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论文范文
1. Introduction It is well known that adaptive backstepping control (ABC) is an effective technique for controlling nonlinear systems in parameter strict-feedback form [1]. For ABC of strict-feedback nonlinear systems without system uncertainties and external disturbances, this issue has been studied by using many control approaches [2–4]. Based on the sliding mode filters, [5, 6] estimated the command derivatives in the design of ABC. Linear filters for derivative generation were considered in [7]. Then, Farrell et al. in [8] introduced a command filtered backstepping control (CFBC) method, in which some new approaches were given to indicate that the virtual tracking errors between the signals of the command filtered and standard ABC methods were of , where represented the frequency of the command filter. Up to now, many command filter control methods have been reported [9–11]. The above literature only addressed the nonadaptive case for nonlinear feedback systems. Design of ABC with complicated situations was given in, for instance, [12–20]. It should be mentioned that the dimension of the input variables of the estimated system must be extended to include the reference trajectory and its first derivatives. However, the aforementioned works studied the approximation problem of the command derivatives, but the resulting implementation does not achieve the theoretical guarantees of the ABC design. That is to say, new approaches are expected to solve this problem. It has been shown that modeling of plant systems is badly affected by system uncertainties, i.e., parameter uncertainties, modeling errors, external disturbances, etc. This strongly motivates the study to design a robust, flexible, and effective controller, which can suppress complexities that demean the exhibition of the plants [21–37]. For backstepping control of nonlinear systems subject to system uncertainties, some control methods have been proposed, for example, in [38–43]. On the other hand, to tackle system uncertainties, scientists and researchers have proposed a lot of intelligent methods such as fuzzy logic systems (FLSs), neural networks, and neurofuzzy systems. In these methods, FLSs have been shown to be most successful and popular [44–48]. Following later advancement in intelligent control techniques, adaptive fuzzy controllers were developed such as fuzzy gain scheduled PID controller, fuzzy model reference adaptive controller, and self-organizing fuzzy controller. Adaptive fuzzy backstepping control (AFBC) methods also have been reported recently, for example, in [2, 4, 38, 45, 49, 50]. In [4], AFBC has been established for fractional-order strict-feedback systems. In [45], AFBC has been given for uncertain nonlinear systems with input saturation. Wang et al. introduced a command filtered AFBC approach for uncertain nonlinear systems, where the “explosion of complexity” problem in backstepping design and chartering phenomenon were solved [49]. However, in their work, external disturbance was not considered, and a complicated second-order filter was used in the controller design. Motivated by above discussion, this paper will investigate the control uncertain Chua’s chaotic system with external disturbance by means of command filtered AFBC. Combining the ABC method and command filter, a robust command filter AFBC is established. The proposed method can guarantee that all signals in the closed-loop system remain bounded, and the tracking errors converge to a small neighborhood of the origin eventually. The main contributions of this paper can be summarized as follows. 
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