Modelling of a magneto-rheological damper by evolving radial basis function networks

Abstract

This paper presents an approach to approximate the forward and inverse dynamic behaviours of a magneto-rheological (MR) damper using evolving radial basis function (RBF) networks. Due to the highly nonlinear characteristics of MR dampers, modelling of MR dampers becomes a very important problem to their applications. In this paper, an alternative representation of the MR damper in terms of evolving RBF networks, which have a structure of four input neurons and one output neuron to emulate the forward and inverse dynamic behaviours of an MR damper, respectively, is developed by combining the genetic algorithms (GAs) to search for the network centres with other standard learning algorithms. Training and validating of the evolving RBF network models are achieved by using the data generated from the numerical simulation of the nonlinear differential equations proposed for the MR damper. It is shown by the validation tests that the evolving RBF networks can represent both forward and inverse dynamic behaviours of the MR damper satisfactorily.

Introduction

Due to their wide dynamic range, low power requirement, large force capacity, and fast response rate to a variable control signal, magneto-rheological (MR) dampers have emerged as newly developed semi-active control devices that have been mass-produced for commercial applications. In particular, MR dampers have found considerable attraction in vibration reduction of bridges, helicopter rotors, truss structures, and buildings. Semi-active control with MR dampers for vehicle suspensions has also been studied by many researchers, and many control strategies such as skyhook, groundhook and hybrid control, $H_{\infty }$ control and model-following sliding mode control have been evaluated in terms of their applicability in practice.

The practical use of MR dampers for control is, however, significantly hindered by their inherently hysteretic and highly nonlinear dynamics. This makes the modelling of MR dampers more important for their applications. In order to characterize the performance of MR dampers, several models have been proposed to describe their dynamic behaviours. These include the phenomenological model proposed by Spencer et al. (1997) based on a Bouc–Wen hysteresis model, neural network model developed by Chang and Roschke (1998) and Chang and Zhou (2002), fuzzy model (Schurter and Roschke, 2000), nonlinear blackbox model (Jin et al., 2001), NARX model (Leva and Piroddi, 2002), viscoelastic–plastic model (Wereley et al., 1998), and polynomial model (Choi et al., 2001), etc. Among these MR models, phenomenological model and viscoelastic–plastic model can accurately describe the dynamic behaviours of the MR dampers, but the corresponding models for the inverse dynamics of the MR dampers are often difficult to obtain due to their nonlinear characteristics. A multi-layer perceptron (MLP) neural network and an adaptive neuro-fuzzy inference system (ANFIS) models can be used to emulate the inverse dynamics of the MR dampers, but the selection of network structure and training data are essential in order to obtain accurate results. In fact, the polynomial model is a convenient and effective choice which can realize the inverse dynamic of the MR damper in an analytical form, and is easy to achieve the desirable damper force in an open-loop control system. However, polynomial model cannot characterize the behaviour of the MR damper favourably at relatively low velocity region since this model does not include variables characterizing the pre-yield property of the damper force. Since an open-loop control is easy to implement and cost-effective comparing with a closed-loop control, it is valuable to develop the accurate inverse dynamic models of MR dampers that are required in the realization of semi-active control.

The artificial neural networks (ANNs) have been effectively applied to model complex systems due to their good learning capability. It is possible to model the dynamic behaviours of the MR dampers by using ANNs. The MLP neural networks have been used to emulate the dynamic behaviours of an MR damper. However, the selection of network structures and training of samples are often complicated tasks but are essential for setting up an accurate MLP model. Moreover, the training speed is normally long due to slow convergence. Instead of using MLP neural networks to emulate the dynamic behaviours of an MR damper, this paper presents an alternative representation for modelling an MR damper in the form of radial basis function (RBF) networks.

The RBF network is a three-layer feedforward network that uses a linear transfer function for the output units and a nonlinear transfer function (normally the Gaussian function) for the hidden units. The input layer simply consists of the source nodes connected by weighted connections to the hidden layer. The net input to a hidden unit is a distance measure between the input presented at the input layer and the point represented by the hidden unit. The nonlinear transfer function (Gaussian function) is then applied to the net input to produce a radial function of the distance. The output units implement a linear weighted sum of the hidden unit outputs. In order to use an RBF network, we need to specify the hidden unit activation function, the number of nodes in the hidden layers and the training algorithm for finding the parameters of the network.

Compared with other types of ANNs, such as MLP neural networks, RBF networks have only one hidden layer, while MLP networks have one or more hidden layers depending on the application task; the hidden and output layers of MLP networks are both nonlinear, while only the hidden layer of RBF networks is nonlinear (the output layer is linear); the activation functions in the RBF nodes compute the Euclidean distance between the input examples and the centres, while the activation functions of MLP networks compute inner products from the input examples and the incoming weights, etc. These characteristics make the RBF networks having more advantages in, e.g., simple architecture and learning scheme, fast training speed (the liner output layer may not be trained), and the possibility of incorporating the qualitative aspects of human experience in the model selection and training. Hence, RBF networks are powerful computation tools and have been used extensively in the systems modelling.

In spite of a number of advantages compared with other types of ANNs, such as better approximation capabilities, simple network structures and faster learning algorithms, the development of RBF networks still have difficulties in selecting the network structure (the number of nodes in the hidden layers, i.e., the number of centres) and calculating the model parameters (e.g., centres, widths and weights). Normally, the training procedure of an RBF network is divided into two phases where the centres and widths are determined first, followed by the calculation of the weights. In order to overcome existing difficulties in developing an RBF network, an evolving RBF network that combines genetic algorithms (GAs) with other standard learning algorithms for an RBF network is presented in this paper to model the nonlinear dynamic behaviours of an MR damper. The structure of the RBF network is selected by a trial and error procedure, which only calculates several cases and compares the sum of squared errors (SSEs) between the true outputs and the network predictions to determine which structure is better. Although it is not an optimal selection for the network structure, it is time-saving because only several cases are calculated and the optimization result of SSEs can guarantee that the obtained SSE is not affected significantly by selecting different network structures, and hence a relative simple structure can be used with respect to the modelling accuracy required. The centres are searched by using GAs instead of using $k$-means clustering algorithm or fuzzy $c$-means clustering algorithm, where the SSE between the true outputs and the network predictions are minimized with respect to the given network structure and the obtained centre locations. This overcomes the drawback of standard RBF network in selecting centres using a clustering approach which is entirely separated from the actual objective of minimization of the prediction error. Finally, a uniform width, which is chosen to be the maximum distance among different centres, is used instead of using different widths for different centres. This will simplify the network structure as an uniform width is sufficient for the RBF network to achieve universal approximation (Chen et al., 1999). The weights that connect the hidden layer with the output layer are determined by calculating the pseudo-inverse matrix instead of using gradient descent optimization algorithm to save learning time. The reason that we do not apply GAs to optimize the network structure and all of the model parameters is that it is a time-consuming procedure when the training data is large in both length and dimension, and a large search space will unavoidably increase the difficulty for GAs to find the optimal results. The application examples presented in this paper show that only using GAs to search for the network centres can obtain better results for the modelling of an MR damper. Therefore, using GAs to search for the network structure, widths and weights is not necessary here.

In this paper, the developed evolving RBF network with four input neurons, which relate displacement, velocity, force and applied voltage, respectively, and one output neuron, which corresponds to either force or voltage, is used to emulate the forward and inverse dynamic behaviours of an MR damper, respectively. Data used for the training and validating of the evolving RBF network is generated from numerical simulation of the nonlinear differential equations proposed for an MR damper by Spencer et al. (1997). By comparing the SSE results for different network structures and different approaches in selecting the network centres, we can see that the resulting evolving RBF networks are shown to satisfactorily represent complicated dynamic behaviours of the MR damper while greatly reducing SSE even when the number of centres is small. Finally, it is validated by simulation that both the forward and the inverse dynamics of the MR damper can be approximated very accurately with the evolving RBF networks.

The rest of this paper is organized as follows. Section 2 introduces the phenomenological model of the MR damper presented by Spencer et al. (1997). The basic structure of the RBF networks and the developed evolving RBF networks are introduced in Section 3. The use of the evolving RBF networks in modelling the forward and inverse dynamic behaviours of an MR damper is presented in Section 4. Conclusions are given in Section 5.