The multilayer perceptron network consists of an input layer, an output layer and usually one or more hidden layers. Figure 1 shows the architecture of MLP network employed for modeling a nonlinear system. It has an input layer of neurons, hidden layers, each hidden layer contains neurons and one neuron in the only output layer. The sigmoid activation function, , is used.

For the nonlinear system (1), if no knowledge about the structure of the nonlinearity of the system is available such system is considered as a “black box” system modeling.

The output of MLP model is given by the following equation:

where is a sigmoid activation function, is the output vector of MLP., , and are the synaptic weights of MLP. is the input vector of MLP.

Among the optimization methods of MLP, the gradient descent method is used in this paper. Optimization of the MLP is to minimize the mean square error E.

where is a function cost.

In this section, 2 hidden layers are taken into account with a single input layer and one output layer, the result of optimization is given by equations (4) to (10):

with:

The RBF consists of only three layers; an input layer, an output layer and usually one hidden layers contains a hidden radial basis function. The RBF model calculates a linear combination of radial basis functions as is given by the following equation:

where is the output vector of RBF. is the synaptic weights of RBF and is a Gaussian activation function:

Compared to the MLP, the RBF contains a very small number of parameters. The purpose of optimizing RBF is to determine, and by minimizing the function cost.

In order to find the minimum of, and two strategies are proposed in the literature for finding the minimum of. The first is based on supervised methods or algorithms using direct time-consuming calculation to determine the minimum of. The second adopts a hybrid scheme (less costly in computation time) to determine the minimum of. Solving these problems can be by various methods such as iterative methods (the Gradient Descent method) and evolutionary algorithms (Genetic Algorithm).

We consider the nonlinear time-varying system described by input-output model:

with:

The trajectory of and are given in figure 3.

The input is sinusoidal signal and it is defined by the following equation:

In figure 4, the time-varying system responses, the MLP model and the optimized RBF model by the GD method are presented. In this simulation figure, the MLP parameters are, , and . The obtained is. However, the RBF parameters are, and. The obtained is .

In figure 5, the time-varying system responses, the MLP model and the optimized RBF model by GA are illustrated. In this simulation the same MLP parameters are taken, while the RBF parameters are, , and. The obtained is.

In these two figures (4 and 5), it is clear that the responses of MLP and RBF models follow the system response although the variation of parameters.

In one hand, the obtained MLP model is found by several tests of parameters and of learning. The large number of MLP parameter increases the difficulty of its use. However, the simplicity of RBF makes modeling is simple and takes much less training time.

In figure 4, the optimized RBF model by the gradient descent method depends on an expensive time of training and depends on different learning rate (, and) while, in figure 5, the parameter of RBF model (and) are finding separately by the GA and the synaptic weights () by the DG method, hence the model is faster than the previous.

To test the effectiveness of the MLP and RBF models we test them on a Continuous Stirred Tank Reactor, CSTR, which is a type of slowly time-varying nonlinear system used for the conduct of the chemical reactions [39-41]. However, the input-output are used in discrete time. A diagram of the reactor is given in the figure 6. The physical equations describing the process are (34) and (35):

where is the height of the mixture in the reactor, (respectively) is the feed of reactant 1 (respectively reactant 2) and (respectively) is the concentration of reactant 1(respectively reactant 2). is the feed product of reaction and its concentration is., , and are consumption reactant rate. They are assumed to be constant. The temperature in the reactor is assumed constant and equal to the ambient temperature. The feed of reactant and the concentration are the input of the process however represents its output. A diagram of the reactor is given in the figure 7.

For the purpose of the simulations the CSTR model of the reactor provided with Simulink-Matlab is used.

In figure 8, the responses of the chemical reactor, the model that produced by MLP and optimized RBF model by the gradient descent method are presented. In figure 9, the responses of the chemical reactor, the MLP model and the optimized RBF model by genetic algorithms are illustrated.

In figures 8 and 9, the responses of the optimized RBF model by the GD method and MLP model follow the response of chemical reactor. Indeed, in figure 8, the MLP model is carried out with 5 neurons in input layer, 25 neurons in first hidden layer, 22 neurons in hidden layer and the learning rate equal to 0.4. However, the RBF model depends only 3 layers, the second and the only hidden layer contains 11 neurons. The used parameters in the GD method are, and. In contrary, in Figure 9, the parameter of

RBF model are optimized separately (and) using GA and () using gradient descent. These parameters are, , and.

The NMSE is given in the table below:

From this table, the NMSE computed in the RBF model optimized by the genetic algorithms method is lower than that found by applying the gradient descent method which proves that the evolutionary algorithms give good accuracy for modeling methods of dynamical systems.