Pleasant Surprises

While developing the principal mathematical results of adaptive inverse control, we would come upon a result from time to time that seemed to us to be a pleasant surprise. Many of these results were desired and were anticipated intuitively. Some were easy to prove and some required a great deal of algebra. We were hoping that the desired results would be true because if they were, it would make the theory simple and the applications easy. Whenever one of the desired results was proven to be true, we said to ourselves: "This is amazing. There must be something right about this approach to adaptive control." The purpose of this brief chapter is to review the pleasant surprises and to summarize the findings of this book.

1. Precise inverse controllers can be devised for minimum-phase plants and, with somewhat delayed response, for nonminimum-phase plants too.
2. The effect of closed-loop response can be obtained in an open-loop feedforward control system by using the feedback inherent in adaptive filtering to find the inverse adaptive controller.
3. Plant disturbance can be optimally canceled using feedback with zero gain around the loop. Best linear least squares plant disturbance canceling can be accomplished without altering the plant transfer function.
4. Plant disturbance canceling can be done independently of plant dynamic control. The optimization of one of these processes is not compromised by the optimization of the other.
5. Achieving an overall system response which is a best least squares estimate of a reference model's response is generally straightforward and natural with adaptive inverse control.
6. If the plant is unstable, it must first be stabilized by feedback. Then the plant and its feedback are subject to adaptive inverse control, treating the plant and its stabilizing feedback as an equivalent plant. The ability to cancel plant disturbance is unaffected by the choice of stabilizing feedback. The ability to achieve a desired overall system dynamic response is also unaffected by the choice of the stabilizing feedback. If the inverse needs delay for its realization, the required delay will not depend on the choice of the stabilization feedback.
7. When creating an inverse controller by the filtered-є or the filtered-X algorithm, small errors in the plant model cause no errors in the controller. Feedback in the adaptive process causes the overall system dynamic response to hover about an equilibrium condition corresponding to an overall dynamic response that is a best least squares match to the reference model response.
8. When adapting an inverse controller with an algorithm other than filtered-є or the filtered-X, errors in the plant model cause errors in the controller. However, these errors are compensated for by second-order errors in the plant dynamics caused by the feedback of the plant disturbance canceler. Thus, the interaction of the multiple adaptive processes in the integrated system cause its overall dynamic response to hover about an equilibrium condition which is a best least squares match to the reference model response. The adaptive feedback provides robust behavior for the overall system.
9. The use of adaptive feedback does not create stability problems, except during startup or during a sudden catastrophic change in plant dynamics when the plant disturbance canceling loop could get out of balance and go unstable. The remedy is to temporarily abstain from plant disturbance canceling by breaking the disturbance canceling loop until the plant model regains a response close to that of the plant.
10. Adaptive inverse control applies readily to the control of MIMO systems. Learning time in a MIMO system goes up only linearly with the number of channels, rather than with the square of the number of channels.
11. Adaptive inverse control applies readily to the control of nonlinear systems, whether SISO or MIMO.
12. Dynamic control of either a minimum-phase plant or a nonminimum-phase plant can certainly be accomplished with adaptive inverse control. But what of a plant having a zero exactly on the unit circle in the z-plane? The inverse of such a plant would need to have a pole on the unit circle, and it would be unstable with either a left-handed or a right-handed impulse response. Here is a case where inverse control should fail. But it does not seem to, and this is surprising. We finish this chapter with a set of experiments making inverse controllers for plants with zeros which are very near the unit circle and in some cases, exactly on it.