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This book introduces a student to Pontryagin's "Maximum" Principle in a tutorial style. How to formulate an optimal control problem and how to apply Pontryagin's theory are the main topics. Numerous examples are used to discuss pitfalls in problem formulation. Figures are used extensively to complement the ideas. An entire chapter is dedicated to solved example problems: from the classical Brachistochrone problem to modern space vehicle guidance. These examples are also used to show how to obtain optimal nonlinear feedback control. Students in engineering and mathematics will find this book to be a useful complement to their lecture notes. Table of Contents: 1 Problem Formulation 1.1 The Brachistochrone Paradigm 1.1.1 Development of a Problem Formulation 1.1.2 Scaling Equations 1.1.3 Alternative Problem Formulations 1.1.4 The Target Set 1.2 A Fundamental Control Problem 1.2.1 Problem Statement 1.2.2 Trajectory Optimization and Feedback Control 2 Pontryagin's Principle 2.1 A Fundamental Control Problem 2.2 Necessary Conditions 2.3 Minimizing the Hamiltonian 2.3.1 Brief History 2.3.2 KKT Conditions for Problem HMC 2.3.3 Time-Varying Control Space 3 Example Problems 3.1 The Brachistochrone Problem Redux 3.2 A Linear-Quadratic Problem 3.3 A Time-Optimal Control Problem 3.4 A Space Guidance Problem 4 Exercise Problems 4.1 One-Dimensional Problems 4.1.1 Linear-Quadratic Problems 4.1.2 A Control-Constrained Problem 4.2 Double Integrator Problems 4.2.1 L1-Optimal Control 4.2.2 Fuller's Problem 4.3 Orbital Maneuvering Problems 4.3.1 Velocity Steering 4.3.2 Max-Energy Orbit Transfer 4.3.3 Min-Time Orbit Transfer References Index
A rigorous introduction to optimal control theory, which will enable engineers and scientists to put the theory into practice.
The calculus of variations is used to find functions that optimize quantities expressed in terms of integrals. Optimal control theory seeks to find functions that minimize cost integrals for systems described by differential equations. This book is an introduction to both the classical theory of the calculus of variations and the more modern developments of optimal control theory from the perspective of an applied mathematician. It focuses on understanding concepts and how to apply them. The range of potential applications is broad: the calculus of variations and optimal control theory have been widely used in numerous ways in biology, criminology, economics, engineering, finance, management science, and physics. Applications described in this book include cancer chemotherapy, navigational control, and renewable resource harvesting. The prerequisites for the book are modest: the standard calculus sequence, a first course on ordinary differential equations, and some facility with the use of mathematical software. It is suitable for an undergraduate or beginning graduate course, or for self study. It provides excellent preparation for more advanced books and courses on the calculus of variations and optimal control theory.
This textbook offers a concise yet rigorous introduction to calculus of variations and optimal control theory, and is a self-contained resource for graduate students in engineering, applied mathematics, and related subjects. Designed specifically for a one-semester course, the book begins with calculus of variations, preparing the ground for optimal control. It then gives a complete proof of the maximum principle and covers key topics such as the Hamilton-Jacobi-Bellman theory of dynamic programming and linear-quadratic optimal control. Calculus of Variations and Optimal Control Theory also traces the historical development of the subject and features numerous exercises, notes and references at the end of each chapter, and suggestions for further study. Offers a concise yet rigorous introduction Requires limited background in control theory or advanced mathematics Provides a complete proof of the maximum principle Uses consistent notation in the exposition of classical and modern topics Traces the historical development of the subject Solutions manual (available only to teachers) Leading universities that have adopted this book include: University of Illinois at Urbana-Champaign ECE 553: Optimum Control Systems Georgia Institute of Technology ECE 6553: Optimal Control and Optimization University of Pennsylvania ESE 680: Optimal Control Theory University of Notre Dame EE 60565: Optimal Control
The theory of optimal control systems has grown and flourished since the 1960's. Many texts, written on varying levels of sophistication, have been published on the subject. Yet even those purportedly designed for beginners in the field are often riddled with complex theorems, and many treatments fail to include topics that are essential to a thorough grounding in the various aspects of and approaches to optimal control. Optimal Control Systems provides a comprehensive but accessible treatment of the subject with just the right degree of mathematical rigor to be complete but practical. It provides a solid bridge between "traditional" optimization using the calculus of variations and what is called "modern" optimal control. It also treats both continuous-time and discrete-time optimal control systems, giving students a firm grasp on both methods. Among this book's most outstanding features is a summary table that accompanies each topic or problem and includes a statement of the problem with a step-by-step solution. Students will also gain valuable experience in using industry-standard MATLAB and SIMULINK software, including the Control System and Symbolic Math Toolboxes. Diverse applications across fields from power engineering to medicine make a foundation in optimal control systems an essential part of an engineer's background. This clear, streamlined presentation is ideal for a graduate level course on control systems and as a quick reference for working engineers.
First truly up-to-date treatment offers a simple introduction to optimal control, linear-quadratic control design, and more. Broad perspective features numerous exercises, hints, outlines, and appendixes, including a practical discussion of MATLAB. 2005 edition.
This is a long-overdue volume dedicated to space trajectory optimization. Interest in the subject has grown, as space missions of increasing levels of sophistication, complexity, and scientific return - hardly imaginable in the 1960s - have been designed and flown. Although the basic tools of optimization theory remain an accepted canon, there has been a revolution in the manner in which they are applied and in the development of numerical optimization. This volume purposely includes a variety of both analytical and numerical approaches to trajectory optimization. The choice of authors has been guided by the editor's intention to assemble the most expert and active researchers in the various specialities presented. The authors were given considerable freedom to choose their subjects, and although this may yield a somewhat eclectic volume, it also yields chapters written with palpable enthusiasm and relevance to contemporary problems.
Rail is potentially a very efficient form of transport, but must be convenient, reliable and cost-effective to compete with road and air transport. Optimal control can be used to find energy-efficient driving strategies for trains. This book describes the train control problem and shows how a solution was found at the University of South Australia. This research was used to develop the Metromiser system, which provides energy-efficient driving advice on suburban trains. Since then, this work has been modified to find practical driving strategies for long-haul trains. The authors describe the history of the problem, reviewing the basic mathematical analysis and relevant techniques of constrained optimisation. They outline the modelling and solution of the problem and finally explain how the fuel consumption can be minimised for a journey, showing the effect of speed limits and track gradients on the optimal driving strategy.
How do you fly an airplane from one point to another as fast as possible? What is the best way to administer a vaccine to fight the harmful effects of disease? What is the most efficient way to produce a chemical substance? This book presents practical methods for solving real optimal control problems such as these. Practical Methods for Optimal Control Using Nonlinear Programming, Third Edition focuses on the direct transcription method for optimal control. It features a summary of relevant material in constrained optimization, including nonlinear programming; discretization techniques appropriate for ordinary differential equations and differential-algebraic equations; and several examples and descriptions of computational algorithm formulations that implement this discretize-then-optimize strategy. The third edition has been thoroughly updated and includes new material on implicit Runge–Kutta discretization techniques, new chapters on partial differential equations and delay equations, and more than 70 test problems and open source FORTRAN code for all of the problems. This book will be valuable for academic and industrial research and development in optimal control theory and applications. It is appropriate as a primary or supplementary text for advanced undergraduate and graduate students.
Geometric control theory is concerned with the evolution of systems subject to physical laws but having some degree of freedom through which motion is to be controlled. This book describes the mathematical theory inspired by the irreversible nature of time evolving events. The first part of the book deals with the issue of being able to steer the system from any point of departure to any desired destination. The second part deals with optimal control, the question of finding the best possible course. An overlap with mathematical physics is demonstrated by the Maximum principle, a fundamental principle of optimality arising from geometric control, which is applied to time-evolving systems governed by physics as well as to man-made systems governed by controls. Applications are drawn from geometry, mechanics, and control of dynamical systems. The geometric language in which the results are expressed allows clear visual interpretations and makes the book accessible to physicists and engineers as well as to mathematicians.