Hybrid model predictive control application towards optimal semi-active suspension

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 modeled 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 optimal control law derived from the corresponding un-constrained system, i.e. "clipped-optimal", remain optimal for the constrained case? If the optimal control law of the constrained system does deviate from its un-constrained counter-part, how different are they? What is the structure of the optimal control law? In this paper, we attempt to answer some of the above questions by utilizing the recent development in model predictive control (MPQ of hybrid dynamical systems. The constrained quarter-car semi-active suspension is modeled as a switching affine system, where the switching is determined by the activation of passivity constraints, force saturation, and maximum power dissipation limits. Theoretically, over an infinite prediction horizon the MPC controller corresponds to the exact optimal controller. The performance of different finite-horizon hybrid MPC controllers is tested in simulation using mixed-integer quadratic programming. Then, for short-horizon MPC controllers, we derive the explicit optimal control law and show that the optimal control is piecewise affine in state. In particular, we show that for horizon equal to one the explicit MPC control law corresponds to clipped LQR. We will compare the derived optimal control law to various semi-active control laws in the literature including the well-known "clipped-optimal". We will evaluate their corresponding performances for both a deterministic shock input case and a stochastic random disturbances case through simulations.