Adaptive Backstepping Control for a Class of Uncertain Nonaffine Nonlinear Time-Varying Delay Systems with Unknown Dead-Zone Nonlinearity

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 758176, 14 pages http://dx.doi.org/10.1155/2014/758176 Research Article Adaptive Backstepping Control for a Class of Uncertain Nonaffine Nonlinear Time-Varying Delay Systems with Unknown Dead-Zone Nonlinearity Wei-Dong Zhou, Cheng-Yi Liao, and Lan Zheng College of Automation, Harbin Engineering University, Harbin 150001, China Correspondence should be addressed to Cheng-Yi Liao; Received 19 February 2014; Revised 29 April 2014; Accepted 2 May 2014; Published 21 May 2014 Academic Editor: Gani Stamov Copyright © 2014 Wei-Dong Zhou 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. An adaptive backstepping controller is constructed for a class of nonaffine nonlinear time-varying delay systems in strict feedback form with unknown dead zone and unknown control directions. To simplify controller design, nonaffine system is first transformed into an affine system by using mean value theorem and the unknown nonsymmetric dead-zone nonlinearity is treated as a combination of a linear term and a bounded disturbance-like term. Owing to the universal approximation property, fuzzy logic systems (FLSs) are employed to approximate the uncertain nonlinear part in controller design process. By introducing Nussbaumtype function, the a priori knowledge of the control gains signs is not required. By constructing appropriate Lyapunov-Krasovskii functionals, the effect of time-varying delay is compensated. Theoretically, it is proved that this scheme can guarantee that all signals in closed-loop system are semiglobally uniformly ultimately bounded (SUUB) and the tracking error converges to a small neighbourhood of the origin. Finally, the simulation results validate the effectiveness of the proposed scheme. 1. Introduction In the past decade, adaptive backstepping design technique has received a great deal of attention since it was pioneered by Kanellakopoulos et al. in 1991 [1]. In [2–4], adaptive backstepping is utilized to construct robust adaptive backstepping controller. The main feature of this approach is that it can handle nonlinear systems without satisfying the matching conditions, but the backstepping design procedure has a shortcoming named explosion of complexity because of the repeated differentiations of virtual controllers. By using dynamic surface control technique, the explosion of complexity shortcoming is overcome [5]. References [6, 7] develop a command filtered backstepping approach which is feasible even when the number of iterations of the backstepping method is large. However, it should be noted that the nonlinear functions are all assumed to be known in the abovementioned methods. Recently, many adaptive backstepping controllers with FLSs or neural networks (NNs) have been developed for nonlinear systems in strict feedback form [8–27]. Owing to the universal approximation property of FLSs or NNs, these control approaches do not require the precise knowledge of system nonlinearities. Nevertheless, the introduced FLSs or NNs may lead to a burdensome computation when the number of the parameters which need to be tuned by online learning laws increases significantly. To handle the inevitable weakness meeting when increasing the number of fuzzy rules or neural network nodes, the optimal weighting vector in FLSs is used as the estimation parameter [8, 9]. In [10, 14, 19, 21, 25, 27], FLSs are utilized to directly approximate the desired control signals instead of the unknown nonlinearities in each backstepping design step. Consequently, the number of parameters needed to be adapted is significantly reduced for only one parameter needed to be estimated online no matter how many fuzzy rules are selected. On the basis of the work in [10], a novel adaptive fuzzy backstepping controller construct method without requirement of the fuzzy basis functions is exploited [22, 23]. Dead-zone characteristic is one of the most common actuator nonsmooth nonlinearities encountered in many industrial processes, which can seriously affect the system 2 performance and indeed make the system unstable. Many controller design schemes are developed for systems with unknown dead zone [2, 3, 15–17, 28–36]. Generally, the dead zone is first treated as a combination of a linear and a bounded disturbance-like term, and then the controller that can achieve a good control performance is designed by adopting robust control technique [16, 17, 28–31]. In [32], a novel two-layered fuzzy logic controller which consists of a fuzzy logic-based precompensator and a usual fuzzy PD controller are developed for controlling systems with dead zone. In [33, 34], by introducing a fuzzy logic deadzone compensator two fuzzy controllers are constructed for motion control system and a DC motor system, respectively. Nevertheless, when there are no suitable rules for the deadzone nonlinearity, this method may be unfeasible for it depends much on operators or experts experience. In [2, 3, 15, 35, 36], the inverse function of dead zone is utilized to compensate the effect of the dead zone. Using this method, an effective control has been achieved, but the shortcoming that the dead-zone parameters are required to be constants is inevitable. Regrettably, although much progress has been made in the fields of controller design for nonlinear systems with unknown dead zone, nonaffine nonlinear systems with unknown dead zone are seldomly investigated. Time delays frequently occur in practical control systems, such as electrical networks and hydraulic systems. Considering that the existing time delays often cause system instability and performance deterioration, to handle the control problem for systems with time delays is an unavoidable issue. Two main tools Lyapunov-Krasovskii functionals and Lyapunov-Razumikhin functions are usually applied to nonlinear time-delay systems [4, 17–25, 37–39]. In [17–19, 22–24], Lyapunov-Krasovskii functionals are constructed to compensate the unknown time delays. Within these schemes, the condition that the unknown time delays are assumed to be unknown constants is too strict. To solve time-varying delays problem, a novel Lyapunov-Krasovskii functionals are designed on condition that the derivative of time delay functions is less than one [20, 25, 30, 37]. In [4, 21, 38], Lyapunov-Razumikhin lemma-based adaptive backstepping control approaches are proposed for nonlinear systems in which the limitation condition on the derivative of time delay is cancelled. In [17, 24, 25], adaptive fuzzy or neural backstepping controllers are designed for a class of nonlinear time-delay systems with unknown control directions. As control direction, that is, the sign of control gain, decide the direction along which the controller parameters are updated, des (...truncated)