Controller reduction with closed loop H infinity performance constraints: A QFT perspective

Abstract

Controller reduction with closed loop H 1 1 performance constraints: a QFT perspective GILLES FERRERES{{ and YANN LE GORREC{Assume that an initial stabilizing controller K 0 … s † , which satis® es various closed loop frequency domain speci® cations, has been a priori synthesized using, e.g. H1 control, · synthesis techniques or closed loop convex synthesis. Remembering that the order of K 0 … s † is typically at least equal to the order of the plant to be controlled, the aim of this paper is to ® nd a stabilizing reduced order controller, which also satis® es the performance speci® cations and which moreover minimizes the open loop bandwidth. To this aim, the structured singular value · is used to translate at each frequency closed loop frequency domain speci® cations into requirements on the frequency response of the controller. The principle of our reduction method is thus very close to the original idea of the SISO QFT design approach, except that the problem of translating closed loop frequency domain speci® cations into open loop ones is much more complex in the MIMO case. The problem reduces to the issue of ® nding a controller, whose frequency response belongs at each frequency to a template. The convexity of these templates greatly facilitates the practical realization of the controller. As a ® nal point, the method is successfully applied to a standard H1 problem, which is extracted from the · Analysis and Synthesis Toolbox of Matlab, namely the synthesis of an autopilot for the space shuttle.

Robust stabilization of an LFT model

“…An alternative is either to use constant scales [6], which do not depend on frequency (but time varying uncertain parameters are now handled), or to use a positivity approach with frequency-dependent multipliers [2,8] : in this last case, Bilinear Matrix Inequalities can be solved either directly (at the price of an extreme computational complexity), or iteratively with LMI solvers. Finally, the QFT approach is now a classical method for SISO systems, but the extension to MIMO plants is not so obvious [3,19].…”

Frequency domain curve fitting: a generalizedμapproach

“…As a "rst example, assume that a high-order plant model is available, and the issue is to compute a reduced order model with the same frequency domain characteristics. As a second problem, consider a classical QFT problem: templates on the frequency response of the SISO or MIMO [8,9] controller have been computed at each point of a frequency gridding (i.e. at a given frequency, each point of the template is known to give a closed-loop response satisfying the frequency domain speci"cations), and the issue is to "nd a controller state-space representation, whose frequency response belongs to these templates.…”