Automatic Loop-Shaping of QFT Controllers Via Linear Programming

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 applications in other areas of control design, their immediate usage here is precluded by the inherent nonconvexity of QFT bounds. Alternatively, these QFT bounds can be over-bounded by convex sets, as done in some 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.