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 10.4 - Plant Disturbance Canceling in MIMO Systems
10.4 PLANT DISTURBANCE CANCELING IN MIMO SYSTEMS Plant disturbance canceling in SISO systems was described in Chapter 8. The same techniques can be utilized, with some modification, to cancel disturbance in MIMO plants. Modification is required in some places in order to ensure that the ordering of signal flow in the disturbance canceling feedback is not commuted. Figure 8.1 shows a SISO plant disturbance canceler using online adaptation to form Q(z), and Fig. 8.3 shows a SISO plant disturbance canceler using an offline process to form Q(z). Disturbance cancelers such as these should generally not be used for MIMO applications. The reason is that the plant disturbance must first be filtered by Q(z) and then applied to P(z) to achieve disturbance canceling. But Q(z) is formed in the systems of Figs. 8.1 and 8.3 as an inverse of The filtered-єalgorithm is used to find [Q(z)] in Fig. 10.19. In order to filter the error vector, it is necessary to have an inverse of the plant [P(z)]. A delayed inverse [ It may be noted that the use of a delayed inverse in Fig. 10.19 often makes a better inverse and otherwise causes no additional problems. The delay must be accounted for, however, when adapting Q(z). This has been done by delaying the input of [Q(z)] by Δ + 1 units of time, Δ units for the delay in the inverse of [ Referring again to Fig. 10.19, we should note that the synthetic noise vector number one used to obtain [Q(z)] should, in principle, be spectrally like the plant disturbance. The synthetic noise vector number two used to obtain [ It is useful to note that if [P(z)] is minimum-phase and if the sampling frequency is high so that z-1 is a very small delay, then the error in generating [Q(z)] will be very small and [Q(z)] will be a close approximation to an exact inverse of [
Figure 10.18 Filtered-є adaptation for a MIMO plant. |
(z), following 
