Wavelet Adaptive Observer Based Control for a Class of Uncertain Time Delay Nonlinear Systems with Input Constraints M.Sharma Medicaps Instt. of Tech. & Mgmt. Indore, INDIA. er.mann24@gmail.com A.Kulkarni Medicaps Instt. of Tech. & Mgmt Indore, INDIA. a.kulkarni17@gmail.com A.Verma IET, DAVV Indore, INDIA. ajayrt@rediffmail.com Abstract— This Paper investigates the mean to design the observer and observer based controllers for a class of delayed nonlinear systems with Lipschitz conditions and unknown dynamics subjected to input constraints. A new design approach of full order wavelet based adaptive observer is proposed. Wavelet neural networks (WNN), which are having superior learning capability in comparison to conventional neural networks, are used as system identification tool. Using the feedback control, based on reconstructed states, the behavior of closed loop system is investigated. The fact that the uncertain time delayed system with observer based control converges to the small neighborhood of the origin is established through Lyapunov-Krasovskii functional. A numerical example is provided to verify the effectiveness of theoretical development Keywords-Adaptive observers; wavelet networks; time delay systems; actuator constraints; Lyapunov- Krasovskii functional I. INTRODUCTION Time delay, either in state or input, and actuator saturation are the two frequently encountered phenomenons which severely limit the performance of real world control systems giving rise to undesirable inaccuracies or even leading to instability [1]. Time delay occurs in physical systems due to so many reasons like finite capability of information processing among various parts of system, inherent phenomenon like mass transfer flow and recycling and /or byproduct of computational delay [2]. Discussions on delays and their effect on stabilization /destabilization effects in control systems have attracted the interest of several investigators in recent years [2-4]. Mainly the results cited in the literature are based on two approaches: Lyapunov-Krasovskii theorems based approach and Razumikhin theorem based approach. The results obtained are having a variable degree of conservatism [3, 4]. Controller design for the systems having actuator saturation is an active area of investigation since last decade. Most of the research in this topic is based on augmentation of baseline controller with additional saturation compensation dynamics [5-7]. Controller design for the systems having actuator saturation as well as time delay is a challenging problem which may lead to a conservative controller design with a small region of stability. In this work, this problem is dealt to relax the conservatism by enlarging the region of stability. Designing of a stable state feedback control algorithm requires the knowledge of the states. In many practical applications, measurement of the states is not feasible and so the unmeasurable states are generally estimated based on the available measurement and knowledge of the physical system. Designing of a stable adaptive observer that estimates the unmeasurable states and unknown system dynamics for a class of nonlinear systems subjected to actuator saturation and/or time delayed state/input, is a very special case of investigation over last few years. Most of the results cited in the literature are based on the norm based uncertainties leading to highly conservative observer design [8-10]. System identification tools like neural networks can be used to relax the conservatism upto some extent. A neural network such as multi layer perceptrons have been proved as efficient approximation tool due to their universal approximation property and has been widely used in the controller design. However there are certain difficulties associated with NN based controller. The basis functions are generally not orthogonal or redundant; i.e., the network representation is not unique and is probably not the most efficient one. Furthermore, the convergence of neural networks may not be guaranteed. Even when it exhibits a good convergence rate, the training procedure may still be trapped in some local minima depending on the initial settings. In addition, approximation errors and external disturbances can not be efficiently attenuated. Hence, performance and even stability may not be guaranteed. Recently by combining the idea of neural networks and the merits of wavelets, a wavelet neural network (WNN) was developed by Zhang and Benveniste [11].Wavelet networks are feed-forward neural networks using wavelets as activation function. In wavelet networks, both the position and the dilation of the wavelets are optimized besides the weights. Due to their space and frequency localization properties, the learning capability of WNN is superior to conventional neural networks. Training algorithms for WNN converge in smaller number of iterations than for conventional neural networks. These WNN combines the capability of artificial neural network for learning ability and capability of wavelet decomposition for identification ability. It has been proved that wavelet neural networks are asymptotically optimal approximators for modeling inaccuracies. Wavelet neural networks are optimal in the sense that they require the smallest possible number of bits to store for reconstructing a function within a precision . Thus WNN based control systems can achieve better control performance than NN based control systems [12,13]. Recently the researchers are inclined towards the designing aspects of Wavelet based adaptive controllers [14,15]. 2009 International Conference on Advances in Recent Technologies in Communication and Computing 978-0-7695-3845-7/09 $25.00 © 2009 IEEE DOI 10.1109/ARTCom.2009.54 863 2009 International Conference on Advances in Recent Technologies in Communication and Computing 978-0-7695-3845-7/09 $26.00 © 2009 IEEE DOI 10.1109/ARTCom.2009.54 863 2009 International Conference on Advances in Recent Technologies in Communication and Computing 978-0-7695-3845-7/09 $26.00 © 2009 IEEE DOI 10.1109/ARTCom.2009.54 863