Observer-Based Adaptive Iterative Learning Control for a Class of Nonlinear Time Delay Systems with Input Saturation

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 645161, 19 pages http://dx.doi.org/10.1155/2015/645161 Research Article Observer-Based Adaptive Iterative Learning Control for a Class of Nonlinear Time Delay Systems with Input Saturation Jian-ming Wei, Yun-an Hu, and Mei-mei Sun Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai 264001, China Correspondence should be addressed to Yun-an Hu; Received 19 May 2015; Revised 14 July 2015; Accepted 22 July 2015 Academic Editor: Xinkai Chen Copyright © 2015 Jian-ming Wei et al. 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. This paper presents an adaptive iterative learning control scheme for the output tracking of a class of nonlinear systems with unknown time-varying delays and input saturation nonlinearity. An observer is presented to estimate the states and linear matrix inequality (LMI) method is employed for observer design. The assumption of identical initial condition for ILC is relaxed by introducing boundary layer function. The possible singularity problem is avoided by introducing hyperbolic tangent function. The uncertainties with time-varying delays are compensated for by the combination of appropriate Lyapunov-Krasovskii functional and Young’s inequality. Both time-varying and time-invariant radial basis function neural networks are employed to deal with system uncertainties. On the basis of a property of hyperbolic tangent function, the system output is proved to converge to a small neighborhood of the desired trajectory by constructing Lyapunov-like composite energy function in two cases, while keeping all the closed-loop signals bounded. Finally, a simulation example is presented to verify the effectiveness of the proposed approach. 1. Introduction Over the past decades, tremendous research efforts have been made aiming at the development of systematic design methods for the iterative learning control (ILC) of nonlinear systems performing control task over a finite interval repeatedly. ILC has become the most suitable and effective control scheme for such repeatable control tasks because of its capacity of achieving perfect tracking by learning mechanism along iteration. Generally, according to the stability analysis tool, ILC can be classified into two categories: traditional ILC [1–4] and adaptive ILC (AILC) [5–10]. The basic principle of traditional ILC is to use information collected from previous execution to form the control action for current operation by a learning mechanism for purpose of improving performances from iteration to iteration. Furthermore, the stability conclusion of traditional ILC is usually obtained by using contraction mapping theorem and fixed point theorem. However, traditional ILC requires for the global Lipschitz continuous condition, which makes it difficult to apply it to certain nonlinear systems. Besides, traditional ILC uses contraction mapping theorem rather than Lyapunov method as the key principle of stability analysis, which makes it difficult to relax the global Lipschitz condition to local Lipschitz or even non-Lipschitz condition and cooperate with the mainstream methods of nonlinear control theory, such as adaptive control and neural control. To relax the constraints of traditional ILC and extend it to a broader range, some researchers tried to introduce the idea of adaptive control into ILC and proposed adaptive iterative learning control (AILC). AILC takes advantage of both adaptive control and ILC, which successfully overcomes the restriction of global Lipschitz condition; thus it enables us to use fuzzy logic systems or neural networks as approximators to deal with nonlinear uncertainties. In general, the control parameters of AILC methods are tuned along the iteration axis, and the so-called composite energy function (CEF) [5] is usually constructed to analyze the stability and convergence property of the closed-loop systems. The past decade has witnessed great progress in AILC of uncertain nonlinear systems [6–10]. In practice, control of systems with time delays has always been a meaning research, since time delay can be often encountered in a wide range of physical systems and devices, such as turbojet engines, aircraft systems, microwave oscillators, nuclear reactors, and chemical processes [11, 12]. The existence of time delays in a system may degrade the 2 control performance and even at worst may become a source of instability. Thus, the investigation of time delay in systems has always been an active topic for control engineers. Consequently, stabilization problem of control systems with time delay has received much attention for several decades and a large number of research results have been reported in the literature that deal with various analysis and design problems [11–16]. However, in the field of AILC, only a few results are available for nonlinear systems with time delays [17–19]. In [17], an AILC strategy was developed for a class of scalar systems with unknown time-varying delay and then extended to a class of high-order systems with both time-varying and time-invariant parameters, where the unknown time-varying parameter was estimated in the iterative learning process. However, the proposed controller in [17] requires that the uncertainties in the system satisfy local Lipschitz condition and nonlinear parameterized condition such that adaptive learning laws can be used to estimate the unknown timevarying parameters. In [18, 19], we designed an AILC scheme for a class of nonlinearly parameterized systems and an RBF NN-based AILC for class of unparameterized systems, respectively, where the systems in two papers are with both unknown time-varying delays and unknown dead zone input. However, all of the aforementioned results are on systems with time delay states. As for systems with time delay outputs, to the best of our knowledge, there are no works reported in the literature. Other than time delay, another challenging problem in control of nonlinear systems lies in the existence of nonsmooth and nonlinear characteristics such as dead zone, hysteresis, saturation, and backlash. Among them, the significance of controller design for systems with saturation can be overemphasized, as any control systems depending on actuators have physical limitations, for example, mechanical actuators and aircraft. The existence of saturation can severely limit system performances and usually leads to undesirable inaccuracies and even instability [20]. Therefore, the control design for nonlinear systems preceded by input saturation is a challenging but worthwhile and necessary issue. For control systems with input saturation, many results have been published in the past several decades [20–26]. To address such (...truncated)