Preface
In this book, methods of adaptive signal processing are borrowed from the field of digital signal processing to solve problems in dynamic systems control. Adaptive filters, whose design and behavioral characteristics are well known in the signal processing world, can be used to control plant dynamics and to minimize the effects of plant disturbance. Plant dynamic control and plant disturbance control are treated herein as two separate problems. Optimal least squares methods are developed for these problems, methods that do not interfere with each other. Thus, dynamic control and disturbance canceling can be optimized without one process compromising the other. Better control performance is the result. This is not always the case with existing control techniques. Inverse control of plant dynamics involves feed-forward compensation, driving the plant with a filter whose transfer function is the inverse of that of the plant. Inverse compensation is well known in signal processing and communications. Every MODEM in the world uses adaptive filters for channel equalization. Similar techniques are described here for plant dynamic control. Inverse control is feed-forward control. The same precision of feedback that is obtained with existing control techniques is also obtained with adaptive feed-forward control since feedback is incorporated in the adaptive algorithm for obtaining the parameters of the feed-forward compensator. Inverse control can be used effectively with minimum phase and non-minimum phase plants. It cannot work with unstable plants, however. They must first be stabilized with conventional feedback, of any design that simply achieves stability. Then the plant and stabilizing feedback can be treated as an equivalent stable plant that can be controlled in the usual way with adaptive inverse control. Model reference control can be readily incorporated into adaptive inverse control. Adaptive noise canceling techniques are described that allow optimal reduction of plant disturbance, in the least squares sense. Adaptive noise canceling does not affect inverse control of plant dynamics. Inverse control of plant dynamics does not affect adaptive disturbance canceling. If initial feedback is needed to provide plant stabilization, the design of the stabilizer has no effect on the optimality of the adaptive disturbance canceler. The designs of the adaptive inverse controller and of the adaptive disturbance canceler are quite simple once the control engineer gains a mastery of adaptive signal processing. This book provides an introductory presentation of this subject with enough detail to do system design. The mathematics is simple and indeed the whole concept is simple and easy to implement, especially when compared with the complexity of current control methods. Adaptive inverse control is not only simple, but it affords new control capabilities that can often be superior to those of conventional systems. Many practical examples and applications are shown in the text. Another feature of adaptive inverse control is that the same methods can be applied to adaptive control of nonlinear plants. This is surprising because nonlinear plants do not have transfer functions. But approximate inverses are possible. Experimental results with nonlinear plants have shown great promise. Optimality cannot be proven yet, but excellent results have been obtained. This is a very promising subject for research. The whole area of nonlinear adaptive filtering is a fascinating research field that already shows great results and great promise. This book was originally published under the title Adaptive Inverse Control. We are grateful to IEEE Press and John Wiley, Inc. for bringing it back into print. We are also grateful to colleagues Gene Franklin, Karl Johan Astrom, Jose Cruz, Brian Anderson, Paul Werbos, and Shmuel Merhav for their early comments, suggestions, and feedback. We are grateful to former Stanford students Steve Piche, Michel Bilello, Gregory Plett, and Ming-Chang Liu who confirmed the results with experiments and who assisted with preparation of the drawings and final manuscript. Bernard Widrow Eugene Walach |
TABLE OF CONTENTS Chapter 3 - Adaptive LMS Filters
Adaptive LMS Filters
3.0 INTRODUCTION The theory of adaptive filtering is fundamental to adaptive inverse control. Adaptive filters are used for plant modeling, for plant inverse modeling, and to do plant disturbance canceling. At every step of the way, adaptive filtering is present. It is important to think of the adaptive filter as a building block, having an input signal, having an output signal, and having a special input signal called the "error" which is used in the learning process. This building block can be combined with other building blocks to make adaptive inverse control systems. The purpose of this chapter is to present a brief overview of the theory of adaptive digital filtering. We will describe several applications for adaptive filters, and will discuss stability, rate of convergence, and effects of noise in the impulse response.1 We will derive relationships between speed of adaptation and performance of adaptive systems. In general, faster adaptation leads to more noisy adaptive processes. When the input environment of an adaptive system is statistically nonstationary, best performance is obtained by a compromise between fast adaptation (necessary to track variations in input statistics) and slow adaptation (necessary to contain the noise in the adaptive process). A number of these issues will be studied both analytically and by computer simulation. The context of this study will be restricted to adaptive digital filters driven by the LMS adaptation algorithm of Widrow and Hoff [1] - [5]. This algorithm and algorithms similar to it have been used for many years in a wide variety of practical applications [6]. We are reviewing a statistical theory of adaptation. Stability and rate of convergence are analyzed first; then gradient noise and its effects upon performance are assessed. The concept of "misadjustment" is defined and used to establish design criteria for a sample problem, an adaptive predictor. Finally, we consider an application of adaptive filtering to adaptive noise canceling. The principles of adaptive noise canceling are derived and some experimental results are presented. These include utilization of the noise canceling techniques to improve results of adult and fetal electrocardiography.
1 Since the parameters of the adaptive filter are data dependent and time variable during adaptation, the adaptive filter does not have an impulse response defined in the usual way. An instantaneous impulse response can be defined as the impulse response that would result if adaptation were suddenly stopped and the parameters were fixed at instantaneous values. |
