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 $H_{\infty}$ 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.