## Soft computing for control of non-linear dynamical systems - Semantic Scholar

The output of m th layer at time t is computed as. It is multilayer feedforward network based on the combination of the input, competitive, and output layers. Model of CPN is instar-outstar. It is three-layer neural network that performs input-output data mapping, that is, producing output in the response to an input vector on the basis of Fuzzy Competitive Learning. In CPN the connection between input layer and competitive layer is instar structure and connection between competitive and output layer is outstar structure.

Counter Propagation Network involves two-stage training process. In first stage input vector is clustered on the basis of Euclidean distance and the performance of network is improved using linear topology. Using Euclidean distance between input and weight vector, we evaluated BMN. In phase II, the desired response is obtained by adapting the weights from competitive layer to output layer [ 27 ]. Euclidean distance between input vector x and weight vector v ij is. The architecture of CPN is shown in Figure 3. FCL is used for adjusting instar-outstar weights and update of BMN for discrete time nonlinear dynamical systems.

CPN is efficient network for mapping the input vector of size n placed in clusters of size m. Like traditional self-organizing map, the weight vector for every node is considered as a sample vector to the input pattern. The process is to determine which weight vector say J is more similar to the input pattern chosen as BMN. First phase is based on closeness between weight vector and input vector and second phase is weight update between competitive and output layer for desired response.

Let v ij be weight of input node i to neuron j and let w jk be the weight between competitive node j and neuron k. We propose a new learning rate calculation method for use of weight update:. Phase I for determination of BMN.

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There are steps for training of CPN as follows. Perform Steps 3 — 8 until stopping criteria for Phase I training fail. The stopping condition may be fixed number of epochs or learning rate has reduced to negligible value. Perform Steps 4 — 6 for all input training vector X.

Euclidean distance is calculated as follows:. Set activation function to x , y input and output layer, respectively. Also update the weights into unit Z J as. Update the weights from node Z J to the output unit as. Decrease the rate of learning s. Test the stopping condition for Phase II i. BPN for learning is not always successful because of its sensitivity to learning parameters. Optimal learning rate always changes during the training process.

Dynamical learning of CPN is carried out using Lemmas 1 , 2 , and 3. Since input to the neural network in this model is bounded, we consider Lemmas 1 and 2 given below. Let all the vectors be column vectors and superscript d p k refers to specific output vectors component.

Let V and W represent the input-hidden and hidden-output layer weight matrix, respectively. The objectives of the network training minimize error function J , where J is.

Consider a three-layer CPN; the network error matrix is defined by the error between differences of desired and FCPN output at any iteration and given as. The objective of the network for minimization of error given in 24 is defined as follows:. Using gradient descent method updated weight is given by. Using 23 — 26 , we have. To obtain minimum error for multi-layer FNN in above equation 28 after simplification, we have.

To demonstrate the effectiveness and merit of proposed FCPN, simulation results are presented and discussed.

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Four different nonlinear dynamical systems and one general benchmark problem known as Box-Jenkins model with time series data are considered. For accessing the performance criteria of FCPN and comparing with Dynamic and Back Propagation Network, we have evaluated various errors as given below. In case of four dynamical models, it is recommended to access criterion such as subset of the following. Given N pair of data points y k , y - k , where y k is output of system and y - k is output of controller, the Maximum Absolute Error MAE is.

The nonlinear dynamical system [ 24 ] is governed by the following difference equation:. The objective of Example 1 is to control the system to track reference output given as sample data points.

Table 1 includes various calculated errors of different NN models for Example 1. Calculated various errors for Example 1. The objective of 44 is to control the system to track reference output given sample data points. Training and testing samples contain sample data points. FCPN network discussed earlier is used to identify the system from input-output data and is described by the equation.

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FCPN identifier needs some prior information concerning the input-output behavior of the system before identification can be undertaken. The weights in the neural network were adjusted. Table 2 includes various calculated errors of different NN models for Example 2. Calculated various errors for Example 2. The nonlinear dynamical system [ 23 ] is governed by the following difference equation:.

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FCPN network equation 47 is used to identify the system from input-output data. Weights in the neural networks were adjusted for every instant of discrete time steps. The objective of 47 is to control the system to track reference output given sample data points. In Figure 6 b , the desired outputs of the system as well as the FCPN model are shown and are seen to be indistinguishable. Table 3 includes various calculated errors of different NN models for Example 3.

Calculated various errors and BFR for Example 3. The nonlinear dynamical system [ 24 ] is governed by the difference equation Generalized form of this equation is Model IV. In this example, the system is assumed to be of the form. Table 4 includes various calculated errors of different NN models for Example 4. Calculated various errors and BFR for Example 4. In this example we have used Box-Jenkins time series [ 32 ], of pairs of data measured from a gas furnace system with single input u t being gas flow rate and single output y t being CO 2 concentration in outlet gas. Training samples and testing samples contained and data points, respectively.

Figure 8 a shows the mean squared control errors of FCPN methods. Figure 8 b shows the performance of the controller with FCPN algorithm. Result shows that FCPN algorithm enable us to appropriate the approximation using fuzzy learning as given in equation 11 of Box Jenkins time series data, based on calculation of BMN. FCPN is a neural network control method which was developed and presented. The key ideas explored are the use of the FCL algorithm for training the weights of the instar-outstar. The performances of the FCPN controller based training are tested using four nonlinear dynamical systems and on time series Box-Jenkins data and compared with the Dynamic Network and standard Back Propagation algorithm.

It can be seen that the FCPN algorithm gives minimum errors as compared to the Dynamic Network and standard Back Propagation Network for all four models of nonlinear dynamical systems and Box-Jenkins time series data.

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Results obtained from FCPN were compared for various errors and it is clearly shown that FCPN works much better for the control of nonlinear dynamical systems. The authors declare that there is no conflict of interests regarding the publication of this paper. Europe PMC requires Javascript to function effectively.

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## Soft Computing for Control of Non-Linear Dynamical Systems

This may be because the snippet appears in a figure legend, contains special characters or spans different sections of the article. Comput Intell Neurosci. Published online Aug PMID: Sapkal , 2 and Dev P. Agarwal 3. Vilas S. Dev P. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.