Smooth weight optimization in loop-shaping design

Smooth variation in the magnitude response of weights facilitates ℋ ∞ loop-shaping design, as it prevents the cancellation of important modes of the system, for example, lightly damped poles/zeros of flexible structures, when the shaped plant is formed based on closed-loop design specifications. For accurate fitting of transfer functions to magnitude data, smooth weights also allow low-order transfer functions to be used when such smooth variations in their magnitude responses are computed point-wise in frequency. In this paper, smoothness constraints for weights, expressed as gradient constraints on a log scale in dB/decade, which is intuitive from a design perspective, are imposed in a weight optimization framework for ℋ ∞ loop-shaping control. This work builds on [A. Lanzon, Weight optimization in ℋ ∞ loop-shaping, Automatica 41 (1) (2005) 1201–1208], where additional constraints are formulated in linear matrix inequality (LMI) form to cast a complete weight optimization framework. The resulting algorithm thus maximizes the robust stability margin and simultaneously synthesizes smooth weights along with a stabilizing controller. A numerical example is given to elucidate the efficacy of the smoothness constraints in ℋ ∞ loop-shaping control.