Elsevier

Neurocomputing

Volume 311, 15 October 2018, Pages 333-343
Neurocomputing

Prescribed performance adaptive fault-tolerant tracking control for nonlinear time-delay systems with input quantization and unknown control directions

https://doi.org/10.1016/j.neucom.2018.05.063Get rights and content

Abstract

This paper addresses the adaptive prescribed performance tracking control problem for a class of nonlinear time-delay systems with actuator fault, input quantization, unknown control directions and disturbances. Neural networks (NNs) are employed to approximate the unknown nonlinear functions and the Nussbaum function is used to deal with the unknown control directions. Then, a novel adaptive prescribed performance controller is designed to reduce the effects of actuator fault, input quantization, NNs approximation errors and disturbances. Compared with the existing results, a new error transformation method is presented, and the knowledge of the quantization parameters and the control directions are unknown in the control design. Furthermore, the proposed control scheme can guarantee the semi-global boundedness of all the closed-loop signals and the prescribed time-varying tracking performance. Finally, simulation results are given to demonstrate the effectiveness of the proposed control method.

Introduction

During the past decades, the performance analysis and control design problem of nonlinear systems have achieved a great deal of progress as most of the practical systems have nonlinear properties. Unknown nonlinear dynamics is an unavoidable issue for the control design of nonlinear systems. Recently, neural networks (NNs) and fuzzy logic systems (FLS) have been employed to approximate the unknown nonlinear dynamics due to the fact that they can approximate any smooth nonlinear function on a compact set to arbitrary accuracy [1], [2], [3]. Many NN and fuzzy control schemes have been proposed for nonlinear systems [4], [5], [6]. For examples, FLS were used in [7] to design an adaptive fuzzy synchronization control method for uncertain complex dynamical networks, and NNs were employed to construct an adaptive scheme in [8].

Quantification is a kind of phenomenon which exists commonly in the controlled systems such as hybrid systems and networked control systems. In the past decades, the quantitative control/filtering problems have attracted considerable attentions [4], [9], [10], [11], [12], [13], [14], [15] as quantification is quite general and has advantages in some aspects. In [12], a hysteretic quantizer was proposed to solve the input quantization problem and avoid oscillation at the same time. Then based on the proposed hysteretic quantizer, an adaptive actor critic tracking control method was investigated in [13] to deal with quantized inputs for nonlinear systems, and the adaptive quantized fault-tolerant tracking control problem was addressed in [15] for uncertain nonlinear systems. As quantization is universal, the study of nonlinear systems with input quantization is expected to be necessary and meaningful.

In addition, the system control directions may be unknown in many control applications. When there is no a priori knowledge of the control directions, the control design problem of nonlinear systems becomes a great challenge, because in such case, we cannot know the controller directions along which the control operates. Hence, the control direction unknown problem has attracted much more attentions [8], [16], [17], [18], [19]. The Nussbaum function proposed in [17] was a popular method employed to tackle the control direction unknown problem. By using the Nussbaum function, the unknown control direction problem was handled in [20]. It is also known that time-delay commonly exists in practical systems, which may cause instability and bad performance. Then, many works have been published to study the time-delay nonlinear systems [6], [8], [21], [22]. Up to now, it is known that only in [9] the adaptive control problem was studied for nonlinear systems with input quantization and unknown control directions, in which the algebraic-loop problem of the control input was raised. Hence, some new methods should be developed to address the control problem for nonlinear time-delay systems with input quantization and unknown control directions, which motivates the present investigation.

In engineering systems, failures are frequently occurred and ineluctable. When the system is failure, fault can lead to the degradation of system performance and instability of the closed-loop systems. Hence, it is significative to research the fault-tolerant control (FTC) problem, and there have been many FTC schemes proposed to solve the actuator fault problems [23], [24], [25], [26], [27]. In order to address the problem of actuator faults for nonlinear systems with unknown nonlinear dynamic, many effective NN and fuzzy FTC methods have been proposed. Through applying NNs or FLS, an adaptive neural FTC scheme was investigated in [28] to tackle the problem of actuator faults for strict-feedback nonlinear large-scale systems and an adaptive fuzzy FTC method was presented in [29] to handle the actuator faults problem for pure-feedback large-scale stochastic nonlinear systems. Therefore, this is becoming a main task to design a FTC scheme to overcome the input quantization and control directions unknown problem simultaneously. As a result, the existing FTC methods are not applicable anymore. Up till the present moment, this problem has not yet been investigated and remains open, which motivates the research of this work.

In the previous works, many control methods were designed only to guarantee the stability of the closed-loop system. While, many practical control systems have the requirement of system performance, so it is of great significance to improve the system performance. Hence, some results on improving the control performance of nonlinear systems have been researched [30], [31], [32], [33], [34], [35]. To tackle performance constraint problem, an error transformation method was presented in [36], [37], in which the performance constraint condition was satisfied by limiting the tracking error within the proposed boundary functions through an error transformation. However, to the best of our knowledge, the published works have investigated the quantized FTC problem of nonlinear systems with performance constraint, but there is no result to handle the adaptive prescribed performance FTC problem for nonlinear time-delay systems with input quantization and unknown control directions.

In this paper, an adaptive NN FTC scheme is proposed for nonlinear time-delay systems with performance constraint. For the considered nonlinear systems, the actuator fault, input quantization, unknown control directions and external disturbances are processed simultaneously. The unknown smooth nonlinear functions are approximated by using NNs and the unknown control directions are handled through the Nussbaum function. A new error constraint transformation is proposed to guarantee the prescribed performance. Then, an addition adaptive control term is presented to reduce the effects of the actuator bias fault, NNs approximation errors, input quantization and external disturbances. The designed control scheme can guarantee the semi-global boundedness of the signals in closed-loop system and the desired time-varying performance of the tracking error. The main contributions of this paper are summarized as follows.

  • (1) A new error constraint transformation is introduced, which does not need the calculation of inverse function and logarithmic function compared with [31].

  • (2) The unknown control direction, loss of actuator effectiveness fault and input quantization are tackled by the Nussbaum function simultaneously in the last step, which can avoid an algebraic-loop problem of the control input compared with [9].

  • (3) The system model and the addressed problem considered in this paper is more complex than the one in [9], [13], and the proposed control method can guarantee that the tracking error satisfies the prescribed performance condition and is bounded by adjustable accuracy boundaries in spite of the effects of the unknown control directions, actuator fault, input quantization and external disturbances.

This paper is organized as follows. First, the problem formulation is given in Section 2. Then, Section 3 is the main results in which the controller and theorem are proposed. Furthermore, an example is given to illustrate the validity of the results in Section 4. Finally, Section 5 is ended by the conclusion.

Section snippets

Problem statement

Consider a class of nonlinear time-delay systems with input quantization in the following forms x˙i=fi(x¯i)+gi(x¯i)xi+1+hi(y,y(tτi))+di(t),1in1x˙n=fn(x¯n)+gn(x¯n)q(u)+hn(y,y(tτn))+dn(t)y=x1where x¯i=[x1,x2,,xi]TRi, i=1,2,,n, x=x¯n=[x1,x2,,xn]TRn and y ∈ R are the system states and output, respectively. fi( · ), gi( · ) and hi( · ) are unknown smooth functions, and τi(t) is an unknown time-varying delay. di( · ) is an unknown disturbance and q(u) is the quantized input with u being the

Control design and stability analysis

In this section, the procedure of designing an adaptive prescribed performance NN fault-tolerant tracking controller for the nonlinear time-delay system (7) is presented.

Numerical simulation

In this section, a practical example is given to show the validity of the proposed scheme. The dynamical model of the two-stage chemical reactor with delayed recycle streams [37], [41] is given as follows x˙1=1H1x1R1x1+1G2L1x2+d1(t)x˙2=1H2x2R2x2+G1L2x1(tτ1)+F2L2u+d2(t)y=x1where x1 and x2 are the compositions, H1 and H2 are the reactor residence times, G1 and G2 are the recycle flow rates, L1 and L2 are reactor volumes, R1 and R2 are the reaction constants, F2 is the feed rate. The system

Conclusions

In this paper, the design of an adaptive prescribed performance NN tracking control approach for nonlinear time-delay systems with actuator fault, input quantization and unknown control directions has been addressed. The unknown control directions are handled by the Nussbaum function and the predefined performance limitation condition is guaranteed through a novel tracking error constraint transformation. Based on backstepping technique and dynamic surface control method, an effective adaptive

Acknowledgments

This work was supported in part by the Funds of the National Natural Science Foundation of China (Grant nos. 61621004, 61703429 and 61420106016), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant no. 2013ZCX01).

Cai-Cheng Wang received the B.S. and M.S. degrees from Northeastern University, Shenyang, China, in 2011 and 2013, respectively. Currently, he is pursuing the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, China. His current research interests include adaptive robust control, fault-tolerant control, and neural networks control.

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    Cai-Cheng Wang received the B.S. and M.S. degrees from Northeastern University, Shenyang, China, in 2011 and 2013, respectively. Currently, he is pursuing the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, China. His current research interests include adaptive robust control, fault-tolerant control, and neural networks control.

    Guang-Hong Yang (SM’04) received the B.S. and M.S. degrees in Mathematics, and Ph.D. degree in control theory and control engineering with Northeast University, Shenyang, China, in 1983, 1986, and 1994, respectively. From 2001 to 2005, he was a Research Scientist/Senior Research Scientist with the National University of Singapore, Singapore. He is currently a professor and the dean with the College of Information Science and Engineering, Northeastern University. His current research interests include fault-tolerant control, fault detection and isolation, cyber physical systems, and robust control. Dr. Yang is a Deputy Editor-in-Chief for the Journal of Control and Decision, an Editor for the International Journal of Control, Automation and Systems, and an Associate Editor for the International Journal of Systems Science, the IET Control Theory and Applications and the IEEE Transactions on Fuzzy Systems.

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