Iterative design of the reduced-order controller for the H∞ loop-shaping method under open-loop magnitude constraints and application to gain-scheduling control

The H_∞ 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. This suboptimal loop-shaping weight design method provides high order weights, emphasizing the second drawback. To resolve these two drawbacks, a reduced-order loop-shaping weight and a stabilizing controller design methods are proposed in this paper. In the proposed method, the weight structure is first fixed, and the weight is decomposed into the frequency-dependent vector and the 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 reduced-order loop-shaping weight design problem is formulated as a generalized eigenvalue minimization problem as well as the Lanzon´s suboptimal loop-shaping weight design method. Moreover, the stabilizing controller is first fixed to the initial stabilizing controller. The initial stabilizing controller is designed for the shaped plant obtained by the reduced-order weight design, by solving linear matrix inequalities. The proposed method can reduce the designer´s burden. The effectiveness of the proposed method is verified experimentally by the gain-scheduling control of a vertical-type inverted pendulum system.