- TITLE: Automatic Loop-Shaping
of QFT Controllers Via Linear Programming
- AUTHORS: Yossi Chait, Qian Chen and C.V. Hollot
- ABSTRACT: In this paper we focus on the following loop-shaping
problem: Given a nominal plant and QFT bounds, synthesize a controller
that achieves closed-loop stability, satisfies the QFT bounds and has
minimum high-frequency gain. The usual approach to this problem involves
loop shaping in the frequency domain by manipulating the poles and zeroes
of the nominal loop transfer function. This process now aided by recently-developed
computer-aided design tools, proceeds by trial and error, and its success
often depends heavily on the experience of the loop-shaper. Thus, for
the novice and first-time QFT users, there is a genuine need for an
automatic loop-shaping tool to generate a first-cut solution. Clearly,
such an automatic process must involve some sort of optimization, and,
while recent results on convex optimization have found fruitful application
in other areas of control design, their immediate usage here is precluded
by the inherent non-convexity of the QFT bounds. Alternatively, these
QFT bounds can be over-bounded by convex sets, as done in some of the
recent approaches to automatic loop-shaping, but this conservatism can
have a strong and adverse effect on meeting the original design specifications.
With this in mind, we approach the automatic loop-shaping problem by
first stating conditions under which QFT bounds can be dealt with in
a non-conservative fashion using linear inequalities. We will argue
that for a first-cut design, these conditions are often satisfied in
the most critical frequencies of loop-shaping and are violated in frequency
bands where approximation leads to negligible conservatism in the control
design. These results immediately lead to an automated loop-shaping
algorithm involving only linear programming techniques, which we illustrate
via an example
- STATUS: submitted to ASME J. Dynamic Systems, Measurement,
and Control; also, an earlier version appears in Procs. Symp on Quantitative
Feedback Theory and other Frequency Domain Methods and Applications,
Glasgow, Scotland, September, 1997, pp. 13-28.
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