Adaptive prescribed performance control for nonlinear pure-feedback systems: a scalarly virtual parameter adaptation app
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ORIGINAL PAPER
Adaptive prescribed performance control for nonlinear pure-feedback systems: a scalarly virtual parameter adaptation approach Chen Wu · Shigen Gao
· Hairong Dong
Received: 24 April 2020 / Accepted: 22 October 2020 © Springer Nature B.V. 2020
Abstract In this paper, an adaptive prescribed performance controller, consisting of a novel scalarly virtual parameter adaptation (SVPA) technique, is developed for a class of single-input and single-output highorder nonlinear pure-feedback systems in the presence of model uncertain yet locally Lipschitz nonlinearities. The objective of this work is to improve the transient and steady performance of pioneering prescribed performance control (PPC) by incorporating a single SVPA mechanism into the virtual and actual controllers, therein, the unknown yet bounded parameters are defined with respect to proper composite system and virtual functions, bringing the gap between pioneering PPC and linearly parameterized approximator-based PPC schemes (including neural networks, fuzzy logic systems, etc.), that is, the computational complexity of proposed method exceeds PPC with one level (caused by introduced single adaptive law) yet maintains low level with comparison to linearly parametrized approximator-based PPC. It is guaranteed that both virtual and actual tracking errors converge transiently to small residual sets characterized by prescribed performance functions and control parameters simultaneously and ultimately converge to zero, which is also proved by rigorously mathematical analysis using Lyapunov stability theorem. The closed-loop C. Wu · S. Gao (B) · H. Dong State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China S. Gao e-mail: [email protected]
signals are kept globally ultimately uniformly bounded, and comparative simulation results are presented to demonstrate the effectiveness and advantages of the theoretical findings. Keywords Adaptive prescribed performance control · Nonlinear pure-feedback system · Virtual parameter adaptation
1 Introduction Prescribed performance control (PPC) method, which was firstly proposed by Bechlioulis and Rovithakis [3], has been successfully extended to various kinds of different applications [26,30,35]. PPC was originally proposed to guarantee “tracking error converge to an arbitrarily small residual set, with convergence rate no less than a prespecified value, exhibiting a maximum overshoot leas than a sufficiently small prespecified constant” [3], which is a preeminent tool dealing with the situation that high accuracy is required in different control systems and has been well applied to various kinds of nonlinear systems. In [4], similar to the PPC scheme, adaptive control, capable of guaranteeing transient and steady-state tracking error bounds, is designed for a class of strict feedback systems using Nussbaum-type gain technique. In [5], a switching robust control Lyapunov function (RCLF)based adaptive state feedback controller is proposed for a class of mu
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