This study investigates adaptive sliding neural network (NN) control for quarter active suspension system with dynamic uncertainties and road disturbances. A Multilayer Perceptron (MLP) neural network is adopted to estimate the unknown dynamics of the system. In addition, sliding mode controller is introduced to compensate the function of estimation error for improving the performance of the system. Furthermore, the uniformly and bounded of closed-loop signals is guaranteed by Lyapunov analysis; the adaptation laws for training of MLP are derived from stability analysis. The simulation results demonstrate that the proposed controller can effectively provide a good ride and makes great trade-off between passenger comfort and handling despite the dynamic uncertainties.

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Let k be a field of characteristic zero, let R be the ring of formal power series in n variables over k and let D(R, k) be the ring of k−linear differential operators in R. If M is a finitely generated D(R, k)−module then d(M ) ≥ n where d(M ) is the dimension of M . This inequality is called the Bernstein inequality. We provide a short proof.