Adaptive neural prescribed performance control for a class of strict-feedback stochastic nonlinear systems with hysteresis input

doi:10.1016/j.neucom.2017.04.017

Neurocomputing

Volume 251, 16 August 2017, Pages 35-44

https://doi.org/10.1016/j.neucom.2017.04.017 Get rights and content

Abstract

This paper studies an adaptive neural tracking control problem for a class of strict-feedback stochastic nonlinear systems with guaranteed predefined performance subject to unknown backlash-like hysteresis input. First, utilizing the prescribed performance control, the predefined tracking control performance can be guaranteed via exploiting a new performance function without considering the accurate initial error. Second, by integrating neural network approximation capability into the backstepping technique, a robust adaptive neural control scheme is developed to deal with unknown nonlinear functions, stochastic disturbances and unknown hysteresis input. The designed controller overcomes the problem of the over-parameterization. Under the proposed controller, all the signals in the closed-loop system are 4-Moment (or 2 Moment) semi-globally uniformly ultimately bounded (SGUUB), and the prespecified transient and steady tracking control performance are guaranteed. Simulation studies are performed to demonstrate and verify the effectiveness of the proposed method.

Introduction

Stochastic disturbances frequently occur in practical applications such as manufacturing processes, flight control systems and robot operating systems. Their existence possibly results in instability and poor performance of the systems. Therefore, stochastic disturbances are necessary factors to be taken into consideration in modeling and analysis in [1]. To ensure the system stability and control performance, thus, the investigation on control design for stochastic nonlinear systems is a meaningful issue and has received increasing attention.

During the past decades, the backstepping-based adaptive control has been extensively studied. Based on backstepping control technique, in [2], a robust constrained control was proposed for multi-input and multi-output (MIMO) nonlinear systems. For the systems with high uncertainty, due to the inherent approximation capability, both fuzzy logic systems (FLSs) and neural networks (NNs) have been proved to be capable of identifying unknown nonlinear functions. In [3], the problem of the system uncertainties was handled by NNs, and the adaptive output-feedback control was investigated for MIMO nonlinear systems. [4] presented an adaptive neural tracking control method for a class of uncertain non-affine nonlinear system. In [5], by utilizing the approximation capability of NNs, the trajectory tracking control was proposed for uncertain autonomous underwater vehicles with the actuator saturation. Recently, many approximation-based adaptive control schemes have also been developed to handle the control problem with stochastic nonlinear systems. For stochastic systems, the main technical obstacle in the Lyapunov design is to solve the It $\hat{o}$ stochastic differentiation terms in [6]. Adaptive fuzzy/neural control methods were investigated in [7], [8], [9], [10] for uncertain stochastic nonlinear systems in the strict-feedback form. Using common Lyapunov function and adaptive fuzzy control scheme, a switched control problem was studied in [11] for stochastic nonlinear systems subject to unknown dead-zone input. When states are unmeasurable, a state observer was developed to design the neural output-feedback controller in [12] for single-input single-output (SISO) nonlinear stochastic systems. Furthermore, [13], [14] developed adaptive fuzzy output-feedback control for MIMO stochastic nonlinear systems in the present of the dead-zone input. Although the above-mentioned control schemes can achieve the tracking control task, especially in the presence of stochastic disturbances, it is difficult and challenging to satisfy certain prespecified transient and steady-state tracking performance for stochastic nonlinear systems with input nonlinearity.

It is well known that non-smooth input nonlinear characteristics, such as saturation [3], dead-zone [11] and hysteresis, are commonly found in a wide range of physical systems and devices. However, the presence of hysteresis input is often ignored in analysis and design of control scheme for simplicity. As the representation of the hysteresis nonlinearity, backlash-like hysteresis is widely used due to its facilitation for the control design [15], [16], [17]. Recently, in order to control uncertain nonlinear systems with unknown backlash-like hysteresis, many adaptive fuzzy/neural control schemes have been developed. In [17], the hysteresis was described by a dynamic equation, and an adaptive fuzzy tracking control method was proposed for strict-feedback nonlinear systems. When state variables were immeasurable, adaptive fuzzy output-feedback control was investigated in [18], [19] for uncertain nonlinear systems with unknown backlash-like hysteresis. In [20], by introducing the disturbance observer, an adaptive neural output-feedback controller was proposed for uncertain nonlinear systems subject to unknown hysteresis. An adaptive neural control was presented in [21] for a class of nonstrict-feedback stochastic nonlinear systems subject to unknown backlash-like hysteresis input. The problem of adaptive tracking control was solved in [22] for uncertain switched nonlinear systems with hysteresis input. Nevertheless, in particular, little attention has been paid to uncertain stochastic nonlinear systems with unknown hysteresis input, which is necessary since stochastic disturbances and the occurrence of hysteresis are unavoidable in practical engineering.

Recently, the problem of the prescribed tracking performance was investigated in [23], and the developed prescribed performance control (PPC) methodology has drawn considerable attention [24], [25]. The PPC can guarantee that the tracking errors asymptotically converge to zero and even are constrained within the prespecified bounds, accompanied by the convergence rate of not less than a certain value. In [26], by combining PPC with dynamic surface control (DSC), a fuzzy prescribed tracking control scheme was studied. When states were unmeasurable, an observer-based adaptive control was presented in [27] for large-scale nonlinear time-delay systems. In [28], based on the designed disturbance observer, a robust tracking control was developed for the wheeled mobile robot. The PPC technique was further extended to MIMO systems [29], [30], [31]. Base on traditional performance function, [32] employed the transformed function to eliminate the limitation of initial error. To the best of the authors’ knowledge, via exploiting a different prescribed performance function, no tracking control method exits for stochastic nonlinear systems with unkown initial errors.

Motivated by the above discussion, we will develop an adaptive neural tracking control scheme for uncertain stochastic nonlinear systems with unknown backlash-like hysteresis. The approximation capability of RBF neural networks is employed to identify the unknown system functions. Based on the backstepping design technique and adaptive control theory, the adaptive neural tracking controller is proposed. It is proved that the proposed controller can guarantee the boundedness of the closed-loop system in probability, and achieve the required tracking performances.

Compared with previous works, the main contributions of this paper are listed in the following: (1) a performance function is introduced to ensure the predefined tracking control performance, and the requirement for the exact initial error is removed. (2) The existence of stochastic disturbances and hysteresis nonlinearity is widespread in practical engineering, and thus the designed controller has a certain practical value. (3) Only one estimated parameter is updated in every step of backstepping process by estimating the norm of the unknown NN weight vectors.

The rest of the paper is organized as follows. Section 2 gives the preliminaries and problem formulation. In Section 3, adaptive neural tracking control scheme is developed and stability analysis is achieved. Two simulation examples are provided to demonstrate the effectiveness of the proposed method in Section 4. Section 5 concludes this paper.

Section snippets

Preliminaries

Consider the following stochastic nonlinear system $\begin{matrix} d x = f (x, t) d t + h (x, t) d ω, \forall x \in R^{n} \end{matrix}$ where x ∈ Rⁿ is the system state vector, f: Rⁿ → Rⁿ, h: Rⁿ → R^{n × r} are locally Lipschitz functions and satisfy $f (0, t) = 0,$ $h (0, t) = 0$ . ω is an r-dimensional independent standard Brownian motion defined on the complete probability space(Ω, F, {F_t}_{t ≥ 0}, P) with Ω being a sample space, F being $σ -$ field, {F_t}_{t ≥ 0} being a filtration, and P being a probability measure.

Definition 1

[33]

For any given positive function V(x, t) ∈ C^{2, 1}, associated

Adaptive control design with prescribed performance

In this section, an adaptive neural control scheme will be developed and the recursive design procedure contains n steps. We choose a Lyapunov function candidate as $V = \sum_{i = 1}^{n} V_{i}$ .

The n-step backstepping design is based on the following changes of coordinate: $\begin{matrix} \begin{matrix} z_{1} = x_{1} - y_{d}, \\ z_{i} = x_{i} - α_{i - 1}, i = 2, \dots, n \end{matrix} \end{matrix}$ where $α_{i - 1}$ is the virtual control signal, which is defined later.

Step 1: Choose a Lyapunov function candidate as follows: $\begin{matrix} V_{1} = \frac{1}{4} ζ_{1}^{4} + \frac{{\tilde{θ}}_{1}^{2}}{2 r_{1}} \end{matrix}$ where r₁ > is the design parameter.

One has $\begin{matrix} L V_{1} = ζ_{1}^{3} (p_{1} (x_{2} + f_{1} - {\dot{y}}_{d} - \frac{\dot{ρ} (t)}{ρ (t)} z_{1} (t))) \end{matrix}$

Simulation studies

In this section, two examples are used to demonstrate the control performance of the developed adaptive controller.

Example 1: Consider the following second-order stochastic nonlinear system. $\begin{matrix} \begin{matrix} {\begin{matrix} d x_{1} = (x_{2} + f_{1} (x_{1})) dt + g_{1} (x_{1}) d ω, \\ d x_{2} = (u (υ) + f_{2} ({\bar{x}}_{2})) dt + g_{2} ({\bar{x}}_{2}) d ω \end{matrix} \end{matrix} \end{matrix}$ where $f_{1} (x_{1}) = 0.5 x_{1},$ $f_{2} ({\bar{x}}_{2}) = x_{1}^{2} \cos^{2} (x_{2}),$ $g_{1} = x_{1} \cos (x_{1}),$ $g_{2} = \sin (2 x_{1} x_{2})$ . $\dot{ω}$ is chosen as the one-dimensional Gaussian white nose with zero mean and variance 1. $u (υ)$ represents the output of the following backlash-like hysteresis: $\begin{matrix} \frac{d u}{d t} = a | \frac{d υ}{d t} | (c υ - u) + B \frac{d υ}{d t} \end{matrix}$

Conclusion

In this paper, the problem of the tracking control is addressed for stochastic nonlinear systems with predefined tracking performance subject to the input backlash-like hysteresis. The approximation capability of RBF neural networks is used to tackle the question of system uncertainties and input nonlinearity, and a performance function and the transformation function are chosen for eliminating the requirement of the exact initial error. By estimating the norm for the unknown NN weight vectors,

Acknowledgments

This work was supported in part by National Natural Science Foundation of China (Grant No. 61374119 and Grant No. 61473121).

Wenjie Si received the B.S. and M.S. degrees in control theory and control engineering from the Zhengzhou University, Zhengzhou, China, in 2008 and 2011, respectively, and the Ph.D. degree in control theory and control engineering from South China University of Technology, Guangzhou, China, in 2015. He is currently an Assistant Professor with the School of Automation Science and Engineering, South China University of Technology, Guangzhou, China. His current research interests include adaptive

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Xunde Dong received M.S. degree in mathematical and applied mathematical from South China University of Technology, Guangzhou, China, in 2010, and the Ph.D. degree in control theory and control engineering from South China University of Technology, Guangzhou, China, in 2014. He is currently an Assistant Professor with the School of Automation Science and Engineering, South China University of Technology, Guangzhou, China. His research interests include distributed parameter system, nonlinear adaptive control, and dynamical pattern recognition.

FeiFei Yang received the B.S. and M.S. degrees from the Department of Electrical Engineering and Automation, Henan Polytechnic University, Jiaozuo, China, in 2008 and 2011, respectively, and the Ph.D. degree from the School of Automation Science and Engineering, South China University of Technology, Guangzhou, China, in 2015. She was an exchange student with the Department of Electronic Engineering, Muroran Institute of Technology, Muroran, Japan, in 2010. She is currently a Post-Doctoral Researcher with the School of Automation Science and Engineering, South China University of Technology, Guangzhou, China. His research interests include pattern-based intelligent control, dynamical pattern recognition, and deterministic learning theory.

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Adaptive neural prescribed performance control for a class of strict-feedback stochastic nonlinear systems with hysteresis input

Abstract

Introduction

Section snippets

Preliminaries

[33]

Adaptive control design with prescribed performance

Simulation studies

Conclusion

Acknowledgments

Syst. Control Lett.

Neurocomputing

Neurocomputing

Neurocomputing

J. Frankl. Inst.

Neurocomputing

Fuzzy Sets Syst.

Inf. Sci.

Neurocomputing

Neurocomputing

Automatica

Inf. Sci.

Neurocomputing

Inf. Sci.

Robust constrained control for MIMO nonlinear systems based on disturbance observer

IEEE Trans. Autom. Control

Adaptive neural control of uncertain nonlinear systems using disturbance observer

IEEE Trans. Cybern.

Output-feedback stochastic nonlinear stabilization

IEEE Trans. Autom. Control

Adaptive tracking control of uncertain switched stochastic nonlinear systems

Nonlin. Dyn.