4.4.1 Planar Truss Structure ...... 71 4.4.2 Rendezvous/Docking Problem ...... 75 4.4.3 Target Pursuing Problem ...... 76 4.4.4 Trajectory Tracking Problem ...... 78 4.4.5 Effect of various g ain s ...... 82 4.4.6 Robustness of fuzzy control system ...... 84 4.4.7 Effect of Inference Strategy ...... 86 4.5 C lo s u r e ...... 87

V Conclusions and Future Research ...... 92

APPENDICES

A Samples of data used for damage identification neural netw orks ...... 98

B Computer Codes ...... 100

BIBLIOGRAPHY ...... 119

vii List of Tables

TABLE PAGE

1 Individual spring damage results ...... 36

2 Fuzzy rules table for SDMs controller ...... 74 List of Figures

FIGURE PAGE

1 Schematic architecture of the intelligent control system ...... 15

2 Management and Organization Level ...... 18

3 Coordination Level ...... 20

4 Execution Level ...... 22

5 Backpropagation neural network ...... 30

6 Three DOF Mass-Damper-Spring System ...... 34

7 Neural network outputs of Case Kmix ...... 40

8 Neural network outputs of Case K m O l ...... 41

9 Neural network outputs of Case K m 45 ...... 43

10 Kabe’s M o d e l ...... 45

11 Neural network architecture for structural damage identification . . . 47

12 NN outputs of Kabe’s modektwo-spring-damage ...... 51

13 NN outputs of Kabe’s model with 0-15-30% training ...... 53

14 NN outputs of Kabe’s model with 0-10-20-30% training ...... 54 15 A four-module truss structure ...... 55

16 NN output for two-member damages of the truss structure ...... 57

17 The modular fuzzy controller ...... 67

18 The normalized universes of discourse and the membership functions 68

19 The adaptive space structure ...... 72

20 The simulation result of docking problem ...... 76

21 Simulation result for pursuing a circular m otion ...... 77

22 Simulation result for pursuing a reflexed m otion ...... 78

23 The simulation result of the reflexed line c a s e ...... 80

24 The simulation result of the hill-n-valley line c a s e ...... 81

25 The simulation result of the circle c a s e ...... 82

26 The simulation result of the wavy curve case ...... 83

27 Comparison of length changes for the circle c a se ...... 84

28 Comparison of length changes for the wavy curve case ...... 88

29 The errors of different input gains ...... 88

30 The trajectories of different output gains for module pairs ...... 89

31 The performance index of different output gain ratio s ...... 89

32 The effect of noise in m easurem ent ...... 90

33 The effect of one stuck actuator ...... 90

x 34 The effect of two stuck actuators ......

35 The errors between different inference implication rules C H A P T E R I

Introduction

1.1 Background Information of Adaptive Structure

The adaptive structure is a new structure design which is required for high perfor­ mance advanced structure. Wada et al. in their overview paper [1] categorized the possible incentive developments in advanced structures. Basically, the nomenclatures cited in this dissertation may follow the similar definitions made in their paper. Nev­ ertheless, in this research, it is attempted to focus on the issues of adaptive structures.

W ada et al. [1] defined the adaptive structures as “those which possess actuators that allow the alteration of system states and characteristics in a controlled manner.”

Miura and Furuya [2] also made a definition that “adaptive structure is a structure that can purposefully vary its geometric configuration as well as its physical proper­ ties.” These two definitions are essentially identical. It should be noted that in their definitions, the adaptive structure does not have sensors. But, the structural system mentioned in the Chapter III requires sensors embedded in the system. By Wada’s definition, such a structural system belongs to the class called “sensory structure” or “smart structure”. However, in this dissertation, it is intentional not to separate these structure forms from the category of adaptive structures because the boundaries

1 2

between these definitions are somewhat unclear.

An adaptive truss structure typically consists of a periodic sequence of the ba­

sic truss modules containing members equipped with the variable-length actuators.

Thus, the adaptive truss structure has the adaptability to perform specified tasks

intelligently or to meet the diverse environment by adapting its geometrical and phys­

ical properties. In addition, the high demand for the Controls-Structures Integration

(CSI) for the future space structure, the adaptive structures which implicitly combine

actuation ability and structural strength seems to be a highly potential candidate.

From the definition, the adaptive structure can be classified into two different

types. One type is known as the variable geometry truss, usually referred to as the

adaptive truss structure. The other type of adaptive structure puts the control effort on changing the mechanical properties of the structure. The concept of adaptive

truss structure was presented in the Mid-80’s. Miura and his colleagues in Japan

[2], Rhodes and his colleagues in the USA [3], and Sincarsin and his colleagues in

Canada [4] are some among a few who presented the prototypes and their various applications. To a larger extent, adaptive structures have already found widespread use in aircraft. Swing wing aircraft such as the F -lll fighter/bomber, the F-14 fighter and the B-1 bomber are some typical examples. Recently, a method of incorporating the mechanism which provides for variable geometry lifting surfaces into the wing substructure or skin, hence developing a true adaptive aeroelastic structure, has been investigated with promising results [5]. This type of adaptive structures is the primary concern of this study. 3

Active changes of mechanical properties of the structure is the other type of adap­

tive structures. Temperature control, electric-rheologica] material, shunted piezoelec­

tric damping, or shape memory alloy are several state-of-the-art developments in this

class. Changes of up to an order of magnitude in damping characteristics of vis­

coelastic materials with a ten degree change in temperature can be expected [6]. If

the viscoelastic material in a structure is replaced by the electro-rheological fluid, the

damping characteristics of the structure can be changed and controlled by applying

different electric fields. Gandhi and Thompson [7] and Coulter et al. [8] have demon­

strated that a change in modal damping by a factor of two to three of the lower

structural modes of a beam. Also, the lower resonant frequencies can be increased by

50-100% if the shear modulus of the electro-rheological fluid which couples the face

sheets of the sandwich plate is increased. Hagood and Crawley [9] have found that the

passive damping can be added to specific modes of vibration by coupling the shunted

piezoelectric damping circuits into mechanical system. Buehler and Wiley [10] intro­

duced a shape memory material often referred to as Nitinol, which has a “memory” of its original position when heated above its transition temperature. Rogers et al.

[11] utilized this property to apply the shape memory alloy on a composite beam.

They found that the mode shapes of the plate can be changed by adjusting the strain energy distribution through controlled actuation of the shape memory alloy wires which are distributed in a grid fashion in a composite plate. 4

1.2 Review of Intelligent Control

There have been many forums, workshops, special issues, and symposiums (e.g. In­ ternational Symposiums on Intelligent Control, Special issues of Institute of Electrical and Electronic Engineers Control System Magazine, American Control Conference,

Conference of Decision and Control, Transactions on American Society of Mechanical

Engineering, etc.) that have devoted time to define an intelligent control system.

Here, Several definitions from the literatures are excerpted as follows:

• Astrom and McAvoy in [12] make the following definition of the intelligent

control system: UA more demanding definition is to say that an intelligent

control system has the ability to comprehend, reason, and learn about processes,

disturbances, and operating conditions.”

• Shoureshi in [13] states that: “A control system with the ultimate degree of

autonomy in terms of self-learning, self-reconfigurability, reasoning, planning

and decision making, and the ability to extract the most valuable information

from unstructured and noisy data from any dynamically complex system and/or

environment.”

• In [14], Saridis postulated the intelligent control as “the process of autonomous

decision making in structured or unstructured environments, based on the in­

teraction of the disciplines of Artificial Intelligence, Operations Research, and

Automatic Controls.” 5

• Passino in [15] uses the following statement: “The physical device called a con­

troller is an intelligent controller if it is developed and/or implemented with

(i) an intelligent control methodology or (ii) conventional systems/control tech­

niques to emulate/perform control functions that are normally performed by

humans/animals/biological systems.” In addition, he also defines “A control

methodology is an intelligent control methodology if it uses human/animal/

biologically motivated techniques and procedures (e.g., forms of representation

and/or decision making) to develop and/or implement a controller for a dynam­

ical system.”

The above definitions can be summarized as:

An intelligent control system possesses high degree of autonomy in terms

of biological/cognitive abilities which are able to accomplish various mis­

sion requirements and complex environments.

Although the term may be described differently, there is a consensus about that intelligence describes the level of autonomy of the control system. That is, an intel­ ligent control system has the ability to reason, learn, make decisions. Specifically, an intelligent control system should have the ability to deal with: a large degree of un­ certainty; qualitative information; huge amount of unstructured data and/or sensory information; highly complex data structure. Thus, the controlled system is expected to: respond to high level of abstraction of human commands; apply sensory data to infer faults and/or operating errors; accommodate heterogeneous (numerical and 6 symbolical) knowledge sources; reconfigure feedback control structure to respond to variation in the system and/or environment; etc. [13]

The ideas of intelligent control were introduced in late 60’s [16]. However, it’s today’s technology that has provided some of the foundations for the developments of an intelligent control system. Knowledge-based control systems such as adaptive elements, fuzzy logic as qualitative reasoning, and artificial neural networks as learn­ ing and action-generating elements are the propellant that made it feasible to realize, conceptualize, and formulate the intelligent control system.

The state-of-the-art research studies on the intelligent control, in general, are based on: expert control systems, fuzzy logic control, and/or neural networks. There has been a lot of progresses about the expert systems applied to the control area. The latter two are not as mature as expert control systems, though both technologies gave their births as early as 70’s. It’s until several recent breakthroughs that the latter two fields would attract a growing attention. Also, these two subjects (fuzzy control and neural networks) are the main tools in this dissertation.

The present progress of expert systems has gained an impressive level of computa­ tional performance on particular problem solving tasks in isolated domains [17,18,19].

The concept has been applied to various platforms, e.g. Firschein et al. described several usages of artificial intelligence techniques to enhance autonomy of the space station [20], Kusiak reported some applications of expert control systems to manu­ facturing processes [21], and Astr5m et al. applied the expert systems to a chemical process control citekjAe. Since the successful applications of expert control systems have been widely reported, there is no intention to exploit this subject in this study.

There has been an explosive number of studies on artificial neural networks since mid-80’s. In the past decade, many variations of neural networks have been developed and the revolving process of the new models is still going on a fast pace. The brief historic review of backpropagation, the most used and common form of multilayer feedforward network, will be described in Chapter III. Here, the most remarkable works related to system dynamics and control are by Werbos [22] on backpropaga­ tion, Widrow and Hoff [23] and Widrow and Stearns [24] on formulation of Adaline,

Narendra and Parasarathy [25] and Chu et al. [26] on system identification and control by neural networks.

Another area that has potential to enhance the intelligent control system is fuzzy logic and fuzzy control. Based on approximate reasoning rather than the exact rea­ soning, fuzzy logic provides high degree of tolerance on imprecision in process. Also, the fuzzy control is able to interpret human’s commands in highly abstract form. This property makes control engineers to apply fuzzy logic design of rule-based controllers for uncertain systems or large complex systems.

Fuzzy logic was introduced in 1965 by Zadeh in U.S., but it was in 1987 the first application of fuzzy control system appeared. It was introduced by Hitachi when they applied fuzzy logic to subway operation control in Sendai, Japan. From 1990,

Japanese industries have been successfully and flourishly introduced many household appliances and consumer electronics equipped with fuzzy logic control. The commer­ cialization of fuzzy logic concept lays down a solid foundation for advanced application 8

to more complicated systems, like the adaptive truss structures.

The progress in the area of adaptive control also attracts a lot of attention. How­ ever, adaptive control algorithms do not sufficiently address problems that arise from

redundancy, nonlinearity in parameters, or distributed sensing and control. Moreover, they may be sensitive to initial conditions and to the adaptation rate [13]. Hence, it is inappropriate to count the adaptive control schemes as part of intelligent control methods. However, the integration of adaptive control techniques and the neural control and/or fuzzy logic becomes a growing topic and is of great interest.

In the issue of robustness, the Hqo theory and quantitative feedback theory(QFT) have demonstrated some satisfactory results in this recent decade. The theories pro­ vide stable controller designs in presence of uncertainties coming from noises in process or parameter variations in system model (system transfer function). Unfortunately, these robust control schemes have been developed basically for linear systems. Also, they require the mathematical model of the system to be available for further design procedure. This requirement greatly reduces the application of such methods to the complex problem or real system.

It is a general belief that the knowledge in human intelligence will boost the devel­ opment of the intelligent control system. A strong evidence shows that the dynamics of various physiological components is in higher-order form. The observation of na­ ture also shows that bifurcations, limit cycles and chaos are crucial components of biological intelligence. Therefore, to advance the intelligent control system, the better understanding of human brain operations, biological neural system and articulatory 9 motor control shall be inevitable.

1.3 Research Objectives

There are several inherent problems of the adaptive structure, as listed below:

1. Complexity: The adaptive structure typically consists of a sequence of basic

structural modules, which indicates a large number of degree of freedom. If the

state space is used for modern control techniques, the dimension of the state

matrix could be up to hundreds, even thousands. This presents a challenging

topic in controlling the structure.

2. Redundancy: Each basic structural module, known as a bay, may contain several

variable-length members and compliant joint mechanisms that allow controlled

changes in configuration. Such a large number of redundant actuators makes

the adaptive structure more dexterous. In the meantime, it also generates a

large number of control possibilities to accomplish the same task.

3. Dynamically varying system: Due to the construction of the variable-length

members, also called the active members, the adaptive structure is able to

change its configuration in large deformation. Such drastic changes make the

state space modeling more difficult.

4. Versatility: Since the adaptive structure can manipulate very articulatory mo­

tion, it may be required to perform several different tasks. This is in contrast to

the anthropomorphic robot arms which typically only perform a specific task. 10

5. Autonomy: One potential use of the adaptive structure is the application in the

space station. For such applications, it is required that the adaptive structure

should be a stand-alone system, i.e., the structure should be highly autonomous.

To tackle all of the above-mentioned problems may present a very complicated

task. The intelligent control technique which can deal with these problems seems to

be an appropriate approach. Thus, the main goals of this dissertation are two major

issues originating from these problems, that is, damage identification problem and

motion control problem for the adaptive structures.

1.4 Dissertation Outline

The dissertation is organized as follows. In Chapter II, the framework and the

architecture of an intelligent control system for the adaptive structural system is overviewed. Two notable features of this system architecture are hierarchical and

modular. This architecture provides an overview picture which the two main issues discussed in this thesis reside in. Under the scope of the intelligent control system, two computational intelligence techniques, i.e. neural networks and fuzzy logic con­ trol, form the foundation of this work. Two main contributions of this dissertation are contained in the following two chapters. In Chapter III, an artificial neural network based identification method to detect and identify the possible damage locations and their severities is suggested. A three-layer neural networks is able to identify the location and the severity of the damages in the sparsely-spaced eigenvalues discrete systems. A new architecture of neural network is also developed to deal with the 11

closely-spaced eigenvalues systems like truss structures. The proposed identification

method can identify multiple structural damages and it only needs a limited number

of response information to carry out the identification process are two advantages

over the existing identification methods.

Since there still exist some difficulties to apply neural control techniques on the

adaptive truss structure, a fuzzy logic control technique is considered. The applica­

tion of the fuzzy logic control to the adaptive structures and the design of such a

control system are presented in Chapter IV. To amend the traditional design weak­

ness of trial-and-error treatment in setting up the control rules, an analytical tool

derived from the inverse kinematic relationship will be adopted in assistance of rule

making. Several options and applications of the fuzzy control system will be studied.

In addition, some observations which are able to aid the engineering design will be

addressed. This work has demonstrated that the fuzzy controller obtained from the

proposed design procedure indeed is capable of accomplishing several mission require­

ments. Also shown in this chapter that the fuzzy controller is effective and reliable. In

the presence of the actuator failures or noises in measurements, the fuzzy controller

successfully achieves the final goals. Its Bimple construction plus the systematical

design procedure proposed in this dissertation provide a very promising future in engineering perspectives.

Finally, conclusions drawn from this study and comments on the future research are given in Chapter V. Two appendices are included in the dissertation. Appendix A lists a couple of samples of data for training and testing the damage identification 12 neural networks. From the samples, one can realize the data flow mechanism of the proposed neural networks. Appendix B lists the computer codes, which is written in

MATLAB specified language, that were developed in this research in dealing with the motion control of the adaptive truss structure. C H A P T E R II

Overview of an Intelligent Control System

Although intelligent control is not the main issue of this dissertation, it is felt that

before we go further to discuss the damage identification technique and the motion

control algorithm, it is better to draw an overall picture of an ultimate control system.

The architecture of the intelligent control system is similar to the one introduced by

Antsaklis and Passino[27], but specially engineered for the adaptive structures.

Consider a general control problem:

How to construct a controller C by given a plant P so that a

certain criteria T holds?

Control engineers normally try to use the available knowledge represented in the plant model P plus extra relevant information (which is often heuristic) to design a controller C. The difference between the conventional approaches and the intelli­ gent control approach is that the conventional mathematical approaches to design the controller often initially ignore the extra relevant information and use it later when it comes time for implementation. On the other hand, the intelligent control techniques (fuzzy or expert control) provide somewhat more formal methods to in­ corporate the extra relevant information, but they often ignore the use of information

13 14 from a conventional model (thus, sometimes it is viewed as “sloppy” )[15].

The objective of this overview is not to try to justify the intelligent control over conventional control techniques. Instead, we would rather to explore the possibility of practicing this novel concept to enhance the performance or to extend the ability of the adaptive truss structure. In this chapter, we will look into the architecture of an intelligent control system. The information flow and message path inside the architecture will be exemplified. The detailed functionality of major modules at each level of the intelligent control system will also be defined and described.

2.1 Architecture of the intelligent control system

An intelligent autonomous controller for the adaptive structure may be represented by a functional architecture shown in Figure 1. The control system is constructed by three hierarchical levels which are, from top to bottom, M anagem ent and Organi­ zation level, Coordination level and Execution level. Each level includes several functional modules where each module may represent the hardware implementation or the software algorithms or the hybrid components. The human/control-system interface resides in the top level of the control system. To the end, the subject to be controlled is connected with the bottom level of the control system.

The highest level, management and organization level, serves as an interface be­ tween human and the control system. It can be viewed as the “brain” of the whole system. The essential responsibility of this level is to replace part of human inter­ actions with the device, e.g. generating a working procedure, switching the compo­ nents, monitoring the status, etc. Also, it is able to decode the human commands and 15

Interface Managementf t Organization Control Executive Level

Control Manager Coordination Level Control Implementatioifiupervisor

Control ft Identification Execution Sensors/Actuators Level

ADAPTIVE STRUCTURES

Figure 1 : Schematic architecture of the intelligent control system 16

translate into machine understandable sequences and operations. The current voice-

activated driver and natural language processing technologies provide some potential

highlights in this matter.

The management and organization level contains modules with high degree of

“intelligence” such as Goal generator , (Long-term)Learner, Performance monitor,

Capability assessor, Planner. More details about these modules will be described in

the next section.

The coordination level includes two panels of modules, Control Manager Panel and

Control Implementation Supervisor Panel. In control manager panel, Task designer,

Task manager, Task planner, (Mid-term)Leamer, and Fault/Damage Identification

modules take part. The control implementation supervisor panel consists of Operation

supervisor, Operation scheduler, Tuner, Information assessor, (Short-term) Learner.

The execution level possesses the modules like Actuators, (Low-level) Controller,

Fault/Damage identification algorithms, Parameter identifier, Data distributor, and

Sensors. All control actions are carried out by this level.

The operations in the lower level are generally numerical processes, requiring the achievement of the system performance like time scale density, bandwidth, system response rate, or decision rate. All these characteristics lead to a change in granularity of models used, or equivalently, to a change in model abstractness. The actions of the actuators require the instructions processed in the order of milliseconds, whereas the time horizons for the planner and the scheduler are in the order of seconds and the subtasks managed by the goal generator may be in the order of tens of seconds 17

up to minutes.

2.2 Detailed descriptions of functional modules

The schematic arrangement of the management and organization level is shown in

Figure 2. The interface accepts the command from the operator and passes it through

the parser or the decipher to translate the command into the machine codes. The

natural language processing technique may be embedded in the decipher which makes

the control system to be able to understand the command in natural language.

The goal generator receives the command from the parser or the decipher and

generates the necessary system calls to activate the consequent modules. According

to the goal obtained from the goal generator, the capability assessor will evaluate

the reachability of such a goal. If it is OK, it will issue a command to notify the

planner. If the goal exceeds the physical limits of the structure, it should issue a

warning message to the console in the interface and halts the system.

The main function of the monitor is to evaluate the current status and health

of the structure. In addition, the high level performance monitoring can be added

into the monitor’s functionality. It predicts what can be reasonably expected to

be accomplished in a certain time. The planner performs high level planning. It

appropriately breaks down the control command into simpler task commands for the

lower level. It also organizes the sequence of these task commands according to the

priorities and logical relationships of such tasks. It can also handle unexpected events

during the process. For example, if there is an obstacle present in the planned path of docking, the planner would generate a new sequence of operations to avoid the OPERATOR

Interface

Decipher Parser

Goal generator I Planner Capability assessor X f ------Monitor Learner

Library

EXECUTIVE ORGANIZER

COORDINATION LEVEL

Figure 2: Management and Organization Level 19

obstacle.

The library can be a huge memory board or a database, but more important,

it can store the vital information such as heuristic rules for references, comparisons

or logistic necessities. The learning ability is essential to develop a true autonomous

system. The learner in this level has significant capability to improve the efficiency by

the past experience and update its capability assessment. The module uses decision

making schemes exclusively. If necessary, it could request, through the interface, additional information from the operation.

The schematic arrangement of the coordination level is shown in Figure 3. The main function of the control manager panel is to accomplish the control tasks given by the management and organization level. It can not only finish predetermined control actions, but also cope with a large degree of failures.

The task manager directs the sequence of task commands to proper channels and modules, and requests the necessary information from the lower levels to prepare for future requirements. The status of the structural system is feedback to the control manager (including current position and orientation, health degree, accumulate per­ formance indices, etc.) continuously. If there are several possible control sequences, the optimizer can decide which sequence will be used by calculating the performance criteria.

The main function of the designer is to develop methodologies to deal with novel situations for which no prior designs have been made. The designer uses decision making under uncertainty to select design algorithms. When the safety issue is taken MANAGEMENT & ORGANIZATION LEVEL

Task Manager Supervisor

(Mid-term) (Short-term) Learner

Tuner

Optimizer Scheduler

Designer Information

CONTROL CONTROL IMPLEMENTATION MANAGER SUPERVISOR

EXECUTION LEVEL

Figure 3: Coordination Level 21

into account, the design may initially suggest a method which preserves the safety of

the system without meeting all the performance specifications.

The supervisor is to carry out the sequence of control actions dictated by the

control manager panel. However, it is able to accomplish predetermined actions and

to deal with limited recognized failure situations. The main function of the scheduler

is to determine, during the performance of a specific control function, if certain con­ ditions are met in order to switch to alternative control laws (and plant models) and to appropriate identification, estimation and FDI (fault/damage identification) algo­ rithms. The scheduler provides the information of the sequence for the adaptation of the active members. For instance, for synchronous control sequence, the new control iteration is halted until all actuations of the active members are completed.

The tuner is to adjust, or fine-tune certain parameters in the control algorithms.

It also may select a new set for the parameters if the parameter identifier sends in new information about the system model. The control algorithms in general can be grouped into two categories: fast motion but less precise, such as configuration control or shape control; and slow motion but precise, such as precision control or vibration control. The different category would use different set of adaptation parameters installed in the tuner.

The main function of the information assessor is to process and to distribute sensor, state and parameter data to the data distributor in the execution level and the supervisor. In this level, learner modules also exist, but with less capability and shorter time frame. 22

Fault/Damage Identification (FDI) is the module designated to detect and identify

the possible system damage or component failure. In other words, it serves as the

health monitoring device in the system and communicates with the FDI algorithms

in the execution level.

COORDINATION LEVEL

Controller Parameter Identifier

Estimator FDIA

Actuators Sensors

ADAPTIVE STRUCTURE

Figure 4: Execution Level

The schematic arrangement of the execution level is shown in Figure 4. Sensors and actuators are the physical devices which accomplish the control operations. The controller executes the control algorithms and issues commands to the actuators. The sensory information is feedback to the data distributor and rerouted to the estimator, fault/damage identification and parameter identifier. The data distributor also sends the data to the information assessor in the coordination level which will split the data 23

into different modules in the upper levels. The main function of the fault/damage

identification algorithm (FDIA) in this level is to detect the structural damage and

identify the damage severity.

2.3 Information flow path

The control commands are passed from the higher hierarchical level to the lower

levels and the responses are returned from the lower levels to the higher levels (not

necessarily the consequent levels). The data buses and message paths are distributed

between modules. The data buses mainly carry the signals and numerical information

generated by the modules whereas the messages paths dispatch the commands and

prompting (including warning and error) messages.

To view how the information flow work in the system, we can use a typical example

to demonstrate. Consider a scenario by taking an adaptive truss structure as the docking device of the future space station. The overall operation example can be

presented here to highlight the information flow of the control system:

1 . The operator or the pilot issues a command like “Begin the docking procedure to

Dock 7.” The user interface interprets the human command (in a highly abstract

symbolic command level) and passes to the M anagem ent and Organization

Level where the control executive module resides.

2 . The command is evaluated by the capability assessor, goal generator, planner

and performance monitor, etc., of the executive level. These modules decide

whether the task is possible to be accomplished and reports the consequent 24

decision to the operator.

3. If the task is possible, the planner then issues a sequence of subtasks to the con­

trol management module in the Coordination Level. This sequence of control

commands could be “Deploy the docking arm (i.e. the adaptive structure)”,

“Move to coordinate (x,y,z)”, “Change the orientation to a,/?, 7 ”, “Fine-tune

to slowly approach the dock”.

4. The sequence of subtasks may still be in an abstract form. The control man­

ager, using its planner, divides say the first subtask, into smaller operations, e.g.

“Open lock mechanism”, “Activate the adaptive members controllers”, “Mea­

sure the distance of the target location”, “Feedback the current status”, etc.

The other subtasks are divided in a similar manner. This information is passed

down to the control implementation supervisor which organizes the operations

and applies the installed control algorithms to accomplish the operations.

5. The operations are executed in the Execution Level. The actuators realize

the numerical control algorithms, and the sensors measure the responses of the

vital information. The sensory information will be fed back to the control imple­

mentation supervisor module, and also be given to the fault/damage detection

module for the sake of safe operation and health monitoring.

The information flow between levels and modules (even the components) are sim­ ilar to one described in Ref.[27]. Specifically, for the case of adaptive structural system, the processes are explained in more details. The lengths of the adaptive truss 25 members are actuated by the control instructions. The sensors pick up the distance and angle information between the docking mechanism and the target point, and feedback to the damage detection and identification and control algorithms via the data distributor.

Since the degree of autonomy in this control design is required to be very high, the higher levels should adapt the performance of the adaptive structural system and interact with the changing environment. Although the command from the pi­ lot/operator may be simple and abstract, the subtasks coming from the higher two levels are successive and dynamical. The hierarchical architecture causes the number of the distinct tasks increase from the high level to the low level. We would say the intelligence increases as one moves up the hierarchical level. The controller has the ability to self-examine the performance, to adapt the change of the environment, to detect the damage of the components, to assure the stability while parts of the system failed, etc. Thus, it is believed that if the controller is realized, such a design will upgrade the structural system with very high degree of autonomy, that is, intelligence. CHAPTER III

Structural Damage Detection and Identification

The construction of optimum designed load-carrying structural systems such as air­ craft structures, turbines, and rotors, etc., has become more complicated. Although they are carefully designed for fatigue loading and inspected prior to service as well as periodically during their operating life, there are instances of cracks or structural dam­ ages escaping inspection. Consequently, the development of the structural integrity monitoring techniques has received increasing attention in recent years. Among these monitoring techniques, it is believed that the monitoring of the global dynamics of structure offers a favorable alternative if the on-line damage detection is necessary.

In a number of studies (Mayes and Davies [28], Cawley and Adams [29], Dimarog- onas and Papadopoulos [30], Gudmundson [31], Shen and Pierre [32, 33], etc.), global dynamic behavior of the damaged structures (e.g. natural frequencies, mode shapes, strain energy, etc.) were calculated based on the given properties such as damage location, damage severity, etc. The question of the estimation of such properties in the case where the behavior of the structure is known, i.e., the inverse problem, is discussed in this chapter. In general, the inverse problem inherently involves the issues of comprehensive search, solvability and uniqueness. Several investigations have been conducted on the inverse procedure of identifying the engineering prop­

26 27 erties of structures from dynamic response information. For example, Gladwell et

al. [34] successfully recovered the coefficients in the differential equation associated

with a Bernoulli-Euler beam from knowledge of the eigenvalues or related spectral data. Shen and Taylor [35] presented an on-line non-intrusive damage identification technique of a vibrating beam. The idea of the procedure is related to methods of structural optimizations in which the damage was identified in a way to minimize one or another measure of the difference between measurements and the corresponding values for dynamic response obtained by analysis of a model for the damaged beam.

Other noteworthy works presented in the literature that utilize modal data to detect the structural damage are Kabe [36], Zimmerman and Kaouk [37], and Zimmerman and Widengren [38].

In addition to the above approaches, a number of schemes were developed based on the measurements of time responses, e.g. Agbabian et al. [39] used a time-domain identification procedure to detect structural changes on the basis of measurements of excitation and acceleration response. Cawley and Adams [29] have found that the change of stiffness, local or distributed, is the primary reason that causes decreasing natural frequencies. Later Cawley and Ray [40] confirmed the observation on the case of a beam with cracks and slots. Identification procedures have also been developed using frequency responses, e.g. Wu et al. [41] and Samman et al. [42].

In general, the goal of these studies was to recover the engineering or damage properties of structures from knowledge of a complete set or a subset of modal and spectral data. In recent years, the applications of neural networks have attracted 28 increasing attention due to its capabilities such as pattern recognition, classification, function approximation, etc. and are well documented in the literature. The basic idea behind the proposed technique in this study is to assess the structural integrity in an on-line mode through a neural network based identification process. There are a few researchers applying neural networks in the field of damage assessment problem.

Teboub and Hajela [43] employed the classification ability of the neural network to identify the damage in composite material beams. Wu et al. [41] utilized the pattern matching capability of the neural network to recognize the location and the extent of individual member damage from the measured frequency spectrum of the damaged structure. Although their results look promising, there are several issues that need to be resolved and improved. The work presented in this dissertation draws on a similar idea but improvises their work to deal with a multiple damages situation. In addition, a new architecture of a neural network is proposed such that l)the number of input data is reduced; 2)it provides better generalization estimations; and 3)it has the ability to detect minor damages.

As an illustration of this concept, the present study examines two discrete multiple degrees-of-freedom (DOF) linear systems using different information representations.

In the example of a three DOF spring-damper-mass system, the changes of eigenvalues of the damaged system are chosen to represent the damage information. An appro­ priate neural network is trained to recognize the features of damage information of the system in which spring members have sustained varying states of damages. With a limited number of training data, the neural network is able to capture the general 29

scope of the damage characteristics. However, this sort of data representation may

not be a proper choice for a real structural or mechanical systems which usually have

closely-spaced eigenvalues. To deal with this class of systems, a dynamic-residual

vector proposed by Zimmerman and Kaouk [37] is adopted to represent the damage

information. Also, a three-subnet configuration of the neural network is proposed to

realize the identification process. The Kabe’s eight DOF spring-mass system [36] is

demonstrated in the present idea. It will be shown that the network is capable of

accomplishing such tasks and its performance is satisfactory for this specific model.

The subsequent section of this chapter presents a brief description of neural net­

works. Numerical results illustrating the use of neural networks in detecting and

identifying damages of a simple discrete system is presented in Section 3.2. In Sec­

tion 3.3, a more challenging model, the Kabe’s model, is sought. In order to deal

with the Kabe’s model, a new type of neural networks is introduced. Then, the new

neural network is applied on a truss structure and the simulation result is shown in

Section 3.4.

3.1 Review of Neural Networks

An artificial neural network is a framework consisting of many numbers of inter­ connected neuron-like processing units. Each neuron unit is stimulated by the sum of the incoming weighted signals and transmits the activated response to the other connected neuron units. Such a network represents an efficient and parallel compu­ tational entity, and will reflect the levels of stimulations by different input signals.

However, the process so far does not involve feedback and relaxation, in other words, 30 it only has the ability of propagating the input information, which can be termed as recognition process. Rosenblatt [44] developed the dynamic weight modification concept into the process of the neural network, which gave the neural networks the ability of “learning”. Rumelhart et al. [45] further provided an excellent algorithm which allows the multilayer neural network to internally organize itself to be able to reconstruct the presented patterns. This method leads to the recent most popular neural network learning scheme called the back propagation algorithm.

Output Level

Direction of Activatior of Error Propagation

Input Level v,______J

Figure 5: Backpropagation neural network

A typical architecture of a multilayer neural network is shown in Figure 5. The input layer receives input patterns and usually does not have processing units in this layer, simply transmits the signals faithfully to the next layer. The hidden layer (or layers), resides between the input layer and the output layer, consists of a certain number of processing units. And then the output layer constitutes the output chan- 31 nels, which also comprises the processing units. Each node in the preceding layer is fully connected to all processing units and the connections are called the weights which represent a different weighting scale of the input signals. The processing unit sums up the weighted signals and activates a response transmitting to the next layer.

The response curve is called the sigmoid function which can be any monotonically increasing nonlinear (or linear) function. From Kolmogorov’s theorem [46] and Cy- bento’s theorem [47], it is known that any nonlinear mapping can be approximated by appropriate combination of weights and sigmoid functions and realized by the multi­ layer neural network. The input pattern is propagated forward and actual responses are obtained. The errors between the desired outputs and the actual outputs then propagate backward through the network, providing a vital information for weight adaptation. The backpropagation algorithm delicately uses this information to adjust the weights such that a “mean-squared” error measure is minimized. This supervised learning algorithm, using a gradient descend optimization scheme, lets the network converge to a minimum in the weight space. The algorithm is summarized as follows:

1. Forward propagation. The processing unit in each layer performs the following

computations:

yt = f( ^ w i3xi - $j) (3.1) «=i where y} is the jth node in any layer, /(•) is a sigmoid function, Oj is a threshold

at the jth node, x; is the incoming signal from the previous layer, Wij is the

connection weight between ith node in the previous layer and jth node in the

current layer and N is the number of input signals. In this thesis, a sigmoid 32

function is used as below

/<*) - (3-2)

where s is the slope of the sigmoidal curve.

2. Calculate the errors at the nodes of each layer. At output layer, such errors are

the differences between the desired output and the estimated output.

3. Calculate the derivative of the system error with respect to the weights, Si, by

the generalized delta rule. At the output layer

Si = /'(*.)($. - *). (3.3)

and at the hidden layer

Si = f \ x i ) SkWki (3.4) fc=i where y, is the desired output value.

4. Update the weights

w !;-u l (k + 1) = + Awl‘- 'JI(k) (3.5)

= + r,S,(k)y]'-'>(k)

+QAur«-,J)(* - 1) (3.6)

where tj is called the learning rate, which is used to control the convergence

speed of the learning process, and a is called the momentum term.

5. Test the convergence criteria. The maximal pattern error is defined by

E p = inax(y, - ys) (3.7) 33

and the mean-squared system error is defined by

£ = (3 8) »=1 jm 1

where Np is the number of the training patterns and JV& is the number of output

nodes. The training process stops if either one of the errors measure is satisfying

the convergence criteria. If not, go back to Step 1.

A larger i] would make the learning faster, yet is more likely to cause the system to be

unstable. However, it can be compensated by a large momentum term. So selecting

ri and a is a tradeoff between learning speed and stability. A rule of thumb for first

trial is to choose Tf small (e.g. 0.2) and let a=0. If the learning process is fluctuating,

add positive momentum term to go on the next trial.

In this work, a backpropagation neural network simulation program called NETS

(Version 2.0) [48] was used for the first example. This program was made available

by NASA Lyndon B. Johnson Space Center. The program was implemented on

an APOLLO DN4500 workstation which has 33MHz processing speed. The second example is done with an efficient simulation code written by Fahlman [49], which is

called QUICKPROP implemented on CRAY Y-MP8/864 supercomputer.

3.2 Three DOF System

For simplicity, a three degree-of-freedom (DOF) model shown in Figure 6 is first analyzed. The equation of motion of a three DOF spring-damper-mass system is formulated as

Mx + C i+ Kx = f (3.9) 34 where

' m , 0 0 ' M = 0 m 2 0 (3.10) 0 0 m 3

‘ Ci + c2 -C i 0 ' C = -C l C2 + C3 - c 3 (3.11)

0 —c3 C3 .

fci + ^ 2 - k i 0 K = - k 2 k2 + k3 - i s (3.12) 0 -fc3 * 3 The task is to utilize the neural network to identify the locations and the severity if damage to one of the springs has occurred. That is, when the stiffness of the spring decreases because of the damage, say A k \, AJfca,or &k3 (in percentage), the goal is to identify which spring and how much the spring stiffness changes from deducing the abnormal frequency responses of the discrete system.

M i

Mi = M5 = 1 M2= 2 q = q = g = 0.02 K1 = 2 K2 = 4 K3= 3

Figure 6: Three DOF Mass-Damper-Spring System

The frequency responses of the three DOF mass-damper-spring system have three spectrums in nature. An intuitive way of choosing the input pattern for the network learning is to use the magnitudes and/or phases of the sampling frequencies in suffi­ ciently large spans of each spectrum or all spectra. However, in order to characterize 35

these spectral properties, a small sampling rate is required, in turn, a tremendous

amount of sampling data will be needed which significantly jeopardizes the efficiency

and accuracy of the neural network training process. Thus, in this study, the changes

of the eigenvalues between the undamaged system and the damaged system are used

as the input pattern. The choice was made based on the following advantages: 1)

The length of input pattern reduces to only three;(compared to 200 nodes in [41])

and 2) The changes of the eigenvalues include two important information of frequency

responses, i.e. natural frequencies and corresponding damping ratios.

The scenario of the proposed method is based on the assumptions that the masses

and damping coefficients, m, and c,, are fixed, and the frequency responses of the

systems are provided from the simulated measurements. The patterns are generated

by calculating the absolute values of the differences between the eigenvalues of the

undamaged system and those of the damaged system. Several cases are taken into

account in our work. In general, these cases can be divided into two categories: single

spring damage and multiple spring damages.

3.2.1 Single Damage Case

In this category, the damage of the spring is simulated by reducing its stiffness from

10% to 90% with 10% intervals. For every varying stiffness matrix, the generalized eigenvalues, which are slightly different from the referenced eigenvalues of the undam­ aged system, of Eq. (3.9) can be calculated by the standard generalized eigenvalue solver. The absolute values of the eigenvalue changes between the damaged and un­ damaged springs are inputs to the neural network, and its outputs are therefore the 36 damaged springs and the corresponding stiffness losses. Then a three-layer neural net­ work was trained by these training patterns with the backpropagation algorithm [48}.

The training process continued until a prescribed convergence criterion was satisfied.

Then, a set of testing data corresponding to 5% to 95% with 10% interval damage states was propagated through the trained network to examine the generalization of the results.

Table 1: Individual spring damage results

% damage A*! A *3 AJfe3 % damage A ki A *3 Afcs 10 0.1026 0.1026 0.1034 5 0.0668 0.0659 0.0678 (2.61) (2.58) (3.40) (33.5) (31.7) (35.6) 20 0.1959 0.1962 0.1957 15 0.1465 0.1469 0.1468 (2.05) (1.90) (2.15) (2.24) (2.07) (2.16) 30 0.3008 0.3003 0.3004 25 0.2481 0.2480 0.2476 (0.26) (0.10) (0.13) (0.74) (0.79) (0.94) 40 0.4028 0.4024 0.4033 35 0.3526 0.3519 0.3526 (0.70) (0.65) (0.82) (0.73) (0.55) (0.73) 50 0.4999 0.5003 0.5009 45 0.4517 0.4517 0.4526 (0.02) (0.06) (0.18) (0.33) (0.38) (0.58) 60 0.5971 0.5977 0.5975 55 0.5480 0.5488 0.5489 (0.48) (0.38) (0.43) (0.42) (0.23) (0.19) 70 0.6995 0.6991 0.6991 65 0.6475 0.6477 0.6475 (0.07) (0.12) (0.12) (0.40) (0.35) (0.39) 80 0.8045 0.8037 0.8044 75 0.7523 0.7515 0.7519 (0.56) (0.46) (0.50) (0.32) (0.20) (0.24) 90 0.8968 0.8976 0.8976 85 0.8535 0.8534 0.8540 (0.35) (0.26) (0.26) (0.35) (0.40) (0.46) 95 0.9321 0.9337 0.9326 Emax 0.004487 0.003892 0.004603 (2.08) (1.72) (1.83)

Table 1 shows the final network outputs of training process and the testing results for single spring damage cases. In each entry, the number in the upper row is the

network output and the number in parenthesis is the relative percentage error. The

convergence criterion for all three simulations is that the maximal pattern error should

be less than 0.5%. The assumed damages are listed in columns 1 and 5 where column

1 indicates the training set and column 5 represents the testing set. The trained

network outputs for each spring damage are tabulated in columns 2 to 4, respectively.

Forty hidden neurons are used for each spring damage and the training process takes

about 80000 iterations to learn the pattern representation. Although the results of

training patterns show the precise estimation of the damage, it is not guaranteed that

for other unknown damages the estimation would have the same accuracy. Therefore,

it is necessary to test the generalization of the neural network. The middle points

between every two consecutive training patterns, which reflect the possible worst

testing representation, are chosen to test the trained network. The testing results are shown in columns 6 to 8. The results all lie in a satisfactory range of desired values. In other words, the neural network not only learns the training pairs, but also establishes an accurate and unique input/output mapping for any possible damage.

The largest discrepancies are those results of 5% and 95% damage cases. This is because that these two testing patterns are beyond the training representative range, which ranges from 10% to 90%. The estimation of both patterns could present a little distortion due to internal extrapolation of trained mapping structure. However, the estimated results are still in acceptable range(absolute error less than 0.02). 38

3.2.2 Multiple Damage Case

In this category, the similar training process that applied in the single damage case is followed. Three ranges of assumed multiple damages were generated. They are:

Case Km ix Wide range variation of spring damages. Each spring has 0-5-10-20-40-

80% damages and then resulting in 216 combinations of spring damages. This

case represents the broad scope of the identification problem.

Case KmOl Short range variation of spring damages. Each spring has 0- 5-10%

damages and resulting in 27 combinations. This case represents the minor

damage identification problem.

Case Km45 Short range variation of spring damages. Each spring has 0-40-45-

50% damages and resulting in 64 combinations. This case would expose the

identification resolution problem.

In each case, according to those damage combinations, the equation of motion was updated first and then generated a set of training patterns.

For multiple spring damages cases, similar to the single spring damage, a three- layer backpropagation neural network was used. The numbers of input and output neurons are both 3. The numbers of neurons in the hidden layer are 100, 60 and 40 for

Kmix, KmOl and Km45 cases, respectively. The convergence criteria for all three cases are that the absolute pattern errors should be less than 3%. The training process for these cases takes about 60000 to 70000 iterations. After the network is trained, several patterns different than the training patterns are randomly chosen to test the 39

performance of the network.

The simulation results for multiple damage cases are depicted in Figures 7-9.

Figure 7, which corresponds to the Kmix case, consists of 236 total patterns, including

216 training patterns and followed by 20 testing patterns. For each testing pattern,

the damage identification error is less than 0.03 except for those points that stand

for 90% damage, in which the errors are still less than 0.05 (which corresponds to

relative percentage error 5-5.5%). Nevertheless, this observation agrees with the result obtained in the single damage case because it is anticipated that the neural network's fair extrapolation ability to dealing the input data beyond the training range.

The result of the multiple damages identification for the KnOl case is shown in

Figure 8. One may observe that it seems to be less accurate than the previous case.

The reason for the lag between the network output and the desired value is due to the 0.03 convergence criterion, which is about 30% of training domain. However, the neural network has captured the feature of the damage information and the results are in agreement with the desired tendency. There are a couple of remedies to improve the result: 1) Toughen convergence criterion, which will require a longer training time;

2) Increase training patterns, which will be demonstrated on the next problem, the

Km45 case. In Figure 8, the last seven patterns are testing patterns. Parenthetically, the desired outputs are chosen to be the actual values in the training process. To compensate for the large relative error due to coarse convergence criterion, the range of the desired outputs can be rescaled from [0,0.1] to, say, [0.3,0.7], which is more sensitive to the neuron's activation output. This adjustment is believed to be able to 40

• • j 1.0 i rarn^± 1est - Desired Output I 0.8 • NN Output ...... _ 0.6 r < 0.4 •i 0.2 0.0 j----- ... i ------0 50 1 0 0 150 200

0.8 0.6 <

0.2 0.0

0 50 1 0 0 150 200

0 0.8

0.2 0.0

0 50 100 150 200 Pattern No. Figure 7: Neural network outputs of Case Kmix 41

0.5 Train < j * Test 0.4 - Desired .Output • NN Out|3ut 1 : 0.5 1 < 0.2 1 : 0.1 • VI ------. . . . ./ L . i* ■ ■ r 0.0 i i i 1 i ------1 0 10 15 20 25 30 35 0.5 Train Test 0.4

0.3 Cnj < 0.2

0.1 0.0

0 510 15 20 25 30 35 0.5 Train Test 0.4

0.3

< 0.2

0.1

0.0

0 5 10 15 20 25 30 35 Pattern No. Figure 8: Neural network outputs of Case KmOl 42 improve the performance of the neural network.

Since the above conjecture indicates that increasing the training patterns could improve the generalization capability of the neural network, the Km45 case was de­ signed and tested for the multiple damage case. The results are very promising and displayed in Figure 9. The front portion of the curve in Figure 9 is 64 training pat­ terns and is followed by 26 testing patterns. One can observe that the neural network indeed produces very good approximations to the desired values, which confirms our conjecture. The pattern errors of testing data are all less than 0.0167 (relative error

3.55%).

From the above simulations of the three DOF discrete system, it can be concluded that the application of the neural network to such a system is highly feasible. However, we should survey the frequency property of this example more carefully. It is under­ stood that since the generalized eigenvalues of this system are sparsely distributed, the drifted eigenvalues due to decreasing spring stiffness will not crossover the lower eigenvalues. This phenomenon provides a simpler input/output mapping relation­ ship and validates the proposed damage information representation. Conversely, for some physical systems which exist with a closely-spaced eigenvalue distribution, the above identification process may have difficulties in establishing proper interpolation of the damage patterns. For such kinds of physical systems, it is necessary to devise a new strategy of pattern representation. In the next subsection, the Kabe’s model, a typical example for a closely-spaced eigenvalue distribution physical system, will be investigated by a novel identification procedure. 43

0.5 0.4

_ 0.3 Train- 0.2 -Desired...Output • NN Output 0.0 0 20 40 60 80 0.5 «• 0.4 ♦*

CN rarn Te 0.2

0.0 0 20 40 60 80 0.5

0.3 f r a n < 0.2

0.0 0 20 40 60 80 Pattern No. Figure 9: Neural network outputs of Case Km45 44

3.3 Kabe’s Model

Kabe’s eight degree-of-freedom spring-mass system is shown in Figure 10. In order to simulate a challenging situation for the structural damage identification, the model was designed to generate the closely-spaced frequencies in both local and global modes of vibration. This is achieved by using very high stiffness and mass ratios as shown in

Figure 10, rather than the standard spring-mass system which commonly uses either or very close stiffness and mass values.

3.3.1 Dynamic Residual Method

Zimmerman and Kaouk [37] proposed a subspace rotation algorithm, which was de­ rived from the generalized eigenvalue problem, to identify the location of a damaged spring in Kabe’s system by only measuring the fundamental modal data. For a mass- spri ng model, the equation of motion can be expressed as

Mx + A'x = 0 (3.13)

The generalized eigenvalue problem of Eq. (3.13) is given by

K i*. = (3.14) where Aj,, and v^ denote the ith eigenvalue and corresponding eigenvector, respec­ tively, of the undamaged structure system. Now consider that the ith eigenvalue and eigenvector of the damaged structure are available. Then define a dynamic residual vector as follows

di = (K - Arft • M) • vdt (3.15) 45

—^\f~ M6 M8 K13 K3 K7 K ll K5 M 3 — t\f\/~

M l =0.001 M2 = 0.002 Mj = 1, j=2,3.....7 K1 = K14 = 1.5 K2 = K4 = K10 = K12 = 1000 K3 = K ll = 10 K5 = K7 = K9 = 100 K6 = K8 = 900 K13 = 2

Figure 10: Kabe’s Model

Inspection of

Zimmerman and Kaouk’s method provides a clever and simple way to detect the damage.

3.3.2 Simulation Result

In this study, the above idea is employed to represent the damage information. The damage identification can be achieved by accomplishing the following tasks: 1 ) to transform the modal data (frequency and mode shape of the damaged system) to the dynamic residual vector; 2 ) to identify the locations of the damaged springs by applying the dynamic residual method; 3) to determine the severity of the damage from measured data. The identification process exhibits two difficulties to be resolved: to translate the inspection of nonzero entries to numerical representation and to quantify the severity of damage from the given information. In light of the context of the problem at hand, a neural network presents itself as a logical tool. To realize these three operations, a neural network that consists of three subnets performing individual tasks is devised (see Figure 11). As shown in Figure 1 1 , the first subnet is constructed of two layers whose inputs are the frequency and mode shape and the weights are corresponding elements of K and M matrices. Note here that the mode shape is normalized with respect to the M matrix such that V TM V — /. The first subnet's output, the dynamic residual vector, is then evaluated based on definition in Eq.(3.15).

Then the dynamic residual vector

3rd SuhNot 1st SubNet 2nd SubN st

= F f f

i

Figure 1 1 : Neural network architecture for structural damage identification

on the input units of the second subnet, where the activation function is defined by

1 if x > 0 / ( x) = agn(x) = 0 if x = 0 (3.16) - 1 if x < 0

The task is similar to a classification problem and is accomplished by a three-layer backpropagation neural network. The number of inputs to the second subnet is 8 in this example and the output would be a 14-bit binary number in which each bit indicates a spring. The bit is 1 for a damaged spring and 0 for an undamaged spring. The subnet is pre-trained by 105 training patterns which are generated by assuming any combination of two damaged springs each with 5% stiffness loss. This 48

5% is chosen such that it is desired that the proposed neural network can identify

at least 5% damage (which represents the resolution). Since the second subnet is

pre-trained, it can make a precise identification of the damage location at once. The

output information would serve as an indicator for retrieving data for the third subnet.

In addition, the outputs could be directly sect to a display device without further

modification.

Finally, the third subnet utilizing the information given by the first two subnets

calculates the severity of the damages. The process can be described in more detail as

follows: at first, the 14-bit number obtained from the second subnet serves as an index

of addresses in a pattern database. The database is prepared in advance by generating

proper patterns corresponding to different combinations of assumed multiple damages.

The number will trigger the third subnet to extract a set of training patterns in the

database at the indexed address. Then the subnet is trained on-line by these training

patterns. Since only a small number of the training patterns are associated with a damage, the on-line training process for the subnet is considered very fast. After

the self-training, the dynamic residual vector d* obtained from the first subnet is fed

into the trained third subnet. Therefore, an estimation of the severity is obtained.

It is important to note that this subnet performs an on-line learning first and then estimates the damage severity. This is different from the second subnet where it is trained in advance and identifies the damage location in real time.

A point should be mentioned in considering the learning process for the second subnet. If only the fundamental modal data is used, the patterns of the signed 49

d-vectors have over one-fifth repeatedness. More specifically, there are 11 pairs in

these 105 training patterns which are identical patterns. This will cause ambiguous

results of the damage location determination and, therefore, the uniqueness of the

damage identification is questionable. To reduce those indistinguishable patterns,

it is suggested that the second modal data, even the third modal data should be

included in the input data for training purpose. If the second modal data is also

used, the identical pattern pairs reduces to 2. Furthermore, if the first three modal

data are used, all training patterns are distinguishable. Nevertheless, the trade-off

is the complexity of the complete network and much longer training process for the second subnet. This observation agrees with the minimal modal data requirement suggested by Kabe [36]. A similar conclusion had been reached by the study of Shen and Taylor [35].

To demonstrate the results of the neural network, two simulations are presented.

In the first simulation, the damages are assumed at &3 and ki3 . The first two modes are used as the input data (the pattern samples are listed in the Appendix A). In the pattern database, the damages of springs are varying from 10% to 50% with

10% increment. Hence, for a two-spring-damage case, 25 patterns are retrieved for training the third subnet. To speed up the learning process, the number of hidden neurons of the third subnet is chosen to be 10. The on-line training was achieved at the stage that the absolute pattern error or the number of iterations are less than the specified values (0.5% and 500, respectively, were adopted in the present case). The input patterns corresponding to 5% to 55% with 10% increment spring damage are 50

generated and propagated through the trained network to test the performance of the

network. The result is shown in Figure 12 where the first 25 patterns are training

patterns followed by 36 testing patterns. The maximal pattern errors for the testing set are 1.7% and 3.2% of &3 and kl3, respectively. One can observe that the network is capable of providing an accurate identification of the damages.

To extend further, a three-spring-damage case is considered. The possible damages occurred at klt ki3 and kM are assumed. As the number of damaged springs increases, the combinatorial problem will be dominant in generating the database. To explore the compromise, two exercises are examined. First, the damage states are given at 0-

15-30%. Thus, 27 patterns for each combination are generated. The simulation result is shown in Figure 13, where the first 27 points are training patterns followed by 64 testing patterns, which represent 2-10-20-28% damage possibilities. It is evident that the discrepancies on the testing results are outstanding. Hence, the damage states are chosen as 0-10-20-30% and 64 patterns per combination are stored in the database.

Two sets of testing data, one corresponding to 5-15-25-35% damages for verifying the generalization and the other corresponding to 3-13-23-28% damages for checking the resolution, are examined. The result is shown in Figure 14 where the first 64 points are training set and followed by 64 patterns of the first testing set and 64 patterns of the second testing set. Except for the points representing 35% damage, the calculated results are pretty good. The abnormal phenomenon for the 35% damage case is expected because the damage severity is beyond the representative domain of the neural network. Moreover, the result of the second testing set shows that the neural 0.6 Train Test • • • • f 0.5 0.4

0.3 0.2 Desired, output NN output 0.0 0 10 20 40 50 60 0.6 Train Test 0.5 0.4

0.2

0.0 0 10 20 30 40 50 60 Pattern No.

Figure 12: NN outputs of Kabe’s modelrtwo-spring-damage 52 network did establish a good resolution of the damage representation. The maximal pattern errors for ki, fcja and k u are 2.3%, 2.8% and 2.5%, respectively (except 35% points).

One point that should be addressed is that the backpropagation neural network has no inherent ability to indicate when it is functioning outside the domain over which it was trained. So the extrapolation performance of the neural network may be fairly poor as shown in Figure 13. This disadvantage may be overcome by using radial basis function networks (RBFN) [50].

3.4 Ttuss Structure

After examining the above two structure models, we are confident to test the damage identification technique on the truss structures. Before applying the proposed method to the trusB structure, there are a few assumptions that have been made for the current truss structure, they are:

1. The truss structure is statically determinate.

2. The truss structure is perfectly pin-jointed.

3. The members are rigid.

For control problem which will be discussed in the next chapter, two more

assumptions are made:

4. The actuators are linearly extensible actuators. 53

0.5 0.4 r 0.3

0.0 0 20 40 60 80 0.5 0.4

K, 0.3 •••« . <

0.0 0 20 40 60 80 0.5 Desired output 0.4 NN output

0.3 j* <

0.0 0 20 40 60 80 Pattern No. Figure 13: NN outputs of Kabe’s model with 0-15-30% training 54

0.6 •* 0.5

_ ° - 4 < 0.3 0.2 0.1 0.0 0 50 100 150 200 0.5

0.4

<

0 50 100 150 200

0.4

0.3

< ” 0.2

0.0 0 50100 150 2 0 0 Pattern No. Figure 14: NN outputs of Kabe's model with 0<10-20-30% training 55

5. The motion is slow enough such that the structural dynamics (or articulatory

dynamics) is negligible.

In this section, the truss structure to study assimilates the plane adaptive truss

structure used in the next chapter, only here the active members are assumed fixed.

The truss structure containing four shear deformation modules is shown in Figure 15.

Figure 15: A four-module truss structure

For the sake of further discussion, the bar elements in the truss structure are numbered from left to right, from bottom to top. The fundamental modal data (first mode) is extracted for the damage identification. The damage location is detected perfectly by using the proposed identification method. For identifying the multiple damage severities, bar element #6 and #11 are assumed to have stiffness degradation.

As same as the discussion in the preceding section, twenty-five training patterns are stored in the database. After on-line training the third subnet, the neural network is fed by the measured modal data. The simulation result is shown in Fig. 16. From the figure we can see that the present method is able to predict the damage severity 56 with a reasonable precision.

It should be noted that the proposed technique is applicable to the very com­ plicated model as long as the system model (in the discrete second-order form) is available. A full order equation of motion for the last illustrative example is used in the simulation result. For more complex structural system, the number of the degree- of-freedom may be very large. However, after several stages of model reduction (e.g.

Guyan reduction [51]), one can obtain a lower order system. Then, the proposed technique can be applied on such a reduced model equation and makes reports on the damage status of the system. But, there may exist some difficulties to trace back the indicated damage information if one tries to pin-point the particular damaged member. This is because the degree-of-freedom of the reduced model represents the distributed physical degree-of-freedoms, so there is no unique answer to pick out the

‘real' damaged member singly from this distributed degree-of-freedom information.

In other words, there are several possibilities of different member damages affecting the same distributed degree-of-freedom. 57

0.6

0.5

0.4 ul CO a 03 0.2

0.1 Desired;, output. NN output 0.0 0 10 20 30 40 50 60 70 0.6

0.5

0.4

J" 0.3 < 0.2

0.1 0.0 0 10 20 30 4 0 50 60 70 Pattern No.

Figure 16: NN output for two-member damages of the truss structure C H A P T E R IV

Fuzzy Logic Control Design

In the proposed adaptive structure models, there usually exist plenty of actuators such that the structures can operate complicated mechanisms for articulation and dexterous maneuvers. In this study, the adaptive truss structure is expected to ac­ complish specified tasks, especially motion control. Thus, in this chapter, we will discuss how to construct a control algorithm for the purpose of motion control for the adaptive truss structures.

Although, the control of the adaptive truss structure may represent a similar approach for the control of the robot arms or the joint-actuated manipulators, the former issue still leaves many problems to be exploited. For example, the active truss members should be capable of undergoing large variations in lengths (in contrast to the piezoelectric type of the active member which is mainly designed for vibration control or precision control) and most significantly, the inverse kinematics for such high degree of redundant adaptive truss structures is required to perform in real time or near real time.

Recently, several approaches have been proposed to solve the redundancy and the inverse kinematics problems, such as the pseudoinverse technique [52]. Indeed, the success of using the pseudoinverse control to solve the inverse kinematics problem has

58 59

been widely recognized. However, the technique relies on an exact inverse kinematic

model to guarantee the satisfactory results. These inverse kinematic equations are

usually derived from a specific model. Although it may be possible to obtain the inverse kinematics equations for most adaptive truss structures, their explicit solutions are usually undesirable due to the structural redundancy. This limitation dilutes the possibility of using the pseudoinverse technique to design a controller for adaptive structures and motivates the search of a controller that does not require an explicit solution of the inverse kinematic equations.

It was our initial intention to apply neural network control on this subject. How­ ever, the up-to-date neural control techniques suffer the problem of requiring long training time such that it may be inappropriate for motion control of the adaptive truss structure. In addition, the adaptive truss structure is a time-varying system that the neural control methods still are difficult to deal with. The conventional practice of neural control is using two neural networks, one of which is for emulat­ ing the system kinematical behavior and the other is for controlling the system, to complete the control operation. However, we observed the difficulty to apply such approaches to the adaptive truss structure.

Conversely, fuzzy control has been proposed as a candidate to accomplish the task of designing a controller for complex and ill-defined space truss systems. For example,

Matsuzaki and his colleagues proposed a series of fuzzy control methodologies to an adaptive planar truss structure [53, 54]. They successfully applied fuzzy rules to control the Bending Deformation Module of the truss structure to trace the moving 60 target. For some reason, the Shear Deformation Module remained iixed during the entire process which is considered inadequate for actual tracking action.

Inspired by their approach, this work attempts to apply the fuzzy logic control to more extended tasks which allow maneuvered Shear Modules of the planar adaptive truss. The control tasks to be considered in this study are to change the configuration of the adaptive truss structure such that the pivoting point or the end-effector of the structure will approach a desired target position, or follow random (unknown) trajectories, or pursue a moving target. The tasks represent some typical maneuvers of the space adaptive truss structure.

Since the adaptive truss structure is usually constructed by a repetitious stack of basic structure modules, such as octahedral, tetrahedral and cubic, in this study, the control problem is broken down to focus on the basic structure module instead of the whole structural system. Moreover, we are utilizing the distance errors as the input data instead of utilizing linear-rate velocities. Thus, a series of fuzzy controllers, each of which governs the basic structural module, based on the distance errors between the pivoting point and the desired target are constructed. The merits obtained from this approach are discussed.

In addition, based on a general inverse kinematic relations of adaptive truss struc­ tures, a systematic way of formulating the fuzzy rule set of each controller is also proposed. A similar concept has been presented in the studies of robot motion con­ trol (Zhou and Coiffet [55] and Nedungadi [56]). We are also considering the issues of how to increase robustness in spite of imperfections in modeling, noises in measure­ 61 ments, possible component failures, and the ability to handle nonlinearities without control system degradation.

4.1 Fuzzy Logic Control Theory

The fuzzy control theory is pioneered by the research of Mamdani and his colleagues

[57], which was motivated by Zadeh's invention of the fuzzy sets [58]. There have been plenty of research works and studies stressed the applications of the fuzzy logic control. It is not intended to do a comprehensive survey on this issue here. Instead, a brief review of the fuzzy control theory is described.

Fuzzy logic, which is the base of the fuzzy control, can be thought as a mathemat­ ical tool closer in spirit to human thinking and natural language than the traditional logical systems. So, the fuzzy logic control is a step toward a reapproachment between conventional precise mathematical control and human-like decision making process.

Each fuzzy controller constitutes of two parts: Inference Mechanism and Knowl­ edge Base. The controller receives the crisp input information, represents the input information as the confidence measure in the fuzzy set, obtains the proper output response information obtained from the knowledge base, then translates it into the crisp output response. The entire infrastructural procedures are referred to as tn- put/fuzzify/rule inference/defuzzify/output steps. By the combinations of different choices on these procedures, the fuzzy logic control will attain a rich domain of linear or nonlinear effects [59].

In designing a fuzzy controller, one must identify the main control variables and determine a rule set which is at the right level of granularity for describing the values 62 of each linguistic variable. The control rule base must be developed using the above linguistic description of the main variables. Sugeno [60] has suggested four methods for doing this:

1. Expert’s experience and knowledge

2. Modeling the operator’s control actions

3. Modeling the process

4. Self-organization

The first two methods are based on the human’s experiences which are mostly ac­ cumulated by trial-and-error practices or learning processes. The last two methods utilize the reaction of the proper manipulations and accrue to a working database. In this work, we will focus on how to utilize the available information at hand, i.e. the kinematics relationships of the adaptive structure, to design a fuzzy rule table.

Fuzzy logic control has been demonstrated to be very effective in cases where high precision is not a major requirement. It is also shown that the fuzzy controller can be so designed to be at least as good as a best-tuned PID controller [61]. However, there are two major shortcomings for the fuzzy control system:

1. The fuzzy control system provides a conservative solution which is far from an

optimal control.

2. There is no analytical approach for the derivation of rules. 63

Recently, Shoureshi and Rahmani [62] has proposed an integration method which

combines the concepts of fuzzy logic and optimal control. However, to tackle the

subject around the adaptive truss structure, it is our intention to investigate the feasibility and applicability of fuzzy logic control on certain missions. It is our belief that the advanced technique involving fuzzy logic control is applicable as long as the current study observing some promising results.

4.2 Kinematics of Adaptive Structures

There exist generalized techniques to develop kinematic description for conventional anthropomorphic manipulators [63]. Those techniques, however, are not all suited for variable-geometry truss manipulator. Naccarato and Hughes developed a gen­ eral kinematic methodology for adaptive truss structures [64]. By their method, a combined kinematic equation consisting of rate-linearized constraint equations will be obtained. For reference to further discussion in the dissertation, the inverse kine­ matics of the adaptive structures is briefly reviewed in this section.

Consider an adaptive truss structure constructed by N linear truss members. All the nodes between members are ideally pin-jointed. The structure is statically de­ terminate and is assumed to move slowly. Consequently, the internal forces, Coriolis forces and the centrifugal forces are neglected. Therefore, only the kinematic relation is involved. There are M truss members which are actuated and rjtn change their lengths. Control of these lengths will result in the determination of the structural configuration as well as the position and orientation of the tipend of the structure.

Define the work vector w with ne degree-of-freedom indicating the position (ze,ye, ze) 64 and/or orientation at the pivoting point with respect to an inertial refer­ ence coordinate system, such as:

“ (*e, ye, 4>Si v, 4 >m)T (4.1)

The member length vector I that indicates the lengths of those active truss members is denoted as:

f = (!,,!*, (4.2)

The region L t where I can move is defined as follows:

L. = {/ = (fi, /3, • ■ • ,/w )T|U « < k < = 1,2, • * •,M} (4,3)

We may state the direct kinematic relationship as

to = f{g) (4.4) where q(t) representing the configuration variables of the structure. In general, the direct kinematic relationship, Eq.(4.4), cannot be inverted directly because of the redundancy. Thus, the differential kinematics is usually resorted to get a rate-linear system as shown below,

to = J(g)g(t) (4.5)

/ = % <«>

For conventional robotic arms, the configuration variables are identical to the actuator variables. However, for the adaptive truss structures, an intermediate set of variables instead of the active member lengths is usually used. For ideally pin-jointed truss 65 structure, the node position vectors are often chosen to be the configuration variables.

Let r denote the congregated nodal position vector, the kinematics for the adaptive truss structure is given as follows;

I = Ji,rT (4.7)

ih = Jw,rr (4.8)

These two equations yield

w = — J l (4-9)

It is known that J, the Jacobian matrix , is usually not a square matrix. Then, a common solution of Eq.(4.9) take the form

/= J+th + (7- J+ J)i. (4.10) where J + is the Moore-Penrose pseudoinverse of the Jacobian matrix, i.e., JJ+ = 7, and I, is an arbitrary configuration velocity that may be used to minimize a per­ formance index [65], avoid obstacles [66] or meet some other objectives [67]. For minimizing a performance index p(/)» the I, can be selected as

i. = - K ^ (4.11) and K is a positive definite gain matrix. Combining the last two equations yields

/ = J +w - (7 - (4.12)

Since the second term is an orthogonal projection on the null space of J so that the velocity vector decreases the performance index without changing the error of the work vector. Note that the active member velocity vector I may change discontinuously because of the bound of the domain L, described in Eq.(4.3). 66

4.3 Fuzzy Controller Design

A process operator, having a specific control goal, would perform visual observation

of the process states, controls and processes outputs. Then, the operator intuitively

assesses the variables and parameters of the process and makes a control decision from

subjective assessment so as to be able to accomplish the assigned control requirement.

As a rule, such a control algorithm is flexible and is better than a control algorithm

obtained from the classical control theory [68]. The fuzzy control algorithm interprets

the similar human-like approach, thus, is felt to do the same or better than the

conventional control methods.

There have been several studies to describe the general procedures of designing

a fuzzy controller. However, few studies emphasized how to construct the fuzzy rule

tables. One of the objectives of this study is trying to provide a systematic way to

formulate the fuzzy rules for adaptive truss structure. One may follow the similar

procedures in the literature to finish the design process. However, a few treatments

specifically suitable for the adaptive truss structures are discussed in the following

design procedure. Also, it should be pointed out that this work is not intended to

discuss all possible variations of fuzzy controller design. Hence, only the most used

formulation is discussed in the dissertation. The design procedure can be applied to

other variations of the controllers.

As mentioned before, instead of designing one fuzzy controller for the entire struc­

tural system, the modular fuzzy controller for the basic structural module is consid­ ered, shown in Figure 17. Then the control system is completed by serial construction 67

Knowledge Base Adaptive Structure Target J Inference Mechanism Fuzzy Controller

Figure 17: The modular fuzzy controller

of the modular fuzzy controllers. This special treatment simplifies the analytical pro­ cess of the model and utterly utilizes the characteristics of the adaptive structures.

The representation of fuzzy partition of the normalized input and output universe of discourses is shown in the Figure 18. The advantages of using the normalized universe of discourse are: (1) standardizing the further discussion of the fuzzy system, thus, the same fuzzy rule may be applied to different ranges of domains; (2) increasing the flexibility for the engineering design. We only need to adjust the gains at each signal channel to adapt various signal ranges. Evenly distributed triangular membership functions are used in this study. Nevertheless, different types of membership func­ tions such as monotonic, trapezoidal and bell-shaped, may be used at the designer’s choice.

The overall fuzzy controller design concept is summarized in the following proce­ dure. 68

Nn N1 iZ P I P n

Input Nn N1 AZ P I P n

Output Figure 18: The normalized universes of discourse and the membership functions

Step 1 - Divide the Input and Output Spaces into Fuzzy Regions

The domain interval of a variable means that most probably this variable will lie in this interval (the value of the variable is allowed to lie outside its domain interval). Divide each domain interval into 2n + 1 regions, i.e, the number of the membership function is chosen as 2n + 1. Assign each region with a linguistic value, say Nn (Negative n), • • •, N\> Z (Zero), Fj, • • •, Pn (Positive n). It is general practice that the number of memberships will be chosen as a middle number. The larger number means high precision where the knowledge base grows proportionally. To keep the compactness of the knowledge base and reach the reasonable precision, it is recommended that n is selected to be 2 ~ 5.

Step S - Generate Fuzzy Rules

Construction of appropriate fuzzy rules is usually the bottleneck in the design 69

procedure. The general practices are using the expert’s knowledge, or experimental data, or the mathematical model if one exists, or on the basis of trial-and-error. It is advised that one should use all available information on hand before continuing the design procedure. In this study, it is felt that the inverse kinematics equations could assist in the design. Some advantages from this approach: ( 1 ) there is an analytical tool to verify the rule generation process, thus, avoiding trial-and-error iteration. ( 2 ) the inverse kinematics equations are referred solely for the purpose of extracting the tendency of actuator action, thus, no explicit solutions are sought. This is the main difference between our idea and the concept from the pseudoinverse technique.

There are two terms in the inverse kinematics equation, Eq. (4.12). In general, one may generate two fuzzy rule sets corresponding to two separate terms. However, since the second term will not affect the error of the work vector, we will retain this term for the next design step. Thus, we have

/ = J +w (4.13) where € 9JWxn*, and ne is the number of coordinates in the work vector. By

A arbitrarily choosing an active member velocity vector / € Lty one can obtain the numerical value of J +. Assign the linguistic values to these numerical values in the inverse Jacobian matrix, denoted by J +. We already have the linguistic value of the work velocity vector, denoted by tn, then the rule associated with a particular u> can be obtained by applying the multiplication of two linguistic arrays, i.e., J + x t u. For 70

example, * * * * ...... f' N2N 2 ' P2 PI Nl X Nl = P 2 N 2 + P1 • Nl + Nl • N2 — Nl (4.14) ...... N 2 The whole fuzzy rule table is completed by the same manner. For the case that there

might be an ambiguous resulting linguistic value, one can make the decision in a way

the adjacent cells should be closely related. In other words, the rule base should be

constructed in a continuous pattern.

Step 3 - Determine I/O Gains

The final step is the choice of the input/output gains. Due to the normality of the

universe of discourse in the fuzzy controller, the gains of input/output signals reflect

the ranges of the domain intervals. However, beyond that, the choice of the gains also affects the stability, steady state error, and the convergent speed of the control system [69]. Nevertheless, only a few papers of such analysis succeeded and they were restricted to very simple systems. It is hoped that there would be more theoretical works devoted to this subject and more beneficial to the engineering design. Thus, in this study, a very primitive guideline for choosing I/O gains is presented.

For the adaptive truss structures, the choice of different I/O gains will affect the convergent speed (i.e. how many steps), smoothness of the trajectory, robustness to the noises, and, if specified, the performance index, etc. FYom our observation, the input gains seem only to have a noticeable effect on the robustness issue. On the other hand, the output gains have dominant effects on speed, smoothness, and steady-state error. Furthermore, the ratio between different fuzzy controller outputs has a major impact on the performance index (this will be seen in the example). 71

Finally, the implication rule and defuzzification strategy should be chosen to de­

termine the combined output value. There are numerous works that have been ad­

dressing this issue (e.g. see a review paper [70]). Denote the membership function

of the fuzzy set A as /i^, the minimum-product implication rule is used in this work.

T hat is,

fit - m in {fitl x x • • ■ /*e„, #*„} (4.15)

product of the premises and minimum of the implication is used to obtained implied

(output) fuzzy set. Here e< are the input fuzzy sets and u is the output fuzzy set.

Then, in the defuzzification process, the center-of-area method is adopted to get the

crisp output value, i.e.

u = (4.16)

4.4 Illustrative Examples

By following the proposed design procedure, one can obtain the fuzzy control system

for the motion control of the adaptive structure in a systematical way. To illustrate the

above design procedure, an adaptive truss structufe is used. For simplicity, only two- dimensional part of the stacked cubic truss structure is selected for demonstration.

4.4.1 Planar Truss Structure

An adaptive truss structure which is similar to the one presented in Murotsu’s paper

[71] is shown in Figure 19. The structure consists of four Shear Deformation Modules

(SDMs), The diagonal member of each module is an active member and restricted 72 to be changed ±40% from their default length (i.e. le = y/2d,i — 1,2,3,4, where d = 50 cm is the length of the fixed-length member).

1. Adaptive member, 2. Shear deformation modules; 3. Pivoting point

Figure 19: The adaptive space structure

Due to this geometric limitation of the structure, the reachable point in the space is confined in a manipulability profile in a two-dimensional work space. To facilitate the simulation, a coordinate system is designated in which the origin is located at the middle of the leftmost truss member with positive rightward and upward. The tipend of the structure, called the pivoting point, is located at 0.5d right to the middle of the the rightmost truss member. Then, the position of the pivoting point can be formulated as a function of the active member lengths, that is,

xe 0-5 + ky/2 - (it)1 ' w = = d x (4.17) . y * .

As mentioned in the previous sections, a series of modular fuzzy controller, each of which governs the basic structural module, is considered. For this model, each pair of conjunctive shear deformation modules is treated as the basic structural module, 73

and thus, has one fuzzy controller. The controller has two inputs and two outputs.

The outputs are the changes in the diagonal active members in lengths. All fuzzy

controllers share the same input information, i.e. the longitudinal and transverse distances between the tipend location and the desired location, denoted by ex = xt —xe and e 2 = yt - ye. Hence, the whole control system consists of two fuzzy controllers.

From Eq.(4.17), the Jacobian matrix can be obtained as 2d(Pe-(j ) J = k = 1,3 (4.18) where k= 1 and 3 indicate the first and the third actuators (from left) in the truss system. It should be emphasized that the analytical form of the Jacobian matrix may not be easily obtained for an other type of structural module, in this case, the numerical approximation can be sought [72].

Five membership functions of the universe of discourse are chosen and assigned as:

Negative-Large(NL), Negative-Small(NS), Zero(Z), Positive-Small(PS), and Positive-

Large(PL). After examining the inverse Jacobian matrix by randomly choosing the active member lengths, the linguistic values are assigned to the J + matrix, for in­ stance, -0.037 0.126 N S PL J + = => J + = (4.19) -0.037 -0.126 N S N L

Then using the second step in the design procedure, the fuzzy rules can be deter­ mined. The fuzzy rule has the following form:

Ri: If ei is NL and e 2 is NL T h e n ui is Z and ix 2 is PL, 74

Rag: If ei is PL and e 2 is PL Then ux is Z and u 2 is NL

For example, Rule 6 is generated by examining the following relations:

' NS \ N S • ' ' NS-NS' + PL-NL' ' ‘ NS' PL 1 => NS NL \ [ NL\ NS ■ NS' + NL • NV PL

where primed variables indicate the linguistic variables of the input data. The com­

plete control rules are listed in Table 2, which constitute the knowledge base in the

fuzzy controllers. Notice that the rule base for should be a mirror image of the

rule base for uj. This particular enforcement, due to the symmetry, guarantees the smoothness and coordinated motion of each pair of SDMs. This phenomenon is dem onstrated in Section 4.4.2.

Table 2: Fuzzy rules table for SDMs controller

C2 e2 NLNS Z PS PL u2 NL NS z PS PL NLZPSPL PLPL NLPLPLPL PS Z NS NS z PS PL PL NS PLPL PS Z NS ei z NL NS Z PS PL ei ZPL PS Z NSNL PS NL NL NS Z PS PSPS z NS NLNL PLNLNL NL NS Z PL Z NS NL NL NL

The next step is the determination of the input/output gains. Since there is a lack of analytical tools to make the decision, the heuristic approach is usually used. The input gains are determined first. The designer could use the reciprocal of a moderate number about the half-range of the domain intervals of the input variables. In this exercise, we choose 1/40 for both input variables. The output gains for each fuzzy 75 controller should be the same to ensure the coordinate motion. However, the output gains for different fuzzy controllers are not necessarily the same. In this case, we choose 4 and 4 for two fuzzy controllers.

The modular fuzzy controller, therefore, is formed. Hence, the fuzzy control sys­ tem is constructed by two modular fuzzy controllers. In the following subsections, several different tasks will be tested with the designed fuzzy control system applied on the planar truss structure.

4.4.2 Rendezvous/Docking Problem

The rendezvous/docking problem is to find the lengths of the active members such that the pivoting point of the structure would reach a prescribed target position. The applications of this kind of adaptive structure could be like the docking bridge, the fuel pump, the gripper, etc.

The simulation results are depicted in the Figure 20. The figure shows four arbi­ trary target positions (denoted by x) and the approaching paths by the controlled truss structure (the convergent criterion is 0.5 cm). One can observe that the control system successfully drives the pivoting point of the structure to the desired target positions as long as the target is located in the manipulability profile. Moreover, the task is carried out within a few steps (in this simulation, the maximum number is

13), which indicates the control algorithm is very effective. The most noticeable point about the effectiveness of the proposed fuzzy rules is shown in the lower-right plot in the figure. When the target is on the horizontal axis, the change amounts of both active members in modular pair of SDMs should be equal. Then, it would result in 76

(150,30) (160,65) 3 0 80

60 20

1 0 20

100 1 2 0 140 100 120 140 160 x x (200,-40) (220,0) 1

-10 0.5

0

-30 -0.5

-1 100 150 200 100 150 200 x X

Figure 20: The simulation result of docking problem

the perfect straight line trajectory. This performance is guaranteed by the mirror image between u* and U 2 of the fuzzy control rules. One should expect that the perfect straight line is not attainable if the fuzzy rules for uj and u 3 are formulated independently.

4.4.3 Target Pursuing Problem

A target pursuing problem is similar to the rendezvous/docking problem. Instead of approaching a fixed target point, the structure is required to pursue a moving target.

Two cases are simulated in this study. First, the target is moving in circular motion 77

20

IS

10

-10

-15 -

90 100 no 1 2 0 130 140 150 X Figure 21: Simulation result for pursuing a circular motion

with constant angular velocity of ^rad/sec. (The center is chosen at (130,0) and the radius to be 15 such that the whole trajectory will reside in the maneuverability profile.) The simulation result is shown in Figure 21. The controlled structural system took about 2 0 steps to intercept the target and then followed the target faithfully. Second test case is assuming the target is moving in a reflexed motion. The target is started at (125,40) with constant speed 0.3606 cm/sec southeast to location

(145,0), then turned southwest with the same speed. As shown in the Figure 22, the controlled structural system is capable of pursuing the target moving in such way.

The above results show that the proposed fuzzy controller successfully accomplished the pursuing mission. It should be emphasized that the present control algorithm is 78 able to pursue random path of moving target as long as ( 1 ) the whole path is within the manipulability profile; and ( 2 ) the moving speed of the target is relatively slow such that the adaptive structure is able to catch up.

40

30

20

»■ 10

-10

90100 120 130 140 150 X Figure 22: Simulation result for pursuing a reflexed motion

4.4.4 Trajectory Tracking Problem

The control objective now is adjusting the lengths of the active members such that the pivoting point would be tracking a desired trajectory.

The controlled structure is tested on four arbitrarily assigned trajectories:

( 1 ) a reflexed line which is represented by

3x + 2\y\ = 570 150 < x < 190; (4.21) with a driving speed of 0.721 cm/sec. 79

(2 ) a hill-n-valley line which is represented by

|x - 160j + y = 20 140 < X < 200; (4.22)

with a driving speed of 0.566 cm/sec.

(3) a circle whose center is at (180,0) with the radius of 30 cm and a driving angular

speed of t / 80 rad/sec.;

and (4) a wavy curve which is generated by a parametric equation of x(t) — 4^/^+170

and y(t) = 30sin(^), where t is the time.

In reality, there exists a control elapsed time for each actuator to complete each

control command. So, in the simulation, the control elapsed time is taken into ac­ count. Refering to Matsuzaki's work [54], the control elapsed time of the actuator is assumed to be a quadratic function of the control command, as defined below:

T _ f \/Ss lul ^ 1 lA oo\ Tcb- { (4,2S) Moreover, the control elapsed times for all actuators are different most of the case.

Therefore, the longest one of all control elapsed times is selected for simulating the actual time spent at each step. It is also assumed that the structure will not receive control command until the previous control motion is completed. Thus, it is referred to as the synchronous control strategy.

The simulation results for these four cases are shown in the Figures 23-26, re­ spectively. In the figures, the real line indicates the approaching trajectory of the controlled adaptive truss structure whereas the dashed line represents the desired trajectory. Recall that the output gains for all actuators are dedicated as 4, Fig­ ure 23 shows the action of the controlled structure tracking the reflexed line. One 80

60

40

2 0

-20

-40 -

150 155 160 170 175 180185165 190 X Figure 23: The simulation result of the reflexed line case

may observe that the motion of the structure is jerky during the starting and end­ ing phases. It indicates that the structure is sensitive to the changes in lengths at large vertical position, which is known as the chatter phenomenon. For low altitude motion, as shown in the Figure 24, the structure did the job very well.

To challenge the effectiveness of the designed controller, two curvilinear trajec­ tories are tested. Figure 25 depicts the result of the controlled structure tracking a constant angular speed circular path. Moreover, the simulation result for tracking the wavy curve trajectory is displayed in the Figure 26. The controlled structure successfully tracks the trajectories and the discrepancies are small.

It should be emphasized that the present controlled structural system is tracking 81

25

2 0 FCS D«iir«d 15

1 0

-10

-15

40 150 160170 190 200 x Figure 24: The simulation result of the hill-n-valley line case

the trajectory on-line, i.e., there is no prior knowledge of the trajectory. If the trajec­ tory is known, by using the inverse kinematics equation, the structural system would be tracking the trajectory exactly. Thus, for comparison, the changes in lengths of the active members for the fuzzy controlled structure and the exact solution obtained from the inverse kinematics equation are superimposed in the Figures 27 and 28 for the circle and wavy curve cases, respectively. As shown in the figures, the resulting re­ sponses of the fuzzy controller is very close to the exact solution. It is demonstrated that the proposed controller is able to drive the adaptive structure to be tracking various and unprescribed trajectories satisfactorily. 82

40 — FCS - - DMired 30

2 0

1 0

-10

-20

-30

160 160 200 220 x Figure 25: The simulation result of the circle case

4.4.5 Effect of various gains

As mentioned in the previous section, the effect of the input gains on the result is not dominant. This can be realized by observing the following case study. Figure 29 shows the tracking errors for four different sets of input gains for pursuing the circular target motion. One can observe that the difference between four curves are not very distinct. However, the input gains as [1/40,1/40] gave the minimal average error, whereas the [1/60,1/40] case resulted in the fastest interception of the target.

It should be pointed out that the output gains cannot be very large for this particular truss structure. When the output gains are larger than 5, the approaching path becomes very jerky, and even larger would cause the failure to converge. Let the 83

30

2 0

1 0

-1 0

-20 FC — Desired traj.

170 175 180 165 100 195 200 205 210 x Figure 26: The simulation result of the wavy curve case

output gain of fuzzy controller be denoted by u^,, where Mi indicates the ith pair of

SDMs. By choosing different output gains, the results would be different. Figure 30 shows the different paths for four different settings of two controllers’ output gains.

As shown in the figure, when the gains increase, the path becomes jerkier. Conversely, as the gains decrease, the number of steps increases. So, this is a tradeoff point which should be determined by the management and organization level and the tuner in the coordination level.

The effect of the different weighting ratio of the two controllers on the performance index is interesting. Define R = UM3/u \tlt and the performance index as

/ ’/ = EIW-UI (4-24) .= 1 84

100 FCS - - Exact98

96 2 & 4

94

92

8 8

8 6

84

82

80, 2 0 40 60 100 1 2 0 140 160

Figure 27: Comparison of length changes for the circle case

where and I,™, are final and initial lengths of the ith active member, respectively.

The performance index is proportional to the manipulation energy when the structural

system moves quasi-statically. The Figure 31 displays the output gain ratio vs. the

performance index. Interestingly, the minimal value occurs when the ratio is equal

to one. That is, when the control effort is evenly distributed to every active member,

the total energy consumption will be the least. Also, in this exercise, when the ratio

is greater than 4.5 or less than 0.22, the controlled structure failed to converge.

4.4.6 Robustness of fuzzy control system

In the real system, the mathematical model of the system may be inaccurate. Also, the measurements of the sensors may be noisy. Namely, there may be uncertainties 85 or unwanted disturbances present in the control process. To see the effects of these disturbances on the fuzzy control system, a couple of experiments are studied, namely considering the cases of noise in measurements and possible actuator malfunctions.

Denoting the noise-to-signal ratio as NSR, the measurements in distance between the pivoting point and the desired target are contaminated by

e\ = e i(l + NSR-6) (4.25)

e'3 = e3(l + NSR-S') (4.26) where 6,6' are random numbers uniformly distributed between -0.5 and 0.5. The simulation result isshown in Figure 32 where three different NSRs are tested. It is not surprising that even with a noise-to-signal ratio as high as 5, the fuzzy control system does accomplish the task.

The second scenario is assuming some of the actuators may have malfunctioned during the maneuvering process. For example, the actuator got stuck while decreasing its length, it is anticipated that the controlled structure would perform well under such situations. Assuming /3 > 1.3/0, i.e. the actuator could not decrease its length any more, the result is shown in Figure 33. Further, in addition to the stuck /3, it is assumed that /3 would stick at l.l/0. Then, the simulation result is shown in

Figure 34. From both figures one can observe that the control system is capable of tolerating partial component failures. But of course, when some actuators were paralyzed, the manipulability region would shrink accordingly. Nevertheless, once the desired target lies in the shrunk region, the controlled structure can reach the target position under control of the fuzzy control system. It is also believed that the 86

imperfection in modeling and the existence of the joint lags, etc. would be tolerated

by the fuzzy control system, although these cases are not investigated in this work.

4.4.7 Effect of Inference Strategy

In the previous study, the product inference strategy is used. However, there are many other proposed inference strategies for the fuzzy logic controller design. To compare the effect of different inference strategies upon the final result, three inference strategies are tested. They are product, minimum and maximum fuzzy inference implication rules. The target is in the circular motion as the same as the one described in the previous subsection. The error histories between the pivoting point and the target for three different inference implication rules are shown in Figure 35.

As shown in the figure, the minimum implication rule and the product impli­ cation rule are able to pursue the moving target while the maximum implication rule produces a large discrepancy. The performances for both product and minimum strategies are almost the same. The reason of the failure of applying maximum impli­ cation rule is that this rule will cause the structural system to overshoot the desired location, thus introducing extrinsic errors. Then the tracking trajectory would look like rugged fluctuation along the target trajectory. This study implies that the infer­ ence implication rule which can result in a smaller output should be chosen as the fuzzy inference strategy. 87

4.5 Closure

Having done the vast variety of the simulation tests, we conclude that the fuzzy con­ troller algorithm is able to accomplish several different tasks satisfactorily. However, this study is not exclusive for all possibilities that the fuzzy controller can do. And also, it should not over-justify that the fuzzy logic control is better than the other control techniques. The important thing learned from this study is that there is al­ ways another alternative to do the same thing. The fuzzy control system provides such luxury to avoid the system modeling on the adaptive structure. This is probably the most important feature on this issue. 88

06

04 Exact 92

0 0 s “ 86

84

80

78. 50 1O0 200 250 300

Figure 28: Comparison of length changes for the wavy curve case

10 - (1/40 1/40] - • (1/eo i/eo] - [1/eo 1/40] (i/4o i/eo]

X ■

20 4060 60 100 1 2 0 140

Figure 29: The errors of different input gains 89

-1 0

-15

-20

-25

-30

-35

-40

90 100 110 1 2 0 130 150 160140 x Figure 30: The trajectories of different output gains for module pairs

1.7 1.6 1.5 1.4 1.3 1.2 1.1

1 0.9 0.8 0 0.5 1 1.5 2 2.5 3 3.5 4 R

Figure 31: The performance index of different output gain ratios 90

140 N /S « 0 120 N /S - 1 N /S - 3

100 N/S - 5

8 0

60

40

20

lO 12

Figure 32: The effect of noise in measurement

(100,40) 40

35 Normal

30

25

>»20

15

10

1 2 0 140 160100 X Figure 33: The effect of one stuck actuator 91

(210,-30)

10

15

-25

-30

- stuck L2 & L3 35

-4 100 120 140 160 180 200 220 x Figure 34: The effect of two stuck actuators

16

14 product minimum

12 - maximum

10 ri

20 40 60 60 100 120 140 ■tap

Figure 35: The errors between different inference implication rules C H A P T E R V

Conclusions and Future Research

The application of the artificial neural networks to the structural damage detection

and identification problem is explored. Since the neural network is capable of learning

the features of the damage information, this study has shown that such an application

is quite feasible and suitable to be the FDIA module in the intelligent control system.

With the limited number of training patterns, the neural network matches not only the training patterns, but also the testing patterns. Since the testing patterns are selected on the midspan between every training pattern pairs, it is concluded that the neural network generalizes the satisfactory results throughout the domain in which the training data spans. However, when the input testing pattern is beyond the representative domain, the neural network may fail to extrapolate such a pat­ tern. Nevertheless, due to the separate network output, the damage location is still accurately identified when the testing data is within or outside the domain.

Two discrete models are first demonstrated. For a three degrees-of-freedom mass- damper-spring system, the three layer back propagation neural network is capable of identifying the single spring damage within 0.5% and the mixed spring damages within 3%. The neural network generalizes the mapping from the frequency changes to damage amounts very well. As a real structural system existing with a closely-

92 93

spaced eigenvalue spectrum, the sample data are no longer uniquely mappable. The

conventional three layer network unavoidably causes some unacceptable error. A new

structural damage identification procedure constructed by an eight*layer network ar­

chitecture is proposed and demonstrated on the well-known Kabe model. The location

of the damaged springs were precisely identified and the corresponding damage sever­

ities were also determined within 3% error. Then, the same technique is applied to

the truss structure and the outcomes are successful.

The proposed approach provides the following advantages: 1) the neural network

can identify the locations and the severity of the damages, either single damage or

multiple damages. 2) Since the network is trained in the design stage, when the network is hardware implemented on the structural system, it can respond to the detection in real time, which make the neural network very suitable for a real-time on-line structure health monitoring device. 3)The training process is computationally easy and is able to be implemented in parallel processors. 4)The network shows the potential usage for structural damage identification.

Some issues remain to be resolved before this approach becomes a truly viable method for structural damage identification. First, for real world complex structures, the degree of freedom of the system could be very large. It is believed that by performing several stages of model reduction, the DOF of the final design model should be a relatively small number. Second, the approach using the dynamic residual vector may be applied to such reduced models. Third, the role of noise in actual modal data measurement should be investigated, where minor damage is concerned, 94 the noise could cause some false estimations. Finally, the appropriate representation for damage information in order to reliably and precisely identify various states of damages requires further investigation. It is anticipated that a neural network based identification technique will be improved if a proper representation can be developed.

Next, we have investigated the development of a fuzzy control algorithm for the motion control of the adaptive truss structure, which can be implemented into the controller module in the execution level. To facilitate the design procedure, a sys­ tematical way to construct the fuzzy rules is introduced. The procedure can be summarized as:

1. Decide the number of membership functions over the universe of discourse.

2. Formulate the fuzzy rule tables by referring to the inverse kinematics relation­

ship of the basic structural modules pair. Construct the control system by the

series of the modular fuzzy controllers.

3. According to the domain intervals of the input/output variables, determine the

input/output gains of the controller. If the performance index is prescribed, the

proper weighting ratio between different fuzzy controller output gains should be

evaluated.

The above procedure provides a satisfactory design of the fuzzy controller for the adaptive structures. In addition, the designer is able to fine-tune the controller by adjusting the design parameters. Although the standard (normalized) fuzzy con­ troller architecture was used, the designer still can improvise the control system, such 95 as changing membership functions, changing inference methods, or changing the de­ fuzzification techniques, etc. The general principle presented here is still applied.

Thus, the proposed method greatly reduces the design process and suitably replaces the conventional ad hoc design method.

A two-dimensional truss structure is tested to illustrate the proposed procedure.

It is shown that the controlled structure is able to reach any target position, or to track any trajectories, or to pursue any moving target in the manipulability profile.

If the manipulation energy performance index is specified, the ratio between different controllers’ output gains should be identical such that the total control effort is min­ imal. The results indicate that the work load should be evenly distributed to each actuator to reach the minimal energy consumption. The control system also tolerates unwanted perturbation such as noise in measurements or some stuck actuators, which is showing a great degree of robustness.

Several interesting discoveries in the design process of the study model are illus­ trated as follows: First, since the adaptive structure is typically consisting of repe­ titious stack of basic structural modules, one can design the controller on the basic structural modules instead of on the entire large complex structure. Second, due to the symmetry of the arrangement of the stacked structural modules, it is advanta­ geous to pair up each of the adjacent modules as one entity. Furthermore, to ensure the coordinated and smooth motion of the structure, the fuzzy rule table for the sec­ ond module should be the mirror image of the first module. This observation is also applied to the tetrahedral stacked adaptive structure. 96

From this work, the applications of the neural networks and fuzzy logic control schemes are very promising. Although the other functions in the framework of the intelligent control system axe not focused in this study, there are many existing tech­ nologies which are suitable for part of (or all) these modules and are able to be adopted in principle. Ultimately, this study intends to show the feasibility of imple­ menting such techniques in the intelligent control system for the adaptive structures.

It also indicates the possibilities in applying computational intelligence algorithms to improve the future control system design.

Since the proposed intelligent control system architecture is modular, one can view the system as the integration of the vast variety of algorithms and schemes and hardware. Once the new technology has been developed, it could be implemented into the system. In this dissertation, the design philosophy is introduced, yet is kept highly flexible. The future work may be plentiful but be categorized into the following directions:

• Standard: the interlinks between various modules should follow the same stan­

dard such that the information flow within the control system can be flowing

without major adjustments or modifications.

• Management: the management technique may be subjective in setting up the

control system. A more efficient and more objective management scheme is in

need to fulfill the expectation of the designer.

• Learning: the neural networks have shown a great deal of potentials in emulating

human’s brain activities. New methodologies have been introduced in a fast pace recently. To outspread the autonomous feature of the intelligent control

system, implementation of new neural networks is worthy of attention.

• Controller: the fuzzy logic control shown in this study indicates a large degree of

robustness and its potential. However, when the precision control is required,

special treatment may be needed. This topic is also a good future research

subject. In addition, an adaptive fuzzy controller which can vary its output

gains dynamically to improve the performance is another study topic. A ppendix A

Samples of data used for damage identification neural networks

For reference purpose, the input/output patterns for the simulation examples pre­

sented in the paper are sampled and listed below.

Example S.h For single damage case, assuming k3 is damaged, the numbers of inputs

and output are 3 and 1, respectively. The training patterns are: 0.024251 0.075986 0.001154 0.1 0.052197 0.144074 0.002107 0.2

The testing patterns would be 0.011685 0.038993 0.000606 0.05 0.037741 0.111006 0.001652 0.15

For multiple damages cases, the numbers of inputs and outputs are 3 and 3. The samples of the patterns are shown below

0.430977 0.190055 0.047652 0.10 0.40 0.20 0.469061 0.516377 0.018584 0.00 0.20 0.90

Example 3.2\ Using A k3 = 10%, A k13 = 10% as illustration, the dynamic residual vector obtained from the first subnet looks like this

dx = [0 - 0.311911 0.311911 0 0 - 0.075872 0 0.075872]

98 99

The training pattern for the second subnet would be like this

Input Output

0 -1 1 0 0 -1 0 1 00100000000010

The training pattern stored in the database has the format as shown below

0 -0.636420 0.636420 0 0 -0.032383 0 0.032383 0.2 0.1 0 -0.636319 0.636319 0 0 -0.075678 0 0.075678 0.2 0.2 A ppendix B

Computer Codes

This appendix lists partial computer codes used in the dissertation. All programs are written in MATLAB format.

1. Rendezvous/docking problem: The following programs are simulating the four- module truss structure controlled by the fuzzy controller.

X ISFUZZl.M X A simulation program for the fuzzy controller of the X adaptive planar structure. X X Environment: SGI IRIS4 + HATLAB 4.0a y. y. Author - - Poyu TSOU, AAE, OSU y, Start — August 3, 1993 X Update — August 3, 1993 [1.1] y. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx X Configuration manipulation routines xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

clear;

X Setup initial configuration (folded platform)

d - SO; Lo ■ sqrt(2) * d; M ■ 4; X Four modules

R * 1.4 * ones(ltH);

100 101

X Target coordinates

Xt«input(*input It — ’); Yt»input(*input Yt — ’); XXt-200;Yt-40;

XPv ■ isconf(R); xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx X Fuzz; controller xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

X First, ve need to setup initial things for fuzzy X controller simulation.

X Specify ranges of membership functions ofinput/output

[Ial.Ibl] ■ imfgen(S); [Oal.Obl] ■ omfgen(S);

X Specify the gains g e ll ■ 40; gel2 > 40; Gu ■ [4444]; XGu ■ input( ' Gu « ') ;

X Rule base

RBI B C 3 4 5 5 5 2 3 4 S 5 1 2 3 4 5 1 1 2 3 4 1 1 1 2 3]; RB2 W C 5 5 5 4 3 5 S 4 3 2 5 4 3 2 1 4 3 2 1 1 3 2 1 1 l] ; for i . 1 : M, Lc(i) ■ R(i) * Lo; d u (i) * 0; end r ■ 0 .5 ; IMAX • 200; for ita r ■ 1 : IMAX,

Pv » isconf(R); dx ■ ( Xt-Pv(l) ) / gall; dy - C Yt-Pv(2) ) / gal2; phistCitar,:) ■ Pv’;

% Chack tha condition to activata BDM if ( norm([Xt-Pv(l) Yt-Pv(2)]) < r ), braak; and

[cfl.idxl] ■ inpcartC dx, Ial, Ibl ); [cf2,idx2] ■ inpcartC dy, Ial, Ibl ); m ■ length(idxl); n - langth(idx2); i i ■ 0; for i « 1 : m, for j » 1 : n, ii ■ ii + 1; Eol(ii) » RB1C idxlCi), idx2(j) ); Eo2(ii) - RB2C idxlCi), idx2(j) ); hil(ii) » cflCi) * cf2(j); and; and for i - 2 : 2 : M, du(i-l) > C0G( Eol, hil, Oal, Obi ) * Gu(i- du(i) - COGC Eo2, h il, Oal, Obi ) * Gu(i); and uhistCitar,:) ■ du; claar h il; claar Eol; claar Eo2; 103

for i ■ 1 : M, Lc(i) ■ Lc(i) + du(i); •nd

R » Lc/Lo;

X Hard limit constraints jl ■ find(R<0.6); ju » find(R>l.4);

if langth(jl)> 0 , R(jl) * 0.6 * onssCsizsCjl)); and if langth(ju)>0, R(ju) * 1.4 * onas(sizaCju)); and and; X for itar

Xdisp([Xt Yt itar R phistCitar,:)]) plotCphistC:,1).phistC:,2),Xt,Yt, ’x’) ss - [*titla("» '(* int2str(Xt) \» int2str(Yt) »)* grid.evalCss),xlabal(’x’) ,ylabal(’y’)

RO » 1.4 * ones(1,4); Rd - R - RO; PI1 ■ normCRd,1); PI2 ■ norm(Rd); PIf ■ normCRd,inf); disp([itar R]) dispC[PI1 PI2 PIf]) xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx %% END OF ISFUZZl.M xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx function [la, lb] ■ imfgan(n) X IMFGEN.M X Ganarata rangas of normalizad input mambarship functions X Usad by fuzzy controllar simulation program. X X Author: Poyu Tsou, AAE, OSU 104

X Update: A pril 29, 1993 X

if (rem(n,2)«0), error('input integer must be an odd number.'); end

nl » n - 1; h - 2.0 / nl; la ■ -1 - h : h : 1 - h; lb » -1 + h :h : 1 + h; Ia(l) - -1; Ib(n) « 1; xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx XX END OF IMFGEN.M xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

function [0a, Ob] ■ omfgen(n) X IMFGEN.M X Generate ranges of normalized output membership functions X Used by fuzzy controller simulation program. X X Author: Poyu Tsou, AAE, OSU X Update: April 29, 1993 X

if (rem(n,2)«0), error('input integer must be an odd number.’); end

nl ■ n - 1; h - 2.0 / nl; 0a « -1 - h : h : 1 - h; Ob ■ -1 + h : h : 1+h; xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx XX END OF OMFGEN.M

function [Ei, Ion] ■ inpcert(xin, ai, bi) X INPCERT 105

X Calculate the certainties for all input membership functions X at input xin. V A X xin input signal X ai left margin of triangle of ith mem. func. X bi right margin of triangle of ith mem. func. ¥A X Ei a certainties vector return to calling program. X if ARG0UT»2, Ei returns only nonzero entries. X Ion an index vector indicating the enabled mem. func. yh X Author — POYU TSOU, AAE, OSU X Update — Apr. 23, 1993

n « length(ai); m ■ length(bi);

i f (n'*m ), error( 'Dimansion inconsistent in [&i] and Cbi] end; for i * i : n,

*/. For Left-Max function

i f ( i» » l) . if (xin < ai(i)), E i(i) - 1; elseif (xin > bi(i)), E i(i) « 0; e ls e Ei(i) - (xin - bi(i)) / (ai(i) - bi(i)>; end

X For Right-Max function

elseif (i««n), if (xin < ai(i)), E i(i) - 0; elseif (xin > bi(i)), E i(i) - 1; e ls e 106

Ei(i) « (xin - ai(i)) / (bi(i) - end

X For middle triangle functions

e ls e if (xin < ai(l) I xin > bi(i)), E i(i) - 0; e ls e Ei(i) ■ 1 - abs(xin - (aiU) ♦ bi(i)) / 2) * 2 / ... (bi(i) - ai(i)); end end end if (nargout ■« 2), Ion * find( Ei "■ 0); Ei > Ei(Ion); end xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx XX END OF INPCERT.M xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx function Bp » C0G(0MFidx, h i, a i, b i) X COG X Defuzzification using center of gravity. Vh X OMFidx index vector of enabled output mem. func. X hi height of each enabled output MF X ai left margin for all output mem. func. X bi right margin for all output mem. func. Vh X Bp defuzzified output YM X Author — POYU TSOU, AAE, OSU X Update — Apr. 23, 1993 X m ■ length(ai); n ■ length(bi); 107 i f (m "■ n ) , error(*Inconsisent dimension of margins for output MFs ... *); end n * length C h i); aon ■ ai(OMFidx); bon - bi(OMFidx); base ■ bon - aon; bb b o .5 * (aon + bon); area • base .* hi .* (ones(size(hi>) - 0.5 * h i);

Bp b bb * area* / sum(area); xmmmmmxxxxmmmmmxxmxmmnmmmmmm n END OF COG.M xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx function ptpos a isconf(R) X ISCONF Calculate the position of the pivot point of the X following Structure 1, o-----o o o-----o-----o \ y. i /i\ i i /i\ i \ y. i / i \ i..t / i \ i > y. i/ r \i 1/ i \i / * y, o — o—o o o—o/ I y, 1 2 n-l n — pivot point y. INPUT: X R change r a tio s of the adaptive members; vector of X length n X y. OUTPUT: X ptpos XY coordinates of the pivot point. X X Note: The fixed length members are assumed to be d“50cm

X Author — Poyu Tsou, AAE, OSU X Update ~~ August 3, 1993 d ■ 50; 108

n • langth(R);

X Chack tha R vactor

if langth(find( R>sqrt(2) I R<0 )) >0, •rrorC’Chang* ratios ara out of allovabla ranga !’); and

X Position of tha pivot point at fully aztandad configuration

zo - C n + 0.5 ) * 50; yo « 0;

xsum a 0; ysum a 0;

for i ■ 1 : n, xsum ■ xsum + R(i) * sqrtC 2-R(i)*R(i) ysum ■ ysum + (-l)*(i-l) * R(i) * R(i); and

dx ■ d * C xsum - n ); dy ■ d * C ysum - (!+(-!)“(n-l))/2 );

ptpos - C xo+dx; yo+dy ]; xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx XX END OF ISCONF.M

2. Trajectory tracking problem: The following code is the main program for sim­ ulating the trajectory tracking task.

X ISFUZZ2.M X A simulation program for tha fuzzy controllar of tha y, adapt iva planar structura — 4 SDMs casa X X Trajactory tracking y...... y. Environment: SGI IRIS4 + MATLAB 4.0a X X Author — Poyu TSOU, AAE, OSU 109

X Start — Saptambar 3, 1993 X Updata — Saptambar 28, 1993 [2.2] x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx X Configuration manipulation routinas xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

claar;

X Satup initial configuration (foldad platform)

d - 50; Lo ■ aqrt(2) * d; H ■ 4; X Four modulas

*/.R ■ 1.4 * onaa(l,M);

X Inquira option for targat trajactory

opt ■ manuC’Chooaa a targat trajactory’Circular*, ... ’Inclinad lina’.’Hill k Vallay', ... ’Wavay curva’);

X Starting point of tha targat trajactory i f C opt ■■ 1 ), Xt « 210; Yt - 0; and if ( opt ■« 2 ), Xt - 150; Yt « 60; and i f ( opt ■■ 3 ), Xt ■ 140; Yt - 0; and i f ( opt ■■ 4 ), Xt - 170; Yt - 0; and 110

X Calling fixed target program to find R isfu zz2a pause xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx X Fuzzy controller xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

X First, ve need to setup initial things for fuzzy X controller simulation.

X Specify ranges of membership functions of input/output X Already defined in ISFUZZ2A.M

X Specify the gains

Xgell ■ 40; Xgel2 m 40; XGu ■ [ 2.5 2 .5 2.5 Gu ■ in p u t(* Gu ■ *

X Rule base

i f ( ■e x i s t ( ’RBl ’ )"■ l ), RBI ■ C 3 4 5 5 5 2 3 4 5 5 1 2 3 4 5 1 1 2 3 4 1 1 1 2 3] ; RB2 ■ C s 5 S 4 3 5 5 4 3 2 5 4 3 2 1 4 3 2 1 1 3 2 1 l l] ; end for i » 1 : M, LcCi) m R (i) < d u (i) - 0; end tc m 0; Ill

X Maximum sim ulation tima if ( opt ■■ 1 ), tmax ■ 320; and if ( opt ■« 2 ), tmax ■ 200; and i f ( opt ■■ 3 ), tmax ■ 150; and if ( opt ■■ 4 ), tmax ■ 320; and

ita r - 0; v h ila C tc <■ tmax ), ita r ■ ita r + 1; LhistCitar,:) ■ [tc R • Lo];

Pv » isconf(R); dx - ( Xt-Pv(l) ) / gall; dy - ( Yt-Pv(2) ) / gal2; phistCitar,:) ■ Pv’; t h is t C it a r ,:) ■ [ Xt Yt ]; a(itar) ■ norm(Pv-thist(itar,:)’);

[cfl.idxl] • inpcartC dx, Ial, Ibl ); [cf2,idx2] » inpcartC dy, Ial, Ibl ); m * langth(idxl); n » la n g th (id x 2 ); i i ■ 0; for i ■ 1 : m, for j » 1 : n, ii ■ ii + 1; Eol(ii) ■ RfilC idxl(i), idx2(j) ); Eo2(ii) - RB2C idxl(i), idx2(j) ); hil(ii) - cfl(i) * cf2Cj); and; and 112

for i - 2 : 2 : M, du(i-l) - COGC Eol. hil, Oal, Obi ) * Gu(i-l); du(i) » COGC Eo2, hil. Oal, Obi ) * Gu(i); and

u h is tC ite r ,:) • du; claar h il; claar Eol; claar Eo2;

for i ■ 1 : M, Lc(i) ■ Lc(i) + du(i); and

R ■ Lc/Lo;

X Hard limit constraints j l « fin d (R < 0.6); ju « find(R>1.4);

if langth(jl)> 0 , R(jl) ■ 0.6 * ones(size(jl)); and if langthCju)>0, R(ju) ■ 1.4 * onas(sizaCju)); and

X Calculate control elapsed time umax * maxC abs(du) ); tneed ■ cat(umax); dhist(itar) ■ tnaad;

X Targat coordinatas are obtained from tha following function

[Xt,Yt,tc] ■ targetlCXt,Yt,tc,tnaad,opt); disp([itar tc a(itar)])

X Dynamical visualization plot(phistCitar,1).phistCitar,2),’o’,thist(itar,l),thistCitar,2),. ’x’, phistC:,1).phistC:.2),thist(:,1),thist(:,2),'—') dravnov 113

end; X for iter xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx XX END OF ISFUZZ2.M xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

function [xt,yt,tc] ■ targetl(xo,yo,tp,dt,opt); X TARGETl.M X A function that calculates thecoordinates of the target X for given current time. X Used by ISFUZZ*.M for trajectory tracking X X Author — Poyu Tsou, AAE, OSU X Update — September 3, 1993

X (1)Circular trajectory X Center at (180,0) with Radius of 30. X (2) A seesaw trajectory X I \(1 5 0 ,6 0 ) X I \ X +...... >— (190,0) X I / X I /(IS O ,*60) X (3) Another seesaw trajectory X I “(160,20) X I / \ X +...... /—\— X I (140,0) \ X I \(1 8 0 ,-4 0 ) X (4) A parametric curve X x(t) ■ 4 * sqrt(t/4) + 170 X y(t) ■ 30 * sin(t/20)

tc * tp ♦ dt;

if ( opt ■■ 1 ), Cx ■ 180; X Center Cy - 0; r ■ 30; X Radius w ■ pi / 80; X Angular velocity in rad/sec q ■ w * tc; 114 xt ■ Cx ♦ r *cos(q); yt ■ Cy + r *sin(q); end X of option 1 if ( opt ■■ 2 ), Vx ■ 0.4; X Horizontal valcoity Vy ■ -0.6; X Vortical valocity (initial)

Dx ■ 40; X half span of x displacsnont t t - Dx / Vx; X turning time yt « yo + Vy * dt; if (tc > tt ), if ( tp <■ tt ), dtl ■ tc - tt; Vx - -Vx; xt ■ 190 * V x * d tl; else Vx ■ -Vx; xt ■ xo + Vx * dt; end else xt ■ xo ♦ Vx * dt; end end X of option 2 if ( opt «■ 3 ), Vx ■ 0.4; X Horizontal velocity Vy ■ 0.4; X Vertical v e lo c ity

Dy * 20; X half-span of y displacement t t « Dy / Vy; X turning time xt ■ xo + Vx * dt; if (tc > tt ), if ( tp <• tt ), d tl ■ tc - t t; Vy - -Vy; 115

yt - 20 ♦ Vy * dtl; •Isa Vy -Vy; yt « yo ♦ Vy * dt; •nd •Isa yt » yo + Vy * dt; •nd

•nd X of option 3

if ( opt ■» 4 ), xt » 170 + 4 * sq rt( tc /4 ); yt ■ 30 * sin ( tc /2 0 ); xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx XX END OF TARGET1.M xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

function t * c«t(dl); 1, CET.H X A function that calculates control alapsad tima X Idll - 0.075(dt)‘2 Idll <« 1 X Idll - 0.5(dt-dtm)“2 + 1 Idll > 1 X X Author — Poyu Taou, AAE, OSU X Update — July 8, 1993

u ■ abs(dl); dtm ■ sqrt( 1/0.075 );

i f ( u > 1 ), t » sqrt( (u-D/0.5 ) + dtm; •Isa t « sqrt( u/0.075 ); end xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx XX END OF CET.M xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

X ISFUZZ2A.M 116

X A simulation program for ths fuzzy controllar of tha X adapt!vs planar structure. X Fixad Targat — Claan Vanion --> find R only

X closaly ralatad to ISFUZZ2.N X Environmant: SGI IRIS4 ♦ MATLAB 4.0a X X Author - - Poyu TSOU, AAE, OSU X S tart — August 3, 1993 X Updata — Saptambar 23, 1993 [2A.1] X

X Configuration manipulation routinas

X Satup initial configuration (foldad platform)

R ■ 1.4 * ones(l,M);

X Targat coordinatas ara obtainad from ISFUZZ2.M

X Fuzzy controllar X ...... X First, va naad to satup initial things for fuzzy X controllar simulation.

X Spacify ranges of membership functions of input/output

[Ial.Ibl] » imfgan(5); [Oal.Obl] * omfgen(5);

X Specify the gains g a ll » 40; gel2 ■ 40; Gu - [ 2 2 2 2 ];

X Rule base

RBI « [34555 2 3 4 5 5 1 2 3 4 5 112 3 4 117

1112 3]; RB2 - [55543 5 5 4 3 2 5 4 3 2 1 4 3 2 1 1 3 2 111]; for i ■ 1 : M, LcCi) - RCi) * Lo; d u (i) ■ 0; •nd r ■ 0.1; IMAX - 100; for ita r « 1 : IMAX,

Pv ■ isconf(R); dx ■ C Xt-Pv(l) ) / g«ll; dy - C Yt-Pv(2) ) / g*12; p h is t f it e r ,: ) » Pv';

% Check the condition to a ctiv a te BDM ed « norm([Xt-Pv(l) Yt-Pv(2)]); if( ed < r ), break; end

[cfl,idxl] ■ inpcert( dx, Ial, Ibl ); [cf2,idx2] ■ inpcertC dy, Ial, Ibl ); m • lengthCidxl); n ■ le n g th (id x 2 ); i i - 0; for i ■ 1 : m, for j ■ 1 : n, ii ■ ii + 1; EolCii) ■ RBlC idxl(i), idx2(j) ); Eo2(ii) • RB2( idxl(i), idx2(j) ); hil(ii) » cfl(i) * cf2(j); 118 end; end fo r i • 2 : 2 : M, du(i-l) ■ C0G( Eol, hil, Oal, Obi ) * Gu(i-l); duCl) • COG( Eo2, hil, Oal, Obi ) * GuCi); end uhistCiter,:) ■ du; clear h il; clea r Eol; clear Eo2; for i ■ 1 : M, Lc(i) ■ Lc(i) ♦ du(i); end

R ■ Lc/Lo;

X Hard limit constraints jl - find(R<0.6); ju » find(R>1.4); if lengthCjl)> 0 , RCjl) ■ 0 .6 * ones(sizeCjl)); end i f lengthCju)>0, RCju) ■ 1.4 * ones(size(ju)); end end; % for iter disp([R ed]) xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx XX END OF ISFUZZ2A.N xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Bibliography

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