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 10.7 - Summary
10.7 SUMMARY In this chapter, means of describing linear multiple-input multiple-output (MIMO) systems have been developed. Block diagrams and flow graphs are useful for this purpose, as are algebraic methods. Adaptive techniques for modeling and inverse modeling were introduced, and they turned out to be very similar to those used with single-input single-output (SISO) systems except that care is exercised not to commute matrix transfer function operators. Formulas for misadjustment and time constant of the adaptive MIMO plant modeling process have been obtained when using dither schemes B or C. For a given level of misadjustment, learning time is the same as for SISO, multiplied by the number MIMO channels, K. It is a surprise that learning time goes up only linearly with K, not with K2 for example. Inverse controls for MIMO plants were devised. One approach was based on an algebraic technique. A second approach was based on the filtered-є LMS algorithm. Both methods work quite well. Cancelation of plant disturbance in MIMO systems is possible. Several methods were explained for this, offline and online. The filtered-є algorithm proved to be quite useful in finding the disturbance-canceling feedback transfer function [Q(z)] from the plant model [ Adaptive inverse control systems were described for MIMO plants. Two different approaches for finding the inverse controller [Ĉ(z)] were demonstrated, both based on the filtered-є algorithm, one offline, the other online. A practical application of adaptive disturbance canceling in a MIMO system is described in Appendix F by Dr. Thomas Himel of the Stanford Linear Accelerator Center. An eight-input, eight-output adaptive canceler is used 24 hours a day for beam control with a two mile long high-power linear accelerator. Beam position is controlled to within a micron. This is a fascinating application. Many of the rules that are invoked in dealing with MIMO systems are applicable to nonlinear systems, such as noncommutability of operators. The next chapter deals with adaptive inverse control of nonlinear plants. The idea of inverse control for nonlinear systems is a strange one, because nonlinear systems do not generally have inverses. The non-linearity invokes even more rules. In the next chapter, we develop techniques like adaptive inverse control for application to nonlinear SISO and MIMO systems. |
(z)]. Care was taken in all these developments to ensure that the ordering of matrix transfer functions was not commuted.
