This chapter is concerned with an adaptive Radial basis function neural network (RBFNN) is studied and implemented for a class of nonlinear discrete-time system with bounded disturbance. Due to immeasurable states and presence of input-nonlinearities like backlash, dead zone and hystersis, the design of controller becomes more challenging. RBFNN is designed to the approximation of such nonlinear system at a relative degree of accuracy, which can be used for adaptation of nonlinear discrete-time systems with or without the presence of nonlinearities. RBFNN employs as a reference model which is useful to closed loop form of pure feedback controller. Based on Lyapunov method it is proven that proposed scheme for discrete-time nonlinear systems is asymptotically stable. Hence, not only stability of proposed system is assured but it is also shown that tracking error of model lies in closed neighborhood of zero. The feasibility of the RBFNN is demonstrated by two examples of nonlinear systems.
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In the last decades, many methods for nonlinear discrete time systems have been proposed for system identification, and it have many applications in control system (Chen, 1989), biological process (Xie, 2006), chemical process control (Watanbe, 1989), signal processing system (Adjrad, 2007) etc. The computational complexity of nonlinear complex dynamical system identification is high in comparison to linear dynamical systems. Some mathematical models were developed by researchers like calculus of variations (Schetzmen, 1980) polynomial identification models (Harnandez, 1993 & Lee, 1998), which gives some degree of accuracy but not efficient. Neural networks are adaptive techniques (Adebiyi, 2014, Bennett, 2013, Sadegh, 1993, Chon, 1997, Sakhre, 2016 & Narendra, 1990) have been extensively studied and used in identification of nonlinear discrete time system and time series forecasting. Various identification problems of industrial systems are solved with the help of neural network and it has been seen that any continuous function can be approximated by neural network to certain degree of accuracy (Adebiyi, 2014).
Self-adaptability and self-learning capability of Neural Network (NN) for tracking of nonlinear system and its computing ability is more resilient (Ticknor, 2013). Neural network using Levenberg-Marquardt training method is well-known for approximation of nonlinear discrete time systems (Zhang, 2002), in categories of neural networks the back-propagation learning based NN, is one of the most popular neural networks (Wang, 2006). In BPNN weights and threshold are randomly initialized, training algorithm for weight connections of a network is adjusted according to minimize our objective i.e. mean square error (MSE) (Yam, 2000). When MSE achieves its goal setting, the connection weights and bias are indomitable, at this stage training process completed. The final results depend on important parameters of NN, initial weight and bias are one of them, which was major challenge for the researcher’s cause of this training result, may give local minima instead of global minima. It is well known that enormous and complex behavior nonlinear discrete time system it is necessary to use of some soft computing techniques and various nature inspired algorithms are applied to control and identification of nonlinear systems and time series forecasting (Theofilatos, 2009, Beligiannis, 2005, Chen, 2004, Kwon, 2007, Zhang, 2011 & Li, 2014). Backstepping feedback control (Tong, 2011) is an adaptive neural network used for nonlinear dynamical systems with delays.