Neural networks are non curly most used in the identification and control systems . They may not be powerful enough to model complex dynamic systems with respect to neural networks curly. Different types of recurrent neural networks have been proposed and have been successfully applied in many fields . The structure of fully connected recurrent neural networks which was proposed by Williams and Zipser , is most often used [27,28] because of its generality.
The second step is learning or in other words, estimating the parameters of the network from examples of input-output system identification. The methods of learning are numerous and depend on several factors, including the choice of error function, the initialization of weights, and the selection of the learning algorithm and the stopping criteria of learning. Learning strategies were presented in several research works that we cite .
The third step is the validation of the neural network obtained using the testing criteria for measuring performance. Most of these tests require a data set that was not used in learning. Such a test set or validation should, if possible, cover the same range of operation given that all learning. In this work, we consider a recurrent neural network Figure 1 for identification of complex dynamic systems to a single input and single output. The architecture of these networks is composed of two parts: a linear model the linear behavior of the system and a non-linear approach to nonlinear dynamics.
Figure 1. Architecture of the considered neural network.
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The output of neural model is given by:. In this section we present three learning procedures of neural network. The learning procedure is to adjust the coefficients of the neural networks considered by minimizing the criterion ; it is necessary to solve the equation:. The term is then written as follows:. The parameters , , are chosen so that the neural model of the system must be stable. In our case, the stability analysis is based on the famous Lyapunov approach .
It is well known that the purpose of identification is to have a zero gap between the output of the system and that of the neural model. Three Lyapunov candidate functions are proposed:. The function satisfies the following conditions:. The term is given by the following equation:.
From Equation 22 , may be as follows:. The term is as follows:. To meet the three conditions of stability of Lyapunov candidate functions proposed parameters , and must verify:. The term can be written as follows:. The choice of initial synaptic weights and biases can affect the speed of convergence of the learning algorithm of the neural network . According to , the weights can be initialized by a random number generator with a uniform distribution between and or a normal distribution.
The proposed Lyapunov-Based used to training dynamic model of the system is presented by the flowchart in Figure 2 , reads as follows:. We fix the desired square error , the parameters , the number of samples , the maximum number of iterations , the number of neurons in the first hidden layer. Calculate the output of the neural network.
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Calculate the difference between the system output and the model. Calculate the square error. Adjust the vector of network parameters using one of the three following relations:. If the number of iterations or , proceed to Step 9. Figure 2. Flowchart of the learning algorithm of the neural network. If , proceed to Step Otherwise, increment and return to Step 2.
The flowchart of this algorithm is given in Figure 2. The neuronal model obtained from the estimation of its parameters is valid strictly used for the experiment. So check it is compatible with other forms of input in order to properly represent the system operation to identify. Most static tests of model validation are based on the criterion of Nash, on the auto-correlation of residuals, based on cross-correlation between residues and other inputs to the system.
According to , the Nash criterion is given by the following equation:. Ideally, if the model is validated, the results of correlation tests and the Nash criterion following results:. In this section, we propose a structure of neural adaptive control of a complex dynamic system and three learning algorithms of a neuronal controller.
In this work, the architecture of the proposed adaptive control is given in Figure 3. The considered neural network is first trained off-line to learn the inverse dynamics of the considered system from the input-output data. The model following adaptive control approach is performed after the training process is achieved.
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The proposed Lyapunov-Base training algorithm is used to adjust the considered neural network weights so that the neural model output follows the desired one. Figure 3. Structure of the proposed adaptive control. The control system consists of using an optimization digital non-linear algorithm to minimize the following criterion:. The minimum of criterion is reached when:. The term defined by:. Using the above equations, the relationship giving the vector minimizing the criterion can be written as follows:.
It is necessary to check the stability of this procedure to adjust the weight of the correction before applying. In this case, the candidate Lyapunov function may be as follows:. According to Equation 32 , the term is written as follows:. Theorem 5. The procedure for adjusting the parameters of neuronal controller can be described by the following equation:.
Theorem 6. This thesis generalizes the multilayer perceptron networks and the associated backpropagation algorithm for analogue modeling of continuous and dynamic nonlinear multidimensional systems for simulation, using variable time step discretizations of continuous-time systems of coupled differential equations. A major advantage over conventional discrete-time recurrent neural networks with fixed time steps, as well as Kalman filters and time-delay neural network TDNN models with fixed time steps, is that the distribution of time steps is now arbitrary, allowing for smaller time steps during steep signal transitions for much better trade-offs between accuracy and CPU time, while there is also still freedom in the choice of time steps after the neural network model has been generated.
In fact, multirate methods for solving differential equations can be readily applied. The use of second order differential equations for each neuron allows for complex oscillatory behaviours even in feedforward networks, while allowing for efficient mappings of differential-algebraic equations DAEs to a general neural network formalism.
The resulting formalism represents a wide class of nonlinear and dynamic systems, including arbitrary nonlinear static systems, arbitrary quasi-static systems, and arbitrary lumped linear dynamical systems. The simplest model for the artificial neural network is the feedforward artificial neural network which was described in Fig.
We had built our algorithm based on this model and added some modifications. For the learning of the neural network we used back-propagation technique. The different is that the output will enter as an input for the system with time delay, also there is a time delay for the input itself so we can extract more inputs Fig3 : Our Model for the system which contains the characteristics for the system we want to identify.
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In our work we used a function which NARX model to enter just the current output is a combination of sigmoid and linear function with the current input to just identify the future Fig. The This function facilitates the identification as the algorithm built was based on that model.
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We results of this function will not be in restricted have used two hidden layers as shown in Fig. The system was trained to predict the pitch angle, and the results had a Neural Network Structure and Error mean square error of 0. Type The Neural network structure we have used for the identification process consists of two hidden layers and the input layer; the first hidden layer consists of neuron and the second one consists of 90 neuron this numbers determined using trial and error this gives us good results.
Back-propagation is a method that is used in the learning process of artificial neural networks for adjusting the weights of the connections between the neurons to guarantee minimal error. Regression Regression is the most used identification method specially in the linear systems. Auto Regression is proposed here  and the Fig 6: Using Regression method to identify the systems using the cross Second system Quadcopter data that we have covariance function which was got from the built in Communication and Aerospace system to estimate the coefficient of the series Technology Center of Zewail City, the obtained of the system equation.
The regression method data are from the IMU. The identification results are shown in Fig. This pervious results were obtained on a pc with I7 intel Processor of 2Ghz clock speed and 6 Gb of Ram in this step we have compared our work with running this algorithm on an embedded system. The embedded system we used is raspberry pi and its processor is 1 Ghz with over clocking. For the quadcopter Data in the I7 processor takes Khattab "Low cost framework for the neural networks architecture and learning parameter identification of unmanned aerial process to make it faster than this identification vehicles" Master Thesis, Aerospace Engineering system which based on the NARX model.
Department, Cairo university