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Control of a Nonlinear Model of a Robotic Arm with a Flexible Joint via the Linearization Technique Consider a single-link robotic manipulator with a flexible joint whose nonlinear mathematical model is given in Chapter 1, Example 1.3 of the book GL96, page 29. It is desired to control the robot arm to hold any reference (nominal-, operating-, trim-, steady-state- set-point) angle in the entire range 0 2. Design a full-state feedback control law for this system using linearization about a set point. As a controller use the “pole placement” (eigenvalue assignment) controller (regulator). To find the feedback gain F via MATLAB use the function “place” as: F=place (A, B, eigenvalues_desired). The project tasks should be achieved in two steps. Step 1. (3 points). Present the results of linearization at steady state points and find the corresponding matrices A and B. Assume that matrix C is identity and matrix D a zero matrix of appropriate dimensions. Use the Simulink state space block and simulate dynamics of the linearized system (see Hint). The linearized system initial conditions that should be set in the neighborhood of the system initial conditions (note x(t0) = x(t0) − xss). The full-state feedback controller u(t) = −Fx(t) should be obtained using the MATLAB function place. Monitor the system response and choose the desired eigenvalues such a good transient response is obtained (small overshoot, short settling and rise times). At the same time observe the magnitudes of the signals in the closed-loop system. Step 2. (7 points). Build the Simulink block diagram of the considered nonlinear robot arm model. Design a full-state space controller that keeps at steady state 1ss = 4 .Simulate the performance of the system response using SIMULINK and plot 1(t) as a function of time until it settles down to its steady state value. Observe also the linearized and actual values of the input signal. Plot all required signals from the MATLAB window using the MATLAB plot function as: plot (out.t, out. signal), see the Simulink block diagram in Figure 9 of the Simulink Short Manual. Consider the time interval of several seconds. Report Format: For this project, every student must submit a typed technical report that should include: Project description, project formulation; basic formulas used and the design steps; SIMULINK block diagrams; graphs of all relevant signals as defined in the project tasks; comments on results obtained; summary; references; appendix with MATLAB codes. PubTeX output 2002.09.03:2253 Chapter One Introduction This book represents a modern treatment of classical control theory of continuous- and discrete-time linear systems. Classical control theory originated in the fifties and attained maturity in the sixties and seventies. During that time control theory and its applications were among the most challenging and interesting scientific and engineering areas. The success of the space program and the aircraft industry was heavily based on the power of classical control theory. The rapid scientific development between 1960 and 1990 brought a tremen- dous number of new scientific results. Just within electrical engineering, we have witnessed the real explosion of the computer industry in the middle of the eighties, and the rapid development of signal processing, parallel computing, neural networks, and wireless communication theory and practice at the begin- ning of the nineties. In the years to come many scientific areas will evolve around vastly enhanced computers with the ability to solve by virtually brute force very complex problems, and many new scientific areas will open in that direction. The already established “information superhighway” is maybe just a synonym for the numerous possibilities for “informational breakthrough” in almost all scientific and engineering areas with the use of modern computers. Neural networks—dynamic systems able to process information through large number of inputs and outputs—will become specialized “dynamic” computers for solving specialized problems. Where is the place of classical (and modern) control theory in contemporary scientific, industrial, and educational life? First of all, classical control theory values have to be preserved, properly placed, and incorporated into modern scien- tific knowledge of the nineties. Control theory will not get as much attention and 1 2 INTRODUCTION recognition as it used to enjoy in the past. However, control theory is concerned with dynamic systems, and dynamics is present, and will be increasingly present, in almost all scientific and engineering disciplines. Even computers connected into networks can be studied as dynamic systems. Communication networks have long been recognized as dynamic systems, but their models are too complex to be studied without the use of powerful computers. Traffic highways of the fu- ture are already the subject of broad scale research as dynamic systems and an intensive search for the best optimal control of networks of highways is under- way. Robotics, aerospace, chemical, and automotive industries are producing every day new and challenging models of dynamic systems which have to be optimized and controlled. Thus, there is plenty of room for further development of control theory applications, both behind or together with the “informational power” of modern computers. Control theory must preserve its old values and incorporate them into modern scientific trends, which will be based on the already developed fast and reliable packages for scientific numerical computations, symbolic computations, and computer graphics. One of them, MATLAB, is already gaining broad recognition from the scientific community and academia. It represents an expert system for many control/system oriented problems and it is widely used in industry and academia either to solve new problems or to demonstrate the present state of scientific knowledge in control theory and its applications. The MATLAB package will be extensively used throughout of this book to solve many control theory problems and allow deeper understanding and analysis of problems that would not otherwise be solvable using only pen and paper. Most contemporary control textbooks originated in the sixties or have kept the structure of the textbooks written in the sixties with a lot of emphasis on frequency domain techniques and a strong distinction between continuous- and discrete-time domains. At the present time, all undergraduate students in electrical engineering are exposed to discrete-time systems in their junior year while studying linear systems and signals and digital signal processing courses so that parallel treatment of continuous- and discrete-time systems saves time and space. The time domain techniques for system/control analysis and design are computationally more powerful than the frequency domain techniques. The time domain techniques are heavily based on differential/difference equations and linear algebra, which are very well developed areas of applied mathematics, for which efficient numerical methods and computer packages exist. In addition, the INTRODUCTION 3 state space time domain method, to be presented in Chapter 3, is much more convenient for describing and studying higher-order systems than the frequency domain method. Modern scientific problems to be addressed in the future will very often be of high dimensions. In this book, the reader will find parallel treatment of continuous- and discrete-time systems with emphasis on continuous-time control systems and on time domain techniques (state space method) for analysis and design of linear control systems. However, all fundamental concepts known from the frequency domain approach will be presented in the book. Our goal is to present the essence, the fundamental concepts, of classic control theory—something that will be valuable and applicable for modern dynamic control systems. The reader will find that some control concepts and techniques for discrete- time control systems are not fully explained in this book. The main reason for this omission is that those “untreated topics” can be simply obtained by extending the presented concepts and techniques given in detail for continuous- time control systems. Readers particularly interested in discrete-time control systems are referred to the specialized books on that topic (e.g. Ogata, 1987; Franklin et al., 1990; Kuo, 1992; Phillips and Nagle, 1995). Instructors who are not enthusiastic about the simultaneous presentation of both continuous- and discrete-time control systems can completely omit the “discrete-time parts” of this book and give only continuous-time treatment of control systems. This book contains an introduction to discrete-time systems that naturally follows from their continuous-time counterparts, which historically are first considered, and which physically represent models of real-world systems. Having in mind that this textbook will be used at a time when control theory is not at its peak, and is merging with other scientific fields dealing with dynamic systems, we have divided this book into two independent parts. In Chapters 2–5 we present fundamental control theory methods and concepts: transfer function method, state space method, system controllability and observability concepts, and system stability. In the next four chapters, we mostly deal with applications so that techniques useful for design of control systems are considered. In Chapter 10, an overview of modern control areas is given. A description of the topics considered in the introductory chapter of this book is given in the next paragraph. Chapter Objectives In the first chapter of this book, we introduce continuous- and discrete-time invariant linear control systems, and indicate the difference between open-loop 4 INTRODUCTION and closed-loop (feedback) control. The two main techniques in control system analysis and design, i.e. state space and transfer function methods, are briefly discussed. Modeling of dynamic systems and linearization of nonlinear control systems are presented in detail. A few real-world control systems are given in order to demonstrate the system modeling and linearization. Several other models of real-world dynamic control systems will be considered in the following chapters of this book. In the concluding sections, we outline the book’s structure and organization, and indicate the use of MATLAB and its CONTROL and SIMULINK toolboxes as teaching tools in computer control system analysis and design. 1.1 Continuous and Discrete Control Systems Real-world systems are either static or dynamic. Static systems are represented by algebraic equations, and since not too many real physical systems are static they are of no interest to control engineers. Dynamic systems are described either by differential/difference equations (also known as systems with concentrated or lumped parameters) or by partial differential equations (known as systems with distributed parameters). Distributed parameter control systems are very hard to study from the control theory point of view since their analysis is based on very advanced mathematics, and hence will not be considered in this book. At some schools distributed parameter control systems are taught as advanced graduate courses. Thus, we will pay attention to concentrated parameter control systems, i.e. dynamic systems described by differential/difference equations. It is important to point out that many real physical systems belong to the category of concentrated parameter control systems and a large number of them will be encountered in this book. Consider, for example, dynamic systems represented by scalar differen- tial/difference equations _x(t) = fc(x(t)); x(t0) = x0 (1.1) x(k +1) = fd(x(k)); x(k0) = x0 (1.2) where t stands for continuous-time, k represents discrete-time, subscript c in- dicates continuous-time and subscript d