International Journal of Neural Systems, Vol. 20, No. 1 (2010) 29–38 c  World Scientific Publishing Company DOI: 10.1142/S0129065710002218 DISCRETE-TIME REDUCED ORDER NEURAL OBSERVERS FOR UNCERTAIN NONLINEAR SYSTEMS ALMA Y. ALANIS CUCEI, Universidad de Guadalajara Apartado Postal 51-71, Col. Las Aguilas, C.P. 45080 Zapopan, Jalisco, Mexico almayalanis@gmail.com EDGAR N. SANCHEZ CINVESTAV, Unidad Guadalajara Apartado Postal 31-438, Plaza La Luna, Guadalajara Jalisco, C.P. 45091, Mexico LUIS J. RICALDE UADY, Faculty of Engineering Av. Industrias no Contaminantes por Periferico Norte Apdo. Postal 115 Cordemex, Merida, Yucatan, Mexico This paper focusses on a novel discrete-time reduced order neural observer for nonlinear systems, which model is assumed to be unknown. This neural observer is robust in presence of external and internal uncertainties. The proposed scheme is based on a discrete-time recurrent high order neural network (RHONN) trained with an extended Kalman filter (EKF)-based algorithm, using a parallel configuration. This work includes the stability proof of the estimation error on the basis of the Lyapunov approach; to illustrate the applicability, simulation results for a nonlinear oscillator are included. Keywords : Reduced order neural observers; recurrent high order neural networks; Kalman filtering learn- ing; discrete-time nonlinear systems; Van der Pol oscillator. 1. Introduction Modern control systems usually require detailed knowledge about the system to be controlled; such knowledge should be represented in terms of dif- ferential or difference equations. This mathematical description of the dynamic system is named as the model. There can be different motives for establish- ing mathematical descriptions of dynamic systems, such as: simulation, prediction, fault detection, and control system design. Basically there are two ways to obtain a model; it can be derived in a deductive manner using physics laws, or it can be inferred from a set of data collected during a practical experiment. The first method can be simple, but in many cases is excessively time- consuming; it would be unrealistic or impossible to obtain an accurate model in this way. The second method, which is commonly referred as system iden- tification, 35 could be a useful short cut for deriving mathematical models. Although system identifica- tion not always results in an accurate model, a satis- factory one can be often obtained with reasonable efforts. The main drawback is the requirement to conduct a practical experiment, which brings the sys- tem through its range of operation. 6, 22 Many of the nonlinear control publications assume complete accessibility for the system state; this is not always possible. For this reason, nonlin- ear state estimation is a very important topic for nonlinear control. 23 State estimation has been stud- ied by many authors, who have obtained interesting results in different directions. Most of those results need the use of a special nonlinear transformation 21 or a linearization technique. 8, 14 Such approaches 29 Int. J. Neur. Syst. 2010.20:29-38. Downloaded from www.worldscientific.com by CENTRO DE INVESTIGACION Y DE ESTUDIOS AVANZADOS DEL IPN (CINVESTAV) SERVICIOS BIBLIOGRAFICOS on 10/19/12. For personal use only.