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 10.3.2 - The Filtered-є Approach
10.3.2 The Filtered-є Approach Adaptive inverse control of MIMO systems can also be done by making use of the filtered-є method of adaptive inverse control introduced in Chapter 7. None of the other adaptive inverse control techniques taught above, including the filtered-X algorithm, can be used with MIMO systems because they all require impermissible commutation of matrix operators. The ordering of matrix operations and the ordering of signal flow through matrix filters occurs in a natural way with the filtered-є approach, leaving it a viable and useful technique for MIMO operation. A filtered-є adaptation scheme for a MIMO plant is shown in Fig. 10.18. In order to adapt the controller [C(z)] COPY, an error vector is required. Ideally this would be є', shown in Fig. 10.18. Since є' is not available, a filtered error vector is used in its place. The filter is [PΔ-l(z)], a delayed plant inverse, estimated from [ A few details about finding the delayed plant inverse filter, especially pertinent for MIMO systems, need to be discussed. Suppose that online inverse modeling is being done (switch left in Fig. 10.18). The inverse modeling signal is then the filtered error vector, as indicated in Fig. 10.18. The reason for using the filtered error for this purpose is to cause the input for [ Analysis of the system of Fig. 10.18 has been attempted, but because of the noncommutability of the matrix operator, no simple expressions for misadjustment, noise in the weight vector of [Ĉ(z)], or range of µ for stability have been obtained so far. The time constants for the adaptation of [Ĉ(z)]are the same as if [Ĉ(z)]were isolated from the rest of the system. The time constants for the adaptation of the plant model [ |
(z)]. Both online and offline means for finding [PΔ-l(z)], the error filter, are indicated in Fig. 10.18. The plant model, [
