Elsevier

Automatica

Volume 106, August 2019, Pages 117-123
Automatica

Brief paper
Finite time command filtered adaptive fault tolerant control for a class of uncertain nonlinear systems

https://doi.org/10.1016/j.automatica.2019.04.022Get rights and content

Abstract

This paper studies the problem of finite time control for a class of uncertain nonlinear systems with unknown actuator faults where the total number of failures is allowed to be infinite. By fusing the techniques of command filter and backstepping control, a two-step design method is proposed to construct the controller, under which the effect of actuator faults can be compensated completely. Moreover, the proposed controller can avoid the “explosion of complexity” and “singularity” problems in the backstepping design framework. Based on the finite time stability criterion, it is proved that both the tracking performance and the closed-loop stability can be ensured in a finite time. Finally, a simulation example of one-link manipulator is given to verify the effectiveness the proposed finite fault tolerant control scheme.

Introduction

Over the last decades, the interest in adaptive fault control problem of nonlinear systems has attracted the attention of many researchers. This is due to the fact that in modern industrial systems, actuators and sensors are vulnerable to faults while system is operating, which often leads to closed-loop instability and even cause catastrophic consequences (Ahmed-Zaid, Ioannou, Gousman, & Rooney, 1991). Therefore it is interesting to develop effective FTC schemes that can handle serious faults and guarantee the stability and performance indexes of the resulting closed-loop system (Blanke et al., 2006, Gao and Ding, 2007, Tao et al., 2001, Yang and Ye, 2010, Zhang et al., 2004). Recently, significant amount of investigations have been already made in the area of actuator fault accommodation control (Cai et al., 2015, Li and Yang, 2016, Tang et al., 2003, Wang and Wen, 2010, Wang and Wen, 2011, Wang et al., 2015, Xing et al., 2017).

Though most contributions on FTC design have been proposed in the aforementioned papers, there are two major drawbacks that need to be addressed, first, that the aforementioned control approaches can only guarantee infinite-time stability, which implies that the desired system performance can be achieved in infinite time. Second, that the derivation of the control law relies on the repeated differentiations of virtual control inputs in the backstepping-based approach, which will lead to the “explosion of complexity” and “over parameterization” problems if the dimension of the system is higher, a difficult task that hampers its wider application.

In industrial applications, the finite time controller is considered as one of the most effective approaches to drive the system trajectories converge to the equilibrium within a finite number of steps. The idea of designing finite control for nonlinear systems has been inspired from the seminal work in Bhat and Bernstein, 1998, Bhat and Bernstein, 2000. Following the seminal work, many results have been developed by using the classical finite time stability theory. For example, the adaptive finite-time control of uncertain nonlinear systems was investigated in the works of Hong, Wang, and Cheng (2006), Huang, Lin, and Yang (2005), Wu, Chen, and Li (2016), Wu, Li, Zong, and Chen (2017), Yu, Yu, Shirinzadeh, and Man (2005) and Zhang, Wang, Su, and Xu (2018). Considering the system model with stochastic noises, we mention the outstanding works (Wang and Zhu, 2017, Wang and Zhu, 2018, Zhu, 2014, Zhu and Wang, 2018), and an effective finite-time control scheme was well developed for a class of high-order strict feedback stochastic nonlinear systems (Wang & Zhu, 2015). Nevertheless, the uncertain functions to be controlled should be bounded by nonnegative C2 functions. Yu, Shi, and Zhao (2018) developed a finite time control scheme with command filtered-based backstepping (Farrell, Polycarpou, Sharma, & Dong, 2009), resulting in practical finite-time stable. However, the system nonlinearities to be controlled should be known a prior, and they have limited efficacy for nonlinear systems with uncertain parameters. Recently, considering the system model with unknown nonlinearities, a finite-time control approach has been proposed for several kinds of nonlinear systems based on finite time fuzzy control (Wang, Chen, Liu, & Lin, 2018). Though the proposed design methods can avoid these problems, it suffers from the “explosion of complexity” and “singularity” problems. Besides this aspect, most of the existing works on the finite time control problem deal with fault free case, there are nevertheless promising results dealing with actuator faults. Due to coexistence of uncertain parameters and actuator faults, the existing approaches for the adaptive finite time control problem cannot be extended directly to the adaptive finite-time FTC problems.

Motivated from these observations, this paper studies the finite-time adaptive tracking control problem for a class of uncertain nonlinear systems with actuator faults. The main contribution of the present paper is therefore threefold: (i) by introducing an intermediate variable in the final step of backstepping, a novel adaptive finite-time controller is presented to compensate for the actuator faults and the controllers constructed need no prior knowledge about the unknown system parameters and the actuator faults; (ii) by introducing the command filtered backstepping control, both “explosion of complexity” and “singularity” problems caused by the repeated differentiations of virtual control inputs in the backstepping-based approach are successfully circumvented; and (iii) instead of estimating the parameters themselves, we estimate the bound of the actuator fault parameters. As a result, the controllers constructed can deal with the infinite number of actuator failures.

Section snippets

System description

Consider a class of parametric strict-feedback nonlinear systems in the following form: ẋi=xi+1+fi(x̲i)+θiTϕi(x̲i),1in1,ẋn=j=1mgjωj+fn(x̲n)+θnTϕn(x),y=x1 where x̲i=[x1,x2,,xi]TRi,i=1,2,,n, and x=x̲nRn are the systems states, ωjR denotes the output of the jth actuator, yR is the output of the system. θiRl and gjR are unknown parameters. fi(x̲i)R and ϕi(x̲i)Rl,i=1,2,,n are known nonlinear functions. Since each subsystem in the parameter strict-feedback system is defined in terms

Finite time controller design and stability analysis

In this section, a command filtered backstepping design method is presented to ensure the stability of the closed-loop systems in a finite time.

Simulation results

To illustrate the efficiency of the finite time controller (44), we consider a one-link manipulator with motor dynamics Dp̈+Bṗ+Nsin(p)=τMτ̇+Hτ=i=12uiKmṗy=x1.where q,q̇, and q̈ denote the position, velocity and acceleration of the link, respectively. τ denotes the motor shaft angle. ui is the control input which represents the motor torque. The system parameters are chosen as D=1, B=1, M=0.05, H=0.5, N=10, and Km=10.

Define x1=q, x2=q̇ and x3=τD. The system (65) is equivalent to ẋ1=x2ẋ2=x3

Conclusion

This paper has developed a finite time control scheme for a class of uncertain nonlinear systems with unknown actuator faults. By fusing the techniques of command filter and backstepping control, a novel finite-time fault tolerant scheme is proposed, which can avoid the “explosion of complexity” and “singularity” problems in the backstepping design framework. By making use of the finite time stability criterion, we can show that both the tracking performance and the closed-loop stability can be

Yuan-Xin Li received the B.S. degree in mathematics and applied mathematics from Qufu Normal University, China, in 2007, the M.S. degree in computational mathematics from the College of Mathematical Sciences, Dalian University of Technology, Dalian, China, in 2009, and the Ph.D. degree in control theory and control engineering from the College of Information Science and Engineering, Northeastern University, Shenyang, China, in 2017.

He is currently an associate professor in the Department of

References (30)

Cited by (270)

View all citing articles on Scopus

Yuan-Xin Li received the B.S. degree in mathematics and applied mathematics from Qufu Normal University, China, in 2007, the M.S. degree in computational mathematics from the College of Mathematical Sciences, Dalian University of Technology, Dalian, China, in 2009, and the Ph.D. degree in control theory and control engineering from the College of Information Science and Engineering, Northeastern University, Shenyang, China, in 2017.

He is currently an associate professor in the Department of Science, Liaoning University of Technology, Jinzhou, China. His research interests include adaptive fuzzy/neural control, fault-tolerant control, event-triggered control and adaptive control of cyber–physical Systems.

This work was supported in part by the Funds of National Science of China (Grant no. 61603166, 61773188), and in part by the Doctoral Research Initiation of Foundation of Liaoning Province, China (No. 20180540047). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Angelo Alessandri under the direction of Editor Thomas Parisini.

View full text