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 2 - Wiener Filters
Wiener Filters
2.0 INTRODUCTION Wiener filters are best linear least squares filters which are used for prediction, estimation, interpolation, signal and noise filtering, and so forth. To design them, prior knowledge of the appropriate statistical properties of the input signal(s) is required. The problem is that this prior knowledge is often not available. Adaptive filters are used instead, making use of input data to learn the required statistics. Wiener filter theory is important to us however, because the adaptive filters used here converge asymptotically (in the mean) on Wiener solutions. Understanding Wiener filters is therefore necessary for the understanding of adaptive filters. Wiener filter theory and adaptive filter theory are fundamental to adaptive inverse control. The idea of best linear least squares filtering was introduced by Norbert Wiener in 1949 [1]. The purpose of this chapter is to explain how Wiener filters work and how they can be designed, given the statistical properties of the input signals. Simple forms of Wiener filters can be made optimal without regard to causality. More complicated Wiener filters can be designed, using the Shannon-Bode [2] approach, to be causal and optimal in the least squares sense. A nice perspective on this is given by Kailath [3]. The discussion of Wiener filters will be made with regard to discrete-time digital filters rather than analog filters. The reason for this is that the modern implementation of Wiener filters and adaptive filters is digital almost everywhere. It should be noted however, that Wiener's original work was analog, dealing with continuous rather than discrete signals and systems. In the discussion to follow, Section 2.1 will describe correlation functions, their transforms, and relations between input and output signals of linear discrete filters driven by stochastic inputs. In Section 2.2, the two-sided (noncausal) Wiener filter will be derived. In Section 2.3, the Shannon-Bode approach to causal Wiener filter design will be given. |
