Adaptive actuator fault tolerant control for uncertain nonlinear systems with multiple actuators

doi:10.1016/j.automatica.2015.07.006

Automatica

Volume 60, October 2015, Pages 92-99

https://doi.org/10.1016/j.automatica.2015.07.006 Get rights and content

Abstract

In this paper, a novel adaptive fault tolerant controller design is proposed for a class of nonlinear unknown systems with multiple actuators. The controller consists of an adaptive learning-based control law, a Nussbaum gain, and a switching function scheme. The adaptive control law is implemented by a two-layer neural network to accommodate the unknown system dynamics. Without the requirement of additional fault detection mechanism, the switching function is designed to automatically locate and turn off the unknown faulty actuators by observing a control performance index. The asymptotic stability of the system output in the presence of actuator failures is rigidly proved through standard Lyapunov approach, while the other signals of the closed-loop system are guaranteed to be bounded. The theoretical result is substantiated by simulation on a two-tank system.

Introduction

Recent years witness the fast development of science and technology, while industrial applications are becoming more complex and larger. Consequently, it is more common to encounter faults within the systems as the number of components increases. Therefore, a large amount of research efforts have been devoted to fault diagnostics and accommodation (Polycarpou and Helmicki, 1995, Thumati and Jagannathan, 2010) in both academic and industrial circles. In particular, substantial attentions have been drawn to fault-tolerant control designs in order to maintain high reliability and robustness for industrial processes against unexpected faults (Mao et al., 2010, Patwardhan et al., 2006, Wang et al., 2007). In these methodologies, a fault detection/estimation mechanism is typically enforced by monitoring the system input/output to alarm the system upon the detection of a fault (Blanke & Thomsen, 2006). Thereafter, by reconfiguring the controller to compensate for the fault dynamics, the stability of the system output is still guaranteed. However, the involvement of additional fault detection system increases the system complexity and also involves a small probability of missed detections.

Among all classes of possible faults, actuator fault has been considered to be one of the most critical challenges to be solved, since system performance can be severely deteriorated by improper actuator actions (Tao, Chen, Tang, & Joshi, 2004). Furthermore, it is very difficult to be isolated from other kinds of faults and thus compensated (Takahashi and Takagi, 2011, Tao et al., 2002). Moreover, some particular types of actuator faults, such as lock in place (LIP) and loss of effectiveness (LOE) (Boskovic & Mehra, 2002), invalidate the fault accommodation designs proposed in Mao et al. (2010), Patwardhan et al. (2006) and Wang et al. (2007), since the effective output of the actuators is eventually not influenced by the designed control signals. To prevent such cases from happening, multiple actuators are often installed to provide redundancy (Steffen, Schiller, Blum, & Dixon, 2013). However, a fault detection system is still required to determine when to switch to the backup actuator and to replace the faulty one.

Meanwhile, most actuator fault-tolerant related literatures only address linear or linearly parameterized plants (Seron et al., 2013, Staroswiecki and Berdjag, 2010, Tang et al., 2007, Tao et al., 2002, Yang et al., 2007) or nonlinear plants with known dynamics (Richter, 2011, Takahashi and Takagi, 2011). But it has been widely recognized that most actual industrial systems are nonlinear by nature and an accurate model is impractical to build. Hence, nonlinear control with learning ability is desired to compensate for the unknown system dynamics, even under the healthy conditions (Krstic, Kanellakopoulos, & Kokotovic, 1995).

To this end, the goal of this paper is to present a novel adaptive learning based fault-tolerant control design for a class of nonlinear systems with totally unknown dynamics and unknown actuator failures. The control design integrates a two-layer neural network (NN) adaptive control law and an automatic switching function mechanism. The NN is employed to approximate the unknown nonlinear system dynamics and any unexpected actuator faults. An online learning algorithm of the NN is also proposed to avoid the offline training phase. In the meantime, the switching function runs for searching all faulty actuators and set the corresponding nominal control signals to zero without human’s intervention. By this means, the influence of the faulty actuators can be reduced and the requirement of any fault detection design is relaxed. The status of the switching mechanism could be further used for fault isolation. Moreover, the output of the system is theoretically proven to track the reference signal even with the occurrence of actuator failures. In addition, to verify the feasibility of our results, the adaptive fault-tolerant control scheme is applied to a two-tank control system under simulation environment.

Section snippets

Problem statement and system diagram

Consider a class of typical uncertain Multi-Input–Single-Output (MISO) nonlinear plants (Zhang, Parisini, & Polycarpou, 2004) with $m$ actuators, $\dot{x} = a (x) + G (x) u$ $y = h (x)$ where $t$ denotes the time index, $x = {[\begin{matrix} x_{1} & \dots & x_{p} \end{matrix}]}^{T} \in R^{p}, y \in R$ and $u = {[\begin{matrix} u_{1} & \dots & u_{m} \end{matrix}]}^{T} \in R^{m}$ are the state, output and control signals respectively. $m$ is the number of the actuators. $a (\cdot) : R^{p} \to R, G (\cdot) = [g_{1} (\cdot), g_{2} (\cdot), \dots, g_{m} (\cdot)] : R^{p} \to R^{p \times m}$ and $h (\cdot) : R^{p} \to R$ , are uncertain nonlinear smooth functions.

Assumption 1

The dynamical system (1) has a known strong relative degree of $n$ and its

Dynamics of filtered tracking error

First of all, it is reasonable to assume that $y_{d}$ is attainable by system (1) when it is healthy. Then, the tracking error can be defined as $e = y - y_{d} .$

Thereafter, the filtered tracking error can be defined as $r \equiv λ_{n - 1} e^{(n - 1)} + λ_{n - 2} e^{(n - 2)} + \dots + λ_{0} e = \sum_{i = 0}^{n - 1} λ_{i} e^{(i)}$ where $λ_{0}, \dots, λ_{n - 1}$ are appropriately chosen constants such that $λ_{n - 1} s^{n - 1} + λ_{n - 2} s^{n - 2} + \dots + λ_{0}$ is Hurwitz (Lewis, Jagannathan, & Yesildirek, 1999). Consequently, $e \to 0$ exponentially when $r \to$ 0. Without loss of generality, let $λ_{n - 1} = 1$ .

Hence, the dynamics in terms of

Theoretical result

In this section, the stability of the entire closed-loop system is demonstrated by the following theorem.

Theorem 1

Consider the closed-loop system consisting of the plant (1) or the equivalent system (2), the controller given by (20), the switching function design in (18)–(19), and the adaptive law in (22). Let Assumption 1, Assumption 2, Assumption 3 hold. In the presence of actuator failures modeled as (3), given a sufficient large $D$ , the tracking error $e$ is asymptotically converging

Simulation studies

In this section, in order to substantiate our design, numerical simulation has been taken on a well-known two-tank system with fluid flow (Zhang et al., 2004). The diagram of the two tank system is depicted in Fig. 2. Suppose the two tanks are identical and are cylindrical in shape with cross section area of $C_{A} = 1.54 m^{2}$ . Besides, the cross section area of the connection pipes is assumed to be $P_{A} = 5 \times 1 0^{- 3} m^{2}$ . The liquid levels of the two tanks are the states $x_{1}$ and $x_{2}$ . The control inputs $u_{1} u_{2}$ , and $u_{3}$

Conclusion

In this paper, a novel adaptive learning based controller is introduced to compensate the unknown actuator failure in a class of uncertain nonlinear systems. Without the need of additional fault detection design, the proposed scheme can automatically switch to a healthy working mode and help isolate the faulty actuators. Extensive theoretical analysis has been carried out to prove the asymptotical stability of the tracking error, while all other signals within the system are guaranteed to be

Acknowledgments

The authors would like to thank anonymous reviewers for their valuable time.

Qinmin Yang received the Bachelor’s degree in Electrical Engineering from Civil Aviation University of China, Tianjin, China in 2001, the Master of Science Degree in Control Science and Engineering from Institute of Automation, Chinese Academy of Sciences, Beijing, China in 2004, and the Ph.D. degree in Electrical Engineering from the University of Missouri-Rolla, MO USA, in 2007.

From 2007 to 2008, he was a Post-doctoral Research Associate at University of Missouri-Rolla. In 2008, he was a

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From 2007 to 2008, he was a Post-doctoral Research Associate at University of Missouri-Rolla. In 2008, he was a system engineer with Caterpillar Inc. From 2009 to 2010, he was a Post-doctoral Research Associate at University of Connecticut. Since 2010, he has been with the State Key Laboratory of Industrial Control Technology, the Department of Control Science and Engineering, Zhejiang University, China, where he is currently an associate professor. His research interests include intelligent control, renewable energy systems, nano-robotics, and system diagnosis.

Shuzhi Sam Ge received the B.Sc. degree from Beijing University of Aeronautics and Astronautics, Beijing, China, in 1986, and the Ph.D. degree from the Imperial College of Science, Technology and Medicine, University of London, London, UK, in 1993.

He is the Founding Director of the Social Robotics Laboratory, Interactive Digital Media Institute, National University of Singapore. He is a Professor in the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. He is also the Director of the Centre for Robotics, University of Electronic Science and Technology of China, Chengdu, China. He has authored or co-authored seven books and more than 300 international journal and conference papers. His current research interests include social robotics, multimedia fusion, medical robots, and intelligent systems.

Dr. Ge is the Editor-in-Chief of the International Journal of Social Robotics. He has served/been serving as an Associate Editor for a number of flagship journals. He also serves as an Editor of the Taylor & Francis Automation and Control Engineering Series. He also served as the Vice President of Technical Activities, 2009–2010, and the Vice President for Membership Activities, 2011–2012, IEEE Control Systems Society.

Youxian Sun received the Diploma degree from the Department of Chemical Engineering, Zhejiang University, Hangzhou, China, in 1964.

In 1964, he joined the Department of Chemical Engineering, Zhejiang University. From 1984 to 1987, he was an Alexander Von Humboldt Research Fellow and a Visiting Associate Professor with the University of Stuttgart, Stuttgart, Germany. Since 1988, he has been a Full Professor with Zhejiang University, where he was elevated to an Academician of the Chinese Academy of Engineering in 1995 and is currently the Director of the Institute of Industrial Process Control and the National Engineering Research Center of Industrial Automation. He is the author or coauthor of 450 journal and conference proceeding papers. His current research interests include the modeling, control, and optimization of complex systems and robust control design and its applications.

^☆: This work is supported by National Natural Science Foundation (NNSF) of China under Grant 61104008, and National High Technology Research and Development Program of China (863) under Grant 2012AA062201. The material in this paper was partially presented at the 2012 International Conference on Machine Learning and Cybernetics (ICMLC), July 15–17, 2012, Xian, China. This paper was recommended for publication in revised form by Associate Editor Gang Tao under the direction of Editor Miroslav Krstic.

¹: Tel.: +86 571 87952994.

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Brief paperAdaptive actuator fault tolerant control for uncertain nonlinear systems with multiple actuators☆

Abstract

Introduction

Section snippets

Problem statement and system diagram

Dynamics of filtered tracking error

Theoretical result

Simulation studies

Conclusion

Acknowledgments

Microelectronics and Reliability

Automatica

Systems & Control Letters

Automatica

Automatica

Journal of Process Control

Automatica

Automatica

Journal of Process Control

Automatica

An introduction to the theory of numbers

Multilayer feedforward networks are universal approximators

Neural Networks

Nonlinear control systems: an introduction

Brief paper
Adaptive actuator fault tolerant control for uncertain nonlinear systems with multiple actuators☆