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Identification and Control of Dynamical Systems Using the Self-Organizing Map Guilherme A. Barreto, Member, IEEE, and Aluizio F. R. Araújo
Abstract—In this paper, we introduce a general modeling technique, called vector-quantized temporal associative memory (VQTAM), which uses Kohonen’s self-organizing map (SOM) as an alternative to multilayer perceptron (MLP) and radial basis function (RBF) neural models for dynamical system identification and control. We demonstrate that the estimation errors decrease as the SOM training proceeds, allowing the VQTAM scheme to be understood as a self-supervised gradient-based error reduction method. The performance of the proposed approach is evaluated on a variety of complex tasks, namely: i) time series prediction; ii) identification of SISO/MIMO systems; and iii) nonlinear predictive control. For all tasks, the simulation results produced by the SOM are as accurate as those produced by the MLP network, and better than those produced by the RBF network. The SOM has also shown to be less sensitive to weight initialization than MLP networks. We conclude the paper by discussing the main properties of the VQTAM and their relationships to other well established methods for dynamical system identification. We also suggest directions for further work. Index Terms—Function approximation, predictive control, selforganizing maps (SOMs), temporal associative memory, time delays, time series prediction.
I. INTRODUCTION
D
YNAMICAL system identification is the discipline interested in building mathematical models of nonlinear systems, starting from experimental time series data, measurements, or observations [1]. Typically, a certain linear or nonlinear model structure which contains unknown parameters is chosen by the user. In general, the parameters should be computed so that the errors between estimated (or predicted) and actual outputs of the system should be minimized in order to capture the dynamics of the system as close as possible. The resulting model can be used as a tool for analysis, simulation, prediction, monitoring, diagnosis, and control system design. Artificial neural network (ANN) models have been successfully applied to the identification and control of a variety of nonlinear dynamical systems, such as chemical, economical, biological or technological processes [2]–[8]. Such achievements are mainly due to a number of theoretic and empirical studies showing that supervised feedforward architectures, such as the Manuscript received November 5, 2003; revised November 20, 2003. This work was supported in part by Funda ao de Amparo Pesquisa do Estado de Sao Paulo (FAPESP) under Grant 98/12699-7 and in part by Conselho Nacional de Desenvolvimento Cient fico e Tecnol gico (CNPq ) under Grant DCR: 305275/2002-0. G. A. Barreto is with the Department of Teleinformatics Engineering, Federal University of Ceará (UFC), Fortaleza-CE 60455–760, Brazil (e-mail:
[email protected]). A. F. R. Araújo is with the Center of Informatics, Federal University of Pernambuco (UFPE), Recife-PE 50740-540, Brazil (e-mail:
[email protected]). Digital Object Identifier 10.1109/TNN.2004.832825
multilayer perceptron (MLP) or the radial basis function (RBF) networks, can approximate arbitrarily well any continuous input–output mapping (see [9] and [10] for surveys). In this paper, we propose a system identification technique which uses the self-organizing map (SOM) [11] for function approximation, instead of the usual supervised ones (MLP and RBF). This technique, called vector-quantized temporal associative memory (VQTAM), shows that the SOM can be successfully used to approximate dynamical input–output mappings, with minor modifications in the original algorithm. The SOM is an unsupervised neural algorithm designed to build a representation of neighborhood (spatial) relationships among vectors of an unlabeled data set. The neurons in the SOM are put together in an output layer in one-, two-, or even three-dimenhas a weight sional (1-D, 2-D, 3-D) arrays. Each neuron vector with the same dimension of the input vector . The network weights are trained according to a competitive-cooperative scheme in which the weight vectors of a winning neuron and its neighbors in the output array are updated after the presentation of an input vector. Usually, a trained SOM is used for clustering and data visualization purposes. The SOM was previously applied to learn static input–output mappings [12]–[14], which are usually represented as (1) depends solely on the current in which the current output input . In this paper, however, we are interested in systems which can be modeled by the following nonlinear discrete-time difference equation:
(2) and are the (memory) orders of the dynamical where at time model. As stated in (2), the system output depends, in the sense defined by the nonlinear map , on the output values and on the past values of the input past . In many situations, it is also desirable to approximate the inverse mapping of a nonlinear plant1
(3) In system identification, the goal is to obtain estimates of and from available time series data , . 1When used for direct-inverse control, the term y(t + 1) in (3) is replaced by a reference signal, r(t + 1), which is usually assumed to be available at time t.
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For the SOM and other unsupervised networks to be able to learn dynamical mappings, they must have some type of shortterm memory (STM) mechanism [15], [16]. That is, the SOM should be capable of temporarily storing past information about the system input and output vectors. There are several STM models, such as delay lines, leaky integrators, reaction-diffusion mechanisms and feedback loops, which can be incorporated into the SOM to allow it to approximate a dynamical mapping or its inverse [17], [18]. In order to draw a parallel with standard system identification approaches, we limit ourselves to describe the VQTAM approach in terms of time delays as STM mechanisms. The remainder of the paper is organized as follows. Section II introduces the VQTAM technique and how it can be used to approximate forward and inverse mappings. Section III deals with a convergence analysis of the VQTAM, showing that the SOM learning rule builds a self-supervised error-reduction scheme for function approximation. In Section IV, we instantiate the VQTAM approach in a variety of complex dynamical modeling and control tasks, namely, time series prediction, identification of SISO/MIMO systems, and nonlinear predictive control. In Section V, several computer simulations illustrate the approximation capability of the VQTAM technique, comparing the obtained results with those produced by MLP and RBF networks and linear models. Section VI discusses the main features of the VQTAM and their relationships to previous unsupervised approaches to dynamical system identification and control. The paper is concluded in Section VII. II. TEMPORAL ASSOCIATIVE MEMORY USING THE SOM As pointed out in the introductory section, MLP and RBF neural networks have been the usual choices for model structures in the identification of nonlinear dynamical mappings. Our goal, however, is to devise a strategy that enables the SOM to approximate input-output mappings. For this purpose, we describe a general procedure in which the input vector to the SOM, , is augmented so that it comprises two parts from now on. , carries data about the input of The first part, denoted the dynamic mapping being learned. The second part, denoted , contains data concerning the desired output of this map, has its dimension ping. The weight vector of neuron , increased accordingly. These changes are written as and
(4)
and are, respectively, the portions of where weight vector which store information about the inputs and the outputs of the mapping being studied. Depending on the variables chosen to build the vectors and one can use the SOM algorithm to learn the forward or the inverse mapping of a given plant (system). For instance, to approximate the forward dynamics in (2) the following definitions apply:
(5) (6)
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If the interest is in the inverse identification, then one defines
(7) (8) During the training stage, the winning neuron at time step is determined based only on (9) For updating the weights, both
and
are used (10) (11)
where is the learning rate and neighborhood function given by
is a Gaussian
(12) where and are, respectively, the locations of neurons and in the output array. The parameters and decay exponentially fast with time and
(13)
and denote their initial values, while and where are the final ones, after training iterations. A trained SOM network can then be used to obtain estimates for the output of the learned mapping directly from the output , as follows: portion of the weight vector, forward case inverse case
(14) (15)
where in both cases the winning neuron, , is found as in (9). steps until an entirely The estimation process continues for new series is built from the estimated values. Note that the learning rules in (10) and (11) follow the usual formulation of the SOM: the first rule acts on the input and the second one acts on the output of the mapping being learned. The underlying idea is that, as training proceeds, the SOM algorithm learns to associate, in a topology-preserving way, the of the mapping with the corresponding inputs outputs . Thus, this technique will be referred to as VQTAM. The VQTAM can be promptly understood as a generalization of the one in [12]–[14] to the more demanding task of modeling dynamical input–output mappings. It is also worth emphasizing the difference between the VQTAM strategy and the usual supervised approach. In MLP is presented to the network and RBF networks, the vector is used at the network output to compute input, while the explicitly an error signal that guides learning [Fig. 1(a)]. The VQTAM scheme instead allows unsupervised neural networks, such as the SOM, to learn to associate or correlate the inputs and outputs of the mapping without explicit computation of an error signal [Fig. 1(b)]. In the next section, we show that the
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Fig. 1. Differences between the (a) supervised and (b) self-supervised approach to system identification.
associative nature of the VQTAM computes indirectly an error signal. Such a property allows us to understand this approach as a self-supervised gradient-based error reduction method.
cooperation among neurons during the learning process, the network convergence can be evaluated through the average expected squared prediction error
III. ANALYSIS OF CONVERGENCE In this section, we present an analysis of the convergence process of the SOM algorithm using the VQTAM approach. More specifically, we want to know if the additional learning rule in (11) allows the SOM to learn an input–output mapping. This analysis, based on [19], is an adaptation for the purposes of the current paper of the one carried out in [11]. Our goal is to verify whether the estimation (or approximation) error really decreases as the SOM training proceeds, converging eventually to a stable state. For the sake of clarity, we assume that the mapping to be approximated is given as in (2) and the only source of information available is the time series of measured input–output data, , . By means of the VQTAM approach the SOM provides an approximation, , of the unknown map. In this case, the SOM computes an estimate , ping , as follows: given an unforeseen vector
(18) where it is assumed that the expectation value is taken over an ; means infinite number of stochastic samples the joint probability density function of the whole data over is a “volume differential” which the estimator is formed, and in the space in which the samples are defined. is unknown and the seSince the probability density quence is finite in reality, we shall resort to Robbins–Monro stochastic approximation technique for the minimization of to , for the parameter . According find an optimal value, to this method, we now work with stochastic samples, , of and the functional , computed from the sequences , as follows:
(19)
(16) where the explicit representation of the winning neuron in (16) emphasizes the associative nature of the VQTAM approach, since it is the element responsible for linking the input and output portions of the learned mapping. Then, we are able to define an absolute value for the estimation error
The underlying idea is to decrease the function at each new step by descending in the direction of its negative gradient with respect to the current parameter vector . The recursive is formula for the parameter vector (20)
(17) Through the definition of (17) we can now demonstrate that the VQTAM scheme is equivalent to an implicit error-based learning procedure. One of the main features of the SOM algorithm is the coand its neighbors operation between the winning neuron . Without loss of through the neighborhood function generality, we assume that the neighborhood function is time-invariant. Thus, the current analysis is valid only for the convergence phase of the SOM training, not for the ordering phase (first thousand iterations of the algorithm). Taking into account
defines the step size and satisfies where the scalar and . Considering (19), the partial derivative in (20) is given by (21) The recursive equation in (20) is then written as (22) Equation (22) is exactly the learning rule in (11). Hence, starting , the sequence from arbitrary initial values,
BARRETO AND ARAÚJO: IDENTIFICATION AND CONTROL OF DYNAMICAL SYSTEMS
will converge to an optimal vector corresponding to a local minimum of . For the 1-D SOM, in particular, this steady-state happens to be a global minimum [20].
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is used to evaluate the accuracy of the model by means of the normalized root mean squared error (NRMSE): (28)
NRMSE IV. CASE STUDIES From this section on, we provide instantiations of the VQTAM method for different modeling and control problems, namely, i) time series prediction; ii) identification of single-input–single-output (SISO) systems; iii) predictive control of SISO systems; and iv) identification of multi-input–multi-output (MIMO) systems. A. Time Series Prediction Consider a scalar time series denoted by , which is described by a special case of (2), the nonlinear regressive model as follows: of order (23) is a nonlinear mapping of its arguments and is a where random variable which is included to take into account modeling uncertainties and/or noise. For a stationary time series, it is drawn from a Gaussian white is commonly assumed that noise process. As expected, the structure of the nonlinear mapping is unknown, and the only thing that we have available is a set of observables: , where is the total length of the time series. The task is to construct a physical model for this time series, given only the data. To do so, we propose to use the VQTAM approach and the SOM to build one-step ahead predictors. For this purpose, the past samples comprises the network input vector , while the output vector is defined as the “next time step” sample . Then, we have (24) (25)
where is the variance of the original time series and length of the sequence of residuals. B. Efficient Linear Prediction Models Through VQTAM
In essence, the SOM is an algorithm for vector quantization. Because of that, the outputs produced by (26) can only assume discrete values, limited by the number of neurons in the network. As will be shown in the simulations, to obtain small prediction errors, the number of neurons should be very large (greater than 400 units). A more efficient approach is to train the SOM as described above using just a few neurons (less than 100 units) in order to have a “compact” representation of the time series encoded in the weight vectors of the network. Then, after training is completed, a linear autoregressive model is fitted to the weight vectors of the SOM rather than to the original input vectors. This technique is detailed next. , is repreA linear AR model of order , or simply sented by (29) where are the coefficients of the model. In geometric terms, fitting an AR model to a time series is equivalent to fit an -di. The basic technique mensional hyperplane to data in to compute the coefficients is the well known least-squares method [21] (30) in which is the prediction vector and is the regression matrix. Using the VQTAM, these entities are constructed through the weight vectors and , , as follows:
The winning neuron is determined on the basis of the input only as in (9), and the weight vectors are updated vector according to (10) and (11). During the prediction (test) phase, is found for each input vector the winning neuron and the one-step-ahead estimate of the time series, at time step is recovered from (26) for which the prediction error or residual is defined as the difference between the “true” value of the next sample of the time series and the estimated value (27) The variance of the residuals, , can be used as an estimate . This estimate of the variance of the white noise, i.e.,
is the
.. . where
.. .
.. .
.. .
(31)
is the number of neurons in the SOM. To write , we used the fact that we are predicting only the next (scalar) sample of the time series. To avoid confusion, from now on we denote this technique as the autoregressive SOM (ARSOM) model, while we refer to the plain VQTAM simply as the SOM model. The main advantages of the ARSOM model with respect to the SOM model are two-fold: 1) smaller prediction error because the hyperplane described by (29) interpolates between the weight vectors of the SOM, and 2) reduction of the computational cost of the prediction task, since much fewer neurons are
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needed to get a small prediction error. If one uses weight vectors (e.g., the most similar to the current input or the neurons in the neighborhood of the winning neuron), the VQTAM can be used to build local AR predictors in a manner similar to [22]. Local models are suitable to model nonstationary time series [23] and, hence, a detailed discussion of this subject is outside the scope of this paper. C. Identification of SISO Systems For univariate time series prediction, we assumed to have observations of one single variable. In this section, we work with a generalization of this task, called identification of SISO systems, involving two univariate time series, one for the input vari, and another for the output variable able, . The goal is to estimate the next output of a given plant based on current and previous values of the input and output variables. Mathematically, we have (32) This equation is equivalent to (2), except that the dimensions of the input and output variables have been adapted for the scalar case. Thus, the application of the VQTAM method to SISO system identification obeys (5) and (6) exactly, replacing the by scalars . Training and prevectors diction follow (9)–(14), accordingly. It is worth emphasizing that the efficient linear prediction approach described in Section IV-B can be equally applied to the identification of SISO systems, reducing the number of neurons needed to get smaller prediction errors. D. Predictive Control Since the SOM and the ARSOM models can be applied to system identification, it is fair to evaluate their performances on model-based control problems, such as nonlinear predictive control (NPC) [7]. As in the field of neural system identification, the field of neurocontrol is dominated by supervised neural models, such as MLP and RBF networks [24]. This is particularly evident for the NPC strategy [25]–[29]. Few works have evaluated the use of self-organizing neural networks in NPC [30], [31]. These, however, have designed the strategies based on highly modified versions of the SOM, limiting the general applicability of their approaches. The VQTAM instead emphasizes the viability of the SOM algorithm, slightly modified to have the input and the output of the desired mapping together in the input of the network, in system identification and control, keeping the training procedure exactly as in the original algorithm. A predictive control strategy comprises basically three components: the system (plant), a feedforward model of the plant, and an optimization algorithm (Fig. 2). The model is used to estimate future outputs of the plant, according to which the op, that timizer computes a sequence of control actions, guides the system output to the desired behavior. The cost function minimized by the optimization algorithm is given by (33)
is the reference signal (setpoint) for the output, is where is the variathe estimated output, and define, respectively, the tion of the control signal, minimum and maximum prediction horizon, is the control is a constant penalizing large variations horizon, and of the control signal. For the present work, the well known Levenberg–Marquardt algorithm [32] is used for the optimization . To simplify the optimization process, we of the functional so that only the control action for the next time adopt step is computed. The main advantage of the predictive control strategy relies on its ability to compute an optimal control action, taking into account estimates of future values of the system output within a given time horizon. This anticipatory scheme results in smoother transient effects caused by sudden changes in the setpoint, and also provides the ability for performing in advance corrective actions against external disturbances. For an efficient use of the VQTAM approach in NPC strategies, the identification scheme discussed in Section IV-C is modified in order to enable the model to predict the system to at once (i.e., in a single step). For output from and are built as follows: this purpose, the vectors
(34) (35) This “all-at-once” approach results in sightly larger prediction errors because it imposes to the network the harder requirement of estimating the values for several time steps into the future. However, it is considerably more stable than the usual recursive approach, in which the plant outputs are estimated one-by-one, feeding the predicted value back to the network input [3]. In general, after running for a while in this recursive mode, the network needs to be reset to avoid instabilities caused by propagation of estimation in time [27]. As usual, the search for winning neurons obeys (9) and the weight updating is governed by (10) and (11). After training is completed, to be used according to the scheme depicted in Fig. 2, the outputs of the model are computed as follows:
.. .
(36)
A summary of the steps of the SOM-based NPC strategy is given below. Step 1) at time step , define the reference signal, , according to (34); and build the vector Step 2) compute the output estimates according to (36); Step 3) choose an optimization algorithm to minimize , given in (33), in order to find an op, taking the plant output as close timal control signal, as possible to the reference signal; Step 4) apply the control signal found in the last step to . control the plant and do Steps 1)–4) are repeated while a reference signal is available.
BARRETO AND ARAÚJO: IDENTIFICATION AND CONTROL OF DYNAMICAL SYSTEMS
Fig. 2. General architecture of a model-based predictive controller.
E. Identification of MIMO Systems As previously said, the VQTAM equations as stated in Section II refers to the more general (and, hence, hard to deal with) case of multi-input–multi-output (MIMO) systems. In MIMO systems, the inputs and outputs at time are vectors, whose components refer to a specific variable or feature measured or observed at that time. To assess the learning ability of the VQTAM approach with regard to MIMO systems, it is chosen the task of modeling and reproduction of complex robot trajectories. This choice allows the performance comparison between the SOM with previous self-organizing approaches [33] The goal of this task is to estimate the joint angles of the PUMA 560 robot arm during the execution of a given trajectory which may contain repeated states. For this purpose, the , of the robot end-effector are asCartesian positions, , sumed as input variables, while the six joint angles are the output variables. Then, by direct substitution, one can write down a version of (2) adapted to the task of robot trajectory modeling and reproduction as follows:
(37) Thus, the vectors
and
are easily defined as before
(38) (39) where the winning neuron is found as in (9) and the weights are updated according to (10) and (11). It is important to note that this formulation allows the SOM to model simultaneously inverse kinematics relationships and temporal aspects of the trajectory. After training is completed, the estimate of the next joint angles of the robot for a given trajectory is as follows: (40) This estimate is used for controlling the robot arm. V. SIMULATIONS The simulations aim to compare the performance of the VQTAM with standard approaches in all the case studies discussed in the previous section. By standard approaches, we
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mean the MLP and RBF networks, as well as linear regression methods. We are particularly interested in understanding the influence of the following parameters of the SOM in the final , number of neurons, number prediction error: model order of training epochs and size of the training data set. The influence of the neighborhood function is also studied. Time Series Prediction: Modeling and prediction of chaotic time series is still a relatively new research topic, dating back to 1987 [34]. The interesting point about this field is that, even under noise-free conditions, a chaotic system shows an apparently random behavior, but may still be modeled using techniques from nonlinear deterministic system identification. However, even with perfect modeling of the dynamical behavior of the system in the noise free case, only short-term predictions are possible due to the extreme sensitivity of chaotic systems to uncertainties in initial conditions. The time series we considered in this paper is obtained from the well known Mackey–Glass differential equation with delay [35] (41) is the time series sample at time step . If where , the system approaches a stable equilibrium point for , enters a limit cycle for , becoming chaotic for , after successive period doubling bifurcations for . Adopting , and , we generated a time series of length 3000 and scaled it between [ 1,1]. The simple Euler method was used to approximate the derivative , . For cross-validation purposes, the first 2000 samples were used to train and the remaining samples to test the models. The first simulation evaluates the influence of the model on the prediction error computed for the test data. order A 1-D SOM with 400 neurons was trained for 250 epochs. Each epoch consists of one presentation of the entire training first samples. The training parameters sequence minus the , , , . were set to The prediction errors were computed for time steps, and the minimum value was found for [Fig. 3(a)]. According to this figure, one can note that absence generates the highest errors, but deep of memory does not imply good performance. For the memory Mackey–Glass time series, the best result was obtained for . Fig. 3(b) shows the first 100 samples estimated by the . SOM with The second simulation is concerned with the influence of the number of training epochs and the size of the training set on the : final prediction error. Two tests were performed using 1) For a fixed number of 250 epochs, the number of the training vectors was increased from 10 to 1000 in increments of 30. For each size of the training set, the prediction error on the test set is computed. This test gives an idea of how much information from the training data should be provided to the SOM in order to perform well in the test set. 2) For a fixed size of the training set, the number of epochs is varied from 1 to 1000 in increments of 10. Roughly speaking, this test reveals how fast the SOM acquires knowledge from the data to perform well on the test
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Fig. 3. (a) Prediction error (NRMSE) versus the memory order (n ). (b) Sequence of 100 first estimated samples of the Mackey–Glass.
data. The results for both tests are shown in Fig. 4. From this figure, one can easily infer that there is no need to train the SOM for more than 150 epochs using a training set of length 850 in order to have small prediction errors. Values higher than these ones do not improve considerably the network prediction performance. The next simulation evaluates the sensitivity of the SOM to the number of neurons and how the ARSOM model can help to reduce this number. For this test, a training set of length 850 . The was used to train a 1-D SOM for 150 epochs, using number of neurons was varied from 5 to 400 in increments of 20. For each number of neurons, the prediction errors generated by the SOM and the ARSOM model were computed. The results are shown in Fig. 5(a). It can be noted that while the prediction errors for the SOM continue to decay exponentially as the number of neurons increases, the error curve for the ARSOM decays much faster, stabilizing in a value around 0.0275 just after as few as 30 neurons have been used. For 30 neurons, the error obtained for the SOM was very high (NRMSE=0.1666) due to the quantization effect process resulting from the use of the plain SOM algorithm. The interpolated output provided by the ARSOM model “smooths” the estimated time series, reducing the prediction error considerably. Thus, one can say that the generalization ability of the ARSOM is much better than that of the SOM alone. The first 100 samples of the Mackey–Glass time series, generated by the SOM and ARSOM algorithms using 30 neurons, are shown in Fig. 5(b). The last simulations compare the SOM and the ARSOM methods with MLP, RBF and standard linear AR models. The coefficients of the AR model were computed using the maximum-likelihood method2 [36]. All these algorithms adopted . A 1-D SOM with 400 neurons was trained for 150 epochs, while the ARSOM model with 30 neurons was trained for 10 epochs only. After some experimentation with the data, the best configuration of the MLP network had seven units in the hidden layer and one unit in the output layer. The transfer function for all neurons in the hidden and output layers is the 2Under the assumption of uncorrelated Gaussian noise, the maximum-likelihood approach is equivalent to the Least Squares method.
Fig. 4. Evolution of the prediction error as the number of epochs and the size of the training set of the SOM increases.
hyperbolic tangent function. The MLP was trained with the backpropagation learning algorithm with adaptive learning rate and momentum. The values for the learning rate and the momentum factor are set to 0.1 and 0.9. The best configuration of the RBF network had 17 Gaussian basis functions whose centers were computed by the -means algorithm [21], and 1 output neuron whose weights were computed by ordinary least squares. The radii (spread) of the Gaussian kernels were set to 0.1. The number of input units of the MLP and RBF is equal . to the dimension of Table I shows the final prediction errors generated by all algorithms for the Mackey–Glass time series and two other well known time series [37], [38]. It is worth noting that the SOM model produced prediction errors of the same order of magnitude of those produced by MLP and RBF networks. The ARSOM had the best performance for the three time series. As expected, all neural models performed better than the linear one. Further comparisons of the SOM with the MLP and RBF networks are presented next. SISO System Identification: Fig. 6 shows measured values of , and the oil pressure the valve position (input variable),
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Fig. 5. (a) Evolution of the prediction error for the Mackey series by increasing the number of neurons. (b) First 100 estimated samples for the Mackey–Glass time series generated by the SOM and ARSOM models, both using 30 neurons.
TABLE I MINIMUM PREDICTION ERRORS (NRMSE) OF THE ARSOM, SOM, MLP, RBF, AND AR MODELS FOR THREE CHAOTIC TIME SERIES
(output variable), , of a hydraulic actuator. It is worth noting that the oil pressure time series shows an oscillative behavior caused by mechanical resonances [5]. The SOM, MLP, and RBF networks were used as nonlinear models to approximate the forward and the inverse mappings of the hydraulic actuator. In the forward identification task, these three neural nets are also compared with the usual linear autoregressive model with exogenous inputs (ARX) (42) is where and are the coefficients of the model and . The the estimated value for the plant output at time step coefficients are computed by applying the least-squares method [1] to the input data directly, not to the weights of the SOM (see Section IV-B). The model accuracy is evaluated through the RMSE
RMSE
(43)
where or depending on the mapping is the length of the estimated series. being learned and The data were presented to the four models without any preprocessing stage. From a total number of 1024 input–output
Fig. 6. Measured values of (a) valve position and (b) oil pressure.
samples, the first 512 samples were used to train the neural networks and to compute the coefficients of the linear ARX model, while the remaining 512 samples were used to validate the four models. A training epoch is defined as one presentation of the
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TABLE II RMSE FOR THE MLP, SOM, WTA, AND RBF NETWORKS IN THE FORWARD AND INVERSE IDENTIFICATION OF THE HYDRAULIC ACTUATOR
Fig. 7. Simulation of the (a) SOM model on test data and (b) evolution of the estimation error with the training epochs. Solid line: simulated signal. Dotted line: true oil pressure.
training samples. For all the simulations, it is assumed and , as suggested in [5]. A 1-D SOM with 500 neurons in a six-dimensional space is simulated. The weights are randomly initialized between 0 and 1, and adjusted for 100 epochs.3 The training parameters are the following: , , , , and . The best result generated by the SOM model during validation (RMSE=0.2051), together with an evaluation of the influence of the number of training epochs in the approximation performance, are shown in Fig. 7. The
0
3Other ranges for random weight initialization, e.g., between 1 and 1 have been tested but they had no significant influence on the final configuration of the weights.
error decays very fast initially, stabilizing around RMSE=0.20 after 40 epochs. The linear ARX model provided a much higher error (RMSE=1.0133). , The MLP network has five input units since one hidden layer with ten neurons and one output neuron. The neurons in the hidden layer have hyperbolic tangent transfer functions, while the output neuron has a linear transfer function. The MLP network is trained through backpropagation algorithm with momentum. The values for the learning rate and the momentum factor are set to 0.2 and 0.9, respectively. The training or a maximum number of 600 is stopped if RMSE training epochs is reached. The RBF network also has five input units, an intermediate layer with neurons with Gaussian basis function, and one output neuron. Following the RBF design in [39], which is specific for regression problems, the number of neurons in the intermediate layer is the same as the number of training samples and, hence, there is a Gaussian kernel centered at every training vector. The intermediate-to-output weights are just the target values, so the output is simply a weighted average of the target values of training cases close to the given input case. The only parameters that need to be tuned are the radii of the Gaussian kernels. In this paper, this parameter was the same for all RBF units, being deterministically varied to evaluate its effect in the accuracy of the approximation. To evaluate the influence of the neighborhood function on the final approximation results, we also simulated a plain winner-take-all (WTA) network. This is equivalent if , and to training the SOM with , otherwise. The results for the four networks in the forward and inverse identification of the hydraulic actuator are shown in Table II. The RMSE values shown for the SOM, WTA, and MLP networks were averaged over ten training runs. For the RBF the radius was varied from 0.1 (minimum RMSE) to 1.0 (maximum RMSE) in increments of 0.1. One can note that the MLP network provides the best results in general. The SOM algorithm, in its turn, produces better results than a RBF network with approximately the same number of neurons. The minimum RMSE for the RBF network occurs for radius = 0.1, increasing in an approximate linear fashion to a maximum for radius = 1.0. An interesting result is that the SOM is less sensitive to weight initialization than the MLP network, as can be seen in the fifth and ninth columns of Table II. This sensitivity is measured through the standard deviation of the RMSE values generated for the 10 training runs. Furthermore, the SOM can adapt online to new incoming data without retraining the entire network as in the standard MLP case. Another difficulty found in designing the MLP network is the occurrence of overtraining. Since the SOM network is in essence
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a type of vector quantization algorithm: the more the SOM network is trained, the more precise is the approximation (in a statistical sense) of the probability density of the training data [11]. However, after some training time, learning in the SOM stabilizes around RMSE=0.20 (Fig. 7) and no substantial reduction in RMSE occurs. The results for the WTA network illustrate that the neighborhood cooperation is crucial for a good performance of the VQTAM approach. The best results produced by the SOM and MLP networks in the inverse identification of the hydraulic actuator are shown in Fig. 8. For these specific cases, the estimation errors were RMSE(SOM)=0.1189 and RMSE(MLP)=0.0566. It is worth noting that in some portions of the estimated time series the errors had high values. This occurs because the inverse mapping may not be unique, an issue that makes it inherently harder to approximate than the forward mapping. Predictive Control: For this simulation, it is considered a nonlinear system, whose model is given by the following secondorder differential equation [7]: (44) where denotes the input and is the output. A total of 1000 input–output pairs to train the SOM were generated by solving (44) for a pulse random binary signal) (PRBS).4 The first 500 samples, used for training, are shown in Fig. 9(a). The remaining samples are for testing the model. A linear model of the plant is used and its coefficients are computed as in Section IV-B, training the ARSOM model with 100 neurons for 10 epochs. The estimated output time series during test is shown in Fig. 9(b). It is worth noting that even using a relatively small number of neurons, the approximation results are quite good due to the interpolated outputs. Having a plant model, it is possible to design a predictive controller. The well known Levenberg–Marquardt algorithm [32] is chosen for the optimization step due to its fast convergence , speed. The other parameters are set as follows: and . the joint simulation of the plant model and the optimizer are shown in Fig. 10 for a given reference signal. It can be noted that the tracking of the reference signal is accomplished as desired, with smooth transients for each abrupt change in the setpoint. MIMO System Identification: For the last simulations, we using the Matlab toolbox generated data pairs ROBOTICS [40], moving the robot end-effector along a at certain time steps figure-eight path. The joint angles are recorded and the resulting Cartesian end-effector positions are computed by means of forward kinematics functions available in that toolbox. The main goal is to study the ability of the SOM to reproduce a learned trajectory with accuracy and without ambiguity caused by the presence of repeated states. The SOM was trained according to the VQTAM approach, and its performance is compared with the recently proposed competitive temporal Hebbian (CTH) network [33], [41] (see 4This kind of input signal is widely used in system identification to force the system to operate at different dynamical ranges or modes.
Fig. 8. Results obtained by the (a) SOM, (b) the MLP, and (c) the RBF for the inverse identification of the hydraulic actuator.
Appendix). The CTH is an unsupervised algorithm designed to store and reproduce robot trajectories. It works by storing the states of a given trajectory in competitive feedforward weights, while their temporal order is encoded in lateral Hebbian weights. Temporal context units allow resolution of ambiguities during trajectory reproduction. In terms of learning, the CTH stores the trajectory states directly into the competitive weights without need for training
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Fig. 9. (a) Input PRBS signal and the corresponding output time series used in the SISO system identification task. (b) Testing the SOM in the identification of the nonlinear plant (open circles denote true values of the output).
Fig. 10.
fu
(t)g.
VQTAM-based predictive controller. (a) Plant output time series y (t) (thicker line) for a given reference signal. (b) Resulting optimal control sequence
(exemplar learning), while the SOM needs recycling the trajectory data over several epochs in order to get a condensed representation of the input vectors (prototype learning). The trajectory used to train the two networks contains 97 states. The SOM used 52 neurons, while the CTH used exactly 97 units. By and . trial-and-error, the memory orders are set to , , The training parameters for the SOM are , , and . For the CTH, we have , , , , and . Table III shows the resulting MSE values for both networks when asked to reproduce the learned figure-eight trajectory. In order to evaluate the generalization ability of the networks, the
initial states were different from those used during training. In this table, one can note that the SOM performs better that the CTH network, no matter if the error is measured in the joint or the Cartesian (world) domain. This result is somewhat predictable since the prototype-based learning of the SOM, even using fewer neurons, provides better generalization than the exemplar-based learning of the CTH. Fig. 11(a), (b) shows the values estimated by the SOM and , of the robot CTH networks, respectively, for the joint 6, arm during the reproduction of the figure-eight trajectory. The corresponding Cartesian values are shown in Fig. 11(c), (d). As expected, the trajectories retrieved by the SOM approach are
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TABLE III MEAN SQUARED ERROR (MSE) FOR THE SOM AND CTH NETWORKS DURING THE REPRODUCTION OF FIGURE-EIGHT TRAJECTORY
closer to the desired values than the ones retrieved by the CTH network. VI. DISCUSSION In this section, we discuss the advantages of the VQTAM method with respect to previous applications of the SOM algorithm to identification and control of dynamical systems. Why the SOM? The first issue to be addressed in this section concerns the choice of the SOM algorithm rather than another competitive neural model to implement the VQTAM approach. Firstly, the SOM was chosen for being a very simple algorithm with wide applicability within the neural network community. Furthermore, a number of theoretic developments are now available which account for many of the computational properties of the SOM [20]. In principle, the VQTAM method can be used efficiently by any topology-preserving competitive neural model, such as the topology-representing network [42] or the growing neural-gas [43], since it was illustrated in Section V that the SOM performs much better than competitive networks that do not preserve topology of the data, such as WTA and CTH models. The topology preservation ability of the SOM is a direct result of the use of a neighborhood function during learning. Temporal Associative Memory: The associative nature of the VQTAM differs from the standard approaches, which are mostly variants of Hopfield and BAM networks (see [44] and references therein), in the following aspects. 1) Commonly, the conventional approaches have prewired weights (i.e., nonadaptive). Thus, changes in the dynamics of the input–output mapping cannot be learned automatically. Instead, the VQTAM is an adaptive method because the associations are learned through time. 2) Hopfield- and BAM-based models build the synaptic memory matrix directly from the input–output data using a correlation-based (Hebbian) learning rule, while the VQTAM builds its memory from the weight (prototype) and . Since the entire set of weight vectors vectors forms a compact representation of the input–output , this approach reduces the computapairs tional costs of the algorithm (less memory are required to store and represent information). 3) The competitive learning process of the VQTAM leads to prototype vectors representing the centroids of quantized regions of the input–output mapping. Hence, the prototype vectors can be seen as filtered versions of the input–output pairs since a moving average process takes place when using (10) and (11). Because of this, the VQTAM is more robust to noise than the standard methods. In fact, this property is inherited from the SOM network. Function Approximation: Recently, clustering techniques have been proposed for function approximation [45]. Since
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system identification is indeed an instance of this broader research field and the SOM can be seen as a clustering method, it is fair to discuss the VQTAM in the light of that previous work. The objective of the function approximation oriented clustering proposed in [45] was to increase the density of prototypes (weight vectors) in the input areas where the target function presents higher variability in the response, rather than just in regions where there are more input examples as done by most of the clustering techniques. The authors argue that this goal is more adequate for function approximation problems because it helps to reduce the prediction error by means of minimizing the output variance not explained by the model. This is exactly what is originally done by the VQTAM approach in a simpler way than that described in [45]. Again due to the influence of the neighborhood function on weight updating, the weight vectors converge to a steady–state (as long as the learning rate is annealed!) in which high-density regions of the input data space are better clustered than low-density comprises past values areas.5 Since the input vector of the input signal, the higher the variability of the signal ampli, the higher the chance of good data clusters tudes within (i.e., well separated data clusters) to be detected by the network. Thus, according to the viewpoint of [45], the SOM is naturally well suited for function approximation tasks. Global Versus Local Modeling: The SOM and the ARSOM models learn an input–output mapping globally, in the sense that only one model structure is used. More recently, some authors have proposed to use the SOM to build several (linear) model structures, each one responsible for approximating a localized region of the system’s state–space, thereby improving the approximation ability of the SOM. One of these approaches, known as local linear mapping (LLM) [46], is similar to VQTAM in that each neuron of the SOM stores an input weight with its corresponding output weight . Additionally, each neuron also stores the local gradient (Jacobian) matrix of this input–output pair calculated during the learning phase. This gradient information is used to produce a first-order expansion in the output space around the representative output leading to an improved estimate of the output, weight , given by (45) Thus, the LLM method is a linear interpolation technique which smooths the quantized output of the original VQTAM. This method has been applied to robotics, system identification and time series prediction [47]–[50]. As discussed in Section IV-B, a simpler and computationally lighter approach to build local models and smooth the output is through the ARSOM model, in which only the weight vectors of the first winners are used to compute the coefficients of a standard AR model (see also [22] for a variant of this method). There is no need for each neuron to store its own Jacobian matrix. In general, local modeling with the SOM produces better results than global modeling for nonstationary signals. 5In SOM theory, this property is called magnification factor, meaning the inverse of point density of the weight vectors [11].
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However, local modeling demands higher computational efforts than global modeling, an issue that may become important in real-time control. Effective performance comparison among different approaches for local modeling is a topic for further research. Unsupervised Error-Based Learning: In Section III, it was demonstrated that the VQTAM method is indeed a self-supervised gradient-based error reduction method. In general, the term self-supervised is used to denote unsupervised methods that have been modified to be applied to supervised problems. One of such methods, named Vector Associative Memory (VAM) has been proposed by [51], who describe self-organizing neural circuits for the control of planned arm movements during visually guided reaching. For the purpose of this paper, it suffices to say that the neural circuits of the VAM comprise 1-D topographic maps with fixed topographic ordering, which was imposed initially. This means that the topographic maps did not develop in the course of learning, which required more prior knowledge about the structure of the set of input data the network had to process. Also, the use of fixed topographic maps did not provide an automatic increase in the resolution of the representation of those movements which have been trained more frequently. Thus, unlike the VQTAM approach, cooperation among neighboring neurons plays no special role in the convergence of the algorithm. Assuming a linear neuron whose output is given by , where is the th input to neuron , the plain , where Hebbian rule can be written as is the learning rate. A modified Hebbian rule was developed by [52] so that, in addition to the usual number of inputs with modesifiable weights, the neuron has also a teaching signal tablished with a fixed negative synaptic weight. In this case, the . This learning rule becomes approach is equivalent to the VQTAM in that the desired output is introduced through the input pathway (Fig. 1) in order to allow the computation of an error signal to guide weight adjustments. However, the VQTAM is more general than the modified Hebconbian rule in two aspects. First, the weight vector necting the desired output, , to neuron is adjustable, which gives more flexibility for the method to extract relevant information from data. Second, as discussed previously, the cooperative learning procedure implemented through the neighborhood function is essential for good accuracy in function approximation. This is not taken account in the modified Hebbian learning rule. Time Series Prediction: Of those discussed in this paper, this is the application domain with the major number of contributions involving the SOM algorithm [50], [53]–[56]. However, most of this methods do not use the representational power of the SOM to build the prediction models. For example, in [54] and [55], temporal variants of the SOM are used to help in the construction of local linear AR models associated to each neuron in the network. After training the network with the entire set of training vectors, the local AR model of neuron is built based solely in the subset of the input vectors for which that neuron was the winner during the training stage. This subset of input data, not the weight vectors of the SOM, is then used to compute the coefficients of the AR model.
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The method described in [53] involves the selection of winning neurons based on a direct (explicit) minimization of the prediction error. This is not equivalent to the VQTAM since the criterion to find the winning neuron during training involves the and , instead of the vectors output vectors and . Thus, one should now write (9) as follows: (46) At first sight, this equation seems to be more intuitive than the original one in (9), since the ultimate goal in prediction is to have the minimum possible error at each time step. However, is not always available the vector of desired outputs during the prediction (test) stage, limiting the application of this method. As a final remark, the approach in [56] is equivalent to the VQTAM for time series prediction [57]. These works were developed independently from each other, with the VQTAM being proposed as a general framework with wider range of applicability. VII. CONCLUSION Identification and control of dynamical systems through neural networks are research fields completely dominated by supervised learning paradigms. Only recently the use of unsupervised neural networks has been studied as a feasible alternative to standard MLP and RBF networks. The goal of this paper is ambitious in the sense that it proposes a common framework for the SOM to approximate nonlinear input–output mappings arising in complex dynamical problems. The simulations shown in this paper illustrated the potential of the VQTAM technique. We demonstrated that the estimation errors decrease asymptotically as the SOM training proceeds, allowing the VQTAM scheme to be understood as a self-supervised gradient-based error reduction method. The performance of the proposed approach was evaluated in a variety of complex dynamical modeling tasks, namely: i) time series prediction; ii) identification of SISO/MIMO systems; and iii) nonlinear predictive control. For all tasks, the simulation results produced by the SOM are as accurate as those produced by the MLP network, and better than those produced by the RBF network. Furthermore, it was verified that the SOM is less sensitive to weight initialization than MLP networks. The ARSOM model always performed better than MLP/RBF-based models. Compared to the SOM, the ARSOM produces smaller prediction errors with considerably lower number of neurons. Also, the number of training epochs needed to train the ARSOM is much smaller than required by the SOM. Finally, we discussed the main properties of the VQTAM and their relationships to other unsupervised methods for dynamical system identification and control. As pointed out in the previous paragraph, this paper focused on the “potentialities” of the VQTAM, since none of the applications were studied in depth. Much more need to be done, both in empirical and theoretical terms, in order to effectively consolidate the proposed technique as a feasible alternative to supervised paradigms. Issues that certainly deserves our attention are residual analysis, computation of confidence intervals for the estimates generated by the SOM and ARSOM models
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Fig. 11. Angular trajectories for the joint 6 (t) generated by (a) the SOM and (b) the CTH networks during the reproduction of the figure-eight trajectory. The solid line denotes the desired trajectory, while the asterisks are the estimated values. Corresponding Cartesian trajectories of the end-effector generated by the (c) SOM and (d) the CTH networks (open circles denote desired values).
and its comparison with those obtained for supervised learning methods, and robustness to noise. A performance comparison of different approaches to local input–output modeling is another topic for further research. APPENDIX Description of the CTH Algorithm: The input vector at time comprises three parts. The first one is the sensorimotor vector (47) The second part is the fixed context usually set to the initial state of the trajectory. It is kept unchanged until the end of the current sequence has been reached. The third part is the time-varying context, an STM mechanism implemented as tapped delay lines
A suitable length of the time window is determined by trialand-error. is compared At each time step, the current state vector with each feedforward weight vector in terms of Euclidean distance as follows: (48)
, and a A fixed context distance, , time-varying context distance, are also computed. While is used to find the winners of the current competition, and are used to solve ambiguities during trajectory reproduction. The output neurons are then ranked as (49) where is the number of output neurons, is the index of the th neuron closest to choice function, defined as follows: if otherwise
, , and or
, is the
(50)
, and the weighting factor is called where neurons, the responsibility function. We consider , , as the winners of the current competition which represents the current and its context. state vector for The activation decay from a maximum value , to a minimum for , as follows: for for
(51)
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where and The function
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are user-defined and is then updated
, for all . (52)
where is called the exclusion parameter and are adjusted for all . Finally, the weight vectors
, (53) (54) (55)
where is the learning rate and is initialized with random numbers. , The lateral weights encode the temporal order of the trajectory states using a simple Hebbian learning rule to associate the winner of the previous competition with the winner of the current competition if otherwise is a gain parameter and where . The output values are computed as follows:
(56) for all ,
(57) where and
and . The function is chosen so that . For , the output values are set to
, for all . Finally, the control signal to be delivered to the robot is computed from the weight vector of the neuron with highest value : of (58) where
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Guilherme A. Barreto (M’98) was born in Fortaleza, Ceará, Brazil, in 1973. He received the B.S. degree in electrical engineering from Federal University of Ceará in 1995, and the M.S. and D.Phil. degrees in electrical engineering from the University of São Paulo, São Paulo, Brazil, in 1998 and 2003, respectively. Currently, he is an Advanced Researcher at the Intelligent Signal Processing Group of the Department of Teleinformatics Engineering, Federal University of Ceará. His main interest involves the development of self-organizing neural models for identification and control of dynamical systems, adaptive filtering, channel equalization, time series prediction, fault detection and diagnosis, robotics, and image processing.
Aluizio F. R. Araújo was born in Recife, Pernambuco, Brazil, in 1957. He received the B.S. degree in electrical engineering from Federal University of Pernambuco, Brazil, in 1980, the M.S. degree in electrical engineering from the State University of Campinas, Campinas, in 1988, and the D. Phil. degree in computer science and artificial intelligence from the University of Sussex, U.K., in 1994. He worked at the São Francisco Hidroelectrical Company for five years and, in 1998, he became an Assistant Professor at University of São Paulo, where, in 1994, he was promoted to Adjunct Professor. Since 2003, he has been with the Federal University of Pernambuco. He has published 22 papers in international or national journals and more than 60 papers in international or national conferences. His research interests are in the areas of neural networks, machine learning, robotics, dynamic systems theory, and cognitive science.
