Fig. 1. A series-parallel architecture of the NARX model used for road profile reconstruction. The network was trained by an expanded Levenberg-Marquardt algorithm (with Bayesian regularization) which computes the new weights new w via the relationship [11] ( ) 1 ( ) T T new old old λ − = + + w w J J I J ε w (2) where I is the identity matrix, ( ) old ε w is an error vector at the previous point, J is a Jacobian matrix (typically consisting of first partial derivatives of the error with respect to the parameters) and λ is a parameter governing the step size. However this training function expands the cost function by searching not only for the minimal error, but also for the minimal error using the minimal weights. Therefore the cost function becomes [7. 9] D W = + ε βε αε (3) where Dε is the sum of squared errors, Wε is the sum of squares of the network weights, and α and β are parameters of the objective function, compromising between fitting the data and producing a smooth network response. This enables the neural network to perform as well on novel inputs as on the training data. T D L IW1 b1 + LW2 b2 g1 T D L LW1 g2 + 1 ∑ - + e(t) Inputs: ( )t u&& Land Rover Defender 110 ( ) rz t % ( ) as network targets rz t ( )t u ( ) t i − u&& ( ) rz t i − Road Inputs Belgian Pave Discrete obstacles 1 Artificial Neural Network Road – Vehicle Test b: biases; IW: input weights; LW: layer weights and TDL: time delay lines 348
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