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By Michael Athans and Peter L. Considerable interest in optimal control has developed over the past decade, and a broad, general theory based on a combination of variational techniques, conventional servomechanism theory, and high-speed computation has been the result of this interest.

We feel that, at this point, there is a need for an introductory account of the theory of optimal control and its applications which will provide both the student and the practicing engineer with the background and foundational material necessary for a sound understanding of recent advances in the theory and practice of control-system design.

We attempt to fill this need in this book. We briefly indicate some of our aims and philosophy and describe the contents of the book in this introductory chapter. In particular, we set the context by discussing the system design problem in Sec. We then discuss Sec. Next, we indicate Sec. Following our statement of purposes, we make some general comments on the structure of the book in Sec.

We conclude this chapter in Sec. A system design problem begins with the statement sometimes vague of a task to be accomplished either by an existing physical process or by a physical process which is to be constructed. For example, the systems engineer may be asked to improve the yield of a chemical distillation column or to design a satellite communication system. As an integral part of this task statement, the engineer will usually be given:.

A set of goals or objectives which broadly describe the desired performance of the physical process; for example, the engineer may be asked to design a rocket which will be able to intercept a specified target in a reasonable length of time. A set of constraints which represent limitations that either are inherent in the physics of the situation or are artificially imposed; for example, there are almost always requirements relating to cost, reliability, and size. The development of a system which accomplishes the desired objectives and meets the imposed constraints is, in essence, the system design problem.

There are basically two ways in which the system design problem can be approached: the direct or ad hoc approach and the usual or standard approach. Approaching the system design problem directly, the engineer combines experience, know-how, ingenuity, and the results of experimentation to produce a prototype of the required system. He deals with specific components and does not develop mathematical models or resort to simulation.

In short, assuming the requisite hardware is available or can be constructed, the engineer simply builds a system which does the job. For example, if an engineer is given a specific turntable, audio power amplifier, and loudspeaker and is asked to design a phonograph system meeting certain fidelity specifications, he may, on the basis of direct experimentation and previous experience, conclude that the requirements can be met with a particular preamplifier, which he orders and subsequently incorporates hooks up in the system.

The direct approach is, indeed, often suitably referred to as the art of engineering. Unfortunately, for complicated systems and stringent requirements, the direct approach is frequently inadequate. Moreover, the risks and costs involved in extensive experimentation may be too great. For example, no one would attempt to control a nuclear reactor simply by experimenting with it. In view of these difficulties, the systems engineer usually proceeds in a rather different way.

The usual, or standard, approach to a system design problem begins with the replacement of the real-world problem by a problem involving mathematical relationships. In other words, the first step consists in formulating a suitable model of the physical process, the system objectives, and the imposed constraints.

The adequate mathematical description and formulation of a system design problem is an extremely challenging and difficult task. Desirable engineering features such as reliability and simplicity are almost impossible to translate into mathematical language.

Moreover, mathematical models, which are idealizations of and approximations to the real world, are not unique. Having formulated the system design problem in terms of a mathematical model, the systems engineer then seeks a pencil-and-paper design which represents the solution to the mathematical version of his design problem.

Simulation of the mathematical relationships on a computer digital, analog, or hybrid often plays a vital role in this search for a solution. The design obtained will give the engineer an idea of the number of interconnections required, the type of computations that must be carried out, the mathematical description of subsystems needed, etc.

When the mathematical relations that specify the overall system have been derived, the engineer often simulates these relations to obtain valuable insight into the operation of the system and to test the behavior of the model under ideal conditions.

Conclusions about whether or not the mathematics will lead to a reasonable physical system can be drawn, and the sensitivity of the model to parameter variations and unpredictable disturbances can be tested. Various alternative pencil-and-paper designs can be compared and evaluated. After completing the mathematical design and evaluating it through simulation and experimentation, the systems engineer builds a prototype, or breadboard. The process of constructing a prototype is, in a sense, the reverse of the process of modeling, since the prototype is a physical system which must adequately duplicate the derived mathematical relationships.

The prototype is then tested to see whether or not the requirements are met and the constraints satisfied. If the prototype does the job, the work of the systems engineer is essentially complete. Often, for economic and esthetic reasons, the engineer is not satisfied with a system which simply accomplishes the task, and he will seek to improve or optimize his design.

The process of optimization in the pencil-and-paper stage is quite useful in providing insight and a basis for comparison, while the process of optimization in the prototype building stage is primarily concerned with the choice of best components.

The role of optimization in the control-system design problem will be examined in the next section. A particular type of system design problem is the problem of controlling a system. For example, the engineer may be asked to design an autopilot with certain response characteristics or a fast tracking servo or a satellite attitude control system which does not consume too much fuel. The translation of control-system design objectives into the mathematical language of the pencil-and-paper design stage gives rise to what will be called the control problem.

A performance or cost functional which measures the effectiveness of a given control action. The mathematical model, which represents the physical system, consists of a set of relations which describe the response or output of the system for various inputs.

Constraints based upon the physical situation are incorporated in this set of relations. In translating the design problem into a control problem, the engineer is faced with the task of describing desirable physical behavior in mathematical terms.

The objective of the system is often translated into a requirement on the output. For example, if a tracking servo is being designed, the desired output is the signal being tracked or something close to it. Since control signals in physical systems are usually obtained from equipment which can provide only a limited amount of force or energy, constraints are imposed upon the inputs to the system.

These constraints lead to a set of admissible inputs or control signals. Frequently, the desired objective can be attained by many admissible inputs, and so the engineer seeks a measure of performance or cost of control which will allow him to choose the best input.

The choice of a mathematical performance functional is a highly subjective matter, as the choice of one design engineer need not be the choice of another. The experience and intuition of the engineer play an important role in his determination of a suitable cost functional for his problem.

Moreover, the cost functional will depend upon the desired behavior of the system. Most of the time, the cost functional chosen will depend upon the input and the pertinent system variables. When a cost functional has been decided upon, the engineer formulates his control problem as follows: Determine the admissible inputs which generate the desired output and which, in so doing, minimize optimize the chosen performance measure. At this point, optimal-control theory enters the picture to aid the engineer in finding a solution to his control problem.

Such a solution when it exists is called an optimal control. To recapitulate, a control problem is the translation of a control-system design problem into mathematical terms; the solution of a control problem is an idealized pencil-and-paper design which serves to guide the engineer in developing the actual working control system.

Before World War II, the design of control systems was primarily an art. During and after the war, considerable effort was expended on the design of closed-loop feedback control systems, and negative feedback was used to improve performance and accuracy. The first theoretical tools used were based upon the work of Bode and Nyquist. In particular, concepts such as frequency response, bandwidth, gain in decibels , and phase margin were used to design servomechanisms in the frequency domain in a more or less trial-and-error fashion.

This was, in a sense, the beginning of modern automatic-control engineering. The theory of servomechanisms developed rapidly from the end of the war to the beginning of the fifties. Time-domain criteria, such as rise time, settling time, and peak overshoot ratio, were commonly used, and the introduction of the root-locus method by Evans in provided both a bridge between the time- and frequency-domain methods and a significant new design tool. During this period, the primary concern of the control engineer was the design of linear servomechanisms.

Slight nonlinearities in the plant and in the power-amplifying elements could be tolerated since the use of negative feedback made the system response relatively insensitive to variations and disturbances. The competitive era of rapid technological change and aerospace exploration which began around mid-century generated stringent accuracy and cost requirements as well as an interest in nonlinear control systems, particularly relay bistable control systems.

This is not surprising, since the relay is an exceedingly simple and rugged power amplifier. Two approaches, namely, the describing-function and phase-space methods, were used to meet the new design challenge.

The describing-function method enabled the engineer to examine the stability of a closed-loop nonlinear system from a frequency-domain point of view, while the phase-space method enabled the engineer to design nonlinear control systems in the time domain.

Minimum-time control laws in terms of switch curves and surfaces were obtained for a variety of second- and third-order systems in the early fifties.

Proofs of optimality were more or less heuristic and geometric in nature. However, the idea of determining an optimum system with respect to a specific performance measure, the response time, was very appealing; in addition, the precise formulation of the problem attracted the interest of the mathematician. The time-optimal control problem was extensively studied by mathematicians in the United States and the Soviet Union. In the period from to , Bellman, Gamkrelidze, Krasovskii, and LaSalle developed the basic theory of minimum-time problems and presented results concerning the existence, uniqueness, and general properties of the time-optimal control.

The recognition that control problems were essentially problems in the calculus of variations soon followed. Classical variational theory could not readily handle the hard constraints usually imposed in a control problem.

This difficulty led Pon-tryagin to first conjecture his celebrated maximum principle and then, together with Boltyanskii and Gamkrelidze, to provide a proof of it.

The maximum principle was first announced at the International Congress of Mathematicians held at Edinburgh in While the maximum principle may be viewed as an outgrowth of the Hamiltonian approach to variational problems, the method of dynamic programming, which was developed by Bellman around , may be viewed as an outgrowth of the Hamilton-Jacobi approach to variational problems.

Considerable use has been made of dynamic-programming techniques in control problems. Simultaneous with the rapid development of control theory was an almost continuous revolution in computer technology, which provided the engineer with vastly expanded computational facilities and simulation aids.

The ready availability of special- and general-purpose computers greatly reduced the need for closed-form solutions and the demand that controllers amount to simple network compensation. Modern control theory and practice can thus be viewed as the confluence of three diverse streams: the theory of servomeehanisms, the calculus of variations, and the development of the computer.

At present, control theory is primarily a design aid which provides the engineer with insight into the structure and properties of solutions to the optimal-control problem. Specific design procedures and rules of thumb are rather few in number. Moreover, since optimal feedback systems are, in the main, complicated and nonlinear, it is difficult to analyze the effects of variations and disturbances. In addition, the need for accurate measurement of the relevant state variables and the computational difficulties associated with the determination of an optimal control often prevent the economical implementation of an optimal design.

We believe, at any rate, that the present theory will become increasingly useful to the engineer. There are several reasons for this belief. First of all, pencil-and-paper or computer designs of optimum systems can serve as comparison models in the evaluation of alternative designs. Secondly, knowledge of the optimal solution to a given problem provides the engineer with valuable clues to the choice of a suitable suboptimal design.

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## Optimal Control Introduction Theory Applications by Michael Athans Falb

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## Optimal Control Introduction Theory Applications by Athans Michael Falb Peter

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## Optimal Control An Introduction To The Theory And Iits Applications

Michael Athans , Peter L. Geared toward advanced undergraduate and graduate engineering students, this text introduces the theory and applications of optimal control. It serves as a bridge to the technical literature, enabling students to evaluate the implications of theoretical control work, and to judge the merits of papers on the subject. Rather than presenting an exhaustive treatise, Optimal Control offers a detailed introduction that fosters careful thinking and disciplined intuition. It develops the basic mathematical background, with a coherent formulation of the control problem and discussions of the necessary conditions for optimality based on the maximum principle of Pontryagin.