Iterative Design of the Reduced-Order Weight and Controller for the $H_{\infty }$ Loop-Shaping Method Under Open-Loop Magnitude Constraints for SISO Systems

The H _infin loop-shaping method is known to be an effective control method. However, it has two drawbacks. The first is that it is difficult to select appropriate loop-shaping weights, and the second is that the resulting controller is very complex. For the first drawback, Lanzon has proposed a suboptimal loop-shaping weight design method. It is formulated as a generalized eigenvalue minimization problem (GEVP). This suboptimal loop-shaping weight design method provides high-order weights, exacerbating the second drawback. To resolve these two drawbacks, a reduced-order loop-shaping weight design method is proposed for SISO systems in this paper. In the proposed method, the weight structure is first fixed, and the weight is then decomposed into a frequency-dependent vector and parameter matrices characterizing the loop-shaping weight. Since the open-loop constraints are represented as linear matrix inequalities with respect to the parameter matrices, the proposed reduced-order loop-shaping weight design problem for SISO systems is formulated as a GEVP, as well as Lanzon´s suboptimal loop-shaping weight design method. The proposed method can reduce the designer´s burden, although it is only valid for SISO systems. The effectiveness of the proposed method is verified experimentally by velocity control of a belt-driven two-mass system.