A Dissertation. entitled. A New Generation of Adaptive Control: An Intelligent Supervisory Loop Approach. Sukumar Kamalasadan

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2 A Dissertation entitled A New Generation of Adaptive Control: An Intelligent Supervisory Loop Approach by Sukumar Kamalasadan Submitted as partial fulfillment of the requirements for the Doctor of Philosophy degree in Engineering at the Department of Electrical Engineering and Computer Science Adviser: Dr. Adel A. Ghandakly Graduate School The University of Toledo August 2004

3 Copyright 2004 This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. ii

4 An Abstract of A New Generation of Adaptive Control: An Intelligent Supervisory Loop Approach Sukumar Kamalasadan Submitted as partial fulfillment of the requirements for The Doctor of Philosophy degree in Engineering at the Department of Electrical Engineering and Computer Science The University of Toledo August 2004 A new class of intelligent adaptive control for systems with complex and multimodal dynamics including scheduled and unscheduled Jumps, is developed. Those systems are often under the challenge of unforeseen changes due to wide range of operations and/or external influences. The underlying structural feature is an introduction of an Intelligent Supervisory Loop (ISL) to augment the Model Reference Adaptive Control (MRAC) framework. Four novel design formulations are developed which evolve from different methods of conceiving ISL, structured into intelligent control algorithms, and then investigated with comprehensive simulation models of a single link flexible robotic manipulator as well as a six degree of freedom F16 fighter aircraft. The iii

5 first scheme is a Fuzzy Multiple Reference Model Adaptive Controller (FMRMAC). It consists of a fuzzy logic switching strategy introduced to the MRAC framework. The second is a novel Neural Network Parallel Adaptive Controller (NNPAC) for systems with unmodeled dynamics and mode swings. It consists of an online growing dynamic radial basis neural network, which controls the plant in parallel with a direct MRAC. The third scheme is a novel Neural Network Parallel Fuzzy Adaptive Controller (NNPFAC) for dynamic Jump systems showing scheduled mode switching and unmodeled dynamics. The scheme consists of a growing online dynamic Neural Network (NN) controller in parallel with a direct MRAC, and a fuzzy multiple reference model generator. The fourth scheme is a Composite Parallel Multiple Reference Model Adaptive Controller (CPMRMAC) for systems showing unscheduled mode switching and unmodeled dynamics. The scheme consists of an online growing dynamic NN controller in parallel with a direct MRAC, and an NN multiple reference model generator. Extensive feasibility simulation studies and investigations have been conducted on the four proposed schemes, and with results consistently showing that the four design formulations developed in this research, for implementing intelligent supervisory loops into the MRAC framework, are feasible, effective and have immense potential for complex systems control. Even though those two systems are specific in nature, they are true representatives of an important and challenging class of dynamic systems that require the new generation of adaptive controllers developed in this project work. iv

6 DEDICATION TO MY WIFE Manya AND TO MY PARENTS P. Krishna Warrier and Kamaladevi v

7 ACKNOWLEDGEMENT I wish to express my profound gratitude to my Advisor Dr. Adel. A. Ghandakly, for his valuable guidance, continuous encouragement, worthwhile suggestions and constructive ideas throughout this dissertation project. His support, pragmatic analysis and understanding made this study a success and knowledgeable experience for me. My sincere thanks to all other advisory committee members for their valuable suggestions and constant encouragement. I am deeply grateful to the department of Electrical Engineering and Computer Science at the University of Toledo for granting Teaching Assistantship throughout my Ph.D. program, which paved the way to achieve my academic goal and complete this work. A special thanks to Dr. Khalid S. Al-Olimat who took lot of pain to read through my report and extended valuable comments. Last but not the least, words and measures can never express my deepest gratitude to my wife. She has been a force of strength all along, and without her it would have been an uphill task for me to complete this work. I am also deeply indebted to my parents. Their incessant support made me to achieve new heights in life and built my character and career. I always honor their love and immutable encouragement in whatever endeavor I undertake in my life. vi

8 TABLE OF CONTENTS Page Abstract. iii Dedication...v Acknowledgement.....vi Table of Contents. vii List of Figures... xi List of Tables......xv Chapters: 1 Introduction Stationary (Constant Parameter) Controllers Dynamic Controllers Gain Scheduling and Self Tuning Regulators (STR): The Model Reference Adaptive Control (MRAC) Framework The Need for a New Generation of Adaptive Controllers An Important and Challenging Class of Complex Dynamic Systems Expanding the Adaptive Controller Performance Requirements Multiple Model Adaptive Control Intelligent Adaptive Control Intelligent Control Intelligent Adaptive Control Techniques Adaptive Controllers with Artificial Intelligence Parallel Neuro-Adaptive Control Higher Level Intelligent Adaptive Controllers Intelligent Controller Techniques Contribution of Artificial Intelligence Fuzzy Logic Techniques Neural Network Techniques...20 vii

9 1.6.4 Intelligent Control Schemes Review on Intelligent Control of Dynamic Systems Dissertation Objectives Proposed Intelligent Supervisory Loop Approach A Fuzzy Multiple Reference Model Adaptive Controller A Neural Network Parallel Adaptive Controller A Neural Network Parallel Fuzzy Adaptive Controller A Composite Parallel Multiple Reference Model Adaptive Controller Feasibility Investigation System Models Feasibility Investigation Systems Single Link Flexible Robotic Manipulator Dynamic Model Physical Details and Mathematical Equations Application to Position Tracking without Structural Complexities Development of Structural Complexities and Challenges Application to Position Tracking with Structural Complexities Six Degree of Freedom (DOF) F-16 Fighter Aircraft Model Physical Details and Mathematical Equations Algorithmic Development of Model in MATLAB and SIMULINK Development of Steady State Trim Conditions Pitch Rate Augmentation Schemes for F16 Aircraft Application to Position Tracking without Structural Complexities Development of Structural Complexities and Challenges Application to Position Tracking with Structural Complexities Conclusions Fuzzy Multiple Reference Model Adaptive Controller Formulation of the FMRMAC Scheme A Multiple Reference Model Adaptive Control Concept The FMRMAC Approach An Important Fault Tolerant Control Feature of the FMRMAC Application to a Generic Linear Jump System Creation of Modes Simulation Results and Conclusion Application to the Robotic Manipulator Position Tracking Simulation Results and Discussion Simulation Results and Discussion Tip Load Pattern Tip Load Pattern viii

10 3.4 Application to the Fighter Aircraft Pitch Rate Command Tracking Development of the Reference Model for each of the trim conditions Development of the Fuzzy Logic Scheme and Knowledge Base Simulation Results Conclusions Neural Network Parallel Adaptive Controller Formulation of the NNPAC Scheme The NNPAC Concept The NNPAC Algorithm Main Features of the NNPAC Scheme Application to the Robotic Manipulator Position Tracking Simulation Results and Discussion Tip Load Pattern Tip Load Pattern Tip Load Pattern Application to the Fighter Aircraft Pitch Rate Command Tracking Simulation Results and Discussion Dynamic Maneuver Dynamic Maneuver Dynamic Maneuver Conclusions Neural Network Parallel Fuzzy Adaptive Controller Formulation of the NNPFAC Scheme The NNPFAC Concept Mathematical Formulation Main Features of the NNPFAC Scheme Application to the Robotic Manipulator Position Tracking Simulation Results and Discussion Tip Load Pattern Tip Load Pattern Tip Load Pattern Application to the Fighter Aircraft Pitch Rate Command Tracking Simulation Results and Discussion Dynamic Maneuver Dynamic Maneuver Dynamic Maneuver Conclusions ix

11 6 Composite Parallel Multiple Reference Model Adaptive Controller Formulation of the CPMRMAC Scheme The CPMRMAC Concept Mathematical Formulation Main features of the CPMRMAC Scheme Application to the Robotic Manipulator Position Tracking Simulation Results and Discussion Tip Load Pattern Tip Load Pattern Tip Load Pattern Tip Load Pattern Application to the Fighter Aircraft Pitch Rate Command Tracking Simulation Results Dynamic Maneuver Dynamic Maneuver Dynamic Maneuver Conclusions Conclusions And Future Work References..227 Appendix A Simulation Programs Appendix B Concepts of Artificial Intelligence Techniques Appendix C Six Degree of Freedom Fighter Aircraft Submodel Data x

12 LIST OF FIGURES Figure 1.1 Block Diagram of System with Gain Scheduling Controller...3 Figure 1.2 Block Diagram of a Self-Tuning Regulator (STR)...5 Figure 1.3 Block Diagram of the Model Reference Adaptive Controller...6 Figure 1.4 Direct Model Reference Intelligent Adaptive Control...13 Figure 1.5 Indirect Model Reference Intelligent Adaptive Control...14 Figure 1.6 Parallel Neuro-Adaptive Control Scheme...15 Figure 1.7 Feedback error Neuro-Adaptive learning control Scheme...15 Figure 1.8 Multiple Model Reference Intelligent Adaptive Scheme...16 Figure 1.9 Concept of Intelligent Supervisory Loop Approach...30 Figure 2.1 Single Link Flexible Manipulator...39 Figure 2.2 Reference trajectory for single link manipulator position tracking...43 Figure 2.3 Manipulator Output with Model Reference Adaptive Control Law...44 Figure 2.4 Structurally Changed Manipulator Output with Adaptive Control Law...46 Figure 2.5 Aircraft three dimensional drawing with axes of control...49 Figure 2.6 Airfoil and the definition of terms...51 Figure 2.7 Functional Algorithmic Flowchart Aircraft Model...60 Figure 2.8 F16 Aircraft SIMULINK Model...62 Figure 2.9 Pitch Rate Command Pattern...66 Figure 2.10 Output Pitch Rate Response with Adaptive Controller...66 xi

13 Figure 2.11 Structurally Changed Pitch Rate Response with Adaptive Control Law...68 Figure 3.1 Proposed way of Intelligent Supervisory Loop...72 Figure 3.2 Domain of subspaces with different parametric values...73 Figure 3.3 Fuzzy Multiple Reference Model Adaptive Controller...74 Figure 3.4 Schematic Diagram of Direct Model Reference Adaptive Control...77 Figure 3.5 Structure of Fuzzy Logic System for the Example...90 Figure 3.6 Functional Relationship for inputs to each output...91 Figure 3.7 Membership Function Details Figure 3.8 Error Patterns with Reference Models 5 / s + 4s + 4 and 5 / s + 10s Figure 3.9 Error Comparison 9a) Case 1, 9b) Case 2 and 9c) Case Figure 3.10 Input-Output Membership function details...98 Figure 3.11 Case 1: Manipulator Position Output Comparison Figure 3.12 Case 1: Position Error Pattern and Fuzzy Switching Scheme Output Figure 3.13 Case 2: Manipulator Position Output Comparison Figure 3.14 Case 2: Position Error Pattern and Fuzzy Switching Scheme Output Figure 3.15 Command Patterns Figure 3.16 Typical Control Systems for Pitch Rate Command Tracking Figure 3.17 Membership function for Inputs and Outputs Figure 3.18 Overall Fuzzy Logic Structure Figure 3.19 Pitch Rate Command Tracking Comparison Dynamic Maneuver Figure 3.20 Pitch Rate Command Tracking Comparison Dynamic Maneuver Figure 4.1 Proposed way of Intelligent Supervisory Loop # Figure 4.2 Neural Network Parallal Adaptive Controller xii

14 Figure 4.3 RBFNN Structure Figure 4.4 Center Movement and Node Growth in RBFNN Structure Figure 4.5 Dynamic RBFNN Network Nodal Generation and Center Movement Figure 4.6 Case 1: Manipulator Position Output Comparison Figure 4.7 Case 1: Control Contribution Figure 4.8 Case 2: Manipulator Position Output Comparison Figure 4.9 Case 2: Control Contribution Figure 4.10 Case 2: Control Contribution Figure 4.11 Case 3: Manipulator Position Output Comparison Figure 4.12 Pitch Rate Command Tracking Comparison Dynamic Maneuver Figure 4.13 Pitch Rate Command Tracking Comparison Dynamic Maneuver Figure 4.14 Pitch Rate Command Tracking Comparison Dynamic Maneuver Figure 5.1 Functional Block diagram with the Intelligent Supervisory Loop Figure 5.2 The Neural Network Parallel Fuzzy Adaptive Controller Figure 5.3 Case 1: Manipulator Position Output Comparison Figure 5.4 Case 2: Manipulator Position Output Comparison Figure 5.5 Case 2: Position Error Comparison Figure 5.6 Case 3: Manipulator Position Comparison Figure 5.7 Sparse Fuzzy Rule Base of the F16 Fighter Aircraft Figure 5.8 Sparse Fuzzy Input Output Relation of the F16 Fighter Aircraft Figure 5.9 Sparse Fuzzy Membership Function Detail of the F16 Fighter Aircraft Figure 5.10 Pitch Rate Command Tracking Comparison Dynamic Maneuver Figure 5.11 Pitch Rate Command Tracking Comparison Dynamic Maneuver xiii

15 Figure 5.12 Pitch Rate Command Tracking Comparison Dynamic Maneuver Figure 6.1 Proposed way of Intelligent Supervisory Loop # Figure 6.2 Composite Parallel Multiple Reference Model Adaptive Controller Figure 6.3 Multi Layer Perceptron Neural Network Architecture Figure 6.4 Block Diagram representation of the Neural Network Corrector Figure 6.5 Case 1: Manipulator Position Output Comparison Figure 6.6 Case 2: Manipulator Position Output Comparison Figure 6.7 Case 3: Manipulator Position Output Comparison Figure 6.8 Case 4: Manipulator Position Comparison Figure 6.9 Pitch Rate Command Tracking Comparison Dynamic Maneuver Figure 6.10 Pitch Rate Command Tracking Comparison Dynamic Maneuver Figure 6.11 Pitch Rate Command Tracking Comparison Dynamic Maneuver xiv

16 LIST OF TABLES Table 2.1 Parametric values of a single link manipulator...42 Table 2.2 Tip Load Variation (Case 2)...46 Table 2.3 Control Deflection Convention...55 Table 2.4 Steady State Trim Conditions for the F16 Aircraft [55]...64 Table 2.5 Initial Control Vector, Throttle Values and Command Patterns...67 Table 3.1 Plant parametric change (Case1, Case2 and Case 3)...89 Table 3.2 Rule Base for each of the Outputs...90 Table 3.3 Rule Base (Flexible Single Link Robotic Manipulator Control)...97 Table 3.4 Tip Load Variation (Case 1)...99 Table 3.5 Tip Load Variation (Case 2) Table 3.6 Proposed Aircraft Maneuver Nomenclature Table 3.7 Selected Reference Model for different operating points Table 3.8 Performance of Adaptive Controller and Static Controller Table 3.9 Initial Control Vector, Throttle Values and Command Patterns Table 3.10 Initial Control Vector, Command Pattern and Throttle position Table 4.1 Tip Load Variation (Case 1) Table 4.2 Growing Dynamic RBFNN Parametric Range Table 4.3 Tip Load Variation (Case 2) Table 4.4 Growing Dynamic RBFNN Parametric Range xv

17 Table 4.5 Tip Load Variation (Case 3) Table 4.6 Initial Control Vector, Throttle Values and Command Patterns Table 4.7 Initial Control Vector, Command Pattern and Throttle position Table 4.8 Initial Control Vector, Command Pattern and Throttle position Table 5.1 Tip Load Variation (Case 1) Table 5.2 Tip Load Variation (Case 2) Table 5.3 Tip Load Variation (Case 3) Table 5.4 Initial Control Vector, Throttle Values and Command Patterns Table 5.5 Initial Control Vector, Command Pattern and Throttle position Table 5.6 Initial Control Vector, Command Pattern and Throttle position Table 6.1 Tip Load Variation (Case 1) Table 6.2 Tip Load Variation (Case 2) Table 6.3 Tip Load Variation (Case 3) Table 6.4 Tip Load Variation (Case 4) Table 6.5 Initial Control Vector, Command Pattern and Throttle position Table 6.6 Initial Control Vector, Command Pattern and Throttle position Table 6.7 Initial Control Vector, Command Pattern and Throttle position xvi

18 CHAPTER ONE INTRODUCTION Control engineering deals with developing proper controllers (control attributes) for fast, accurate and effective guidance of static or dynamic systems (system characteristics) to perform according to a given set of specifications or requirements (desired performance). If the systems characteristics are known and are more or less static in nature, the controller design is less challenging. There are several control techniques developed based on linear system theory, which can control such systems even when their characteristics change to certain extend. On the other hand, control of dynamic systems is often hard as such controllers should accomplish that task, especially if the system is difficult to model or vary rapidly. This can be quite challenging as it implies that the controller should have the ability to assess system conditions on a continuous basis, which calls for powerful control techniques. And such controllers become complex in nature, which in turn may hamper their speed and accuracy. Moreover, most practical systems do exhibit time varying response and behave differently at different times. This is mainly due to system nonlinearities, failure or drift of parameters, presence of external disturbance, and/or environmental changes. 1

19 2 1.1 Stationary (Constant Parameter) Controllers Stationary or constant parameter controllers, deal with plants, which can be modeled with fixed parameter models. They are used to meet specific and predictable operating conditions of the plant. As such, the performance of such stationary controllers would normally deteriorate, and may even become unstable under dynamic conditions that were not predicted a priori. Subsequently, a different approach in the form of Robust Control [1] and Optimal Control [2] emerged, which developed fixed controllers with a novice form of tolerating a limited range of variations in plant parameters. Research in this direction has got a long history and by now there is strong theoretical knowledge on the design of such controllers. In spite of this, since these controllers are designed on the basis of a priori operating domain, they inevitably fail as the system continues to depart from that domain. 1.2 Dynamic Controllers In contrast to the above-mentioned techniques, dynamic controllers are those, which change their control characteristics in real time. Unlike nonlinear control design techniques, these controllers utilize several techniques and structures to cope with systems with complex dynamics. There are three main approaches in this class of controllers, namely a) Gain scheduling controllers b) Self Tuning Regulator (STR) and c) Model Reference Adaptive Control (MRAC). It has been proven, and as outlined below, that these approaches are especially effective when the plant parameters are uncertain, unknown or rather time varying.

20 Gain Scheduling and Self Tuning Regulators (STR): Gain scheduling is a divide and conquer approach for the design of nonlinear control systems such as aerospace to process control. The classical gain-scheduling theory originates from the 1960s, and there has recently been a considerable increase in interest in gain scheduling with many new results obtained. Conventional gain schedule approach typically involves 1. Linearization of a nonlinear plant about a number of equilibrium points. 2. Designing a linear controller for each of the plant linearization. 3. Combining the linear controllers to obtain the nonlinear controller. The conventional gain scheduling however is developed on the linearized trim conditions and is not fully robust in nature. Figure 1.1 shows a block diagram of this representation. Regulator Parameters Gain Schedule Auxiliary Measurements Command Signal Regulator Control Signal U Process Output y Figure 1.1 Block Diagram of System with Gain Scheduling Controller As can be seen the regulator parameters are changed due to the change in offline gain-schedule design block with respect to the auxiliary measurements. The controller is thus dynamic in nature.

21 4 One improved version of the gain scheduling technique is the velocity-based gain scheduling approach [3]. In this approach a linear system (the velocity-based linearization') is associated with every operating point of a nonlinear system (not just the equilibrium points). A family of velocity-based linearizations is therefore associated with the nonlinear system. This family embodies the entire dynamics of the nonlinear system and so does an alternative representation. It is emphasized that this representation is valid globally and does not involve any restriction to the vicinity of the equilibrium points. A design of a gain-scheduled control of a linear parameter varying system with an application to flight control is demonstrated in [4]. In this approach a gain-scheduling control with a fewer number of linear controllers is developed which is applicable to Linear Parameter Varying (LPV) systems. The authors showed that this design technique is very effective and produced promising results. However, when the system changes its domain drastically during operation, such approach needs to be modified to the one based on intelligent systems. A self-tuning regulator (STR) is a controller that automatically finds its parameters in the control law. Another name or synonym is self-adjusting controllers. The self-tuning regulator is a class of adaptive controllers which is used when it is assumed that the process to be controlled has constant but unknown parameters [5, 6]. However, the self-tuning regulators can also be used in an adaptive context. Figure 1.2 shows the block diagram of a self-tuning regulator. As can be seen from the figure, the regulator controls the process and the parameter estimation mechanism constantly monitors the plant input and output. This will enable for delivering a dynamic model of the plant every time instant such that the control value of the regulator changes based on

22 5 this model(s). This approach thus enables for having a parameter varying dynamic model to control the plant and will be very well adaptive in nature. There are several research work conducted using this approach which provides better results than the stationary controller without using nonlinear control design techniques. Design Calculations Parameter Estimation Regulator Parameters Command Signal Regulator Control Signal U Process Output y Figure 1.2 Block Diagram of a Self-Tuning Regulator (STR) The Model Reference Adaptive Control (MRAC) Framework A Model Reference Adaptive Controller (MRAC) is another structure in adaptive control, which is especially effective when the objective is to track the output with a command signal or trajectory [7, 8]. Figure 1.3 shows a block diagram representation of the MRAC. This control scheme can also be used for controlling unknown linear timeinvariant plants, both stable and unstable. Two algorithm variations are common in implementing the MRAC scheme. First is the direct model reference adaptive controller in which the adaptation is carried out using the implicit development of the plant model structure. In the second variation, the model of the plant is developed explicitly. To date the theory behind this technique, and its formal stability proofs have been so matured that

23 6 there are several important designs using MRAC in the industry with special processors or microcontrollers for their implementation. Reference Model Error e - + Regulator Parameters Adjustment Mechanism Command Signal Regulator Control Signal U Process Output y Figure 1.3 Block Diagram of the Model Reference Adaptive Controller 1.3 The Need for a New Generation of Adaptive Controllers The above-mentioned class of dynamic controllers is very useful and effective when the system changes its parameter values or the changes are slow in nature. This is because the controller is adjusted using an adjustment mechanism monitoring the output and the plant auxiliary measures in the case of STR, or the error between the outputs of the plant and the reference model in the case of the MRAC. In the case of gain scheduling the controller uses several offline linear system models. However, when there is a sudden change in the parameter values, which is the case of component failure or in fact any functional changes due to internal or external influence, such dynamic

24 7 controllers often fails or won t be able to perform within a certain performance criteria. The main reason for such a failure is that, for an important class of systems as described in the following section, there are discontinuous changes in the plant structure or functional property, which are not often accommodated in such designs. Furthermore, these schemes are generally limited to linear systems with changing parameters, and do not lend to perform well for time varying nonlinear systems. Thus there is a need to have adaptive controllers with higher level of capabilities, sophistication and therefore effectiveness. The importance in developing such adaptive controllers has already been recognized and there are several research directions along this venture An Important and Challenging Class of Complex Dynamic Systems There is an important class of the complex dynamic systems that shows multi modality, often subjected to sudden jumps, prone to unforeseen disturbances and shows unmodeled dynamics. They are systems, which are highly nonlinear in nature and often seen in practical cases. Two most notable such systems are a) Aircraft control b) Control of industrial robots. In order to control such systems, during system operation the controller should be able to not only adapt but also possess the strength of learning the system changes. This will become even more important if the control requirement is precise and accurate in nature. This means in such situations there is a need for higher-level adaptive controllers to control such systems. These controllers should be intelligent, adaptive and should possess the online learning feature and most importantly autonomous in nature.

25 8 1.4 Expanding the Adaptive Controller Performance Requirements There are several ways to enhance the adaptive controller performance and include the features of learning and intelligence other than being autonomous in nature. Some of them are as follows; 1. Improve the model structure by explicitly or implicitly modeling the plant with offline designs and using these different models to develop different controllers online. 2. Develop intelligent based solutions using techniques such as feedback linearization, and dynamic inversion and fully using Artificial Intelligence (AI) approaches such as neural network and fuzzy systems. 3. Develop intelligent based solutions partially using AI techniques and then combining them with traditional control design techniques such as adaptive controller and other linear or nonlinear controllers. Based on these classifications, the research work in the area of high level adaptive controllers, which are autonomous and intelligent in nature, can be included in three main categories: 1) Multiple Model Adaptive Controllers (MMAC) 2) Intelligent Adaptive Controllers (IAC) and 3) Intelligent Controllers (IC). A brief review is given below for each category Multiple Model Adaptive Control One of the drawbacks in the MRAC is the identification of a static model of the plant. This creates controller instability when the plant moves from one operating point to another and shows multi modality. Multiple model adaptive control is higher-level

26 9 adaptive control approach explicitly identifying the plant models offline. Thus, this approach enables for development of mathematical models of the plant over all these operating ranges using stable offline design. Further these models are used to generate separate controllers depending on the modes of operation of the plant with the help of switching device. Reference [9] shows the concept of such controller and the way to switch and tune the adaptive control schemes, which combine fixed and adaptive models. The mathematical results in the continuous time domain are tested also for stability and then used to control the linear time-invariant systems. In similar context reference [10] proposed the adaptive control of discrete-time systems using multiple models. This paper also outlines a procedure for switching between a finite numbers of controllers to improve the performance. Theoretical results are then validated using computer simulations. In a stochastic representation reference [11] develops a multiple model adaptive controller using a residual correlation Kalman filter bank with a specific application to flight control. The main focus in this paper is the development of an estimation algorithm using Kalman filter residuals, and thus to assign conditional probabilities for each of the modeled hypotheses. Reference [12] developed a general methodology for the design of adaptive control system, which can learn to operate efficiently in dynamic environment. The concept developed in [9] and [10] is extended with NN based models in order to apply it to nonlinear systems. As an extension to [11], reference [13] develops a separate algorithm to reduce the computational complexities involved in the probabilistic based development of MMAC. The paper demonstrates a

27 10 way of reduction in the computational complexities and makes it robust even though the development still provides hard switching. These research work and several others indicate the complexity in switching and mathematical computation, which clearly emphasize the need for some intelligent based solutions for these higher-level adaptive controllers Intelligent Adaptive Control Intelligent adaptive control is a multidisciplinary field that encompasses adaptive control and such intelligent system techniques viz., Fuzzy logic, Artificial neural networks, and Genetic algorithms. Given the fact that most of the practical systems are highly nonlinear in nature, there is a need to develop a next generation of Adaptive Controllers that incorporate the fundamental qualities of Intelligent Systems: High adaptivity to significant unanticipated changes and so learning is essential Ability to exhibit a high degree of autonomy in dealing with changes Ability to deal with significant complexity which leads to certain sparse types of functional architectures such as hierarchies, and Ability to adapt to changing process The goal of such control is to design robust, adaptive and learning controllers operating in an uncertain environment with very limited mathematical knowledge of the controlled process principles. The ability of such controllers to control complex systems is quite imaginable, considering the fact that the system changes with respect to time is learned and the controller will be able to effectively contribute to its objective. Considerable effort has been already gone into the development of such controllers due to

28 11 the advancement in the nonlinear and computational control theory. Further, growth in the field of artificial intelligence concepts as well as advancement in the Adaptive control theory has enhanced the strength of the Intelligent Adaptive Control [14]. Section 1.5 compiles the important research works in these directions of intelligent adaptive control Intelligent Control The objective of the intelligent control is similar to that of the intelligent adaptive control. However, the main difference of such controllers when compared to intelligent adaptive controllers is in the way these objectives are accomplished. Unlike the intelligent adaptive controller, in which the adaptation process is accomplished with the help of a traditional mathematical control law, in this approaches both the adaptation and learning is accomplished with the artificial intelligence techniques. Several research directions have already established in this area to accomplish the autonomous control objective and many theoretical issues such as the stability of the system when fully controlled using artificial intelligence technique have been discussed in the literature. Section 1.6 is dedicated for reviewing the research work in this direction. 1.5 Intelligent Adaptive Control Techniques Intelligent adaptive control applications deal with merging the computational capability of Artificial Intelligence techniques with the adaptive control theory. The research work in this direction can be classified into four schemes. Scheme one is based on Neural Network (NN) architecture in an MRAC structure, and therefore is compitable

29 12 with nonlinear systems. The second scheme has evolved from the indirect model reference adaptive controller. The scheme has an NN identifier and controller. In the third scheme, the NN is used in parallel with a traditional controller in order to compensate for any error and deal with the system s uncertainty. This scheme can be thought of as a feedback error learning structure. The fourth scheme, which has been recently proposed in [15], develops an NN identifier for the nonlinear plant and switches the reference model accordingly, and thus changes the control parameters in a multiple model adaptive control structure. The scheme gained a lot of interest and is suited for multimodal systems. Details of these schemes are presented next Adaptive Controllers with Artificial Intelligence This scheme has been evolved from the direct model reference adaptive control. As mentioned in [15], the typical structure of such scheme is as shown in figure 1.4. In this scheme the reference model is used to generate a desired trajectory y m, and plant output y should follow. The tracking error ε = y-y m is the deviation of the plant output form the desired trajectory. In an MRAC system, the controller parameters are optimized by an adjustment mechanism. Minimizing the error between the system and the reference model makes the adjustments. These adjustments continuously adapt the controller to optimally control the system based on the reference model. In the nonlinear case, the linear controller gain is replaced with the nonlinear neural network. Thus the neural network dictates the adjustment mechanism. The tracking error is used to adjust the weights of the neural network. Further the controller parameter evolves from the ANN and controls the nonlinear network. It is not possible to directly adjust the control

30 13 parameters based on the tracking error in this case as the nonlinear plant lies in between the controller and the error. Thus this method is not practically implementable. Reference Model Delay Lines r Neural Network U Non linear Plant y p - + Σ y m ε Figure 1.4 Direct Model Reference Intelligent Adaptive Control The second scheme developed in [15] evolves from the indirect model reference adaptive controller. As the main difference between the direct and the indirect model reference adaptive controller is the explicit identification part, unlike the first scheme it can be seen that this scheme utilizes the explicit identification. Figure 1.5 shows the details of this scheme. As mentioned before since there is an explicit way of identifying the plant online, this scheme is implementable. In this scheme, using the input-output behavior of the plant, an online identification is developed to capture the plant behavior. With the resulting identification model, the parameter of the controller is then adjusted. The identification model consists of neural network and the dynamic elements of the plant. One such setup developed in [15] uses the dynamic backpropagation through a system consisting of neural networks

31 14 and linear dynamic systems. The identification error ε i (which is the error between ỳ p and y) and the output error ε is fed to the identification NN and the control NN generates the controller output with respect to ε. The identification model in this case can be used to compute the partial derivatives of a performance index with respect to controller parameters created by neural network through identification and tapped delay lines. Reference Model Delay Lines r Control NN (N c ) U Non-Linear Plant y p - + Σ y m e c Identification NN (N i ) - y p Σ + e i Figure 1.5 Indirect Model Reference Intelligent Adaptive Control Parallel Neuro-Adaptive Control In the parallel controller structure the neural network is used to compensate the control signal, which is provided by a conventional controller such that the plant output can track the desired output as close as possible. Basic structure of the feedback error learning scheme or the parallel controller is shown in figure 1.6. This type of neurocontroller is used to adjust the control input U 2 to the plant which is the output of the conventional controller, such that the plant output y could follow a desired reference

32 15 signal r as precisely as possible. The function of the parallel type neuro-controller thus is to adjust a conventional control input U 1 if it cannot provide good results. In this scheme the neural network is learned online by looking at the error between the output and the reference signal. Kawato [16] proposes one of the modifications to this parallel structure, which can be thought of a special case of the parallel control scheme. In this scheme the feedback error, which is the output of the feedback controller is fed back through the neural network layers over thousands of the learning cycles until convergence is achieved. The neural network then takes control as the dominant controller for the plant eliminating the effect of the conventional controller. The schematic diagram of such controller is shown in figure 1.7. r Traditional Controller U 1 + U Plant y Neural Network U 2 Figure 1.6 Parallel Neuro-Adaptive Control Scheme Neural Network r + - Feedback Controller e U 1 U 2 U Plant y(t) Figure 1.7 Feedback error Neuro-Adaptive learning control Scheme

33 Higher Level Intelligent Adaptive Controllers As discussed, when there is a drastic change in the plant characteristics, the framework of the MRAC fails to control the plant efficiently. This is mainly because the plant exhibits different modes of operation, which are often referred as multi modality [14]. If appropriate reference models for the plant at these modes are provided, then exact reference model tracking may be achieved. Such a concept leads to multiple model reference control. Further, if an effective Switching algorithm switches the reference models and in turns the control value based on the changes in system dynamics. Thus this scheme incorporates the details of MRAC and the other controller. However, if the decision maker and identifier are non linear in nature which can be accomplished providing an AI based technique then each model will be a nonlinear model and could exactly represent the plant. Such a scheme can be visualized as in Figure 1.8. This is different from the traditional MMAC in the fact that the controller has intelligence in it even though it is static in nature. Σ Nonlinear Plant Identifier Σ Controller 1 Switching Block Controller 2 : : Controller n Decision Maker Figure 1.8 Multiple Model Reference Intelligent Adaptive Scheme

34 Intelligent Controller Techniques Extensive research have been performed off late on the intelligent controller techniques especially to control highly complex system as the ability of the AI techniques has been theoretically proven and such controllers have become increasingly feasible due to the enhancements in the computer technology. In this section the contribution of the AI techniques on intelligent control and some relevant applications are compiled Contribution of Artificial Intelligence One of the main features of an intelligent control is its capability to learn any system dynamics. This is achieved by using computational algorithms in the form of artificial intelligence techniques. They are also termed as soft computing techniques. Two main and developing constituents of the AI techniques are artificial neural network and fuzzy systems. It can be seen that these two main branches under AI will have a significant contribution in intelligent control too. Thus it is necessary to discuss the basic concepts of these two theories Fuzzy Logic Techniques Fuzzy Systems due to its simplicity in dealing with complex processes have been used in control as a way opposed to the complex mathematical equations. There are several works in which fuzzy systems are used as a controller basically as a self-tuning PID controller and off late with a way to learn and change the rule base. There are some applications, in which the fuzziness of such systems has been used for switching the reference models. The main interests in the proposed research are to use these to switch

35 18 the reference models dynamically online. In this section the details of fuzzy logic switching and a brief literature review are conducted. In Appendix B, the fuzzy logic concepts are detailed. Fuzzy Logic Switching Fuzzy logic switching is one method for switching multiple models in a model reference adaptive control. The advantage of fuzzy logic switching is related with the capability of fuzzy logic when compared with the conventional control system. Fuzzy systems have the ability of representing systems, which are complex and otherwise cannot be represented by mathematical means. Thus such representations are closely related to real world problems [17]. Fuzzy rules developed are simple in nature and don t require much detailed calculations. Just knowing the plant behavior, which in turn can be extracted from a knowledge base, can represent the fuzzy inference engine. Once the nature of the plant is established, through a learning algorithm these rules can be modified even if there is a substantial change in plant behavior. Thus the fuzzy system can be made adaptable for the changing plant environment Fuzzy switching provides a soft switching environment, which interacts with different reference models in multiple model adaptive algorithms. This approach doesn t require mathematical complexity, which is inherent in switching dynamic algorithms. Further this scheme provides dynamic switching environments, which is advantageous when compared to static multiple model algorithms.

36 19 Fuzzy Logic Applications There are several applications on fuzzy logic control and fuzzy switching system. In 1986, Fu and Barmish [18] proposed the idea of switching between controllers and further a number of papers have appeared. The switching strategy used in these efforts consists of switching from one controller to another in a pre-defined sequence until one is found which achieves the control goal. Monitoring the growth of a performance index, which is dependent directly on the plant response, makes the decision as to when the next controller in the sequence must be activated. Another kind of parallel controllers was proposed by Åstrom et al [19]. In this approach, a knowledge-based system is involved. The system consists of a controller and process in an ordinary feedback loop. The control block consists of much control algorithms and the identification block also consists of many identification and supervision algorithms. The knowledge-based system acts in such a way to decide which algorithm in each block to be used and when to use it, and subsequently interacts with the operator. Al-Olimat and Ghandakly [20] used the same criterion, which is the identification error to switch between the controllers or to assign a weighting factor, by using fuzzy logic switching. In [21], a fuzzy logic based switching control scheme is proposed for an energy capacitor system, in order to enhance the stability of electric power system. In the method developed, the real power flow signal is used to generate switching control to regulate charging and discharging of the power. It was shown that the fuzzy control signal effectively improved the damping of generator oscillations. The above work is implemented in laboratory for a generator system. In another effort [22], a methodology based on fuzzy logic reasoning is developed in an effort to simplify the

37 20 development of such switching algorithms. The input to the algorithm is the error, which gives weight age to fuzzy controller. The summation of all the individual weighted control value gives the overall control to the process. To demonstrate the effect, authors presented the case of a universal stepper motor controller, which comprises of multiple analog non-linear controller and fuzzy switching algorithm Neural Network Techniques Neural networks are biological networks of nerve cells in brain. Neurons are the nucleus. Neurons continuously process and transmit information to each other. A neuron has three major regions: the cell body (soma), the axon and the dendrites. Soma provides the support functions and the structure of the cell. The axon is a branching fiber, which carries signals away from the neurons. The dendrites consist of more branching fibers which receive signals from other nerve cells [23]. Artificial Neural Network (ANN) is an artificially created parallel computing technique based on the biological neural network. One of the main strength of ANN is that it can effectively approximate any nonlinear function. Given the fact that most of the real world applications are nonlinear in nature and dynamically changing, these computational paradigms can be used to approximate any nonlinear function by learning. This can be very beneficial for control of dynamic systems. Thus the ability of the neural network can be applied as a part of intelligent control. Furthermore, any traditional controller needs to develop a prior mathematical model for control. These mathematical models are normally based on the linearization of the system and might not reflect physical changes or the true behavior of the system. On the contrary neural networks have the ability to learn from the input-output functions,

38 21 which provide simpler and accurate solutions to complex control problems. In recent years NN applications in control have been successfully implemented. There are several research works done in this direction. The ANN concepts, the qualities of ANN and application areas are discussed in Appendix B Intelligent Control Schemes Considering the characteristics of neural network training, neuro-control schemes can be divided into two groups; offline and online. In the first classification, the neural network is trained offline and used along with the system as a control. In the second, neural network is trained online and used along with another traditional controller or by itself to control a system [23]. Further based on the methods in which the AI techniques are implemented, intelligent control schemes can be classified as 1. Emulator and Controller Neuro-Control 2. Series Neuro Control 3. Self Tuning Control The schematic diagrams and the details of those schemes are discussed in Appendix B Review on Intelligent Control of Dynamic Systems In this section various intelligent control application to control dynamic systems are reviewed. The review covers only the relevant research areas under these control applications. In reference [24] a neural network based intelligent aircraft dynamics control is proposed. The approach is based on adaptive backstepping concept and neural networks

39 22 Controller. The main controller is designed with adaptive backstepping approach with the assumption that all aerodynamic coefficients are fully understood. This controller is then used to track aerodynamic commands. The controller has proven to reduce the tracking error exponentially to a compact set. Further, the size of the set is made arbitrarily small by tuning the design parameters by reducing the controller gain. An adaptive controller based on neural networks compensates the aerodynamic modeling errors. The closedloop stability of the error states and the parameters of the neural networks are also examined in this approach by the Lyapunov theory. Renowned contribution to the research in flight dynamics neural control comes from the research group of A J. Calise [25]. They proposed a control scheme based on inversion of a linearized plant model (feedback linearization) combined with a multilayer perceptron neural network that compensates adaptively for the effects of inversion error. The feedback linearization concepts and the proposed methodology have been proven in the following application range. Systems operating in the regimes characterized by highly nonlinear aerodynamics. Systems displaying multi time scale behavior and thus rapidly varying nonlinear dynamics. Systems characterized by a high degree of uncertainty and Systems demanding the maintenance of a certain level of the handling qualities even after failures in the actuation channels. Proven to be advantages in improving performance of an existing gain-scheduling autopilot, designed for an agile anti air missile both in conditions between design points and beyond the field covered by the gain scheduling.

40 23 There are certain drawbacks for the approach based on feedback linearziation concept though. They can be summarized as Direct application of feedback linearization to aircraft control requires the second and third derivatives of uncertain aerodynamic coefficients and does not guarantee internal stability for non-minimum phase systems. Another approach to design flight control laws with feedback linearization is to separate the flight dynamics into fast and slow dynamics by two time scale properties [26, 27]. This method is justified only if there is sufficient time scale separation between the inner and outer loops dynamics because the fast states p, q and r are used as control inputs in the outer loops system. Hence the states p, q, r in the inner loop should be much faster than the states α, β and ф in the outer loop. Moreover in most flight control research, the gain of the innerloop controller is set much larger than that of the outer-loop controller and it is assumed that the aircraft dynamics satisfy this property. This doesn t guarantee closed-loop stability. Another difficulty associated to the application of feedback linearization to a flight control system is that a complete and accurate aircraft dynamic model including aerodynamic coefficient is required. It is difficult to identify accurately aerodynamic coefficients because they are nonlinear function of several physical variables. An attempt to augment an existing proportional-integral controller through adaptive NN has been investigated by Krishnakumar and Kulkarni [28]. They have developed an inverse adaptive neural controller for a nonlinear engine model: The NN is

41 24 trained to allow the controller to keep stationary performance in the presence of big engine changes. The uniqueness of their approach as they propose is in the NN architecture, which is fully forward connected; the learning rule for online tuning of the weights is the Standard Back-Propagation Algorithm (SBPA). Another work is done by the research group of Napolitano [29], who focused the last ten years of their research activity on fixed-wing aircraft neural control. The stress is on the role of using extended back-propagation algorithm (EBPA), which was demonstrated to be particularly effective in terms of accuracy and learning speed for the online, learning of NN s. Applications concerning restructurable flight control systems during both actuator and sensor failure are addressed here. The reconstruction of the control law is performed online to bring the aircraft back to a new equilibrium condition, after failure due to vital control surfaces and failure of sensors. In contrast to the above, in [30], the author adopt a predictor-corrector control scheme which is based on the specialized learning belongs to the direct inverse adaptive control strategy. The neural controller is designed according to the reference model adaptive inverse scheme. The online training has been performed through a modified version of a standard, recursive prediction error method based on a Proportional Derivative Performance Index (PDPI), which enhances the algorithm convergence features. The PDPI has been proved to make the plant response track the reference signal perfectly, reducing the plant output error. In [31], authors proposed a feasibility study of using ANN as control systems for modern, complex aerospace vehicles via an aircraft control design. The problem considered is designing a controller for an integrated airframe/ propulsion longitudinal

42 25 dynamics model of a modern fighter aircraft to provide independent control of pitch rate and airspeed responses to pilot command inputs. The model of desired dynamics is used as a command generator, and a multilayer feed forward neural network is trained to control the vehicle model within the physical limitations of the actuator dynamics. The above demonstration is done on a linearized representation of the neuro-controller. In [32], a discrete-time lateral-directional controls law for a high-performance aircraft using neural network is proposed. The control law structure is composed of feedback and filter components formulated in the form of three-layer feed forward neural network whose parameters are adjusted by a gradient descent algorithm to provide stabilization about the aircraft center of mass and asymptotic tracking of pilot command input. In an effort to combine the classical and intelligent control, reference [33] picks up the most effective elements of old and new design concepts with the promise of producing better control systems. In such an effort, a nonlinear NN control system is developed and tested on a flight control implementation. The results developed show that the effect of NN to control the nonlinear systems especially in control of the aircraft dynamics is promising. A hybrid sliding mode control of an aero elastic system is developed in reference [34]. The idea behind the control law is that the aero elastic system is treated as two subsystems, which have a separate control target expressed in terms of sliding surface. Then a hybrid-sliding surface is utilized which includes two subsystems information in order to make both the subsystems move in the sliding surface. Better control performance of a pitch angle of the aero elastic system is demonstrated.

43 26 An adaptive critic based neural network approach is demonstrated in reference [35] in order to study and develop an optimal control law. The control scheme consists of successively adapting neural networks in the form of an action network and a critic network, which has the states as inputs and co-states as the outputs. In [36] seven different nonlinear control laws for multi axis control of a high-performance aircraft are compared using simulation. These control approaches are fuzzy logic control, back stepping adaptive control, neural network augmented control, variable structure control and indirect adaptive versions of model predictive control. These controls are used on a six degree of freedom aircraft model with sufficient modeling of engine, aerodynamics and actuators. In [37], a direct adaptive neural-network control is presented for a class of affine nonlinear system with unknown nonlinearities. This approach is a full neural network adaptive control with suitable selection of the parameters. It is shown that the parameter selection and the system form are very important for the effectiveness of the controller. In [38], a neural network observer based adaptive controller design for robotic manipulator is proposed. This work is the synthesis of the output feedback control approach with an observer and the NN based adaptive control approach. In reference [39], NN estimator based controls of mobile manipulators are proposed with online NN estimators. The ability of the controller is shown using four DOF manipulator arm and illustrated in comparison with a conventional robust control. Reference [40], shows a NN based advanced control of robotic manipulators which includes a linear combination of a set of off-line trained NN s and an update law of the linear combination coefficients to adjust robot dynamics and payload uncertain

44 27 parameters. It is shown that the control error converges to zero whose size is evaluated and depends on the approximation error of the NN bank and the design parameters of the controller. In [41], a robust adaptive neural control for a class of strict feedback nonlinear systems is proposed. The proposed approach is suitable for completely unknown virtual control coefficients and unknown nonliearities and approximated by linearly parameterized neural network. Simulation results are shown to validate the theoretical results. In [42], an adaptive controller, which provides good transient and steady state response, is presented. It was shown that when the error between the system and the model is forced to a value other than zero, the minimum phase condition can be met. The control consists of an LTI compensator together with a switching mechanics to adjust the compensator parameters. A hybrid fuzzy logic flight controller synthesis via pilot modeling is attempted in reference [43]. The proposed technique combines the control capabilities of fuzzy logic with the learning capabilities of the genetic algorithm. This technique was implemented and tested in an offline engineering simulation using a wide envelope F/A-18 longitudinal model. A design methodology is proposed for the optimal control of nonlinear systems with Takagi-Sugeno fuzzy systems and its application to spacecraft control is shown in reference [44]. The authors claimed that the controller is robust with respect to a class of input uncertainties. A fuzzy model based adaptive control algorithm for a class of continuous-time nonlinear dynamic systems is presented in [45]. In this, the fuzzy model consists of linear fuzzy local models that are combined using fuzzy inference mechanism to model a class of nonlinear systems. A fuzzy adaptive method for intelligent control is

45 28 presented in [46]. The proposed technique is used for time-varying systems with an application to agriculture. An adaptive output feedback control for nonlinear system using neural network is attempted in [47]. The developed controller is used for highly nonlinear and uncertain system that does not rely on state estimator. The proposed technique is applied on a theoretical example. A neural network based adaptive control of flexible robotic arms is proposed in [48]. The paper is a study on RBFNN as key components of adaptive controllers aimed at controlling flexible robotic arms. The neural network basically approximates the inverse dynamics of the arms. The stability and experimental details of an adaptive output feedback controller with an application to robotic arm is attempted in [49]. A nonlinear observer is used to develop joint velocities and position error. Experiment results are performed to validate the effectiveness of the proposed technique. A combined PID/adaptive controller for a class of nonlinear systems is proposed in reference [50]. The controller is robust and the technique is based on the dominant second order linear equivalent model with the unmodeled dynamics, which is possibly nonlinear and time varying. A theoretical example is used to demonstrate the effectiveness. An adaptive control of a class of nonlinear systems with nonlinearly parameterized fuzzy approximator is presented in reference [51]. In this, various adaptive fuzzy control schemes have been developed to deal with nonlinear systems whose dynamics are poorly understood. Based on these investigations it can be found that, the intelligent adaptive control techniques have captured great deal of attention recently. Moreover, there is definitely a

46 29 need to develop higher-level controllers for that important class of systems with challenging and complex dynamics. These investigations also show that developing such controllers is still an ongoing process. 1.7 Dissertation Objectives In this research, a novel concept for the control of system classes mentioned under section is established. The main idea is to develop intelligence based adaptive control framework using artificial intelligent techniques and the MRAC main structure with a novel supervisory loop approach. The resulting controllers will be especially for complex dynamic systems, which show behavioral change (parametric, functional or both) due to sudden changes in system dynamics, sudden environmental impacts, failure of system components, or some combination thereof. Such controllers will provide for an autonomous performance with a crucial capability to learn, adapt and deal with complexities. Four such intelligent adaptive control schemes will be developed and their feasibility investigated using two important and challenging systems, namely a single link robotic manipulator, and an F16 aircraft. For this purpose, a MATLAB based detailed simulation models for those two systems are developed and investigated with the four proposed schemes Proposed Intelligent Supervisory Loop Approach As mentioned, the design propositions developed in this dissertation are based on a novel concept of the intelligent supervisory loop approach. Figure 1.9 shows the concept behind the supervisory loop approach. As in Figure 1.9, when the plant operates, the adaptive controller changes and adapts to the operation with suitable signals from an

47 30 adaptive mechanism. These changes are established monitoring the error between the plant output and the reference model output. However, when the plant modes change, parameters move drastically or when the plant is susceptible to internal / external disturbances, then the performance of the adaptive control deteriorates and it will become unstable. The main reasons for these failures are a static reference model structure and/or the inability to develop a significant appropriate control value. Intelligent Supervisory Loop Intelligent Module Adaptive Controller Reference Model Reference Output Adaptive Mechanism Error Input Signal Adaptive Control Law Σ System Under Plant Output Desired Intelligent Module System States Intelligent Supervisory Loop Figure 1.9 Concept of Intelligent Supervisory Loop Approach Thus the main motivation of this dissertation project is to investigate possible Intelligence based solutions to the fundamental structural problems that exist in the adaptive controllers. These problems are due to a) designer s a priori choices, such as the

48 31 choice of a MODEL as required in either of the two schemes b) Inability to control systems with changing dynamics when such changes are drastic or multimodal in nature. To accomplish this goal, the proposed intelligent supervisory loop approach in Figure1.9 is developed to monitor the plant changes and keeping track of some of the auxiliary parameters and variables, and a) changing the structure of the reference model (reference model generation) or b) augmenting the controller value. If these changes and augmentation can be done effectively then these intelligent adaptive controllers can be used to control that class of system, which shows dynamic behavior. The advantages of using this proposed approach are: 1. Only input/output and auxiliary measurements needed 2. Provides CONTROLLER that adapts 3. Provides online learning scheme 4. Reduces the computational complexity (other AI schemes) 5. Supervises the adaptive controller Based on this proposed approach four main controller designs are developed which control systems depending on their characteristics. The specific objectives of these controllers development are presented in the following subsections A Fuzzy Multiple Reference Model Adaptive Controller In this design, the objective is to develop a novel Fuzzy Multiple Reference Model Adaptive Controller (FMRMAC). This controller is used to control a system, which shows sudden parametric Jumps and multimodal in nature. As the reference model structure in the MRAC scheme is a bottleneck in controlling such systems there is

49 32 a need to provide an intelligent supervisory loop to generate suitable stable reference model structure(s) during the plant movement in these operating domains. The proposed scheme should have a fuzzy inference engine as a part of the intelligent supervisory loop which generates the reference model structure in a Multiple model reference adaptive control, at every time instant thereby providing a smooth change in the functional relation between plant and reference model. Based on this positive change in the reference model, the adaptive control will be able to adapt and control the system even though when the system moves drastically due to sudden parametric Jumps and is multimodal in nature. Moreover, due to the fuzzy switching mechanism, the change provides a soft movement in reference model structure by alleviating the problems associated with hard switching. To meet this objective, first, the theoretical formulation of the problem needs to be developed. Further, some good reference models from the plant known modes of operation need to be established using the linear systems theory. Then a fuzzy inference engine will be developed based on the fuzzy rules formulated using these reference models. Chapter three is dedicated for the development of this novel design A Neural Network Parallel Adaptive Controller In this design, the objective is to develop a novel Neural Network Parallel Adaptive Controller (NNPAC). This controller is used to control a system, which shows mode swings and is susceptible to unmodeled dynamics and disturbances. At these plant conditions, the MRAC scheme will eventually fail to control such system. This is because the plant model is an offline design. When the system does not belong to this model,

50 33 there should be an online learning scheme to understand the changes in the system to develop a suitable controller. Thus, there is a need to provide an intelligent supervisory loop to augment the adaptive controller working as a parallel controller during the online plant changes. The proposed scheme thus has a learning intelligent supervisory loop, which learns the plant characteristics online at every time instant and augments the adaptive control law monitoring the output and certain auxiliary states or measuring variables. This learning controller is simple and operates at the entire range of plant operation and on the other hand is able to record the history. Based on the parallel controller structure, the intelligent adaptive control will be able to adapt and control the system even though when the system shows mode swings and is susceptible to unmodeled dynamics and disturbances. Moreover, due to the parallel structure the controller does not need to be trained offline by reducing the need to have an offline knowledge base. To meet this objective, first the theoretical formulation of the problem needs to be developed. Further, a suitable single reference model based on the plant mode swings needs to be established using the linear systems theory. Then an online growing dynamic NN learning controller needs to be developed. This will then be combined with the adaptive control law to work together in order to act as a parallel controller. Chapter four is dedicated for the development of this design A Neural Network Parallel Fuzzy Adaptive Controller In this design, the objective is to develop a novel Neural Network Parallel Fuzzy Adaptive Controller (NNPFAC). This controller is used to control a system, which shows mode swings, scheduled mode switching, sudden plant jumps, and is susceptible to unmodeled dynamics and disturbances. When there are mode swings and systems that are

51 34 susceptible to unmodeled dynamics which in turn show functional uncertainties, the MRAC scheme will eventually fail to control such systems. This is because the plant model is an offline design and when the system does not belong to this model, there should be a learning scheme to understand the changes in the system online to develop a suitable controller. Thus there is a need to provide an intelligent supervisory loop to augment the adaptive controller working as a parallel controller during the online plant changes. Further, in the event of the drastic plant movement, sudden Jumps and scheduled mode switching, the reference model structure also needs to be generated if possible softly in the positive direction, such that the adaptive control will perform effectively. This as described before needs to be done in the form of the second intelligent supervisory loop to monitor the reference model. The proposed scheme thus has a learning intelligent supervisory loop, which learns the plant characteristic online at every time instant and augments the adaptive control law monitoring the output and certain auxiliary states or measuring variables. Further, a fuzzy inference engine (even though sparse in nature when compared to the FMRMAC) needs to be developed to monitor the reference model structure and provides generation of the reference models. To meet this objective, first, the theoretical formulation of the problem needs to be developed. Second, some good reference model based on the plant mode swings needs to be established using the linear systems theory. Then a fuzzy inference engine needs to be established with certain developed methods from the control theory. Further, an online growing dynamic NN learning controller needs to be developed. This will then be combined with the adaptive control law to work together in order to act as a parallel controller. Chapter five is dedicated for the development of this design.

52 A Composite Parallel Multiple Reference Model Adaptive Controller In this design, the objective is to develop a novel Composite Parallel Multiple Reference Model Adaptive Controller (CPMRMAC). This controller is used to control a system, which shows mode swings, unscheduled mode switching, sudden plant jumps, and is susceptible to unmodeled dynamics and disturbances. When there are mode swings and systems that are susceptible to unmodeled dynamics which in turn show functional uncertainties, the MRAC scheme will eventually fail to control such systems. Thus there is a need to provide an intelligent supervisory loop to augment the adaptive controller working as a parallel controller during the online plant changes. Further, in the event of the drastic plant movement, sudden Jumps and unscheduled mode switching the reference model structure also needs to be generated in the positive direction such that the adaptive control will perform effectively. This needs to be done in the form of the second intelligent supervisory loop to monitor the reference model. And thus, the proposed scheme has a learning intelligent supervisory loop, which learns the plant characteristics online, and augments the adaptive control law. In the case of reference model generation for an unscheduled mode switching, the intelligent loop learns the system mode changes in real time, which is done through an online neural network learning strategy. Thus, and unlike in the fuzzy multiple reference model generation scheme, a neural network offline reference model generator with an online neural network corrector is in the center of this proposed scheme. To meet this objective, first, the theoretical formulation of the problem needs to be developed. Second, some good reference model based on the plant mode swings needs to be established using the linear systems theory. Then an offline neural network needs to be trained for these sparse reference model switching. The output of

53 36 this neural network is then augmented with an online neural network corrector, monitoring the output error. Further, an on line growing dynamic NN learning controller needs to be developed. This should then be combined with the adaptive control law to work together in order to act as a parallel controller. Chapter six is dedicated for the development of this design Feasibility Investigation System Models An important aspect of this research is to test these developed algorithms on challenging close to practical nonlinear simulation models. This is because; the interest all along is to develop intelligent adaptive control solutions to control a class of highly complex practical systems. Thus there is definitely a need to develop challenging simulated models, extracting practical system data and testing them with traditional adaptive controllers. Further the proposed effects such as multi modality, un modeled dynamics, unforeseen changes and disturbances should be created artificially on these system models. Thus as a modeling objective, two challenging practical system models viz., single link flexible robotic manipulator model and six DOF F16 fighter aircraft model are to be developed and tested completely on all these propositions making them ready to test the developed intelligent adaptive control algorithms. In the next chapter, this challenging nonlinear dynamic model development is discussed in details in order to accomplish this objective successfully.

54 CHAPTER TWO FEASIBILITY INVESTIGATION SYSTEMS This chapter is dedicated to the development of detailed simulation models for the following two important challenging dynamic systems that are directly relevant to the need for the new generation of adaptive control proposed in this dissertation: 1. A single link flexible robotic manipulator 2. A six Degree of Freedom (DOF) F16 fighter aircraft model These simulation models are developed with the highest degree of modeling details and with real practical data, in order to generate realistic structural as well as dynamic challenges for feasibility investigation of the proposed techniques. Provisions are included to introduce a wide range of internal and external changes to system signals and functions as necessary to simulate different study scenarios and cases. From feasibility investigation viewpoint, there are two important aspects in the model development suitable for the analysis of the proposed algorithms. First the model should be nonlinear and the values for model parameters and variables need to be extracted from practical tests in order to make the model as close to physical systems as possible. Secondly, the developed system model needs to be challenged with a variety of realistic dynamic conditions, in order to investigate system performance as it experiences unmodeled dynamics, sudden Jumps and/or mode switching. Initially the single link robotic manipulator model is developed. 37

55 38 Subsequently, some test results applying the direct model reference adaptive control law on this model to track a command signal are demonstrated. Dynamic changes are further introduced in the developed model in order to create the challenges and the details of these changes are discussed. Some test results are then illustrated to demonstrate the deterioration in the effect of the model reference adaptive control law to track the reference trajectory in presence of these disturbances. Similarly the F16 fighter aircraft dynamic nonlinear model is also developed. The main thrust in this development is to use the practical system data from technical report [52] and generate a highly nonlinear dynamic realistic model suitable for testing the proposed intelligent control algorithms. To this effect several sub models and the main six DOF models are generated using MATLAB and SIMULINK. Further using a simplex routine algorithm, the steady state trim conditions are developed and subsequently structural challenges are generated. Thus the objective of this chapter is not only to demonstrate the nonlinear simulation models for testing the controller, but also to show the introduction of the structural complexities into it and the subsequent performance deterioration when controlled by a traditional adaptive controller. This should enable a wide range of investigations for the proposed intelligent controllers introduced in the coming chapters, with a highly nonlinear structurally challenged model. 2.1 Single Link Flexible Robotic Manipulator Dynamic Model The robotic arm has the position zero x and y axes denoted as x 0 and y 0 as can be seen in the figure 2.1. Depending on the angular movement based on the required

56 39 trajectory there will be deflection, and the arm settles to new axes denoted as x 1 and y 1 at every time instant. This angular deflection is represented by the angle θ. y 1 y 0 α x 1 θ x 0 Figure 2.1 Single Link Flexible Manipulator The movement of the arm basically depends on the mass and the weight of the link and the respective dynamic model shows the equations along with the parameter values. Further the arm tip will have the effect of the tip load which is varied at time instant denoted as the Load Torque. Due to this, physically the arm tip may not track the trajectory, which is denoted, by the angle α. In order to create this effect in the robotic manipulator dynamic model, the load torque is changed at different time instant online while the controller value in the form of the input voltage to the manipulator changes at each instant depending on the output error. The important physical parameter, which relates the mass and the weight of the manipulator, is thus its angular position θ and the angular velocity. This along with the links vibration mode constructs the basic mathematical equations. However, the velocity and vibration mode in turn depend on the physical properties of the link such as mass,

57 40 gravity, moment of inertia, elasticity and the length of the link. One of the sub-models is the DC motor model, which is used to drive the link with its torque and speed. The physical parameters involved in this modeling are the DC motor torque, resistance, inductance, gear ratio and efficiency. Thus the complete model consists of five first order differential equations Physical Details and Mathematical Equations Based on the above physical description, a nonlinear mathematical model with actual physical parameters is extensively described in [53, 54]. The remaining part of this section is used to describe the mathematical equation of the model, its physical relation and the parameter set utilized to generate a comprehensive nonlinear model. Further this mathematical model is generated in algorithmic form using MATLAB and SIMULINK. The system dynamics can be represented as, y ( 1 ) = θ ( t ) y y ( 2 ) ( 3 ) = = θ ( t ) φ ( t ) (1-4) y ( 4 ) = φ ( t ) Where the y(1) to y(4) are Robotic link s angular position, Link s angular velocity, Link s vibration mode, Link s vibration mode first derivative respectively. Differentiating, y(1) = θ ( t) = y(2) y(2) = θ ( t) y(3) = φ( t) = y(4) y(4) = φ( t) (8-12)

58 41 Emerging from above the complete model of a single link manipulator is as follows. y ( 1) = y(2) 1 y( 2) = [ r1 ( τ + r2 r3 r4 ) r5 ( r6 + r7 r8 )] (13-16) r r r 1 9 y ( 3) = y(4) y( 4) = r r r r y( 5) = in a b y L Where: a [ r ( r + r r ) r ( τ + r r )] [ V R y(5) nk (2)] ( K y(5) nk y(2 T ) Tm τ = nη ), r 1 = ρζ 1, r m 2 L = [ gρ cos( (1) )], 2 2 y r = ρζ y(3) y(4) (2), y d ( (1) ) r4 = gρy( 3) ζ 3 sin y, r 5 = ρζ 2, f 4 r = ( ζ, 2 6 ρy 3) y(2) 1 ( (1 ) ζ 3 r =, 7 gρ cos y ) r = EIy( ζ, 8 3) 4 L r = ρ ρy(3) Where: ζ 1 Ra and L a : D.C motor resistance and inductance respectively

59 42 K m : D.C motor torque constant and Tf and K b : Back e.m.f. constant K d : Torque friction and Damping coefficient respectively, Tm : Tip Load Torque n andη : Gear ratio and gear efficiency respectively, E : Young s modulus of elasticity I : Link s area moment of inertia with respect to its neutral axis, L : Length of the link g : Earth s gravity, ρ : Mass per unit length of the link ζ1, ζ 2, ζ 3 and 4 ζ : Link s area moment of inertia with respect to all other axes. The values of the parameters are given in Table 2.1. Table 2.1 Parametric values of a single link manipulator R a =8.33 n=65.5 ρ= L a = η=0.66 ζ 1 = K b = E=68.9e 9 ζ 2 = T f = I= e -12 ζ 3 = K d = e -4 L=1 ζ 4 = K m = g= Application to Position Tracking without Structural Complexities Although the objective of developing the above detailed mathematical nonlinear model is to use it as a simulation model for testing the ability of proposed intelligent adaptive controllers on the position tracking with tip load torque, the model needs to be tested first with an adaptive law. In this section, the effectiveness of the direct model reference adaptive control law on position trajectory tracking utilizing this mathematical model is discussed. Figure 2.2 shows the desired trajectory, which needs to be tracked by the output of the system model with the controlled input voltage delivered from the control law. It is worth noting that, this trajectory is used through out the simulation results for all the developed algorithms detailed in the coming chapters.

60 43 As an initial step a direct model reference adaptive control law which forms a part of the developed intelligent adaptive control laws is used to test the controlled input voltage so that the output of the nonlinear simulation model could track this desired trajectory. This is done without any tip load torque variation just to test the model. Figure 2.3 shows the test results, which clearly indicate that with a suitable reference model structure; the existing model reference adaptive control law enables for the output of the nonlinear system to track the model output precisely. The suitable reference model structure utilized, is a second order reference model as stated in equation (17) with a natural frequency ω, as a constant set at five and damping coefficient ς fixed at 0.7. n 2 n ω Wm ( s) = (17) 2 2 s + 2ςω + ω n n Figure 2.2 Reference trajectory for single link manipulator position tracking

61 44 Figure 2.3 Manipulator Output with Model Reference Adaptive Control Law Development of Structural Complexities and Challenges As described and demonstrated in the previous section, an adaptive controller performs well to control nonlinear systems for a case of position tracking. The proposed intelligent adaptive controllers are developed for a system, which is susceptible to internal and external disturbances, unforeseen dynamics, sudden Jumps and scheduled and unscheduled mode switching. Thus there is a need to artificially create these structural complexities and challenges in the simulation model preparing it to be close to a practical system prone to such difficulties.

62 45 These structural impacts are created in the following ways. 1. Changing the Tip Load Torque of the robotic manipulator model online at randomly selected time intervals so that this is similar to modal switching. 2. Changing the internal structure of the mathematical equations randomly so that the unmodeled dynamics are generated. 3. Augmenting the mathematical equations with a random signal to create external disturbances. 4. Combining the various disturbances and changing the frictional torque value to develop a structurally challenged system for testing various algorithms. The details of each structural change are discussed in the following chapters where the algorithm application cases are described. Then the performance of the proposed designed controllers is tested on the simulated model for each case Application to Position Tracking with Structural Complexities As discussed in section the structural impacts are created in the nonlinear simulated model in several ways. However, it is important to test the structurally impaired model with a traditional model reference adaptive controller in order to establish the fact that this adaptive controller scheme is not sufficient and is often prone to instability. This is important, as it is required to see on all conditions that the single reference model adaptive controller performs bad or even fails so that it can be declared that the simulated model is complete and ready for testing the intelligent adaptive controllers. Figure 2.4 shows a sample case of the performance of the adaptive controller for output position tracking when the model is subjected to a tip load variation in the

63 46 form as shown in table 2.2. As can be seen the traditional adaptive controller fails to perform as required. It is also worth noting that the model is tested for all the structural complexities and found to have similar effect when controlled using a single reference model adaptive controller, which proves the model completeness and readyness to test the higher level controllers. Table 2.2 Tip Load Variation (Case 2) Time Range (sec) Load Torque (Nm) Figure 2.4 Structurally Changed Manipulator Output with Adaptive Control Law 2.2 Six Degree of Freedom (DOF) F-16 Fighter Aircraft Model In this section a nonlinear six DOF F16 fighter aircraft model based on [52] and [55] is developed. Initially the reason behind choosing F16 aircraft nonlinear model is

64 47 discussed. Further, the building blocks of the proposed aircraft model, the control surfaces and the mathematical 6-DOF F16 Fighter aircraft model are developed. In order to apply the intelligent controller techniques, the steady state trim conditions from this dynamic model need to be extracted. This is discussed in section Then, the adaptive controller is tested using the developed model, with and without structural complexities. Thus this section will include the objective of developing the nonlinear aircraft model with subtasks described in each sub section. Aircraft dynamics are highly nonlinear in nature. The rigid body six DOF equations of the aircraft contain non-linearity in the form of inertia constant and the state equations, which contains simultaneous nonlinear terms. Further, the control surface deviation directly affects the aerodynamic forces and moments that are highly nonlinear in nature and used to control the aircraft motion. The aircraft maneuver influences the aerodynamic forces and moments. This can be seen as an external influence. Moreover, the external disturbance in the form of atmospheric temperature and pressure, which vary with the speed and altitude, will affect the aircraft states and the aerodynamics. Thus it is well evident that such a system by its characteristics calls for complex control especially in the form of intelligent control. Due to these reasons in order to demonstrate the effect of the techniques developed, F16 fighter aircraft control is selected as one of the candidate system Physical Details and Mathematical Equations Above description now calls for a real prototype of an aircraft to test the techniques or an equivalently competent aircraft dynamic model, which depicts the

65 48 behavior of an aircraft. Because of the high cost of building and flight-testing of a real aircraft, the best way to approach this issue is to develop an aircraft dynamic model. The mathematical model is used then in conjunction with computer simulation, to evaluate the performance of the modeled aircraft and hence will finally deliver a pragmatic design. The dynamic models can also be used to test the control system effects on several modes of operation. Moreover, the technological development in simulators obtaining wind tunnel data enables for development of close to practical dynamic models. Building Blocks of the Aircraft Model The aircraft flight mechanics can be aggregated from the object body (fuselage and wing), engines, aerodynamics, gravity, atmosphere and winds. A three dimensional aircraft drawing showing the details of controls and axes is as in figure 2.5. The aircraft body can be detailed by rigid body dynamics, and assembling its equations of motion, which constitutes the dynamic states. The engine(s) basically can be developed by engine model that in turn provides the engine thrust. The aerodynamics will contribute to the development of aerodynamic forces and moments build up based on the coefficients and reflects in the rates of moments. The gravitational constants effect the moments and the atmospheric conditions dictated by the aircraft maneuver and reflect the change in the dynamic pressure and Mach number. Finally, but very important the control surfaces and actuators will control the aircraft maneuver based on the pilot inputs. These constitute the building blocks.

66 Figure 2.5 Aircraft three dimensional drawing with axes of control 49

67 50 As mentioned earlier, the aircraft flight mechanics can be aggregated from the object body (fuselage and wing), engines, aerodynamics, gravity, atmosphere and winds. Based on this aggregation, dynamics and kinematics, basic aerodynamics, aerodynamic forces and moments, atmosphere model, engine model, controller used in F-16 aircraft and actuator modeling are discussed next. Dynamics and Kinematics The force in the translational direction, which in effect creates the velocity, mentions the dynamics of the aircraft. The velocity vector can be divided into three components based on three axes. Each axis has the effect of the moments on the other two axes, a gravitational effect developed based on the kinematics components and the force in each direction. Thus the velocity vector components will constitutes the three states of the dynamics [55]. The kinematical equation shows the Euler angles of the motion. They are the effect of the moments and the attitude of the aircraft. In the x-axis it is related to the moment P and the effect of the other two angles with moments Q and R. In the y- axes it is related to the moments Q and R and the angle in the x direction. Further the z-axis component of the angle will have the effect of Q and R along with the angles in the x and y directions. Basic Aerodynamics The aerodynamic data represents the aircraft as a whole. However, the shape of airfoil determines its aerodynamic property [55]. So to gain insight about the aerodynamic data for the aircraft as a whole the airfoil structure is discussed as in figure 2.6. The chord line is a straight line drawn from leading edge to the trailing edge and is

68 51 the reference line for describing the shape. The mean line (or camber line) is a line drawn from leading edge to trailing edge midway between the two surfaces and the difference between the mean line and the chord line shows the amount of camber. The shape of the upper and lower surfaces, the amount of camber, the thickness and the leading edge radius are combined to determine the aerodynamic properties and the useful speed range. The flow field around the airfoil is represented by the streamlines shown in figure 2.6 [55]. As can be seen the angle that the chord line makes with the free-stream velocity vector is the airfoil angle of attack, usually denoted by α (referred as alpha). The aerodynamic force is conventionally divided into two components called the lift and the drag as shown. Lift is defined as perpendicular to air stream vector and the drag parallel to it. They normally increase as alpha is increased. A location, which is important from the stability point, is the aerodynamic center (ac) of the airfoil. The ac is a point at which the aerodynamic pitching moment tends to be invariant with respect to alpha. It is normally one quarter back from the leading edge and may be slightly above or below the chord line. Stream Lines Mean Line Chord Line Camber Lift α Free stream Direction Figure 2.6 Airfoil and the definition of terms M Drag

69 52 Aerodynamic Forces and Moments The aerodynamic forces and moments have the components due to aerodynamic effects and to the engine thrust. The aerodynamic transitional forces V T can be written as, U VT V T cosα cos β T T T T V = = V = S V = S 0 VT sin β (18) W 0 V sinα cos β T Where α and β are angle of attack and side slip angle respectively, V T the total velocity and U, V and W are components of V T [55]. Subsequently from above α and V T ( U + V + W ) tan = W / U,sin β = V / V T = (19) The forces and moments acting on the complete aircraft are defined in terms of dimensionless aerodynamic coefficients as shown below. Drag, D = q SCD,lift, L = q SCL,sideforce, Y = q SCY, rolling moment, L q = S C b l, Pitching moment, M = q S ccm and yawing moment, N q S bc N = (20) Where q = free stream dynamic pressure, S is wing reference area, b is wing span and c = wing mean geometric chord. Atmosphere Model Atmospheric model actually calculates the Mach number (based on the temperature and the velocity), dynamic pressure (based on the air density and velocity), and the static pressure (based on the temperature and air density). The relevant equations are discussed in section

70 53 Engine Model The NASA data for the F-16 model is an after burning turbofan engine model, derived from the practical turbofan engine used in F-16, in which the thrust response is modeled as a first-order lag. The model consists of a lag time constant, which is a function of the actual engine power level (POW) and the commanded power (CPOW) [52]. This time constant is calculated in an algorithmic function, whose value is the rate of change of power, while the state variable (power) represents the actual power level. The throttle gearing relates the commanded power level to the throttle position (0-1.0) and is a linear relationship apart from a change of slope when the military power level is reached at 0.77 throttle setting. The variation of engine thrust with power level, altitude, and Mach number is generated in the algorithmic development in the form of engine thrust function. Controllers used in F-16 Aircraft and Actuator Modeling An aircraft has three axes: 1. Lateral 2. Longitudinal 3. Normal (Vertical) It can rotate around each of these three axes and these rotation motions are termed as Roll- motion around longitudinal axis, pitch-motion around the lateral axis and yawmotion around the normal axis. These control deflections and surfaces are shown in the aircraft diagram as in Figure 2.5. Main controllers in F-16 aircraft are similar to that in any conventional aircraft termed as aileron, rudder and elevator. These control surfaces are flap like surfaces, which can be deflected back and forth at the command of the pilot.

71 54 The ailerons are located at the trailing edge of the wing and the elevator at the trailing edge of the horizontal stabilizer. The rudder is located at the trailing edge of the vertical stabilizer. The aileron controls the roll or lateral motion and is therefore often called the lateral controls. When the ailerons are deflected the down going aileron increases the camber of one wing and the up-going aileron decreases camber on the other wing. This causes the roll rate to increase away from the wing with the greater lift. As long as there exists a net moment (lift times distance) between the two wings the aircraft will roll faster and faster. In the F-16 aircraft, deflection of conventional wing-mounted ailerons and the differential deflection of the horizontal stabilator control the roll motion. The elevator controls the pitch or the longitudinal motion and thus is often called as the longitudinal control. In order to fly the total pitch moment must be zero almost at all the time. Momentary nose up and nose down moments are required to get the aircraft attitude changing. However, a quick return to zero moment must be initiated subsequently. The main wing produces positive lift and the tail produces the negative lift. In the F-16 aircraft a symmetrical deflection of the horizontal tail (stabilator) is used for Pitch Control. The rudder controls the yaw or the directional motion and thus is called as the directional control. The rudder looks like a vertical fin as can be seen in the aircraft diagram. It is also used to control the directional motion of the F-16 aircraft. Further specific to F-16 aircraft, there is a leading-edge flap deflection and trailing edge flap deflection. Mach number and Angle of attack automatically guide the leading edge flap

72 55 deflection as mentioned. Actuators control the deflections of the surfaces and they are modeled similar to that in [52]. The details are as follows. The stabilator deflection is controlled by an actuator, which is modeled as a firstorder lag of sec, with a rate limit of 60 deg/sec. The surface deflection limit was +/- 25 deg. The leading edge flap deflection actuator was modeled as a first-order lag of sec, with a rate limit of 25 deg/sec with a maximum flap deflection of 25 deg. The roll control system is provided using both aileron and differential-tail deflection at a ratio of four deg of δ a per one deg of δ d '. The surface actuators for them are modeled as sec first order lags with rate limits of 60 deg/sec for the differential tail and 80 deg/sec for the ailerons. The surface deflection limits were +/ deg and +/-21.5 deg for the differential tail and ailerons, respectively. The rudder actuator was modeled as a sec first-order lag with a rate limit of 120 deg/sec. The total rudder travel was limited to +/- 30 deg. The signs convention used in the model for the above mentioned control deflections are summarized in table 2.3 [55]. Table 2.3 Control Deflection Convention Control Deflection Sense Effect Elevator Trailing Edge down Positive Negative Pitching moment Ailerons Right-Wing trailing Positive Negative Rolling edge down moment Rudder Trailing Edge left Positive Negative Yawing moment, positive rolling moment

73 56 Mathematical Model The mathematical model developed here uses the wind-tunnel data from NASA- Langley wind tunnel tests on a scale model of an F-16 airplane [52]. This data applies to a speed range up to Mach number=0.6, and were used in NASA-piloted simulation to study the maneuvering and stall/post-stall characteristics of a relaxed static-stability ο airplane. The data cover a very wide range of angle of attack ( 20 ο to90 ), and of ο sideslip angle ( 30 ο to30 ). This model is restricted to reduce range of α in ο between ( 10 to 45 ο ) since the dynamic modeling in the post stall region is not allowed by present state of art technology and the aircraft has insufficient pitching moment control for maneuvering at angles of attack beyond about ο 25 [52]. The data table for each of the sub models is compiled in Appendix C. As mentioned before, F-16 aircraft have a leading edge flap deflection, which is influenced by an angle of attack, dynamic pressure, and in turn Mach number. In [52] this is modeled by the following equation δ = 1.38(2S / S ) * α 9.05( q/ ) (21) lef P s Where: S is the wing area, α is the angle of attack P s is the static pressure q is the free-stream dynamic pressure However the effect of the leading edge flaps deflection in the speed range of Mach Number less than 0.6 is very small [52]. Thus the dependence of Mach number

74 57 on δ lef is eliminated. Further these independent tabular data for the leading edge flap deflection is incorporated in the aerodynamic tables extracted from [52]. The aerodynamic look up table, which then developed, is used as the backbone for the control surface deflections. The tabular form as indicated by [55] is constructed from steady state flight trim conditions and the corresponding dynamic modes. The authors have claimed that this developed model is close to the full (50-lookup-table) model developed in [52]. The elements of the state vector will comprise, respectively, the components of the velocity vector V B, the vector of Euler Angles ф, the angular rate vector ω B and the position vector P NED. Therefore, the state vector is X T [ U V W φ θ ψ P Q R p p h] = (22) N E Where h is the altitude in the NED frame. as follows. Based on these developments, the equations of motions of the aircraft model are Force Equations U = RV QW g sin θ + F / x m V = RU + PW + g sin φ cos θ + F / y m W = QU PV + g cos φ cosθ + Fz / m (23-25) Kinematic Equations φ = P + tanθ ( Qsinφ + R cosφ) θ = Q cosφ Rsinφ ψ = ( Q sinφ + R cosφ) / cosθ (26-28)

75 58 Moment Equations P = ( c1r + c2p) Q + c3l + c4n (29-31) 2 2 Q = c5 PR c6 ( P R ) + c7m R = ( c8 P c2r) Q + c4 L + c9 N Navigation Equations p N = U cosθ cosψ + V ( cosφ sinψ + sinφ sinθ cosψ ) + W (sinφ sinψ + cosφ sinθ cosψ ) p E = U cosθ sinψ + V (cosφ cosψ + sinφ sinθ sinψ ) + W ( sinφ cosψ + cosφ sinθ sinψ ) h = U sinθ V sinφ cosθ W cosφ cosθ (32-34) The constants c s are defined by 2 1 = ( J y J z ) J z J xz Γ c, Γ c 2 = ( J x J y J z ) J xz Γ 4, c 5 = ( J z J x ) / J y, c 6 = J xz / J y, c 7 = 1/ J y, Γc 8 = J x J x J y ) + J Γc 3 = J z, c = J xz 2 ( xz c = Γ 9 J x where x z 2 xz Γ = J J J (35) The force and moment components (Fx,Fy,Fz,L,M,N) in the 6-DOF equations are broken into aerodynamic and thrust contributions as explained before. Further a table of engine thrust values versus altitude and speed (or Mach Number) is needed with the throttle setting as the parameter. The total control input vector can be written as [ thtl el ail rdr] U T = (36)

76 59 Where the elements in U T are throttle setting, elevator, aileron and rudder deflections respectively Algorithmic Development of Model in MATLAB and SIMULINK In this section the algorithmic development of the six DOF F16 Fighter aircraft model in MATLAB and SIMULINK based on [52] and extracted Aerodynamic, Engine Thrust and Atmospheric tables from [55] are discussed. Initially the steps involved in the development are established. Further a functional block diagram representation is shown and then individual functions developed related to the tables are detailed. Finally the SIMULINK block diagram for the developed aircraft model is shown. The steps involved in the algorithm development is as follows Compute the Mach number and dynamic pressure from standard atmosphere models. Then compute engine thrust for use in the force equations. Compute the aerodynamic coefficients for the force equations; compute U, V and W from V T, α and β. Compute the aerodynamic coefficients for the moment equations, using α and β if necessary, and then evaluate the moment equations. Evaluate the kinematics equations and the navigation equations. Block Diagram representation of the Functional Flow Chart The developed algorithm to deal with all the aircraft building blocks using MATLAB is according to the wind tunnel data from [52]. A block diagram representation of the functional flow chart dealing with this aircraft dynamic model is as shown in Figure 2.7.

77 60 Computing Air data Outputs: - Mach numbers, Dynamic Pressure Inputs: -Velocity, Altitude Computing Engine Model Outputs: - Engine Thrust Inputs: -Power, Altitude, Mach number Aerodynamic look-up table and coefficient buildup Outputs: - Aerodynamic Force (Cxt, Cyt, Czt) & Moments (Cnt, Clt, Cmt) coefficients Inputs: -Control Variables (elev, ail, rdr) and (α, β) Control Vector State Equations Force Equations Derivative Inputs: -Moment Rates (P, Q, R), Velocity (UVW), Kinematics (Phi, Theta) and Aerodynamic Force coefficients Outputs: - Vt, α and β Derivatives Kinematic Equations Derivative Inputs: -Moment Rates (P, Q and R), Kinematics (Phi and Theta) Outputs: - Φ, θ and ψ Derivaties Moments Equations Derivative Inputs: -Moment Rates (P, Q, R), Aerodynamic Moment Coefficient (Clt,Cmt.Cnt) and Inertia Constants Outputs: - Moments Derivatives Navigation Equations Derivative Inputs: -Moment Rates (P, Q, R), Aerodynamic Moment Coefficient (Clt,Cmt.Cnt) and Inertia Constants Outputs: - Moments Derivatives Figure 2.7 Functional Algorithmic Flowchart Aircraft Model

78 61 Development of Aerodynamic and Engine tables The aerodynamic and the interpolation algorithms are developed in MATLAB based on the ones developed in [55] which provide values for the body-axes dimensionless aerodynamic coefficients for the F-16 model at arbitrary values of the independent variables. The angle of attack range of the tables is from 10 deg to 45 deg with 5 deg increments and the sideslip angle range is from 30 deg to 30 deg with either 5 deg or 10 deg increments. The interpolation algorithm interpolates linearly between the data points and extrapolates beyond the table boundaries to some extent. Development of Atmospheric table The calculation steps of various atmospheric components in the mathematical model are as follows. The atmospheric temperature is expressed as T=519.0* ( E -5 *altitude). Further if the altitude is greater than feet then T=390.0 The atmospheric density is expressed as, ρ=2.377*e -3 *( E -5 *altitude) 4.14 Further the mach number is calculated as Mach=V t /(Sqrt(1.4*1716.3*T), Where V t =Velocity The dynamic Pressure Q BAR =0.5*Rho*V t *V t and the static pressure P S =1715*Rho*T Development of Engine Thrust tables The engine thrust tables are developed based on [55], which performs the table value using Mach number, Power and Altitude. Engine thrust is calculated based on the following relation If P3 < 50 T = T + T T )( P / 50) idle ( mil idle 3 else if P3 >=50 T = Tmil + ( Tmax Tmil )( P3 50) / 50 (37)

79 62 Where P3 = 1/ τ T ( P2 P3 ) and P = P 3 3 dt (38) The three thrusts, T mil, T max and T min are calculated from a look up table, which is a function of input power P 1, altitude and Mach number. Further P 2 and P 3 are functions of P 1 (Input Power) SIMULINK model The SIMULINK model shows the graphical representation of the MATLAB codes, which enables us for linking the simulation modules easily and to ease and create a user friendly interaction between modules this approach, has been developed. The proposed graphical representation is shown in figure 2.8. This figure shows only the main model blocks, which include all the sub building blocks. Figure 2.8 F16 Aircraft SIMULINK Model Development of Steady State Trim Conditions Steady state aircraft flight can be defined as a condition in which all of the motion variables are constant or zero. This means the linear and angular velocity components are

80 63 constant (or zero) and all acceleration components are zero. Steady state flight conditions need to be developed as these conditions serve as the initial conditions of testing the aircraft dynamics from a steady state operating point and also to design the reference model structure at various operating points. Having defined the steady state condition it can be seen that [55] Steady state flight means P, Q, R, U, V, W ( orv T, β, α) = 0, keepingu = const With the following additional constraints according to the flight condition: Steady Wings-Level Flight: φ, φ, θ, ψ 0( P, Q, R 0) Steady Turning Flight: φ, θ 0( ψ turnrate) Steady Pull-up: φ, φ, ψ 0( θ pulluprate) Steady Roll: θ, ψ 0( φ rollrate) While a pilot finds it easy to put an aircraft into a steady-state flight condition, the mathematical model requires the solution of the simultaneous nonlinear equations. These conditions are determined by solving the nonlinear state equations for the state and control vectors, which make the state derivatives identically, zero. A convenient way to do this is to utilize readily available numerical algorithm to reduce a cost function from the sum of the squares of the derivatives above. Such a simplex algorithm developed by [55] for getting the steady state trim conditions is utilized. Table 2.4 shows the trim conditions generated using this routine for five operating conditions.

81 64 Table 2.4 Steady State Trim Conditions for the F16 Aircraft [55] Conditions Variables Nominal X cg = 0. 3C X cg = 0. 3C X cg = 0. 3C X cg = 0. 38C ψ = 0.3rad / sec θ = 0.3rad / sec V T (ft/sec) α(rad) β(rad) -4.0E E E E E -5 φ(rad) θ(rad) P(rad/sec) Q(rad/sec) R(rad/sec) Thtl(0-1) El(deg) Ail(deg) -1.2E E E E -4 Rdr(deg) -6.2E E E Nominal Condition: h=0 ft, q=300 psf, X cg =0.35C, φ = θ = ψ = γ = 0 In the table 4.2, first column indicates a steady state flight with center of gravity as 0.35 of the mean chord length. This is nominal condition of aircraft flight. In the second column it can be seen that the center of gravity is forward and thus termed as forward cg condition, similarly column three indicates an aft cg condition. Column four indicates a steady state turn condition and column five indicates a steady state pull up condition both with a forward cg. It can be seen that aft cg condition normally de stabilizes the aircraft movement. These steady state flight conditions are used as the conditions and named as C 1 -C Pitch Rate Augmentation Schemes for F16 Aircraft Normally when the aircraft is under manual control (not in auto pilot mode) the only automatic control needed is the stability augmentation control. However, in the case of high-performance military aircraft where the pilot may have to maneuver the aircraft to its performance limits and also to perform tasks such as precision tracking of targets,

82 65 specialized control augmentation systems are needed. These control systems provide the pilot with selectable task tailored control laws which enable for maneuvering the aircraft easily. One common mode of operation of such control augmentation in the pitch axis is called pitch-rate command system. When the situation requires precision tracking of a target, a deadbeat response of the pitch-rate command is well suited to the task. This control approach is also suitable for approaching and landing. The Pitch Rate Augmentation scheme for F16 aircraft is utilized to test the intelligent adaptive controllers throughout as such a control definitely calls for precision target tracking at various operating conditions of the aircraft. In this controller, a pitch rate command is tracked by the system output changing the input elevator controller deflections keeping other controller deflections un attended and maintaining it at the trim conditions set point or dynamically changing them for structural impaired ness Application to Position Tracking without Structural Complexities On all the flight trim conditions mentioned above, suitable reference models are developed and detailed in the Chapter 3. They are discussed when applied to test the developed intelligent control algorithms. Before creating some structural complexities to enable the developed models for testing the higher level control a sample case of pitch rate command tracking in the longitudinal axis with a conventional model reference adaptive controller are as shown in Figure The controller changes the elevator and keeping all the other control deflections unchanged. The pitch rate command, which needs to be tracked, is as shown in Figure 2.9. As can be seen, the adaptive controller with the suitable reference model tracks the pitch rate very well.

83 66 Figure 2.9 Pitch Rate Command Pattern Figure 2.10 Output Pitch Rate Response with Adaptive Controller Development of Structural Complexities and Challenges This section shows how this model is challenged with structural complexities. There are mainly three areas, which are chosen for incorporating these complexities.

84 67 1. Random changes in the controller deflections other than the elevator (viz., rudder and aileron). 2. Changes in the throttle value throughout the time period in a random fashion. 3. Application of different pitch rate command pattern requirement The changes are developed such that the nonlinear aircraft simulation model shows the structural challenges such as mode switching unmodeled dynamics, sudden Jumps and internal and external disturbances. All these structural impacts are discussed when testing the intelligent controller on each dynamic maneuver in respective chapters Application to Position Tracking with Structural Complexities At this point it is necessary to show that the conventional adaptive controller is insufficient to track the pitch rate in presence of the structural complexities. To this effect a sample demonstration is illustrated in figure 2.11 which shows the inability of the adaptive controller to track the pitch rate command in presence of the structural complexities incorporated into the model as shown in table 2.5. All other conditions are also tested in the same manner and similar results are obtained. Thus it can be concluded that the simulated model is ready to be used to test the higher level controllers with all the challenges embedded into it. Table 2.5 Initial Control Vector, Throttle Values and Command Patterns U 1 = Elevator U 2 = Aileron U 3 = Rudder *10^-7 6.2*10^-2 Throttle Value in Percentage Time (sec) < 10 < 20 < 40 < 100 <150 < 180 Pitch Rate ( deg/sec) Throttle (%) Control Vector Pattern Time (sec) < 10 < 20 < 40 < 100 < 150 < 180 Aileron (deg) -1.2*10^-2-1.2*10^-2-1.2*10^-2-1.2*10^-2-1.2*10^-7-1.2*10^-7 Rudder(deg) 6.2*10^-2 6.2*10^-2 6.2*10^-2 6.2*10^-2 6.2*10^-7 6.2*10^-7

85 68 Figure 2.11 Structurally Changed Pitch Rate Response with Adaptive Control Law 2.3 Conclusions The development of detailed simulation models for two important challenging dynamic systems, which are directly relevant to the need for the new generation of adaptive control, has been presented. MATLAB simulation of the two system models have also been developed and used throughout this project for investigating the feasibility and effectiveness of the proposed intelligent controller schemes. These simulation models have been developed with the high degree of modeling details and with real practical data in order to generate realistic structural as well as dynamic challenges for feasibility investigation of the proposed techniques. Provisions are included to introduce a wide range of internal and external changes to system signals and functions as necessary to simulate different study scenarios and cases. The nonlinear single link robotic manipulator model has been developed based. A traditional model reference adaptive control law with suitable reference model is used

86 69 to change the input voltage value of the DC motor model such that the position output changes. This is then used to track a desired trajectory to test the ability of the adaptive controller and it is found to be effectively controlling the system. In order to create the structural complexity, this model is impaired by some system changes internally as well changing the load torque and frictional torque. It was well observed that the traditional model reference adaptive controller fails to control the system in all developed scenarios indicating a need for higher level adaptive control. A detailed 6 DOF F16 fighter aircraft model has also been developed based on the physical system wind tunnel data developed in [52] and the modeling detailed described in [55]. The traditional direct model reference adaptive control law was used to control the elevator deflections for each of the trim conditions developed using the model and the output pitch rate response is judged when compared to the pitch rate command pattern. In order to achieve good results, a suitable third order reference model is established at each of the trim conditions. It was demonstrated that the adaptive control law performs well in all the conditions. Further, the structural complexities and challenges are incorporated in this simulation model, by changing the unattended controller deflections, throttle value pattern and the pitch rate command pattern. This is done to create the propositions such as unmodeled dynamics, sudden Jumps and scheduled and unscheduled mode switching. The adaptive controller performance was then investigated in presence of such model disturbances and it was thus established that the controller fails to perform when used with a single reference model showing the multimodality and also the other structural disturbances.

87 CHAPTER THREE FUZZY MULTIPLE REFERENCE MODEL ADAPTIVE CONTROLLER In this chapter a novel fuzzy logic based multiple reference model adaptive controller schemes for multimodal and dynamic Jump system is proposed. The proposed scheme consists of a fuzzy logic switching method within the Model Reference Adaptive Control (MRAC) framework without using any explicit identifier. The switching scheme is used for generating appropriate reference models on line so that effective overall performance of the adaptive controller is achieved. The scheme is based on Takagi-Sugeno [56] fuzzy system and produces a soft way of generating the reference model, combining a group of weighted reference models effective at each modal operation which can therefore be implemented as a Fuzzy Multiple Reference Model Adaptive Controller (FMRMAC). Following a rule base, the fuzzy switching scheme effectively monitors changes in plant operating conditions and mode changes due to any sudden Jumps in the plant. A fuzzy inference engine then fires appropriate rules, which gives fuzzified output values. Defuzzification is then performed on line, monitoring the plant auxiliary states or derived measurements. The main contribution of such approach is that it can be performed online and is very well suitable for applications that show sudden movements viz. Jumps in the plant operating conditions. 70

88 71 Unlike static multiple model algorithms for switching (non-interacting individual modelbased filters) or switching dynamic algorithms (susceptible to numeric overflow), this scheme provides an interactive multiple model environment with soft switching. The scheme is computationally feasible, effective and efficient. Further this method can be enhanced by additional learning strategy to modify the rule base depending on the expansion of the plant operating range. Moreover, due to its ability to functionally represent the modes at each control interval from the combination of the modes obtained based on the developed rules; the scheme is shown to yield a fault tolerant controller. Following the details of the proposed scheme, simulation results of application to a generic linear Jump System, as well as to two practical examples of multimodality systems will be presented. 3.1 Formulation of the FMRMAC Scheme In this proposed scheme, the Intelligent Supervisory Loop (ISL) is incorporated into the traditional MRAC framework by utilizing a fuzzy logic-switching scheme in order to generate the plant reference model at every control interval. The theme behind the approach can be realized using Figure 3.1. As illustrated, if the operation of a system under consideration can be modeled using one reference model, then the adaptive controller can be utilized for controlling the system to track the desired reference model output. It is well known that the model reference adaptive controllers can very well control the system with the adaptation mechanism adapting the control law based on the output error. However, if the reference model is rigid and the system moves from the operating point far from this reference model then the controller will be stressed and

89 72 finally fail to control the plant. The main purpose of the intelligent module is to alleviate this rigidness in the reference model. Adaptive Controller Input Signal Control Law Adaptive Mechanism Error System Under Consideration Plant Output Reference Model Reference Output Fuzzy Reference Model Generator Intelligent Supervisory Loop Figure 3.1 Proposed way of Intelligent Supervisory Loop The intelligent module looks at the plant output and the auxiliary measurements/states and generates an appropriate reference model at each control interval, and thereby provides a moving reference structure with respect to the plant without losing its desired characteristics and becoming unstable. Thus it acts as an ISR, supervising the plant controller closed loop A Multiple Reference Model Adaptive Control Concept Let a system be characterized by n reference models which conveniently fits in the range of system parametric space S. This n reference model can be thought of n

90 73 subspaces in the predefined domain N. Suppose the plant parametric variation can be captured fully by these reference models pertaining to that domain. For every parametric vector change {(a i ) i1-n, (b i ) i1-n } there exists a subspace in the n finite reference model ranges which fully or partly determines the plant parametric change. This concept is outlined in Figure 3.2. If any one particular reference model in a spanned subspace can describe a plant desirable performance at one of the plant specific modes, such a subspace partitioning is referred as hard partitioned. However, a real system operation often exhibits continuous transitions from one subspace to another, and at the same time, it is not normally desirable to switch the subspaces or modes on a prespecified fashion abruptly. Therefore, there is a need to smoothly change the reference model along with the plant dynamics in order to intelligently control the plant. This is often effective along the imaginary boundaries of the subspace where the hard switching of one reference model to another deteriorates or makes the system unstable. Thus for a system which switches or jumps its operating modes in a scheduled fashion or at any unforeseen circumstances, the Multiple Reference Model Adaptive Control will be more effective than its single reference model counterpart [54]. : : Subspace n Subspace a. Subspace b Predefined Domain N Figure 3.2 Domain of subspaces with different parametric values

91 The FMRMAC Approach The proposed Fuzzy Multiple Reference Model Adaptive Control [FMRMAC] design approach is illustrated using figure 3.3. Fuzzy Multiple Reference Model Auxiliary Inputs & Output f( x) = r ri µ i 1 T = M ϑ= Φ* ϑ r µ i= P i i= 1 y mi ( t) = ( Φ * ϑ / R )* S( t) i i m y mi (t) + Command Signal Θ = T T [ θ θ ] T k 0 1 θ 2 - Σ ui T = Θ ω i yi ( t) = Ai X ( t) + b U ( t mode( t) i mode( t) y i (t) Adaptive Controller Multi Modal System Figure 3.3 Fuzzy Multiple Reference Model Adaptive Controller The scheme consists of a model reference adaptive controller that derives its control law using a fuzzy reference model. The generation of the reference model is initiated by a fuzzy switching algorithm, which performs online at every time instant. Therefore, the main ingredients of the proposed scheme are direct MRAC, multiple models and fuzzy switching algorithm. The following sections present the technical design details of the scheme. At first the overall problem statement is developed for the case of a system, which shows multimodality. Then the three important aspects in the

92 75 overall structure; the direct model reference adaptive control, the fuzzy switching scheme, and the reference model generation are discussed. The system to be controlled will have input U and output y i. The objective is to make the control error e c = y mi -y i tends to zero asymptotically, where y mi is the output of the reference model at a specific mode and y i is the corresponding system output. Let the system be represented in state space form by y i ( i + mod e( t ) imod e( t) t) = A X ( t) b U ( t) (1) Where: y i (t) is the plant output at a specific mode, U(t) is the control Input, X(t) is the state vector [X 1 (t) X n (t)] T Є R n A i (t)=[a 1i,a 2i,.,a ni ] T Є R nxn and b i takes values from the set of H constant elements as indexed by subscripts i and i Є {1,2,,H}. Thus, the parameter vector can be represented by the triple {(A i,b i,c i ),..,(A H,b H,c H )}, which changes its values depending on the modes of operation. Let the above set denote the scheduled jump parameters for each mode specified by the parameter index i. The mode variable mode (t) takes the form mapped into any of the values in the domain i Є {1,2 H} and correspondingly A mode(t) and b mode(t) are time varying. The mapping of mode (t) be denoted by mode (t) = γ[x (t-d)] where d represents the time delay. The above-mentioned system is a spatial multimodal type because the dynamics are scheduled through the states with a nonlinear mapping γ. As the system mode

93 76 changes, the system parameters jump or move drastically, which in turn may cause unstable performance if a conventional adaptive controller is used. A structural change in the reference model is required if there is no explicit identification procedure available for the plant. This is accomplished in the proposed approach through fuzzy logic multiple reference model generation, which in turn allow direct model reference control law to operate effectively under different plant operating conditions. Direct Model Reference Adaptive Control The main adaptive control framework of the scheme is the direct model reference adaptive control, as outlined in Figure 3.4 and can be formulated as follows [57]. Consider an unknown linear SISO plant described by the following differential equations x y p p = A = h p T p x x p p + b p u (2-3) Where: u is the input, y p is the plant output, x p is the n th state vector, Ap is an n x n matrix A reference model that represents the desired trajectory may be described by the following differential equations: x y m m = Ax = h T m x + br m (4-5) The aim is to find u such that the output error e = y p y m (6) tends to zero asymptotically for arbitrary initial conditions, and an arbitrary piecewise continuous and uniformly bounded reference signal r. The control structure for such scheme is

94 77 T u = Θ ω (7) T T T Θ = is the control parameter vector and ω = [ ω 1 ω ], T T where, [ ] T k θ 0 θ1 θ 2 r y p 2 is the regressor vector. The regression vectors are updated online based on the following equation Figure 3.4 Schematic Diagram of Direct Model Reference Adaptive Control ω Λ + Lu (8) 1 = ω1 ω 2 = Λω 2 + Ly p (9) Where Λ is a stable matrix of order ( n 1) x( n 1) such that the determinant si Λ = Z (s) and the vector L is defined as L t = [0 0 1]. Further, the control signal m u is structured as k = sgn( K p ) er (10) θ = sgn(k ) (11) p ey p θ = sgn( ω (12) T T 1 K p ) e 1

95 78 θ = sgn( ω (13) T T 2 K p ) e 2 Thus in the direct MRAC approach, equations 2 to 13, are used in the control process. It is also seen from these equations that the control law is adjusted using the adaptive adjustment mechanism in response to plant parametric changes. In this sense any parametric uncertainty is controlled effectively by this scheme. However, when there is a drastic change in the plant characteristics, this framework of the MRAC will most probably fail to control the plant efficiently. This is mainly because the plant exhibits different modes of operation, which are often referred as multi modality [14]. If appropriate reference models for the plant at these modes are provided, then reasonable reference model tracking can be achieved under a wide range of operating conditions. Selection of the Multiple Reference Models This is the proposed design procedure for the selection of the multiple reference models and the subsequent generation of Fuzzy Multiple Reference Models (FMRM), based on the system operation at different operating region. To select the suitable multiple reference models, first the order of the reference model under consideration needs to be selected. Further linear system model under steady state conditions should be developed. Once the linearized model is obtained overall closed loop system performance has to be assessed with the help of say for example PID controller. The stabilized system response will contain the behavior of the system in the specific operating point. The reference model at that operating point can then be developed based on this closed loop response. This process of developing the reference model has to be done at each of the operating points or steady state trim conditions in order to develop the different mode

96 79 reference models. Using these suitable reference models for the different well defined modes, the FMRM scheme is then developed using the plant auxiliary variables, a fuzzy structure (function) and an appropriate switching method, as given below. Development of the FMRM Scheme Consider a fuzzy system output denoted by a function f (Ω) and represented as f ( Ω) = r r i i= 1 r i= 1 µ µ i i (14) The above mentioned fuzzy system has r rules and µ is the membership function of the antecedent of i th rule given by the input Ω Where: m Ω R Ω is a vector containing the relevant auxiliary states/developed measurements, Assume that this fuzzy system is constructed in such a way that the µ 0 for all relevant auxiliary states and the parameter r i is the consequent of i th rule. Then the abovementioned function can be written in the parameterized form as follows, i r i= 1 i f ( Ω) = r i= 1 r µ i i= 1 r µ i i = M T Pϑ = Φ * ϑ (15) Where: M 1/ χ1( Ω) 1/ χ 2 ( Ω) = : 1/ χ m 1( Ω) m Ω R, p1,0... p1, m 1 T P = : : : and p r,0... pm 1

97 80 [ µ µ r] T ϑ = and M T P r 1... i= 1 µ i Φ = Thus the function approximation by fuzzy scheme is equal to the product of a parameter vector Φ and weight matrix ϑ. Now let us consider the reference model in state space form by m ( t) = Ammod ( ) X ( t) + e t m bmmod e( t) S( t) X (16) and y mi (t)=c T X m (t) Where: y mi (t) is the reference model output and S(t) is the command signal The reference model in the transfer function form will then be W ( s) = y ( t)/ S( t) = K * Z / R (17) mi mi mi mi mi From Equation (17), it can be seen that the numerator and the denominator are functions of state variables X and the location of these poles and zeros are further influenced by the modes of operation of the plant. In order to include these modal transitions, equation (17) has to be combined with equation (15). This can be rewritten as W s) = f ( Ω) *( K * Z / R ) (18) m ( i mi mi mi Thus the changes in the system dynamics can be mapped through auxiliary states to the changes in system polynomial roots or the poles/zero combinations. Considering the above facts reference model transfer function is written as a function of the fuzzy logic output, which is y m i T ( t) = ( M Pϑ )* Wmi* S( t) ( Φ * ϑ )* Wmi* S( t) (19) i i i i i

98 81 Thus for a constant command signal, the output m (t) will be y t) = ν ( Φ, ϑ, W ) (20) m ( i i i m i Where: y i y m i (t) is the reference model output for the i th mode, Φ i is the parameter vector developed by the fuzzy system depending on the system operating points, ϑi is the membership function weights and W mi is the corresponding reference model transfer function. It can be seen that the membership function weights act as a performance index function in modifying the reference model output. Based on each modal transition of the system, the parameter vector Φ * ϑ, which is the fuzzy system, output changes. Subsequently the reference model output changes such that the closed loop system provides a stable output with the roots on the left half S plane. This movement in the reference model secures the system from becoming unstable. Moreover, the modal transitions are smooth in nature, which reduce the transients during the system mode changes. More importantly, as the changes in the fuzzy parameter vector Φ generate the reference model, this method avoids a pre defined calculation like a performance index as in the case of the identified plant model approaches as the appropriate fuzzy rules firing selects an appropriate reference model. The time required for the calculation and settlement of the parameter values is minimum. This helps the control parameter identification and adaptation to perform efficiently toward plant environment changes.

99 82 Two Proposed Fuzzy Switching Methods: In order to develop the knowledge base, the system input/output measurements are performed for each of the steady state operating regions, and using those as the inputs, the rule base is developed. These rules should fire appropriately to get the suitable reference model for the entire operating range. To achieve this task, the number of inputs, membership functions, defuzzification and outputs are selected. It is worth noting that in all this, the switching scheme is dependent on a reasonable a priori knowledge of the system operating conditions and range of modality changes. Two methods of designing the fuzzy switching scheme are presented next based on the two different methodologies to develop stable desired closed loop system trajectories. The fact that the closed loop system trajectories pattern changes with change in the plant operating modes gives us a relation between desired trajectory and plant operating mode. This relationship is exploited to develop a fuzzy reference model scheme. a) Method 1 In this method the fuzzy logic rules are developed in such a way that the system closed loop performance at each mode is stable by changing the natural frequency of the reference model. This will provide the dominant poles and zeros of the reference model transfer function at each operating point. For this purpose, consider the reference model transfer function as in (21), which shows the desired system response. G( s) W mi ( s) = (21) 1+ G( s) H ( s) Let p( s) n( s) G ( s) = and H ( s) = q( s) d( s)

100 83 Where: p(s), q(s), n(s) and d(s) are polynomials in S. Then the closed loop transfer function will be W mi K( s + z1)( s + z2 )...( s + zm ) ( s) = (22) ( s + p )( s + p )...( s + p ) 1 2 n Considering that the poles consist of real and a pair of complex-conjugate pair, the closed loop response of the system to a unit step input will then be, W mi ( s) = q s K ( s + p ) m i= 1 r i j= 1 k = 1 ( s + z ) ( s 2 i + 2ς w s + w k k 2 k ) (23) Equation (23) represents a pair of complex conjugate poles, which yields a second order term in s for each set of complex pole pairs. Thus the location of the poles and/or zeros of the closed loop system is changed depending on the system operating modes looking at the dominant closed loop poles. All the closed loop poles mentioned in (23) play an important role in the system transient response. The one, which has the dominant effect, is termed as the dominant closed-loop poles. Let these dominant complex poles for k=1 be s 2 + 2ς ω s + ω Where: ω 1 is the natural frequency and ς 1 is the damping ratio From (23), the roots of the characteristic equation for the dominant poles being S, S = ς ω + / ω ς 1 (24)

101 84 Equation (24) represents the closed loop system desired response. By changing these roots it is obvious that the desired system response can also be changed. In this approach, at each mode of system operation the desired closed loop response with the plant and the controller in the form of ς and ω is obtained first. Further reference model roots are developed to match with the corresponding roots of the characteristic equation. Thus equation (21) can be rewritten as W m i ( s) = Ψ(, ς ) (25) ω n Combining equation (20) and (25) we get y m i ( t) = Λ( Φ, ϑ, ω, ς ) (26) Where: i i n λ = [, ϑ, ω, ς ] is the input parameter vector Φ i i n However, from equations (14) & (15) it can be seen that Φ, n i ϑi is dependent on the system auxiliary states Ω. Thus the proposed fuzzy relationship give the parameter vector λ as the output and rule weight set based on the system auxiliary states. Thus, in this method a fuzzy system approach has been formulated to realize this nonlinear relationship. Fuzzy decision rules are prepared based on the auxiliary inputs and the outputs are the reference model variable parameters λ. The inputs are the command signal and an auxiliary output of the system, and the decision is preformed by fuzzy rule, which has the following form: R i : IF x is A i AND y is B i THEN z is C i Where x and y are the inputs, z is the output and the subscript i indicate the i th rule. Each of the fuzzy rules thus has two parts. The IF part which is called the premise and the THEN, which is termed as the consequent part. The former is used to describe the

102 85 system within certain mode, which then trigger certain fuzzy rules. Corresponding to each of these modes the fuzzy parameter vector Φ and membership function weights ϑ i will be changed. Subsequently, the crisp set of output vector λ is established. This process is continued for each time instant. i b) Method 2 The second method is inspired from [58], where the desired feedback gain matrix is formulated for each mode of operation such that the eigenvalues and eigenvectors of the original closed loop system are recovered. This gain matrix will serve as the static control gain for each of the system operating points under consideration. Corresponding to this gain matrix, the desired closed loop system is derived. This acts as the desired reference model structure for each mode of operation. Further, these poles and zeros, which are adjusted based on eigenvalues for each mode of operation, act as the fuzzy outputs where the system auxiliary states comprise the input vector. Suppose there are H modes of operation. Thus for each mode of operation, the closed loop system gain matrix will change and correspondingly the desired system response is identified with corresponding active poles and zeros. Based on that fuzzy rules have been developed as follows R i : IF x is A i AND y is B i THEN z is C i1. C in & D i1. D im Where: 1to n corresponds to the number of poles based on the roots of the characteristic equation and 1to m corresponds to the number of active zeros,

103 86 It has been shown in [58] that considering a system, which has undergone large variations due to some failure, or operating mode change, a new system model can be represented as in equation (27). With a reconfigurable control gain matrix an output gain matrix K can be established such that the maximum number of closed-loop eigenvalues of the reconfigured system is the same as that of the original system. Mathematically it is rewritten as m i λ = λ( A + B K C ) = λ (27) m m m m i Where: λ i, is the original eigenvalues Thus the gain matrix will satisfy m m i i m m m m i λ v = ( A + B KC ) v (28) Thereby the eigenvalues and corresponding eigenvectors of the original closed loop system can be recovered even when the system operating modes change or a failure occurs. Instead of synthesizing the gain matrix K, in this method the reference model is moved such that the v m i is close to the original eigenvector v i' (the corresponding closed-loop eigenvector of the original system) as possible. Based on these, a new reference model structure is developed at each control interval by changing the roots of the characteristic equations. Once these offline studies are conducted the corresponding reference model changes at each mode are established and the fuzzy system is developed based on recovering these poles and zeros. For both methods one and two above, it is to be noted that the extraction of the dominant closed loop poles (method 1), and the regeneration of the eigenvalues and

104 87 corresponding eigenvectors (method 2), are part of the offline analysis. However, the following steps are used for developing the fuzzy inference engine: 1. Prepare the closed loop response of the system either, by calculating the natural frequency and the damping constant or, by finding the eigenvalues and vectors for each mode of operation of the system. 2. Find the natural frequency and damping ratio (method 1) or the corresponding poles and zeros from the eigenvalues (method 2) of the modes. 3. Develop the fuzzy inference engine using the command signal and the auxiliary states as the premise and the outputs corresponding to the values as in step two by projecting the input corresponding to each mode. Based on the projection set up the number of rules, which should fire for each mode corresponding to an input output mapping. 4. Establish the membership function weight matrix for each mode. 5. Depending on step 4, establish the membership functions for each rule and create the knowledge base. 6. Test the condition offline for the operating modes in order to retain the desired closed loop poles of the system An Important Fault Tolerant Control Feature of the FMRMAC Consider the case that known faults are used to develop certain modes of operation. Further, let us assume that nominal conditions are the rest of the modes. If a case occurs that at some instant, certain mode occurs which is a combination of faulty modes and a nominal condition, and the reference model developed by the fuzzy system

105 88 is a fuzzified combination of models in the model set M. Let each mode be represented by a set of fuzzy rules. Then the model set will be the combination of the each model shown in equation (29) M = N i= 1 µ im i (29) The membership function weights ϑ i in equation (15) represent the combination of fuzzy outputs of mode transition, which will in turn affect the development of model combination at each time instant. In this sense, equation (29) is exactly represented by the output of the fuzzy logic system. Considering the proof [59, 60] that if the controllers at the vertexes are stable, the states are developed, and the control is applied on the basis of these states, then the controller will also be stable. In the proposed scenario the membership function weights act as the probabilistic function of each model and the controller used is of the stable direct model reference adaptive controller type. Thus from (29) and applying the design aspects in [59, 60], the proposed scheme is fault tolerant in nature. 3.2 Application to a Generic Linear Jump System The proposed FMRMAC scheme is first tested on a linear Jump System artificially creating the changes in the transfer function of a linear plant in S domain. For simulation purposes, consider a system with the transfer function s s 10. This is a second order system with unstable open loop characteristics. A stable direct model reference adaptive control law is developed to control this system as derived before. Considering a suitable reference model the system response for a unit step command is

106 89 observed. It is identified that the plant tracks the command signal with acceptable error for a specified time. The initial condition vector of the parameter θ is changed to destabilize the controller initially. Thus the error shows an initial variation and settles down subsequently. This stable adaptive control acts well for this normal plant operating range. Comparing the error patterns with two best references s 2 5 and + 10s + 25 s 2 5, + 4s + 4 it was observed that the tracking error is better with the reference model s s + 25 [61] Creation of Modes In order to show the plant parametric changes and the effect of reference model structure in tracking, three cases have been performed on this theoretical system and the corresponding fuzzy logic-switching scheme is developed based on method 1. The details of each modes of the plant for these cases and the reference model output by the fuzzy system are as shown in table 3.1. Table 3.1 Plant parametric change (Case1, Case2 and Case 3) Cases Time T < 40 T < 70 T <100 1 Plant Structure 1/ s s 10 1/ 2 + 3s / s + 9s 1 Ref. Model from FLSS 2 5 / s s / s / s s + 2 Plant Structure 1/ s 2 + 9s 30 1/ s / s + 3s 2 Ref. Model from FLSS 2 5 / s s / s / s s + 3 Plant Structure 1/ s s 20 1/ s / s + 9s 3 Ref. Model from FLSS 2 5 / s s / s / s s + s 30 s s 30 s s 30 s The proposed fuzzy logic switching in this example consists of two inputs, two outputs and nine rules. The basic structure of the switching scheme is as in figure 3.5. Initially a rule base has been developed based on the performance of the system under study. The rule base in this example consists of three input linguistic terms in the form of Small (S) Medium (M) and Large (L). In order to show the effect of the proposed scheme, both the inputs are taken directly as the plant parameter values. In physical

107 90 process these linguistic inputs will be plant outputs or auxiliary parameters. The output values are the parametric changes in the reference model directly. Figure 3.6 shows the functional relationship of the inputs with each of the outputs, which demonstrate the functional relationship with plant structure, and that of the reference models. The input and output membership functions are as shown in figure 3.7. Input functions are trapezoidal and the outputs are triangular. There are nine rules, which generate two output values. Table 3.2 shows the set of rules for each of the outputs. These rules have been developed analyzing the simulated response of the system. Further min operation implication method and centroid defuzzification method has been employed to generate crisp values from these fuzzy outputs. Figure 3.5 Structure of Fuzzy Logic System for the Example Table 3.2 Rule Base for each of the Outputs a0/a1 (Output1) Small Medium Large Small Medium Vlarge Vlarge Medium Medium Large Medium Large Medium Large Large a0/a1 (Output2) Small Medium Large Small Medium Large Medium Medium Small Small Medium Large Medium Small Small

108 91 Figure 3.6 Functional Relationship for inputs to each output Figure 3.7 Membership Function Details

109 Simulation Results and Conclusion The simulation is carried out for three cases by changing the transfer function modes of operation and by adjusting the parameters as shown in table 3.1. First step was to find out the effect of the previously diagnosed best reference model s s + 25 during these cases to find out whether this reference model still acts as the best one. Figures 3.8 indicate the error patterns with reference models s 2 5 and + 10s + 25 s s + 4 respectively. It can be seen that unlike as observed before with normal operating conditions the reference model with structure s s + 4 shows improved error pattern. This proves in order to track those parametric changes effectively the model structure has to be changed. The whole process of changing the reference model can be thought of hard switching in which the structure of the reference model is switched knowing the plant parametric change and as well moved to a completely new value. Thus, this fixed reference model is used for performance comparison in all the cases. Case 1 In the first case, as in table 3.1(Case 1), the newly found best-fixed reference model s s + 4 is utilized and the performance of the transfer function in this mode is assessed. The next step is to implement the fuzzy switching scheme, which has been developed earlier. In order to change the reference model structure softly or smoothly online, the plant parameters are used as the fuzzy inputs in this case and the corresponding parametric values of the reference model structure is obtained from a fuzzy logic scheme every time instant. Table 3.1 also shows the reference model structure

110 93 obtained from Fuzzy Logic Switching Scheme (FLSS) during each plant parametric change. It can be seen that the FLSS changes the reference model structure based on fuzzy rules fired appropriately. Figure 3.9a shows the error pattern comparison between FLSS and fixed reference model for the time period (39-44) seconds with an error scale ranging from ( ) while switching occurs. Case 2 In this case the plant parameters have been changed as shown in table 3.1(Case 2). Initially the previously identified best single reference model has been used to represent the whole operating change and the performance has been assessed. Further, proposed FLSS is applied to determine the changes in the reference model structure and analyze the performance. Table 3.1(case 2) also shows the fuzzy outputs in the form of the reference structure. Figure 3.9b shows the error pattern comparison for a time from (69-74) seconds and the error scale ranging from ( ). It is observed that the parameter values of reference model have been changed during the time span and provides better tracking by FLSS when compared to single reference model. Case 3 In this case the plant pattern shows a complete change in the structure as can be seen from the table 3.1(Case 3) including the FLSS outputs in the form of reference model structure. Figure 3.9c shows error pattern comparison between FLSS and the single fixed reference model. Here only the switching instant around (66-75) seconds has been shown as a comparison. The error scale ranges from ( ). It is observed that

111 94 the proposed scheme was able to switch the reference model structure effectively in all three cases. It can also be seen that the error have reduced to more than half in Case 1 and Case 2 while by one tenth in Case Figure 3.8 Error Patterns with Reference Models 5 / s + 4s + 4 and 5 / s + 10s + 25 Thus the proposed Fuzzy Multiple Reference Model Adaptive Control produces an effective and efficient way of reference model switching. The scheme provides a functional relationship with the plant and reference model structure, creating a multiple model adaptive control environment especially during drastic plant parametric change. This test also confirms that the scheme can be used for scheduled switching in which certain auxiliary inputs are monitored to keep track of unforeseen changes along with the plant output.

112 Figure 3.9 Error Comparison 9a) Case 1, 9b) Case 2 and 9c) Case 3 95

113 Application to the Robotic Manipulator Position Tracking After assessing the ability of the scheme on the previous generic theoretical example, it is tested for the control of position tracking on the single link manipulator dynamic nonlinear model as described in Chapter 2. This is a practical system, and thus the objective will be to utilize plant auxiliary parameters or derived variables as the input to the fuzzy system unlike as tested in the theoretical example before. Moreover, varying a physical parameter the Load Torque as opposed to varying the coefficients of the plant transfer function performs the creation of mode changes. Thus the case studies performed are termed as load torque patterns. In order to facilitate the Fuzzy Multiple Reference Model generation in this case, the fuzzy logic scheme is formulated using method 1 discussed and based on offline studies. The switching scheme consists of two inputs, one output and twenty-five rules. The rule base for this case study consists of five input linguistic term membership functions in the form of Very Small (VS), Small (S), Medium (M), Large (L) and Very Large (VL). These inputs are auxiliary parameters such as the tip load and a 1. The output values are the changes needed in the natural frequency of the second order reference model structure ( ω ), which is generated by the rule base. These rules have n been developed analyzing the simulated response of the system. Further min operation implication method and centroid defuzzification method has been employed to generate crisp values. The reference model representation has the following specific form, 2 n ω Wm ( s) = (30) 2 2 s + 2ςω + ω n n

114 97 Table 3.3 Rule Base (Flexible Single Link Robotic Manipulator Control) Load Torque ω n VS S M L VL VS S S M L S S S S M L S M L S M L S L S S M L VL VL S S M L VL a 1 In this example the value of ς is set to 0.7. The value of ω will be evaluated every control interval, dependent on the two auxiliary inputs mentioned above. The first step is to determine the range of the inputs and outputs. By simulating and studying the process it was found that for the range of load torque [0,24], the closed loop system response was best by keeping the range of ω [0,12]. The process of dividing these ranges into fuzzy n membership function and the input and output membership functions are as shown in figure After the division of the membership function, the rule base was created depending on the system operation modes, which is divided into five operating region of VS, S, M, L and VL. The rule base created for this specific example is shown in Table 3.3. Considering the proposed approach for the following manipulator tip load, the fuzzy mapping is thus used for the generating multiple reference models Simulation Results and Discussion n Several cases have been conducted to assess the effectiveness of the proposed scheme to track the desired command signal in case of variation in the load torque of the manipulator tip arbitrary at different times, of which two most important ones are presented next. In each of these cases, tables show the proposed variation in the load torque. Further the output angular position for the desired time span is plotted for traditional MRAC and proposed scheme along with the reference models output.

115 98 Correspondingly, the angular position error with the output of the reference model and the variation in the fuzzy output ( ω ) in case of the MFRMAC is also shown. Please note that on all cases the ω for the single reference model is kept as five. n n VS S M L VL µ(load Torque) Load Torque VS S M L VL µ(a1) a 1 VS S M L VL µ(ωn) ω n Figure 3.10 Input-Output Membership function details

116 Simulation Results and Discussion Several cases have been conducted to assess the effectiveness of the proposed scheme to track the desired command signal in case of variation in the load torque of the manipulator tip arbitrary at different times, of which two most important ones are presented next. In each of these cases tables shows the proposed variation in the load torque. Further the output angular position for the desired time span is plotted for traditional MRAC and proposed scheme along with the reference models output. Correspondingly, the angular position error with the output of the reference model and the variation in the fuzzy output ( ω ) in case of the MFRMAC is also shown. Please n note that on all these cases the ω for the single reference model is kept as five. n Tip Load Pattern 1 In this Case the tip load of this manipulator is changed at different times, as in table 3.4. Table 3.4 Tip Load Variation (Case 1) Time Range (sec) Load Torque (Nm) Figure 3.11 shows the position trajectory plot and the output of the robotic arm with the proposed scheme and the single reference model adaptive controller. Figure 3.12 show the position error pattern and the change in ω at every control interval for the proposed scheme. It can be seen that the single reference model adaptive controller is clearly unstable and was unable to control the system using the fixed reference model. n

117 Tip Load Pattern 2 In this case the tip load of this manipulator is changed at different times, as in table 3.5. Figure 3.13 shows the position trajectory plot and the output of the robotic arm with the proposed scheme and the single reference model adaptive controller. Figure 3.14 shows the position error pattern and the change in ω at every control interval for the proposed scheme. In this case also the single reference model adaptive controller is unstable and was unable to control the system using the fixed reference model. Table 3.5 Tip Load Variation (Case 2) Time Range (sec) Load Torque (Nm) n

118 101 Figure 3.11 Case 1: Manipulator Position Output Comparison

119 102 Figure 3.12 Case 1: Position Error Pattern and Fuzzy Switching Scheme Output

120 103 Figure 3.13 Case 2: Manipulator Position Output Comparison

121 Figure 3.14 Case 2: Position Error Pattern and Fuzzy Switching Scheme Output 104

122 Application to the Fighter Aircraft Pitch Rate Command Tracking The proposed scheme is now applied to a pitch rate tracking of fighter aircraft. In order to show the effect of varying command patterns two different command patterns viz., (P 1 -P 2 ) are used for the tracking as shown in figure Based on these, table 3.6 shows each dynamic maneuver, which combines the conditions developed in Chapter 2 and patterns along with the actuator and sensor disturbances. The offline development strategy of the suitable fixed reference model is described next. Table 3.6 Proposed Aircraft Maneuver Nomenclature Maneuver classifications Combination of Conditions C and Patterns P Dynamic Maneuver 1 C 1 P 1 Dynamic Maneuver 2 C 2 P 1 Dynamic Maneuver 3 C 3 P 1 Dynamic Maneuver 4 C 4 P 1 Dynamic Maneuver 5 C 5 P 1 Dynamic Maneuver 6 C 1 P 2 Dynamic Maneuver 7 C 2 P 2 Dynamic Maneuver 8 C 3 P 2 Dynamic Maneuver 9 C 4 P 2 Dynamic Maneuver 10 C 5 P 2 Figure 3.15 Command Patterns Development of the Reference Model for each of the trim conditions A static approach for a pitch-rate command augmentation system is shown in figure 3.16 [55]. With this approach the reference command signal is tracked by the pitch

123 106 rate so that the controlled variable is the pitch rate. A possible method of developing a static control approach for a pitch rate command control is to develop a linearized and decoupled longitudinal model for the aircraft and developing the gains by plotting a root locus. Such a design is explained in [55]. The closed loop transfer function of such system with a step response and steady state nominal trim condition has fast rise time and overshoot of almost 20 % as shown in equation (31). This transfer function indicates that there is a pair of short-period poles at S = / j3 which is a complex pair and some additional poles at S=-10.7, S=-13.7 and S=-1.02 respectively. This information is utilized to develop a reference model for this condition as well as for other conditions ( s )( s )( s ) q / r = (31) ( s )( s )( s )( s / j303) + Reference - Σ e Controller + Σ - - F 16 Aircraft Kα α q Kq Figure 3.16 Typical Control Systems for Pitch Rate Command Tracking The reference models thus developed are as shown in table 3.7 for each of the conditions mentioned in Chapter 2, table 2.4. In order to create a multimodal system, contingencies are developed in the form of changing set point on the un-attended control. As explained in chapter 2 the control actuators are the elevator (U 1 ), rudder (U 2 ), and aileron (U 3 ) deflections and the specific one for the pitch rate control is the elevator control (U 1 ). For each of proposed dynamic maneuver the set points of the un-attended actuator deflections (rudder and ailerons) are varied throughout the time period.

124 107 Moreover, the throttle power pattern was also changed. The demonstrated cases detail these variations for each dynamic maneuver. Please note that the purpose of the MRAC controller in each case is to track the pitch rate command by controlling U 1. Table 3.7 Selected Reference Model for different operating points Conditions Variables Nominal X cg = 0. 3 C Reference Models 90s s s + 115s Ref: I 110s s s + 115s Ref: II X cg = 0. 3C X cg = 0. 3C X cg = 0. 38C ψ = 0.3rad / sec θ = 0.3rad / sec 235s s + 40s + 608s Ref: III 10s s s + 115s Ref: IV 132s s s + 115s Ref: V Further the direct model reference adaptive control with each of the developed reference models from table 3.7 at all the dynamic maneuvers with contingencies are simulated for pitch rate control. Table 3.8 Performance of Adaptive Controller and Static Controller Model Reference Adaptive Controller Cases Controller Performance with Selected Reference Models on each cases Ref: I Ref: II Ref: III Ref: IV Ref: V Dynamic Maneuver 1 Dynamic Maneuver 2 Dynamic Maneuver 3 Dynamic Maneuver 4 Dynamic Maneuver 5 Dynamic Maneuver 6 Dynamic Maneuver 7 Dynamic Maneuver 8 Dynamic Maneuver 9 Dynamic Maneuver 10 Static Controller X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Failed Acceptable Best Abnormal

125 108 The results are compiled as in Table 3.8. It is observed that the direct MRAC works really well and better than that of the traditional controller and in most cases the static controller failed. This is due to the creation of the actuator dynamics and uncertainties. However, it is observed that the MRAC controller works well with the relevant reference models as shown in the table but failed to respond with other reference models. These results gave us an interesting conclusion for the thought. The situation arises when the changing operating conditions call for movement of the system in the different modes and call for change in the control value or as we thought before the change in the reference model structure. The next step is to develop a fuzzy logic scheme and create a knowledge base so that the reference model changes can be introduced online Development of the Fuzzy Logic Scheme and Knowledge Base The proposed fuzzy logic scheme in this case is created using the design of method 2 as described in section The fuzzy structure consists of four inputs and one output. The output value is the zeros of the reference model and the inputs are dynamic pressure (dp), angle of attack (alpha), force values in the Y axis (Be) and error between the output with command signal (error). These inputs can be represented as Ω = [ d,, F, e]. The membership pattern consists of five, six, five, and three membership functions for these inputs respectively and five membership functions for the output. There are total of 90 rules, which have been developed based on the input and the membership function. The details of the input and output membership function are shown in figure Figure 3.18 shows overall fuzzy scheme in a block diagram form. p α y

126 109 Figure 3.17 Membership function for Inputs and Outputs

127 110 Figure 3.18 Overall Fuzzy Logic Structure Simulation Results The overall scheme consists of the nonlinear aircraft model along with the fuzzy switching algorithm which evaluates the change in the reference model at each time instant. Further, the model reference control law changes the elevator control value and keeping the other control vector elements unchanged initially and then changed during each case. Performance evaluation for this scheme has been conducted for each of the cases and the output values of the pitch rate and some relevant auxiliary outputs are observed. It has been noted that the proposed method shows considerable improvement when compared to adaptive control based on best-fixed reference model for each of the cases. It is also worth noting that the other reference models failed or performed badly as shown in table 3.8. Two of such cases are described here. These cases have been selected as they show two distinct observations as far as the performance based on adaptive control with reference model and the static controller.

128 111 Dynamic Maneuver 1 The details of this maneuver are as discussed above. Table 3.9 shows the initial control vector settings where U1 = δ h, U 2 = δ a and U3 = δ r. The throttle value and the initial control vector are changed during the time span of 180 seconds. The control is applied only to the elevator and U 1 is varied based on the proposed control law with the deflection limit of 50 deg. At each control interval, the auxiliary vector Ω = [ d,, F, e] is used to generate the desired reference model. The output pattern at the total time span and during switching is as shown in figure As can be seen from the figure, the pitch rate output based on adaptive control law with fuzzy reference model shows better tracking when compared to adaptive control law with best fixed reference model especially during the switching period (40 seconds). Please note that the reference model p α y used for this case is s s + 115s s from table 3.8. It should be noted that the adaptive control with the best-fixed model shows a considerable improvement when compared to the abnormal pattern indicated with a static controller. This intelligent controller shows even better values especially while switching. Table 3.9 Initial Control Vector, Throttle Values and Command Patterns U 1 = Elevator U 2 = Aileron U 3 = Rudder *10^-7 6.2*10^-7 Throttle Value in Percentage Time (sec) < 10 < 20 < 40 < 100 <150 < 180 Pitch Rate ( deg/sec) Throttle (%) Control Vector Pattern Time (sec) < 10 < 20 < 40 < 100 < 150 < 180 Aileron (deg) -1.2*10^-2-1.2*10^-2-1.2*10^-2-1.2*10^-2-1.2*10^-7-1.2*10^-7 Rudder(deg) 6.2*10^-2 6.2*10^-2 6.2*10^-2 6.2*10^-2 6.2*10^-7 6.2*10^-7

129 112 Dynamic Maneuver 10 Table 3.10 shows the initial control vector settings where U1 = δ, 2 = δ and U3 =. Unlike in the first throttle value and this initial control vector are the same δ r h U a during the time span of 180 seconds. The control is applied only to the elevator and U 1 is varied based on the proposed control law. At each control interval, the auxiliary vector Ω = [ d,, F, e] is used to generate the desired reference model. The output pattern at p α y the total time span and during switching is as shown in figure As can be seen, the pitch rate output based on adaptive control law with fuzzy reference model shows better tracking when compared with adaptive control law with best fixed reference model in this case, especially during the switching period (40 seconds). Please note that the reference model used for this case is s s + 115s s from table 3.8. Similar results are obtained for all the cases we tested in the knowledge based of the fuzzy scheme. It is noted that the creation of knowledge base is a critical factor in designing the operating regime and thereby for effective control. Table 3.10 Initial Control Vector, Command Pattern and Throttle position Initial Control Vector U 1 = Elevator U 2 = Aileron U 3 = Rudder *10^ Command Pattern Time (sec) < 10 < 20 < 40 < 100 <150 Pitch Rate ( deg/sec) Throttle (%) Control Vector Pattern Time (sec) < 10 < 20 < 40 < 100 < 150 Aileron (deg) -6.2*10^-4-6.2*10^-4-6.2*10^-4-6.2*10^-4-6.2*10^-4 Rudder(deg)

130 Figure 3.19 Pitch Rate Command Tracking Comparison Dynamic Maneuver 1 113

131 114 Figure 3.20 Pitch Rate Command Tracking Comparison Dynamic Maneuver 10

132 Conclusions A Fuzzy Multiple Reference Model Adaptive Control scheme for multimodal and Jump systems has been developed. The controller concept, methodology, and control law, as well as the design methods of the fuzzy switching scheme is established. The scheme feasibility and effectiveness have been investigated through application to a generic system as well as two important and challenging practical systems; Position Control of a single link flexible robotic manipulator and the pitch rate control of a fighter aircraft. Results of all investigation show that the proposed FMRMAC scheme performs well when both traditional non-adaptive and single reference model adaptive controllers fail. The scheme provides soft switched fuzzy reference model and is found to be stable, especially at the modal boundaries when the hard switching mathematical approach fails. Further this scheme is computationally feasible, and fault-tolerant. It is to be noted that, the following requirements are important in order to design such control scheme: Possible creation of the offline knowledge base. The creation is not dense meaning the number of rules is not large so that it is not dimensionally complex. Unmodeled dynamics of the system is bounded The output of the reference models should not deviate much from the reference signal so that the design specification and precision control is achieved. For the last two requirements, even though they can be established on many systems, the associated offline fuzzy design can be difficult and may create heavily dense rule base. If the system modal changes are scheduled, then such design will be easy as the

133 116 modes of operation are known, which will allow the designer to develop a firm knowledge base for the prescribed operating range. On the other hand, for some classes of multimodal systems it is important to consider possible further development of this scheme to meet such challenges. This is because it is difficult or even impossible for the designer to study the system under consideration off line in order to create an adequate knowledge base. The proven research contributions in using the artificial neural network to approximate any nonlinear functions enables us for thinking if this parallel distributed artificial networks can be used to dynamically inverse the plant non-linearity allowing the adaptive control approach to perform well even when there are unmodeled dynamics. Further if a neural network control can be used to work in parallel with the adaptive controller, then it may also be used to correct the reference model(s) and the referenceinput signal deviation. More ambitiously this approach may also be able to control the systems when there is mode swings with a suitably selected single reference model. In the next chapter a growing dynamic online Radial Basis Function Neural Network (RBFNN) parallel adaptive controller is presented such that the RBFNN controller augments the model reference adaptive controller. The main reason in such augmentation is to precisely track the reference trajectory, and to linearize the system nonlinearities, while taking care of the unmodeled dynamics with a single reference model, even when there are vast mode swings.

134 CHAPTER FOUR NEURAL NETWORK PARALLEL ADAPTIVE CONTROLLER This chapter presents a novel neural network based intelligent adaptive controller. In this scheme the Intelligent Supervisory Loop (ISL) is incorporated into the traditional MRAC framework by utilizing an online growing dynamic Radial Basis Function Neural Network (RBFNN) structure in parallel with the MRAC. The idea is to control the plant by a direct MRAC with a suitable single reference model, and at the same time respond to the plant multimodal dynamics by on line tuning of an RBFNN controller. The parallel RBFNN controller is designed in order to precisely track the system output to the desired command signal trajectory, regardless of system multi modality and/or unmodeled dynamics. The updating details of the RBFNN width, centers and weights are derived in order to ensure error reduction and for improved tracking accuracy. The importance of the proposed scheme is in its ability to perform effectively even when the plant mode swings without using multiple model concept or a multiple reference model adaptive controller if a suitable reference model structure can be established. Further, the parallel controller will be able to precisely track the reference trajectory even with unmodeled dynamics. Unlike the model reference adaptive control in which the plant follows the reference model output, the proposed scheme is capable of preparing the plant output to follow the desired trajectory itself. 117

135 118 Moreover, the RBFNN structure should be able to avoid the Dimensionality Problem inherent in such architecture and is dynamic and growing in nature. The chapter details the Neural Network Parallel Adaptive Controller (NNPAC) starting with the method of incorporating the intelligent supervisory scheme in this case, the parallel controller concept and the mathematical formulation. The theoretical results are then validated by conducting simulation studies based on a nonlinear dynamic system model for the position control of the single link flexible manipulator undergoing several mode swings due to the tip load variation and the pitch rate control of the fighter aircraft. 4.1 Formulation of the NNPAC Scheme It is important now to examine the method of incorporating the Intelligent Supervisory Loop (ISL), in this approach, before discussing the mathematical formulation and design details. The theme behind the approach can be realized using figure 4.1. As illustrated, if the operation of a system under consideration can be modeled using one reference model, then the adaptive controller can be utilized for controlling the system to track the desired reference model output. The Model Reference Adaptive Controller can very well control such system with the adaptation mechanism adapting the control law based on the output error. However, if the reference model is rigid and the system moves from the operating point far from this reference model then the controller will be stressed and finally fail to control the plant. Further, if there are unmodeled dynamics, then the controller performance may deteriorate or it may become unstable, as this new system dynamics will be different from the original modeling the developed

136 119 reference model. The main purpose of the intelligent module is to produce an additional control value which in-turn linearizes the system by canceling the system nonlinearities. The intelligent module in this case is a growing dynamic online RBFNN Controller. Adaptive Controller Reference Model Adaptive Mechanism Reference Output Error Input Signal Adaptive Control Law Σ System Under Consideration Plant Output Desired System States RBFNN Neural Network Controller Intelligent Supervisory Loop Figure 4.1 Proposed way of Intelligent Supervisory Loop #2 Thus if there is any unmodeled dynamics during the course of operation of the system or the system mode swings, the online growing dynamic RBFNN controller cancels the nonlinearities and create a stable linear system. Due to this, ideally there is no need to have multiple reference models and there is no need for those required conditions that were mandatory for the algorithm developed in the last chapter. On the other hand at the initial stage while the RBFNN controller is learning, the adaptive control stabilizes the system dynamics. The only criteria thus will be to have a system model known at the beginning of the control and obviously should not have any unmodeled dynamics at that time. The intelligent module monitors the plant output, senses the required system states

137 120 and provides a desired control value at every time instant and thereby canceling the system nonlinear dynamics. Moreover it will generate the control value such that the output of the system tries to precisely track the desired trajectory. Thus it acts as an ISL supervising the plant-controller closed loop. The reason behind selecting an online neural network, the parallel structure intelligent supervisory loop and the capabilities of RBFNN neural network when compared to multilayered perceptrons are discussed before presenting the mathematical formulation and design details The NNPAC Concept The parallel controller originates from the fact that a neural network can be used to learn from the feedback error of the plant controller closed loop. The plant, which is assumed to undergo vast dynamic changes is, controlled by an adaptive controller that tracks the reference model output. When there is unmodeled dynamics during plant operation, or if there exists a steady state error between the reference model output and the reference trajectory, then the RBFNN controller learn and develops a controller such that this error is made to zero asymptotically. This in fact is carried out continuously such that the output error is zero and precision tracking is achieved. Thus in a parallel structure, controllers aid each other s contribution. It is worth noting that in this case, instead of moving the reference structure the additional controller augments the controller dynamics. The advantage of using the online RBFNN Network is that there is no need for any offline design and subsequently about the system knowledge. However, due to the parallel structure, initially the system needs to be kept under a stable state by the direct

138 121 adaptive controller when the RBFNN controller learns about the plant dynamics. The use of RBFNN ensures that the change in the dynamics acts linearly to the modification in the weights of the network when compared to the multilayer perceptrons. Moreover, the addition of the centers creates a growing network, which increases the nodes depending on the plant dynamics. The corresponding weights in the network will act as a memory module for each operating point such that when the plant revisits the previous operating points, these weights are activated. Thus it will learn grow and at the same time overcome the dimensionality problem The NNPAC Algorithm The block diagram of the Neural Network Parallel Adaptive Controller (NNPAC) is as shown in the figure 4.2. Suitable Single Reference Model Direct Model Reference Adaptive Controller x m y m = = a Cx m m x m + b m r y mi (t) - Command Signal Θ = T T [ θ θ ] T k 0 1 θ 2 ui T = Θ ω U mr + x ( t ) = af ( x ) + bg ( x ) U ( t ) y ( t ) = h ( x ) + y i (t) Desired States U nn + Dynamic System - RBFNN Controller + Figure 4.2 Neural Network Parallal Adaptive Controller r

139 122 The nonlinear system dynamics are represented in continuous systems by the following form. x = af ( x) + bg( x) U y = h( x) (1-2) This scheme is restricted to a case of Single Input Single Output (SISO) control, noting that the extension to Multiple Input Multiple Output (MIMO) is possible. Let the nonlinear functions a, b and h be unknown with parametric uncertainty and the output y(t) tracks r(t) when t >=0 and all other states x are bounded. The following assumptions are also made: The systems relative degree r* is known, The system internal dynamics, I(z 1, z 2 ) are Lipschitz in z 1 and z 2, the system is minimum phase that is, systems zero dynamics are globally exponentially stable, there is a lower bound for the unknown functions for all the values of x and due to the first two assumptions y=f(x)+g(x)u(t), the reference inputs and their derivatives are bounded and known, and finally, only the system output is available for measurements. In order to realize the control laws based on the model reference adaptive control, assume that the nonlinear functions f(x) and g(x) are known and the sgn (b) is known and function g(x) > c > 0 1 x R and some constant c >0. Let f(x) is bounded for bonded x. It is desired that x tracks the state x m of the reference model given by x m = am x + m bm r (3) for any bounded reference input signal r and in turn the input signal. The control law is Where U = U mr + U (4) nn

140 123 U = θ T *ω (5) mr r θ = [ k θ 0 θ1 θ 2 ] and y ω = p ω 1 ω 2 U nn = node j= (exp ( d / r ))* w (6) j j j Where d j = Nm k = 1 2 ( X C is the distance and r j and W j are radius and weights of k ( j, k ) ) each node respectively. The controller design concept is illustrated using the following second order system, which can be expanded to higher order system comfortably: x1 = x2 x 2 = g( x1 ) * U + f ( x1, x2 ) (7-8) and let the output, y = x1 Differentiating y = x1 = x 2 = g( x1) * U + f ( x1, x2 ) (9) This can also be represented as y = D x, x, ) ( 1 2 U ( 1 1 x2 1 Thus U = g x ) ( x 2 f ( x, )) (10) 1 Which is the same as U = D ( x1, x2, x 2 ) Suppose a controller U d can be established, which should track a desired value of signal say x 2d then the controller equation can be written as U d 1 = g( x1 ) ( x 2d f ( x1, x2 )) (11)

141 124 with x 2d = x 2. Thus it is possible to have a system response equals to the desired value if the controller U d can effectively inverse the system dynamics. In other words the controller U should track the system such that e = 0. However due to system dynamics, the error equation has to be written as: e = ( x x) =0 (12) d Thus the controller U should be rewritten as ( U = g x1 ) ( x 2d f ( x, x ) + BV ) (13) Where V is the new controller input which stabilizes overall system dynamics thus creating the e t =x d -x =0, and B is the linear coefficient for V. This new controller V enables the system dynamics for settling to zero and tracks the output asymptotically. The nonlinear system can be linearized, which can be written in canonical form as below x1 = x 2 x 2 = C * X + V (14-15) The stability of the system and the adaptability is achieved then by an adaptive control law V tracking the system states x to a suitable reference model as in equation (3) such that the error e=x-x m =0 asymptotically where V = U mr. The neural network control law now becomes U d 1 = D ( y, x2d, x 2d ) (16) Where y is the plant output. From the above discussion it can be seen that the input to the neural network should be Θ = x2 d x d (17) y 2

142 125 This linearized controller V is represented as model reference adaptive controller output value and U d as the neural network output, which captures the inverse dynamics of the plant. A direct MRAC is utilized with the proposed scheme as outlined in figure 4.2. The error equation will have the form C*e+ε=0 where ε is the neural network approximation error. Further the neural network weights are adjusted in order to reduce the gradient of the output error so that the nonlinear dynamics are kept bounded. The model reference adaptive controller identifies the plant parametric value online and effectively control even though there is any parameter drift. Moreover, the MRAC will stabilize the output trajectory with the desired value especially at the initial stages, which allows the NN to learn the plant dynamics online. Added to all that, unlike certain linear controllers like designing the static gains such as integral, derivative and proportional are not needed if MRAC is used as a base line controller which is of great advantage especially when there is changing operating conditions. Development of the RBFNN Controller The artificial neural network control algorithm is based on the Radial Basis Function Neural Network (RBFNN). As shown in equation (17) the inputs to the neural network are the desired system states, its derivatives and the plant output. The RBFNN then approximate the system non-linearity and develops a control value such that the error between the desired trajectory and the plant output is forced to zero asymptotically. The derivations for the centers, weights and radius of the RBFNN structure are done in such a way that the trajectory error is zero. The proposed online RBFNN structure then constantly monitors the plant deviations either in the form of modal swings or due to the unmodeled dynamics by achieving the task of a developing linear parametrically

143 126 changing system, which can be controlled using a direct model reference adaptive controller. Basic Structure of the RBFNN The RBFNN is one of the most popular types of neural network because of its accuracy and simplicity. Like the feed forward neural network structure, it has three layers, the input layer, the node (hidden) layer and the output layer. Figure 4.3 shows a block diagram of three-layered RBFNN [62]. I 1 ϕ 1 I 2 I 3 ϕ 2 ω 1 ω 2 Σ y I n ϕ 3 : : : : ϕ m ω 3 ω n Output Input Layer Hidden Layer Figure 4.3 RBFNN Structure Input layer uses the incoming data and distribute it to the hidden layer without performing any mathematical calculation. The hidden layer is made of elements called nodes. Each of these nodes consists of two parts, a center and a nodal radius. Center is basically a vector that defines a single, unique point in the network state space. Each center has an area of influence around it termed as nodal radius. Each node calculates the nodal distance and basis function. The basis function calculation determines the value of the output from each node, which can be classified as individual nodal output. Finally the

144 127 total nodal output is calculated by a weighting factor, which symbolizes the ability of the RBFNN to learn the function, which it represents. The network learns by having known the input/output pairs running through it adjusting the weights. In the offline learning the weights are adjusted applying the input/output pairs continuously for a period of time. However, in the online training the pair is applied at each time instant and the network is learned during the course of time. Design Steps for the Proposed RBFNN Controller The proposed RBFNN structure is used along with an MRAC to capture the functional non-linearity through online training. Following are the design steps involved in the proposed RBFNN. a) Selection of the Number of Nodes and Centers A growing dynamic RBFNN is proposed that will start with 3 nodes. The centers for these nodes will be initialized to zeros except for the last center, which will be 0.1 as in order to calculate the radius with these centers at least two different centers are needed. Thus the RBFNN initially is novice and doesn t know about the systems, which is being approximated or controlled. Further each time a new input is passed on by the input layer, the distance and the outputs are calculated using equation (18) 2 ( j, k ) j Oj = exp( ( X C ) / r ) (18) k Where j=1: node and k=1: number of Input vector elements, c the center, r the radius and X the inputs. If the new input vector falls close to any one of the existing centers it will result in the output value extremely close to unity. Then the value of the centers is adjusted to a new position such that both inputs are within their range. This is as shown in figure 4.4a. On the other hand, if none of the input vector values fall near the center, then

145 128 the norm will have a value and the output value falls below one. In this case the node number will be increased with additional center and radius located at the new inputs as in figure 4.4b. Thus this criterion is used to see how far the new input elements come close to the existing centers. The above procedure is done online at every time instant and the RBFNN grow and at the same time dynamic monitoring the input vector value, which needs to be tracked. Existing node movement New node Figure 4.4 Center Movement and Node Growth in RBFNN Structure b) Distance Measurement The distance measurement is Euclidean in nature based on Euclidean norm. The calculation of the distance is based on equation (19) dj = Ni k = 1 2 ( X k C (19) ( j, k ) ) Where N i = number of input vector elements and j is the number of nodes At first the distance between each input vector element is calculated with each center and finally the norm of them is found to get the distance with respect to each node for all the developed nodes.

146 129 c) Calculation of the Radius The radius is calculated from each center as follows. Assume there are three nodes A, B and C and the radius of B needs to be found. Initially the distance of node B with nodes A and C is found out. Subsequently the maximum of the distance multiplied by 0.6 gives the radius. This can be represented as Ni 2 R= for j=1:n (max( dj = ( X k C ) )) * 0. 6 (20) k = 1 ( j, k ) d) Selection of Basis Function Most common used basis function is the Gaussian basis function. The function for the Guassian function can be expressed using the equation below 2 2 Oj = exp( d j / r j ) (21) The main reason to choose this function is due to the fact that Guassian basis function gives the most weight to points that are close to the nodal center and exponentially less weight to the points, which move, farther from the node centers. e) Weight selection and online adjustment In order to sufficiently capture any non-linear function the weights must be chosen correctly. To get the correct weights the neural networks are trained to learn the function, which is being approximated. However to train the weights, data regarding the inputs and the outputs need to be collected. The weights are initially adjusted to zeros. This means at the first instant the output obtained from the neural network will be a random value. At each instant of the new value of the input/output vector is formulated, the weights are updated. This updated law is based on the gradient descent technique according to the following equations

147 130 e = r t y i W * = W + U * e * d (22) j j p t j Where W j * is the new value of the weights and W j the old value. U p is an update rate, which is pre defined and d j is the distance for each calculated value based on the equation discussed in the previous section. Proposed Adaptive Algorithms and Solution for the Curse of Dimensionality One of the main drawbacks on the RBFNN is its large size when used during an offline or an online learning especially to control a changing dynamic system. This is termed as Curse of Dimensionality. It is a known fact that as the system moves in its operating domain there is no need to use all the nodes at the same time as only some may be needed at a particular instant. Taking advantage of this fact, reference [14] formulated problem for a time varying system, the input to the RBFNN which is basically the system response and auxiliary inputs, will exists only in some regions of the input space. Unlike as discussed in [14] where the nodes are made active in this region, in the proposed approach an algorithm based on [62] is developed, which is basically a modified version of Clustering Algorithms. Initially the centers are chosen randomly as described before. Let this initial centers be represented as C i ' where (i=1, 2..m). Further based on a learning rate, the centers are modified every instant of time according to following equation. Ci ( k) = Ci ( k 1) + update *[ X ( k) Ci ( k 1)] (23) Based on the new centers, the radius is calculated each time when the centers are varied using the equation

148 131 Nm Ri = abs( Where: k= 1 (( C ( k,1) C i j ( k,1)) 2 ) *0.6) (24) i' represents current center, j represents the next center and k represents the total number of input vector elements. Correspondingly the nodes are increased. Thus the system will grow eventually. Figure 4.5 shows the effect of the growth in the node and the center movement in reducing the Curse of Dimensionality and the node comparison with the static network. Static Network Nodal Region Active Nodes Center Movement Number of nodes required for a Static Network y(t) Figure 4.5 Dynamic RBFNN Network Nodal Generation and Center Movement However, at any time instant if the system operating condition approaches a previously established value, then those corresponding centers are used. Due to the above-mentioned dynamic RBFNN approach the curse of dimensionality issue is addressed. One important aspect to be considered is the network approximation error. In order to address the issue of pre described approximation error in the network-bounded region, a method to increase the rate of convergence near the minimum based on Chen, Khalil [63] is developed. The following equation shows this algorithm

149 132 Let error e t *(t) be applied as the input to the dead-zone function D(e*), then 0 D( e*) = e * d e * + d 0 0 if if if e * d e * > d 0 e * < d t t t 0 0 (25) Further the output of the dead-zone function is used in the weight update rule as follows Wj * = Wj + update * D( e *)* e * d (26) t t j Confluence of ANN with Adaptive Control As can be seen from equation (14) and (15) once the system is linearized, then an adaptive controller will be able to control the plant to a lesser extend even when there is still a change in the plant dynamics. In this essence the adaptive controller makes sure that system dynamics are stable and follow the reference mode output thus acting as a stable controller if not precise. The task of preparing the plant output to track the trajectory and dynamically developing the control value to address the unmodel dynamics and mode swings are accomplished by the neural control. The proposed adaptive controller in this case is the direct model reference adaptive controller, which is derived in the previous chapter Main Features of the NNPAC Scheme The features of the developed algorithm are as follows: Able to control systems, which show the mode swings Precisely tracks the plant output to a desired signal Performs even in presence of unmodeled dynamics

150 133 The neural network algorithm is online and is growing and dynamic in nature A parallel control structure, which learns and adapts from the plant output. 4.2 Application to the Robotic Manipulator Position Tracking The developed NNPAC algorithm is first tested for the control of position tracking on the single link manipulator dynamic nonlinear model as described in Chapter 2. Several cases have been used for testing the ability of the proposed controller. In the following analysis two important cases are demonstrated. In the first one, the NNPAC algorithm is compared with a single reference model adaptive control. In the second case the proposed controller is used to test with FMRMAC developed in the previous chapter with artificially created unmodeled dynamics. Finally the controller is tested for the same load pattern and the unmodeled dynamics with a different reference model to show the effectiveness Simulation Results and Discussion As described several cases have been conducted to assess the effectiveness of the proposed scheme to track the desired command signal in case of variation in the load torque of the manipulator tip arbitrary at different time instant of which two most important ones are presented next. In each of these cases tables below show the proposed variation in the load torque. Further the output angular position for the desired time span is plotted for the proposed scheme and the traditional MRAC along with the reference models output. Correspondingly, the control contribution for the Umr and Unn for the

151 134 proposed scheme and the node number variation pattern is also shown. The reference model representation in all the cases has the following specific form, 2 n ω Wm ( s) = (27) 2 2 s + 2ςω + ω n n In this example the value of ς is set to 0.7. In the first case the value of ω for the single reference model is kept as five Tip Load Pattern 2 In Case 1 the tip load of this manipulator is changed at different time instants as in table 4.1. Figure 4.6 shows the position trajectory plot and the output of the robotic arm with the proposed scheme and the single reference model adaptive Controller in which ω is kept as 5. It can be seen that the traditional single reference model adaptive controller failed to respond and subsequently became unstable at around 7 seconds. As can be seen from figure 4.7, there has been significant contribution on the control value of RBFNN and thus the output tracks the trajectory well using the proposed controller. It is also worth noting that the node number varies in the case of proposed controller scheme starting at four nodes and ending at seven nodes. The details of the maximum and minimum values of the online RBFNN parameters such as centers, radius, distance and weight at the end of the eight seconds are as shown in table 4.2. n n Table 4.1 Tip Load Variation (Case 1) Time Range (sec) Load Torque (Nm)

152 135 Table 4.2 Growing Dynamic RBFNN Parametric Range RBFNN Parameters Center Radius Distance Weight Max.Value -2.8,446.8, Min.Value , , It can be seen that the single reference model adaptive controller is clearly unstable and was unable to control with the fixed reference model Tip Load Pattern 3 In Case 2, the tip load of this manipulator is changed at different time instant as in table 4.3. Table 4.3 Tip Load Variation (Case 2) Time Range (sec) Load Torque (Nm) Moreover, additional to to the load variation as shown in Table 4.3, in order to create the unmodeled dynamics, a random signal is added to the angular position output throughout the simulation time of eight seconds. In this case the single reference model adaptive controller failed to perform and showed instability with the second order reference model structure as shown. Further the NNPAC is compared with the FMRMAC. Figure 4.8 show the position trajectory plot and the output of the robotic arm with the proposed scheme and that multiple reference

153 136 model adaptive controller. It can be seen that the FMRMAC failed to perform and the proposed intelligent controller performed well in this condition. Figure 4.9 shows the controllers (Adaptive controller and RBFNN controller) contribution and the nodal change pattern is as in figure Table 4.4 shows the RBFNN parametric values for this case. Table 4.4 Growing Dynamic RBFNN Parametric Range RBFNN Parameters Center Radius Distance Weight Max.Value Min.Value These are two distinct cases at which the traditional MRAC and multiple model adaptive controller failed. Thus the main advantage of this novel intelligent controller is its ability to control systems with model swings and unmodeled dynamics. Further its shown that the output of the plant tracked perfectly the desired trajectory. This is effective when there is a need for precision tracking, as the reference model(s) output will have a steady state error with the desired pattern always. In all these cases the suitable reference model used in this parallel controller have a natural frequency ω equal to seven. Before proceeding further there is a need to check the performance of the proposed controller with a different reference model. n Tip Load Pattern 4 In Case 3 the tip load of this manipulator is changed at different time instant as in table 4.5.

154 137 Table 4.5 Tip Load Variation (Case 3) Time Range (sec) Load Torque (Nm) Figure 4.11 shows the position trajectory plot and the output of the robotic arm controlled by NNPAC with the adaptive controller having a reference model of natural frequency ω n equal to three. This case is performed to show the effect of the proposed scheme with a different reference model. As can be seen that the parallel control becomes no longer stable when a different reference model is chosen. Thus it is apparent that the issue of reference model selection is critical for the performance of the controller.

155 Figure 4.6 Case 1: Manipulator Position Output Comparison 138

156 Figure 4.7 Case 1: Control Contribution 139

157 Figure 4.8 Case 2: Manipulator Position Output Comparison 140

158 Figure 4.9 Case 2: Control Contribution 141

159 142 Figure 4.10 Case 2: Control Contribution

160 143 Figure 4.11 Case 3: Manipulator Position Output Comparison

161 Application to the Fighter Aircraft Pitch Rate Command Tracking The proposed controller scheme will now be applied on a 6-DOF dynamic F16 fighter aircraft model for the precision tracking of pitch rate command. The dynamic maneuvers created in Chapter 3 based on the modeling details in Chapter 2 will be used to test the developed controller. Three most relevant cases are discussed and the ability of the controller for precision tracking is then compared to the FMRMAC developed in Chapter 3. Further from the results, the merits and demerits are discussed Simulation Results and Discussion The following Maneuvers are chosen specifically to show the ability of the NNPAC. Dynamic Maneuver 1: This is used to test and compare the ability of the proposed controller with the Fuzzy Multiple Reference Model Adaptive Controller (FMRMAC). Dynamic Maneuver 7: This maneuver is used as the FMRMAC in this case a severe oscillation especially at the period where the dynamic model is shaken in an effort to create the unmodeled dynamics Dynamic Maneuver 3: This maneuver is used to show the issue related to the selection of suitable reference model in the case of the NNPAC. Two such reference models are selected and the performance is compared with the FMRMAC These simulation case studies are discussed next.

162 Dynamic Maneuver 1 The detail of this maneuver is as discussed before. Table 4.6 shows the Initial Control vector settings whereu1 = δ h, U 2 = δ a and U3 = δ r. The throttle value and this initial control vector are changed during the time span of 180 seconds. The control is applied only to the elevator and U 1 is varied based on the proposed control law with the deflection limit of 50 deg. The output pattern at the total time span and during switching is as shown in figure As shown, the pitch rate output based on NNPAC shows better tracking than the FMRMAC especially when there is a change in the command pattern. The figure also depicts comparison of FMRMAC with best-fixed reference model direct Adaptive Control. The difference is increasingly visible at the switching period (40 seconds). Please note that the reference model used for this case is s s + 115s s from table 3.8 and this is used as the best suitable fixed reference model for NNPAC. Thus in this simulation example the effect of the RBFNN controller aiding the adaptive control has been proven. Table 4.6 Initial Control Vector, Throttle Values and Command Patterns U 1 = Elevator U 2 = Aileron U 3 = Rudder *10^-7 6.2*10^-7 Throttle Value in Percentage Time (sec) < 10 < 20 < 40 < 100 <150 < 180 Pitch Rate ( deg/sec) Throttle (%) Control Vector Pattern Time (sec) < 10 < 20 < 40 < 100 < 150 < 180 Aileron (deg) -1.2*10^-2-1.2*10^-2-1.2*10^-2-1.2*10^-2-1.2*10^-7-1.2*10^-7 Rudder(deg) 6.2*10^-2 6.2*10^-2 6.2*10^-2 6.2*10^-2 6.2*10^-7 6.2*10^-7

163 Dynamic Maneuver 7 The detail of this maneuver is discussed before. Table 4.7 shows the Initial Control vector settings whereu1 = δ h, U 2 = δ a and U3 = δ r. Unlike in the first the throttle value and this initial control vector are the same during the time span of 150 seconds. The control is applied only to the Elevator and U 1 is varied based on the proposed control law. The output pattern at the total time span and during switching is shown in figure This case is similar to have an un modeled dynamics and the performance of the FMRMAC deteriorated at around 50 seconds while the NNPAC controller precise tracking having error within the limits. Please note that in this case too the same suitable reference model structure as in Dynamic Maneuver 1 is utilized. Table 4.7 Initial Control Vector, Command Pattern and Throttle position U 1 = Elevator U 2 = Aileron U 3 = Rudder *10^-8-8.3*10^-7 Command Pattern Time (sec) < 10 < 20 < 40 < 100 <150 Pitch Rate ( deg/sec) Throttle (%) Control Vector Pattern Time (sec) < 10 < 20 < 40 < 100 < 150 Aileron (deg) -7.0*10^-4-7.0*10^-4-7.0*10^-4-7.0*10^-4-6.2*10^-8 Rudder(deg) 8.3*10^-4 8.3*10^-4 8.3*10^-4 8.3*10^-4 8.3*10^ Dynamic Maneuver 3 The details of this Maneuver are as shown in Table 4.8. In this case, the NNPAC with Ref 1 is compared with the FMRMAC. As can be seen from figure 4.14, even though both controllers performed well during the change in the command pattern at around 40 seconds, when subjected to system changes due to dynamics at around 85 seconds the

164 147 FMRMAC shows deterioration in its performance. The figure also shows that when the suitable reference model is changed from Ref 1 to Ref 2 the NNPAC shows transient overshoot at 40 seconds. This is in agreement with the earlier observation on the robotic manipulator control. Table 4.8 Initial Control Vector, Command Pattern and Throttle position U 1 = Elevator U 2 = Aileron U 3 = Rudder *10^-7-4.3*10^-6 Command Pattern Time (sec) < 10 < 20 < 40 < 100 <150 <180 Pitch Rate ( deg/sec) Throttle (%) Control Vector Pattern Time (sec) < 10 < 20 < 40 < 100 < 150 < 180 Aileron (deg) -5.1*10^-7-5.1*10^-3-5.1*10^-3-5.1*10^-4-5.1*10^-7-5.1*10^-7 Rudder(deg) 4.3*10^-6 4.3*10^-3 4.3*10^-3 4.3*10^-3 4.3*10^-6 4.3*10^-6

165 Figure 4.12 Pitch Rate Command Tracking Comparison Dynamic Maneuver 1 148

166 Figure 4.13 Pitch Rate Command Tracking Comparison Dynamic Maneuver 7 149

167 Figure 4.14 Pitch Rate Command Tracking Comparison Dynamic Maneuver 3 150

168 Conclusions A new Neural Network Parallel Adaptive Control (NNPAC) scheme has been presented for the control of complex systems, which show mode swings and are prone to unmodeled dynamics. The concept, control methodology, derivations of the control law and design methods of the RBFNN controller is established. The method is novel in the sense that the growing dynamic RBFNN structure is presented as an augmentation to the direct model reference adaptive control law. Further it can learn and adapt with respect to changing and unmodeled dynamics. The scheme has been investigated with two physical dynamic systems; the position control of the Single link manipulator and the pitch rate control of a fighter aircraft. Results show that the scheme performs well when its traditional single reference model adaptive control counterpart deteriorates in performance or even becomes unstable. Further the proposed scheme performs well for precision tracking of the output with a desired trajectory unlike the FMRMAC presented in the last chapter. More importantly it performed well even when there is unmodeled dynamics as demonstrated in some cases. Extensive offline simulations to produce a large plant knowledge base are also not essential for this controller, and therefore the requirement for offline knowledge base associated with the FMRMAC presented in the previous chapter is alleviated by this new NNPAC scheme. The NNPAC algorithm presented is effective and efficient as it can learn dynamically as compared to the static learning methodology adopted in the FMRMAC algorithm. Moreover it tracks the command signal precisely without the need of offline design and is more stable for unmodeled dynamics. However, and as demonstrated in the investigations above, the selection of a suitable reference model is

169 152 crucial for the scheme s effectiveness and overall system stability, especially in the presence of plant Jumps or mode changes. It has been also demonstrated Case 3, figure 4.11 (the reference model structure was changed with the change in ω n to three from seven) that in the presence of some challenging modal changes, and even when the designer can select a suitable reference model, the system may go unstable. One of the main reasons for this instability is the inability of the adaptive control law to control the plant when there is sudden mode switching. Even though the parallel RBFNN algorithm learns the system changes and tries to react, the system movement is so fast that the adaptive control law looses its effectiveness as it is stressed to its limit for such control. This scenario clearly shows that there is a need to change the reference models to control such drastic changes in dynamics, which strengthens the original thought. These modal changes can be classified as: Scheduled Modal Switching: - If the mode switching is scheduled (mode changes are known a priori), then a fuzzy scheme with sparse rules can be developed. Unscheduled Modal Switching: - In the case of mode switching that is not scheduled, and if there is unforeseen external impact on the system, then there is a need for dynamic reference switching to deal with such structural changes. An offline neural network with an online dynamic RBFNN network will be developed for such switching. In the next chapters, two separate algorithms are presented to address these issues.

170 CHAPTER FIVE NEURAL NETWORK PARALLEL FUZZY ADAPTIVE CONTROLLER This chapter introduces an intelligent adaptive controller, which consists of an online growing dynamic Radial Basis Function Neural Network (RBFNN) structure, in parallel with a Fuzzy Multiple Reference direct Model Adaptive Controller (FMRMAC). This intelligent controller uses a direct MRAC loop with a fuzzy multiple reference model, and responds to the system vast changing dynamics through a parallel online dynamic RBFNN loop. In the presence of the scheduled mode switching and sudden plant Jumps, the fuzzy multiple reference model generation enables for a changing reference model structure based on an offline knowledge base. The update details of the RBFNN width, centers and weights are developed to ensure the error reduction and for improved tracking accuracy. The importance of the proposed scheme is its ability to perform effectively even when the plant mode switches, using Fuzzy Multiple Reference Model (FMRM) concept and to precisely track the reference trajectory even with unmodeled dynamics. Unlike the model reference adaptive control in which the plant follows the reference model output, the proposed scheme pursues the desired trajectory directly for the plant output to follow. The proposed Neural Network Parallel Fuzzy Adaptive Controller (NNPFAC) scheme concept and algorithm development is detailed in this chapter. 153

171 154 Simulation investigations of the proposed scheme are then presented, using two applications, the position control system of the single link flexible manipulator undergoing several modal switching and plant Jumps due to the tip load variation, as well as the pitch rate control system of the F16 Fighter Aircraft. 5.1 Formulation of the NNPFAC Scheme Figure 5.1 gives a functional block diagram of the proposed controller. The scheme is a combination of the algorithms presented in the previous chapters which enables us for controlling the system showing multi-modality and sudden Jumps and at the same time susceptible to unmodeled dynamics. In the presence of unmodeled dynamics and modal swings the parallel controller with RBFNN augmenting the stable direct model reference adaptive controller controls the system effectively as seen in the last chapter. However when the system mode switches and the plant shows sudden Jumps, the reference model structure needs to be changed. If these modal changes are known a priori, a fuzzy inference engine can generate multiple reference model structure. Thus in the presence of schedule modal changes this proposition effectively controls the system. There are two intelligent loops; the first generates a changing reference model structure, the second one that augments the stable direct model reference adaptive controller. If the operation of a system under consideration can be modeled using one reference model, then the adaptive controller can be utilized for controlling the system to track the desired reference model output. The MRAC can very well control such system with the adaptation mechanism adapting the control law based on the output error.

172 155 However, if the reference model is rigid and the system modal swings and sudden Jumps occur then the controller will be stressed and may fail to control the plant. Under this scenario the fuzzy multiple reference model generation is necessary. Intelligent Supervisory Loop Fuzzy Reference Model Generator Adaptive Controller Reference Model Adaptive Mechanism Reference Output Error Input Signal Adaptive Control Law Σ System Under Consideration Plant Output Desired System States RBFNN Neural Network Controller Intelligent Supervisory Loop Figure 5.1 Functional Block diagram with the Intelligent Supervisory Loop Further, if there are unmodeled dynamics then the controller performance may deteriorate or it may become unstable, as this new system dynamics will be different from the originally modeled reference model. The second is an intelligent supervisory loop that is designed to produce an additional control action to compensate for the first loop deficiencies.

173 The NNPFAC Concept The parallel controller structure is based on the fact that a neural network can be used to learn from the feedback error of the plant controller closed loop. The plant is controlled by an adaptive controller, which tracks the reference model output. Further when there are unmodeled dynamics during plant operation or if there exist a steady state error between the reference model output and the reference trajectory, then the RBFNN controller learn and develops a controller such that this error is made to zero asymptotically. This in fact is carried out continuously such that the output error is zero and precision tracking is achieved. However in the presence of sudden plant Jumps and mode switching as seen in the last chapter the adaptive controller fails to perform with a single reference model as it is often stressed to its limit with only one reference model. Additionally, the designer choice of a fixed reference model can also be critically deficient as the system operates under multimodality conditions. Therefore, there is still a need under such conditions to generate the fuzzy multiple reference model by fuzzy logic scheme. It is worth noting at this stage that as pointed out in the NNPAC scheme presented in chapter 4, an RBFNN parallel controller effectively controls the mode swings. This means that the rule base does not need to be dense as in the case of the FMRMAC as the primary aim of the fuzzy reference model will be to cover the modal regions with the help of the knowledge base. As a result, there can be drastic reduction in the number of rules as will be shown in the pitch rate control of F16 aircraft investigation, where the reduction in the number of rules was from ninety to sixteen rules.

174 157 Thus the designer has the flexibility to design a sparse set of rules, which cover the operational range Mathematical Formulation Another block diagram of the NNPFAC scheme is given in figure 5.2. Fuzzy Multiple Reference Model Auxiliary Inputs And Output f ( x) = r ri µ i 1 T = M ϑ = Φ * ϑ r µ i= P i i= 1 y mi ( t) = ( Φ * ϑ / R )* S( t) i i m y mi (t) Direct Model Reference Adaptive Controller - Command Signal Θ = T T [ θ θ ] T k 0 1 θ 2 ui T = Θ ω i + x = Ai mod e ( t) y ( t) = h( x) i f ( x) + b mod ( ) g( x) U i e t y i (t) + Desired States + Dynamic System - RBFNN Controller + r Figure 5.2 The Neural Network Parallel Fuzzy Adaptive Controller Let the nonlinear system to be controlled have input U and output yi and can be represented as x = Ai mod e ( t) y ( t) = h( x) i f ( x) + bi mod e( t) g( x) U (1-2)

175 158 Where: y i (t) is the plant output at a specific mode, U is the control input, X is the state vector where [X 1 (t) X n (t)] T Є R n A i (t)=[a 1i,a 2i,.,a ni ] T Є R nxn and B i takes values from the set of H constant elements as indexed by subscripts i and i Є {1,2,,H}. Further, based on the change in values, let f(x) and g(x) be two nonlinear functions such that they are continuous in nature. The derivation of the problem is done in two stages. In the first stage, system is assumed to be operating in one mode with mode swings and unmodeled dynamics. Then in the second stage the additional intelligent module needed in the presence of the mode switching is assessed. First assume that the system is operated at one mode without any mode switching. Under such condition equation (1) & (2) can be written as x = Af ( x) + bg( x) U y( t) = h( x) (3-4) Assuming SISO control and a second order system the above generic equation can be written as x1 = x2 x 2 = F( x1, x2, U ) or (5-7) x 1 = F( x 1 1, U ), x Assuming the function F is invertible, then

176 159 U = F[ x, x1, x1] 1 (8) Let U d is the desired value of the control, then U d = F[ x1, x1d, x d 1d ] (9) [ d d d Further assuming x 1 =y, U = F y, y, y ] (10) d Thus in the presence of a control U d the system can be effectively controlled through learning if it is invertible. An RBFNN controller is proposed for controlling the nonlinear system with unmodeled dynamics, which learns the system dynamics online and then generates the value of U d. As can be seen from equation (10), the inputs to RBFNN controller are thus the output of plant y and other desired states. The output and trajectory error e 1 is thus reduced by a gradient learning technique asymptotically and system learns during the course of time generating the value U d. The error dynamics during system learning can then be written as e = F[ x1, x1, U ] F[ x1, x1d, U d ] (11) d Applying the taylor series expansion and collecting the first order terms, equation (11) is e A( t) e + B( t)[ U Ud] + N( t)ε (12) Where: F( x, U ) A ( t) = X T d, U x d F( x, U ) B ( t) = X T d, U d (13-14) U When the RBFNN controller is able to learn the system characteristics successfully then U=U d and the error dynamics will be

177 160 e A( t) e + N( t)ε (15) Further in the presence of perfect learning the value ε will be zero. Thus equation (15) will be e A( t) e (16) It can be seen from (16) that the error derivative is dependent on A(t) which is a function of the states and control value. An adaptive control law is used to reduce this error asymptotically to zero. Thus the total control law is U = U mr + U nn (17) Where U = θ T *ω (18) mr r θ = [ k θ 0 θ1 θ 2 ] And y ω = p ω 1 ω 2 U nn = node j= (exp ( d / r ))* w (19) j j j Where d j = Nm k = 1 2 ( X C is the distance and r j and W j are radius and weights of k ( j, k ) ) each node respectively. The epsilon ε value in the error equation (15) is the neural network approximation error. Further the neural network weights are adjusted in order to reduce the gradient of the output error so that nonlinear dynamics are kept bounded. The model

178 161 reference adaptive controller identifies the plant parametric value online and effectively control even though there is any parameter drift. Moreover, the MRAC will stabilize the output trajectory with the desired value especially at initial stages, which allows the NN to learn the plant dynamics online. Added to all that, unlike certain linear controllers like designing the static gains such as integral, derivative and proportional are not needed if MRAC is used as a base line controller which is of great advantage especially when there is changing operating conditions. RBFNN Control Algorithm For the completeness of the proposed algorithm, the RBFNN design details discussed in the previous chapter are rewritten next. Design Steps The proposed RBFNN structure is used along with a MRAC to capture the functional non-linearity through online training. Following are the design steps involved in the proposed RBFNN. a) Selection of the Number of Nodes and Centers A growing dynamic RBFNN is proposed that will start with 3 nodes. The centers for these nodes is initialized to zeros except for the last center, which is kept as 0.1 as to calculate the radius with these centers at least two different centers are needed. Thus the RBFNN initially is novice and doesn t know about the systems, which is being approximated or controlled. Further each time a new input is passed on by the input layer, the distance and the outputs are calculated using equation (20)

179 162 2 ( j, k ) j Oj = exp( ( X C ) / r ) (20) k Where j=1: node and k=1: number of input vector elements, c the center, r the radius and X the inputs. b) Distance Measurement The distance measurement is Euclidean in nature based on Euclidean norm. The calculation of the distance is based on the equation (21) dj = Ni k = 1 2 ( X k C (21) ( j, k ) ) Where N i = Number of input vector elements and j is the number of nodes. At first the distance between each input vector element is calculated with each center and finally the norm is found for all the developed nodes to get the distance with respect to each node. c) Calculation of the Radius The radius is calculated from each center as follows. Assume there are three nodes A, B and C and the radius of B needs to be found. To this effect initially the distance of the node B with nodes A and C is found out. Subsequently the maximum of the distance multiplied by 0.6 gives the radius. This can be represented as Ni 2 R= for j=1:n (max( dj = ( X k C ) )) * 0. 6 (22) k = 1 ( j, k ) d) Selection of Basis Function Most common used basis function is the Gaussian basis function. The function for the Guassian function can be expressed using the equation below 2 2 Oj = exp( d j / r j ) (23)

180 163 The main reason to choose this function is due to the fact that Guassian basis function gives the most weight to points that are close to the nodal center and exponentially less weights to points which move farther from node centers. e) Weight selection and online adjustment In order to sufficiently capture any non-linear function, weights must be chosen correctly. To get the correct weights the neural networks are trained to learn the function, which is being approximated. However to train the weights, data regarding the inputs and the outputs need to be collected. The weights are initially adjusted to zeros. This means at the first instant the output obtained from the neural network will be a random value. At each instant of the new value of the input/output vector is formulated, the weights are updated. This updated law is based on the gradient descent technique according to the following equations e = r and t y i W * = W + U * e * d (24) j j p t j Where W j * is the new value of weights and W j the old value. U p is an update rate, which is pre defined and d j is the distance for each calculated value based on the equation discussed in the previous section. Details of the adaptive ness of the RBFNN algorithm and the solution to the curse of dimensionality are as described in Chapter 4. In the second stage in the presence of the mode switching the parameter vector can be represented by the triple {(A i,b i,c i ),..,(A H,b H,c H )}, which changes its values depending on the modes of operation. Let the above set denote the scheduled jump parameters for each mode specified by the parameter index i. The mode variable mode (t) takes the form mapped into any of the values in the domain i Є {1,2 H} and

181 164 correspondingly A mode(t) and b mode(t) are time varying. The mapping of mode (t) be denoted by mode (t) = γ[x (t-d)] where d represents the time delay. The above-mentioned system is a spatial multimodal type because the dynamics are scheduled through a nonlinear mapping γ. As the system mode changes, system parameters jump or move drastically, which in turn may cause unstable performance if a conventional adaptive controller is used. A structural change in the reference model is required if there is no explicit identification procedure available for the plant. This is accomplished in the proposed approach through fuzzy logic multiple reference model generation, which in turn allow direct model reference control law to operate effectively under different plant operating conditions. Details of the MRAC derivations and the generation of the fuzzy inference engine are as detailed in Chapter 3. One feature of the proposed controller is its ability of the controller to effectively control the system in the presence of mode switching with sparse set of fuzzy rules. This is attributed to the RBFNN structure, which learns the system even in the presence of mode swings. Thus even though the designer is left to develop rules the number of rules and knowledge base information about the system can be very little Main Features of the NNPFAC Scheme The features of the developed algorithm are as follows: Able to control systems, which shows to mode swings Able to control system which are subjected to sudden Jumps and mode switching Precisely tracking the plant output to a desired signal Performs even in presence of unmodeled dynamics

182 165 The neural network algorithm is online and is growing and dynamic in nature A parallel control structure, which learns and adapts from the plant output. 5.2 Application to the Robotic Manipulator Position Tracking The developed NNPFAC algorithm is first tested for the control of position tracking on single link manipulator dynamic nonlinear model as described in Chapter 2. Several cases have been used for testing the ability of proposed controller. In the following analysis three important cases are demonstrated. In the first one the NNPFAC algorithm is compared with FMRMAC algorithm developed before. In the second case the proposed controller is used to test with NNPAC developed in the previous chapter with artificially created unmodeled dynamics. After assessing the improvement in the output position tracking, in the third case the proposed scheme is compared with the NNPAC algorithm for a different reference model structure other than suitable reference model Simulation Results and Discussion In each of these cases, tables below show the proposed variation in the load torque. Further the output angular position for the desired time span is plotted for the proposed scheme and the traditional MRAC along with the reference models output. Relevant graphs for each of the cases are also shown. The reference model representation in all these cases has the following specific form,

183 166 2 n ω Wm ( s) = (25) 2 2 s + 2ςω + ω n n In this example value of ς is set to 0.7. The value of ω is changed depending on the fuzzy switching scheme. n Tip Load Pattern 5 In this test tip load of this manipulator is changed at different time instant as in table 5.1. Table 5.1 Tip Load Variation (Case 1) Time Range (sec) Load Torque (Nm) Demonstration of this example is to show the ability of proposed design in the presence of unmodeled dynamics. The comparison is with the FMRMAC developed in Chapter 3. At the load torque variation as shown in this table the FMRMAC went unstable and NNPFAC controller presented in this chapter successfully controlled the plant. The position output comparison is as shown in figure 5.3. The effectiveness of the proposed controller to track the position output in presence of unmodeled dynamics and the failure of FMRMAC is notable from the figure. The next two cases are to demonstrate the ability of the proposed controller in case of mode switching. It is shown that how the NNPAC controller developed in the previous chapter fails or shows deteriorated performance when compared to NNPFAC Tip Load Pattern 6 In Case 2 the manipulator tip load is changed at different time instant as in table 5.2.

184 167 Table 5.2 Tip Load Variation (Case 2) Time Range (sec) Load Torque (Nm) The ability of proposed controller with changes in the reference model to control this case when compared to the NNPAC algorithm is shown in figure 5.4. Both the controllers perform well and the proposed scheme showing better tracking ability when compared to the NNPAC algorithm. The position error pattern is as shown in figure 5.5, which clearly indicates this effect. It is worth noting that the value of the wn for suitable second order reference model structure when used with NNPAC algorithm is kept as seven Tip Load Pattern 7 In Case 3 the tip load of this manipulator is changed at different time instant as in table 5.3. Moreover, between 6 to 6.5 seconds the frictional load torque is augmented with a value of It can be seen that the tip load variation in this case is the same as that of the tip load variation 5. However this case is to demonstrate the ability of proposed algorithm and the ineffectiveness of NNPAC algorithm when reference model structure is not the suitable one. Here the wn values for suitable second order reference model structure when used with NNPAC algorithm is kept as three. Table 5.3 Tip Load Variation (Case 3) Time Range (sec) Load Torque (Nm) The output performance of the proposed scheme when compared to the NNPAC algorithm is as shown in figure 5.6. As can be seen that the parallel control become no

185 168 longer stable when a different reference model is chosen. Thus it is apparent that the issue of reference model selection is critical for performance of the controller.

186 Figure 5.3 Case 1: Manipulator Position Output Comparison 169

187 Figure 5.4 Case 2: Manipulator Position Output Comparison 170

188 Figure 5.5 Case 2: Position Error Comparison 171

189 Figure 5.6 Case 3: Manipulator Position Comparison 172

190 Application to the Fighter Aircraft Pitch Rate Command Tracking The proposed controller scheme will now be applied on a 6-DOF dynamic F16 fighter aircraft model for the precision tracking of pitch rate command based on the dynamic maneuvers created in Chapter 3. Figure 5.7 Sparse Fuzzy Rule Base of the F16 Fighter Aircraft One important issue, which is to be checked, in this case is the density of the offline knowledge base needed. To this effect it is worth nothing that the fuzzy rule base has been reduced with sixteen numbers of rules when compared ninety rules in FMRMAC. Figure 5.7 shows the total number of rules developed. It was demonstrated in the last chapter that the NNPAC algorithm is capable of effectively controlling the plant in the presence of mode swings. If that is the case when the reference model structure in a

191 174 single modal region can be defined then the RBFNN parallel algorithm can effectively control the system. Thus these rules are created in such a way that it covers the modal regions but not effectively cover each modal point under the regions. Figure 5.8 and figure 5.9 shows the nonlinear knowledge base functional surface relation between fuzzy inputs and the output and the corresponding membership view respectively.

192 Figure 5.8 Sparse Fuzzy Input Output Relation of the F16 Fighter Aircraft 175

193 176 Figure 5.9 Sparse Fuzzy Membership Function Detail of the F16 Fighter Aircraft