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 5 - Inverse Plant Modeling
Inverse Plant Modeling
5.0 INTRODUCTION Control concepts taught in Chapter 1 involve the use of adaptive plant inverses as controllers in feedforward control configurations. Pursuing these ideas, our next step is the development of general techniques for finding inverses of plants that need to be controlled. We shall restrict our development to apply only to stable plants. If the plant of interest is unstable, conventional feedback should be applied to stabilize it. Then the combination of the plant and its feedback stabilizer can be regarded as an equivalent stable plant. The subject is discussed in detail in Appendix D. Only linear, single-input single-output (SISO) systems will be treated here. Nonlinear and MIMO systems will be discussed subsequently in Chapters 10 and 11. The plant generally has poles and zeros. The inverse of the plant therefore should have zeros and poles. If the plant is minimum-phase, that is, has all of its zeros inside the unit circle in the z-plane, then the inverse will be stable with all of its poles inside the unit circle. If the plant is nonminimum-phase, then some of the poles of the inverse will be outside the unit circle and the inverse will be unstable. In general, not knowing whether the plant is minimum-phase or not, there is uncertainty about the feasibility of making a plant inverse. This uncertainty can for the most part be overcome and excellent inverses can be made in practice by using the appropriate adaptive inverse modeling techniques. We shall begin with a discussion of inverse modeling of minimum-phase plants, then discuss inverse modeling of nonminimum-phase plants. Model-reference inverse modeling will be described next. The effects of plant disturbance will then be considered, and online and offline inverse modeling will be illustrated. Finally, the effects of gradient noise upon the plant inverse model will be analyzed, leading to expressions for noise in the weights of the inverse model and for the variance of the dynamic system error of the entire control system. |
