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Fig. 11. Angular trajectories for the joint 6 g(t) generated by (a) the SOM and (b) the CTH networks during the reproduction of the figure-eight trajectory. Th solid line denotes the desired trajectory, while the asterisks are the estimated values. Corresponding Cartesian trajectories of the end-effector generated by the (« SOM and (d) the CTH networks (open circles denote desired values).

Figure 11 Angular trajectories for the joint 6 g(t) generated by (a) the SOM and (b) the CTH networks during the reproduction of the figure-eight trajectory. Th solid line denotes the desired trajectory, while the asterisks are the estimated values. Corresponding Cartesian trajectories of the end-effector generated by the (« SOM and (d) the CTH networks (open circles denote desired values).

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Fig. 1. Differences between the (a) supervised and (b) self-supervised approach to system identification. cooperation among neurons during the learning process, the net- work convergence can be evaluated through the average ex- pected squared prediction error associative nature of the VOQTAM computes indirectly an error signal. Such a property allows us to understand this approach as a Self-supervised gradient-based error reduction method.
Fig. 1. Differences between the (a) supervised and (b) self-supervised approach to system identification. cooperation among neurons during the learning process, the net- work convergence can be evaluated through the average ex- pected squared prediction error associative nature of the VOQTAM computes indirectly an error signal. Such a property allows us to understand this approach as a Self-supervised gradient-based error reduction method.
Fig. 2. General architecture of a model-based predictive controller. E. Identification of MIMO Systems
Fig. 2. General architecture of a model-based predictive controller. E. Identification of MIMO Systems
Fig. 3. (a) Prediction error (NRMSE) versus the memory order (7,,). (b) Sequence of 100 first estimated samples of the Mackey-Glass.
Fig. 3. (a) Prediction error (NRMSE) versus the memory order (7,,). (b) Sequence of 100 first estimated samples of the Mackey-Glass.
Fig. 4. Evolution of the prediction error as the number of epochs and the size of the training set of the SOM increases.
Fig. 4. Evolution of the prediction error as the number of epochs and the size of the training set of the SOM increases.
MINIMUM PREDICTION ERRORS (NRMSE) OF THE ARSOM, SOM, MLP, RBF AND AR MODELS FOR THREE CHAOTIC TIME SERIES (output variable), y € *, of a hydraulic actuator. It is worth noting that the oil pressure time series shows an oscillative be- havior caused by mechanical resonances [5]. The SOM, MLP, and RBF networks were used as nonlinear models to approxi- mate the forward and the inverse mappings of the hydraulic ac- tuator. In the forward identification task, these three neural nets are also compared with the usual linear autoregressive model with exogenous inputs (ARX)
MINIMUM PREDICTION ERRORS (NRMSE) OF THE ARSOM, SOM, MLP, RBF AND AR MODELS FOR THREE CHAOTIC TIME SERIES (output variable), y € *, of a hydraulic actuator. It is worth noting that the oil pressure time series shows an oscillative be- havior caused by mechanical resonances [5]. The SOM, MLP, and RBF networks were used as nonlinear models to approxi- mate the forward and the inverse mappings of the hydraulic ac- tuator. In the forward identification task, these three neural nets are also compared with the usual linear autoregressive model with exogenous inputs (ARX)
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.
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.
Fig. 6. Measured values of (a) valve position and (b) oil pressure. samples, the first 512 samples were used to train the neural net- works 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
Fig. 6. Measured values of (a) valve position and (b) oil pressure. samples, the first 512 samples were used to train the neural net- works 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
RMSE FOR THE MLP, SOM, WTA, AND RBF NETWORKS IN THE FORWARD AND INVERSE IDENTIFICATION OF THE HYDRAULIC ACTUATOR TABLE II
RMSE FOR THE MLP, SOM, WTA, AND RBF NETWORKS IN THE FORWARD AND INVERSE IDENTIFICATION OF THE HYDRAULIC ACTUATOR TABLE II
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.
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.
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 o the nonlinear plant (open circles denote true values of the output).
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 o the nonlinear plant (open circles denote true values of the output).
Fig. 10. VQTAM-based predictive controller. (a) Plant output time series y(t) (thicker line) for a given reference signal. (b) Resulting optimal control sequence {u*(t)}.
Fig. 10. VQTAM-based predictive controller. (a) Plant output time series y(t) (thicker line) for a given reference signal. (b) Resulting optimal control sequence {u*(t)}.
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. 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. TABLE III
Related topics:
Time Series PredictionMultilayer PerceptronAssociative MemoryTime FactorsFunction approximationSelf Organized MapMIMO System
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