On-line learning of fuzzy rule emulated networks for a class of unknown nonlinear discrete-time controllers with estimated linearization

Recently, the linearization of a class of unknown discrete-time dynamic systems has achieved considerable topics for the controller design. The unknown functions after system linearization have been estimated by several methods including artificial intelligence techniques such as neural networks, fuzzy logic systems and neurofuzzy networks. In a number of published articles, the issues of system theoretic analysis have been introduced and addressed in the topics of stabilization, tracking performance and the bounded parameters. For all of these cases, the results are validated in the domain around the equilibrium point or state (9; 11). These methods of linearization including local linearization, Taylor series expansion and feedback linearization impose Lipschitz conditions (4; 6; 10; 14; 18). The closed-loop system stability and tracking error have been analyzed in the case of neural network adaptive control (5; 7) but during the learning phase the stability and convergence can not be ensured because of the special conditions. The system stability or bounded signals analysis has been verified (1; 13) and references therein. However, these nonlinear systems under control should be obtained in the format as y(k+ 1) = f (k) + g(k)u(k) when y(k) and u(k) are the system output and the control input at time index k, respectively and f (k) and g(k) are unknown nonlinear functions. The small learning rate is often defined to solve the stability problem but the convergence is very slow. The discrete-time projection has been introduced for adaptive control systems in (16). The node number of multi-layer neural networks can take more effect of closed-loop stability and tracking performance. In (15), the unknown nonlinear part has been compensated by neural networks and the closed-loop system stability has been also guaranteed for a class on discrete-time systems. Nevertheless, this algorithm needs the renovation when the operating point is changed. In the case of robust system, the dead-zone function has been applied for feedback linearization systems (8) but this control algorithm are only limited for the system with slow trajectory tracking. In this chapter, we discuss about the controller for a class of nonlinear discrete-time systems with estimated unknown nonlinear functions by Muti-input Fuzzy Rules Emulated Networks (MIFRENs). These nonlinear functions are occurred when 16