The optimal control problem of a quarter-car semi-active suspension has been studied in the past. Considering that a quarter-car semi-active suspension can either be modelled as a linear system with state dependent constraint on control ( of actuator force) input, or a bi-linear system with a control (of variable damping coefficient) saturation, the seemingly simple problem poses several interesting questions and challenges. Does the saturated version of the optimal control law derived from the corresponding un-constrained system, i.e. "clipped- optimal'', remain optimal for the constrained case as suggested in some previous publications? Or should the optimal deviate from the "clipped- optimal'' as suggested in other publications? If the optimal control law of the constrained system does deviate from its unconstrained counter- part, how different are they? What is the structure of the optimal control law? Does it retain the linear state feedback form ( as the unconstrained case)? In this paper, we attempt to answer some of the above questions by utilizing the recent development in model predictive control (MPC) of hybrid dynamical systems.
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