This chapter proposes a new modeling and control scheme for uncertain strict-feedback nonlinear systems based on adaptive fuzzy wavelet network (FWN) and dynamic surface control (DSC) approach. It designs adaptive FWN as a nonlinear-in-parameter approximator to approximate the uncertain dynamics of the system. Then, the proposed control scheme is developed by incorporating the DSC method to the adaptive FWN-based model. Stability analysis of the proposed scheme is provided and adaptive laws are designed to learn all linear and nonlinear parameters of the network. It is proven that all the signals of the closed-loop system are uniformly ultimately bounded and the tracking error can be made arbitrary small. The proposed scheme does not require any prior knowledge about dynamics of the system and offline learning. Furthermore, it eliminates the “explosion of complexity” problems and develops accurate model of the system and simple controller. Simulation results on the numerical example and permanent magnet synchronous motor are provided to show the effectiveness of the proposed scheme.
Top1. Introduction
Recently, approximator-based adaptive backstepping control approaches have been widely applied for the control of a wide class of uncertain nonlinear systems in (Yang and Zhou, 2005, Wang, Chen, and Dai, 2007, Tong, C. Li, and Y. Li, 2009, Tong and Li, 2009, Chen and Zhang, 2010 and Wang, Liu, Zhang, X. Chen and C.L.P. Chen, 2015). In these studies, many approximators such as neural networks (NNs), fuzzy systems (FSs) and wavelet functions are used to approximate the unknown nonlinear dynamics of the system and then adaptive backstepping technique is applied to provide a systematic framework for the controller design. The developed approaches handle wide class of uncertain nonlinear systems. For example, they can apply to the uncertain nonlinear systems that their uncertainty does not satisfy the matching condition, or their uncertainty cannot be linearly parameterized, or their uncertainty is completely unknown. However, they suffer the “explosion of complexity” problem which is caused by the repeated differentiations of virtual control inputs in the backstepping control design.
To overcome this problem, Swaroop, Hedrick, Yip, and Gerdes (2000) proposed a dynamic surface control (DSC) approach that introduces a first-order low-pass filter at each step of the backstepping design procedure to eliminate the “explosion of complexity” problem. After (Swaroop et al. 2000), several adaptive approximator-based DSC schemes have been developed in (Wang and Huang, 2005, Zhang and Ge, 2008, Xu, Shi, Yang, and Sun 2014, Tong, Y-M. Li, Feng, and T-S Li 2011a and Tong, Yu. Li, Yo. Li, and Liu, 2011b).
Wang and Huang (2005) developed a radial basis function NN-based adaptive tracking control scheme for a class of uncertain nonlinear systems. Then, the adaptive NN-based DSC scheme of Wang and Huang (2005) was extended to a general class of pure feedback SISO systems in (Zhang and Ge, 2008). Composite adaptive tracking control for a class of uncertain nonlinear systems in strict feedback form was studied by (Xu et al. 2014). The authors of Tong et al. (2011a) and Tong et al. (2011b) developed an adaptive fuzzy backstepping dynamic surface control approach for a class of MIMO nonlinear systems and uncertain stochastic nonlinear strict-feedback systems. Adaptive fuzzy output feedback control was designed for uncertain nonlinear systems with unmodeled dynamics using DSC technique by Liu, Tong, and Chen (2013). Some extensions and applications of approximator-based DSC scheme can be found in (Li, Tong, and Feng, 2010a, Li, Wang, Feng, and Tong, 2010b, Yu, Shi, Dong, Chen, Lin, 2015, H. Wang, D. Wang, and Peng, 2014; Y. Li, Tong, T. Li, 2015 and Chang and Chen, 2014).