H_∞ Control Co-Design for Uncertain Polytopic Systems

Control co-design (CCD) refers to approaches that fully integrate plant and control system interactions, using an optimization-based methodology where physical and control system designs are addressed simultaneously. In this process, the design of physical systems and their controllers are typically interdependent tasks. This study explores a bi-level (nested) control co-design approach integrated with robust H_∞ control for the combined design of both the physical system and its controller. While the nested approach is well-established in the literature, with the linear quadratic regulator commonly used for the controller optimization, this work introduces a novel approach by focusing on minimizing the H_∞ norm/ guaranteed cost as the controller optimization problem instead. The proposed method seeks to bridge the fields of control co-design and robust control, extending the application of control co-design to systems subject to disturbances and parametric uncertainties. It assumes that any uncertain system parameter can be described as a subset of a polytopic domain, and that a feedback-stabilizing control can be synthesized to ensure the H_∞ norm/guaranteed cost of the system is bounded, thus minimizing the impact of exogenous inputs on the system’s output. The synthesis conditions are demonstrated through linear matrix inequalities and an adaptation of traditional Lyapunov stability conditions. To illustrate the method presented, this study revisits two previously addressed control co-design problems in the literature: a scalar plant and an active car suspension system. The results indicate that the integration of CCD with robust control strategies not only guarantees the system performance and disturbance rejection but also provides a systematic approach for managing uncertainties within a polytopic framework.