Eiji Miyazako, March 1997 (in Japanese)

This paper presents two methods of designing deadbeat servomechanisms
using multi-rate controllers which achieve faster settling time
than ordinary single-rate controllers.
According to the single-rate deadbeat control theory,
single-rate type-one deadbeat servomechanisms require at least
*n+1* samples or *(n+1)T* settling time
to track a reference signal completely if taking account of all
possible initial states.
The dimension of the plant is *n* and the sampling period is *T*.
How can we achieve a faster settling time with the same sampling period?
To answer this question is an interesting problem.

In the case of single-rate control, schedules of sampling and zero-order-hold must be the same. But, the sampling and hold operators in themselves are different devices. It is desirable to choose their schedules independently of each other so that we obtain the best control performance.

There are some studies about comparison between multi-rate and single-rate controllers. About strong stabilization, multi-rate controllers can perform better than single-rate ones. But, in terms of robust stabilization, the control ability of multi-rate controllers is equal to that of single-rate controllers. As for deadbeat regulators, the author has shown that multi-rate controllers require less number of samples than single-rate ones to achieve the same settling time as single-rate ones.

In this paper, the author uses the settling time of deadbeat servomechanisms as a control measure to compare the ability of multi-rate control with single-rate one. The author proposes a method of designing multi-rate type-one deadbeat servomechanisms with ``lifting'' technique. It is shown that multi-rate controllers can achieve the same settling time with less number of samples. The minimum settling time is clarified and a necessary and sufficient condition for achieving it is derived. Since this design method only takes account of sampling instants, there sometimes remains ripples between them. The author also proposes a design method of ripple-free deadbeat servomechanisms. The settling time of the two servomechanisms proposed is compared with the minimum settling time of single-rate control. This comparison shows that multi-rate controllers can enhance control performance.

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