In this paper, a two-degree-of-freedom model of a semi-actively suspended vehicle is used as a starting point in the design of an optimal suspension. The optimization is performed with respect to a quadratic performance index reflecting suspension design constraints and ride quality requirements. Two closely related mathematical descriptions of the model are given, one leading to a linear and the other to a bilinear system of differential equations, with an additional inequality constraint reflecting the passivity of the semi-active device. Since the resulting stochastic optimization problem does not allow for a closed-form analytical solution, a numerical method is proposed as an approximate solution. The justification for the method is based on a recent existence theorem from stochastic optimal control theory. Illustrative simulation results of the optimization are presented.

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