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This paper proposes the use of Group Method of Data Handling (GMDH) technique for modeling Magneto-Rheological (MR) dampers in the context of system identification. GMDH is a multilayer network of quadratic neurons that offers an effective solution to modeling non-linear systems. As such, we propose the use of GMDH to approximate the forward and inverse dynamic behaviors of MR dampers. We also introduce two enhanced GMDH-based solutions. Firstly, a two-tier architecture is proposed whereby an enhanced GMD model is generated by the aid of a feedback scheme. Secondly, stepwise regression is used as a feature selection method prior to GMDH modeling. The proposed enhancements to GMDH are found to offer improved prediction results in terms of reducing the root-mean-squared error by around 40%.

System modeling is instrumental for designing new processes, analyzing existing processes, designing controllers, optimizations, supervision, and fault detection and diagnosis. Nonetheless, it is not always possible to have an explicit mathematical expression for nonlinear systems. Hence, the use of pattern recognition based on input and output patterns of a system is a viable alternative for system identification and modeling [

Polynomial classifiers have been used for various applications of classification and regression. It has been shown that they outperform other classical modeling techniques such as neural networks [

However, the use of polynomial classifiers becomes prohibitive from computational and storage standpoints when modeling a highly nonlinear systems. Moreover, numerical instabilities may arise due to the need of high order polynomials. Subsequently, the technique of Group Method of Data Handling (GMDH) was introduced [

In this paper, a variety of GMDH-based techniques are proposed for modeling magnetorheological (MR) dampers and compared against previously published modeling techniques using neural networks and fuzzy logic, and polynomial model [11-13]. An MR damper is filled with a fluid electrically controlled by a magnetic field. The damping characteristics are continuously controlled by varying the power of an electromagnet. MR dampers are semi-active control devices. They have unique characteristics such as low power requirement, fast response rate, mechanical simplicity, low manufacturing and maintenance cost, compactness, and environmental robustness [

The organization of this paper is as follow. In Section 2 we give an overview of generic system identification and modeling using GMDH. In Section 3 we describe the MR data generation process and propose the use of GMDH for modeling the generated data. Enhanced GMDH-based solutions for MR modeling are proposed in Section 4. Modeling results are discussed and compared in Section 5. Finally, the paper is concluded in Section 6.

GMDH is a modeling technique that provides an effective approach to the identification of higher order nonlinear systems. It was first introduced by A. G. Ivakhnenko [

GMDH uses a multilayer network of second order polynomials (quadratic neurons) to characterize the complex nonlinear relationships among the given inputs and outputs of a system. Each quadratic neuron has two inputs and a single output. If the two input variables are [x_{1}, x_{2}] the output of each quadratic neuron is calculated as described in Equation (1):

where, are the weights of the quadratic neuron to be learnt.

Consider a set of training data comprised of N feature vectors each of which is d-dimensional, and a corresponding set of response variables (i.e. targets, {y_{k}; k = 1∙∙∙N}). Accordingly, a GMDH network can be constructed by considering the combinations of all possible input pairs. Each pair of the d dimensions will then be the input to a quadratic neuron of the first layer of the network.

selection criterion per layer is required in order to keep the network complexity feasible. More specifically, neuron selection is based on a regularity criterion defined by

where g_{k} is the output of the m^{th} neuron on the L^{th} layer for the k^{th} feature vector. Consequently, a neuron is selected if its value is below a certain threshold. The threshold is determined based on the maximum and minimum values of denoted by and respectively such as

where α is a parameter between 0 and 1.

The L^{th} layer is retained if is smaller than, otherwise, layer number would be the last layer in the network (i.e. the output layer). In this layer only the neuron that corresponds to is retained. For further illustration,

In this work, the mathematical model of the MR damper proposed by Spencer Jr. et al. [

where “x” and “f” are the displacement and the force generated by the MR damper respectively; “y” is the internal displacement of the MR damper; “u” is the output of a first order filter whose input is the commanded voltage, “v”, sent to the current driver. In this model, the accumulator stiffness is represented by k_{1}, the viscous damping observed at large and low velocities are denoted by c_{0} and c_{1} respectively; k_{0} controls the stiffness and large velocities, x_{0} is the initial displacement of spring k_{1} associated with the nominal damper force due to the accumulator; γ, β and A are hysteresis parameters for the yield element, and α is the evolutionary coefficient. A set of typical parameters of the 2000 N MR damper is presented in

To produce a useful model of an MR damper, the input data must include information in the entire operating range of the system, and cover the spectrum of operation in which the damper functions. Usually, the limits of the input signals are based on the characteristics and applications of the MR damper. Advance knowledge of the input signals enables the generation of a more useful training dataset. The range of the voltage signal is between 0 and 2.5 V. The upper limit (i.e. 2.5 V) represents the saturation voltage of the damper and is obtained experimentally. The saturation voltage implies a voltage level at which no further increase in the yield strength of the damper is exhibited. Similarly, the displacement signal varies within ±2 cm. Matlab is used to generate 4-second worth of simulation data according to equations 4 to 10. This corresponds to 8000 samples, considering that the sampling rate is 2000 Hz.

In the forward model, the damper force is predicted from the applied voltage. The GMDH has 11 inputs and one single output. The inputs are comprised of three displacement samples, three velocity samples, and three voltage samples taken at times n, n − 1 and n − 2. The remaining inputs are two history samples of the force at times n − 1 and n − 2. The output of the model is the damping force at the current time n,. Out of the generated 8000 data points, 4000 of which are randomly selected for the training set while the remaining 4000 points are used for testing.

After completing the GMDH training, the eleven input variables are reduced to 6 and the irrelevant input variables are automatically eliminated. The relevant inputs of the network are found to be the displacements at times n and n − 2, the voltages at times n and n − 2, and the forces at times n − 1 and n − 2. The final structure of the forward MR damper model is shown in

The prediction accuracy of the GMDH model is evaluated by computing the force root means square error

(RMSE) defined as, where

and are the actual and the predicted forces respectively. The GMDH model described above yielded a force RMSE of 6.44 N. Plots of the time sequences of the predicted and the actual forces along with the prediction error are shown in

ment between the actual and the predicted force time sequences indicating the adequacy of the GMDH model. The horizontal axis is labeled in seconds where 4 seconds are equivalent to 8000 samples.

In the inverse model, the output of the system is the predicted command voltage. To model the inverse MR model, another GMDH network with 11 inputs and one single output is implemented. The inputs to the model are comprised of three displacement samples, three velocity samples, and three voltage samples taken at times n, n − 1 and n − 2. The remaining inputs are two history samples of the voltage at times n − 1 and n − 2.

The output of the model is the command voltage at time n,. After completing the GMDH training, the 11 input variables are reduced to 6, and the rest of the input variables are automatically eliminated. The relevant input variables to the inverse model are shown in

The GMDH model of the inverse MR damper yielded a voltage RMSE of 0.03 V. Plots of the time sequences of the predicted and the actual voltages along with the prediction error are shown in

It is worthwhile to mention that we have presented the results in Figures 4 and 6 in the 6th International Symposium on Mechatronics and Its Applications (ISMA’09) [

While the GMDH models described above resulted in satisfactory prediction results there seems to be further room for improvement to reduce the corresponding RMSE values. As such, two different GMDH-based systems are proposed. In the system, we propose the use of a two-tier architecture in which the estimated force/voltage is augmented to the feature set and the training is carried out again in a second tier. In the second system, we apply the technique of stepwise regression to reduce the dimensionality of feature vectors prior to the evaluation of the

GMDH network.

The two-tier identification architecture is illustrated in

The prediction results of the force and the voltage of the MR damper using the proposed two-tier approach are shown in Figures 8 and 9 respectively. The reported experimental results show that such architecture noticeably improves the accuracy of the identification process.

In the second proposed system, stepwise regression is

used as a preprocessing step to reduce the dimensionality

of the feature vectors [17,18] using the training dataset. The outcome of this step is the indices of the retained dimensions of the feature vectors. This arrangement is further illustrated in the block diagram shown in

Alternatively, the stepwise regression procedure can

be implemented after the polynomial expansion in each neuron in the GMDH network. The stepwise regression procedure is elaborated upon next for completeness.

Stepwise regression is a widely used regressor variable selection procedure. To illustrate the procedure (as described in [16,17]), assume that we have K candidate variables and a single response variable y. In identification, the candidate variables correspond to the elements of the feature vectors and the response variable corresponds to either the actual force or voltage. Note that with the intercept term β_{0} we end up with K + 1 variables. In the procedure, the regression weights are iteratively found by adding or removing variables at each step. The procedure starts by building a one variable regression model using the variable that has the highest correlation with the response variable y. This variable will also generate the largest partial F-statistic. In the second step, the remaining K − 1 variables are examined. The variable that generates the maximum partial F-statistic is added to the model provided that the partial F-statistic is larger than the value of the F-random variable for adding a variable to the model, such an F-random variable is referred to as fin. Formally the partial F-statistic for the second variable is computed by:

. Where denotes the mean square error for the model containing both x_{1} and x_{2}. is the regression sum of squares due to β_{2} given that β_{1}, β_{0} are already in the model.

In general the partial F-statistic for variable j is computed by:

If variable x_{2} is added to the model then the procedure determines whether the variable x_{1} should be removed. This is determined by computing the F-statistic

. If f_{1} is less than the value of the F-random variable for removing variables from the model, such an F-random variable is referred to as f_{out}.

The procedure examines the remaining variables and stops when no other variable can be added or removed from the model. Note that in this work we experiment with a maximum P-value of 0.05 for adding variables and a minimum P-value of 0.1 for removing variables.

Again, note that the elements of the expanded feature vectors are examined using the aforementioned procedure during the training stage of the identification system. The indices of the retained elements of the expanded feature vectors are stored and passed on to the testing or stage. Only the feature vector elements corresponding to the indices found from the training stage are retained.

In this section we compare the results obtained using the three proposed methods described above (GMDH, twotier GMDH, and GMDH with stepwise regression) with two previously published techniques. These two modeling techniques are neural networks (NN) and adaptive neurofuzzy inference systems (ANFIS) [11,12,19]. The results using ANFIS are taken from [

The comparison of the prediction results for both force (forward model) and voltage (inverse model) is being done using RMSE values as described above. Figure12 shows the forward model performance (force RMSE values) of the proposed and reviewed methods. Likewise,

On the other hand,

Lastly, unlike neural networks and neuro-fuzzy modeling, the proposed GMDH-based techniques can be viewed as layered second order polynomial networks, where the processing function of the node is a polynomial rather than a sigmoid function. Therefore, the optimization in the proposed GMDH-based techniques is based on a series of least-squares fittings rather than iteration-based optimization as in back propagation training of neural networks or ANFIS.

In this paper, we investigated the use of GMDH networks for modeling MR200 damper in the context of system identification. We have also introduced two enhanced versions of GMDH networks. Firstly, two-tier architecture was used where the predicted values in the first tier are fed to the network for a second tier training. Secondly, a hybrid of GMDH and stepwise regression in which feature selection is done by stepwise regression prior to GMDH training. Modeling the MR200 damper is done in forward and inverse modes where force and voltages are being predicted respectively. The GMDH-based modeling has been compared to two modeling methods (i.e. NN and ANFIS). GMDH with stepwise regression is found to offer significant reduction of about 40% in RMSE for both forward and inverse models.