Incorporating smoothness into weight optimization for ℋ∞ loop-shaping design

Smoothness constraints are formulated for weights in ℋ∞ loop-shaping design in order to ensure smooth variations in their magnitude response. Smoothness in the magnitude response of weights prevents the cancelation of lightly damped poles/zeros of the plant when the shaped plant is formed by cascading the nominal plant with the weights. It also allows fitting by low-order transfer functions when the smooth variations are computed point-wise in frequency. Gradients of weights, expressed in dB/decade, are used to formulate the smoothness constraints in LMI form as additional constraints to those on the singular values and condition numbers of weights in [Lanzon, 2005, Weight optimization in ℋ∞ loop-shaping, Automatica, 41(1): 1018–1029]. The resulting solution algorithm maximizes the robust stability margin while simultaneously synthesizing smooth weights and a stabilizing controller subject to the shaped plant lying within a specified region that depicts the closed-loop design requirements. Keywords: ℋ∞ loop-shaping; Smoothness constraint; Weight optimization; Robust control.