Parametric Multi-objective Optimization for Simultaneous and Nested Control Co-Design Formulations with Tube-based Model Predictive Controllers

Abstract Control co-design (CCD) aims to jointly optimize physical systems and controllers to achieve superior system-level performance compared to traditional sequential design. However, practical challenges, such as handling uncertainty, ensuring stability and feasibility, and enabling design exploration of multiple criteria over varying requirements, limit its application. This paper introduces a parametric multi-objective optimization framework for CCD problems based on tube-based model predictive control (MPC), which improves closed-loop performance while maintaining constraint satisfaction under stochastic disturbances through constraint tightening. By integrating parametric optimization, the proposed approach captures how optimal designs vary with respect to parameters (e.g., control limits), allowing efficient trade-off analysis and decision-making without re-solving optimization problems. Simultaneous and nested CCD formulations are developed and demonstrated on a numerical example and an active suspension system. The CCD solutions dominate most of designs solved by control-only and sequential strategies. In addition, quantitative results, evaluated by the parametric hypervolume indicator (pHVI), show that the CCD approach yields higher-performing and more robust solutions than other strategies.