Optimal Control

Linear Quadratic Methods

Brian D. O. Anderson, John B. Moore

• Publisher: Dover Publications
• Year: 2014
• Print ISBN: 9780486457666
• eBook ISBN: 9780486151571
• Topic: Technology & Engineering
• Subtopic: Engineering General
• Index: Technology & Engineering

This augmented edition of a respected text teaches the reader how to use linear quadratic Gaussian methods effectively for the design of control systems. It explores linear optimal control theory from an engineering viewpoint, with step-by-step explanations that show clearly how to make practical use of the material.
The three-part treatment begins with the basic theory of the linear regulator/tracker for time-invariant and time-varying systems. The Hamilton-Jacobi equation is introduced using the Principle of Optimality, and the infinite-time problem is considered. The second part outlines the engineering properties of the regulator. Topics include degree of stability, phase and gain margin, tolerance of time delay, effect of nonlinearities, asymptotic properties, and various sensitivity problems. The third section explores state estimation and robust controller design using state-estimate feedback.
Numerous examples emphasize the issues related to consistent and accurate system design. Key topics include loop-recovery techniques, frequency shaping, and controller reduction, for both scalar and multivariable systems. Self-contained appendixes cover matrix theory, linear systems, the Pontryagin minimum principle, Lyapunov stability, and the Riccati equation. Newly added to this Dover edition is a complete solutions manual for the problems appearing at the conclusion of each section.

Part 1. Basic Theory of the Optimal Regulator

1

Introduction

1.1 LINEAR OPTIMAL CONTROL

The methods and techniques of what is now known as “classical control” will be familiar to most readers. In the main, the systems or plants that can be considered by using classical control ideas are linear and time invariant, and have a single input and a single output. The primary aim of the designer using classical control design methods is to stabilize a plant, whereas secondary aims may involve obtaining a certain transient response, bandwidth, disturbance rejection, steady state error, and robustness to plant variations or uncertainties. The designer’s methods are a combination of analytical ones (e.g., Laplace transform, Routh test), graphical ones (e.g., Nyquist plots, Nichols charts), and a good deal of empirically based knowledge (e.g., a certain class of compensator works satisfactorily for a certain class of plant). For higher-order systems, multiple-input systems, or systems that do not possess the properties usually assumed in the classical control approach, the designer’s ingenuity is generally the limiting factor in achieving a satisfactory design.

Two of the main aims of modern, as opposed to classical, control are to de-empiricize control system design and to present solutions to a much wider class of control problems than classical control can tackle. One of the major ways modern control sets out to achieve these aims is by providing an array of analytical design procedures that facilitate the design task.

In the early stages of a design, the designer must use his familiarity with the engineering situation, and understanding of the underlying physics, to formulate a sensible mathematical problem. Then the analytical design procedures, often implemented these days with commercial software packages, yield a solution—which usually serves as a first cut in a trial and error iterative process.

Optimal control is one particular branch of modern control that sets out to provide analytical designs of a specially appealing type. The system that is the end result of an optimal design is not supposed merely to be stable, have a certain bandwidth, or satisfy any one of the desirable constraints associated with classical control, but it is supposed to be the best possible system of a particular type—hence, the word optimal. If it is both optimal and possesses a number of the properties that classical control suggests are desirable, so much the better.

Linear optimal control is a special sort of optimal control. The plant that is controlled is assumed linear, and the controller, the device that generates the optimal control, is constrained to be linear. Linear controllers are achieved by working with quadratic performance indices. These are quadratic in the control and regulation/tracking error variables. Such methods that achieve linear optimal control are termed Linear-Quadratic (LQ) methods. Of course, one may well ask: why linear optimal control, as opposed simply to optimal control? A number of justifications may be advanced; for example, many engin...

Table of contents

• Cover
• Title Page
• Copyright Page
• Preface
• Contents
• Part I. Basic Theory of the Optimal Regulator
• Part II. Properties and Application of the Optimal Regulator
• Author Index
• Subject Index
• Solutions