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In this paper, an algorithm that gives the best achievable performance bound on a given control problem is proposed using the H ? loop-shaping design framework. In view of standard design requirements, the robust performance is maximized at low and high frequencies while keeping the robust stability margin above a specified level, and the robust stability margin is directly improved at mid frequencies (around crossover). The proposed frequency-dependent optimization problem is cast in an LMI framework. The resulting solution algorithm simultaneously synthesizes loop-shaping weights and a stabilizing controller that achieve the maximum performance for a given level of robust stability margin corresponding to sufficient gain and phase margins of the closed-loop system. � 2012 John Wiley & Sons, Ltd.

This paper proposes a state-space solution to weight optimization problem in @?"~ loop-shaping control. A pointwise-in-frequency weight optimization framework is transformed into convex optimization searches that are independent of frequency. The introduced optimization problem therefore avoids gridding of the frequency space and consequently, the inaccuracies attributed to fitting transfer functions to magnitude data. In this optimization problem, the order of the weight is specified 'a priori', thus facilitating the synthesis of low-order controllers, which is desirable from an implementation perspective. The proposed solution algorithm simultaneously synthesizes a robust stabilizing controller and a loop-shaping weight (pre-compensator) that maximize the robust stability margin subject to constraints on the performance and the singular values of the weight. Three numerical examples are given to illustrate the effectiveness of the technique.

This paper proposes a design framework that maximizes the robust performance of a closed-loop system via weight optimization in ℋ ∞ loop-shaping control. In line with design objectives, frequency-dependent optimization problems are formulated in LMI framework to maximize robust performance at low and high frequencies, and directly improve the robust stability margin at mid frequencies. The philosophy of the work thus gives the best robust performance that can be achieved on a particular problem once a level of robust stability margin is demanded. Loop-shaping weights and controller are simultaneously synthesized from the proposed algorithm.

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.

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.