Elsevier

Automatica

Volume 97, November 2018, Pages 48-57
Automatica

Brief paper
Output feedback adaptive sensor failure compensation for a class of parametric strict feedback systems

https://doi.org/10.1016/j.automatica.2018.07.014Get rights and content

Abstract

In this paper, an adaptive output feedback compensation control scheme is proposed for a class of nonlinear systems with unknown sensor failures. For estimating the unmeasured states, a novel switching-type adaptive observer is constructed, in which the observer gains are tuned in a switching manner. To compensate for the failure effects on transient performance, a new error signal which contains an adaptive compensation coefficient is introduced into backstepping procedure. It is shown that the proposed controller can guarantee the closed-loop system is globally uniformly ultimately bounded, and system output converges to an adjustable neighborhood of the origin. Simulation results are presented to illustrate the effectiveness of the proposed scheme.

Introduction

In practical control mechanisms, sensors are thought to break down as frequently as actuators, and sensor failures often bring serious and even disastrous situations. For reliability and safety reasons, sensor failure compensation has long been an active issue in the control community. In early stage, one typical design method for control of systems with sensor failures is based on the sensor redundancy, which is rendered by the measurements from multiple sensors. Such designs have been employed to some practical systems, such as hovercraft power system, aircraft systems Guo and Nurre (1991), Zhao et al. (1994). However, in many applications, sensor redundancy may not be available. Recently, considerable analytical redundancy based control approaches for tolerating sensor failure have been developed for linear and nonlinear systems, such as robust control Aouaouda et al. (2014), Dong and Yang (2015), Yang and Ye (2007), descriptor system approach (Gao & Ding, 2007), sliding mode control (Liu, Cao, & Shi, 2013) and adaptive compensation schemes (Li & Tao, 2009).

Since 1990s, global adaptive control of uncertain nonlinear systems has received great attention (Ionanou & Sun, 1996). Apart from them perhaps the most significant one is the development of adaptive backstepping control approach for a class of nonlinear systems in a triangular structure, called parametric strict-feedback systems (PSFSs). The reasons lie in its advantages such as the transient performance can be established and improved with explicit tuning of design parameters, and those nonlinear systems without satisfying the matching conditions can be dealt with effectively. A number of important results have been summarized in Krstic, Kanellakopoulos, and Kolotovic (1995). In recent years, the studies on adaptive backstepping control have been extended from PSFSs to more general classes of nonlinear systems, such as nonlinearly parameterized systems Lin and Qian (2002), Long et al. (2015), Ye (2003), parameter-varying systems Chiang and Fu (2014), Marino and Tomei (1993), time-delay systems (Zhou, Wen, & Wang, 2009). In addition, some important robustness issues have also been addressed, for example, robust control with input saturation Fischer et al. (2013), Gong and Yao (2000), Wen et al. (2011), dead-zone (Zhang, Xu, & Zhang, 2014) and hysteresis (Zhou, Wen, & Zhang, 2004). The problem of adaptive actuator failure compensation for PSFSs was investigated in Tang, Tao, and Joshi (2003). To improve transient performance when failure occurs, the prescribed performance technique is incorporated into backstepping procedure (Wang & Wen, 2010). Furthermore, to remove the restrictions that the total number of failures is finite, a bound estimation approach is proposed in Wang, Wen, and Lin (2015). However, the effect of sensor failure has not been addressed with this approach, although it is of both theoretical and practical importance. The main challenges to find an adaptive solution to the problem of compensating for sensor failures lie in that all the state variables are unavailable for feedback design such that the standard backstepping technique (Krstic et al., 1995) is no longer feasible, and the control approaches obtained via failed measurement feedback cannot guarantee the adaptive systems achieve a desired transient performance.

In this paper, escaping from the framework of tuning function based backstepping design, a novel adaptive output-feedback failure compensation scheme is proposed for PSFSs with sensor failures. To circumvent the obstacle caused by unmeasured state variables, a switching-type adaptive state observer is designed where observer gains are tuned online in a switching manner according to the proposed logic switching rules. In controller design, a new error signal which contains an adaptive failure compensation coefficient is introduced into the backstepping produce. As a result, the effects of sensor failures can be compensated for and the transient system performance can also be tunable by adjusting design parameters. To the best our knowledge, this is the first adaptive backstepping scheme capable of tolerating unknown sensor failures, and this also enlarges the nonlinear systems currently studied by using backstepping approach.

Section snippets

Preliminaries and problem formulation

Consider the SISO nonlinear system which can be linearly parameterized by the parameter θ: ẋ1=x2+θTf1(x1)ẋi=xi+1+θTfi(x1,x̄i)ẋn=β(u+d)+θTfn(x1,x̄n)y=x1where i=2,,n1, x̄i=[x2,,xi], for 2in, x=[x1,x̄nT]T is the state and yR is the measured output by the sensor, uR is the input of system, f1(x1)Rp and fi(x1,x̄i)Rp for i=2,,n, are known smooth nonlinear vector-valued functions, θRp is unknown parameter vector, β is the known control gain, and d(t)R denotes the external disturbance

Adaptive state observer design

Noting that all states x1,,xn in system (4) are not available for feedback design, therefore, a state observer should be established to estimate the states, and then an adaptive output feedback failure compensation scheme is investigated based on the designed state observer.

The state observer is designed for (4) as follows xˆ̇1=ρˆxˆ2+θˆ1Tf1(κˆχ1)+k1(yFxˆ1)xˆ̇i=xˆi+1+θˆiTfi(κˆχ1,xˆ̄i)+ki(yFxˆ1)xˆ̇n=βu+θˆnTfn(κˆχ1,xˆ)+kn(yFxˆ1)where i=2,,n1, xˆ1 is the estimate of χ1, xˆ̄i=[xˆ2,,xˆi]T is

Adaptive backstepping controller design

In this section, the observer-based adaptive output feedback failure compensation control scheme will be developed by using backstepping technique. The global boundednessof all the signals in the resulting closed-loop system will be achieved. First, combining (4)–(6), the system can be taken as ė=Ae+B1ρ̃xˆ2+FT(κˆ1χ1,xˆ̄n)Θ̃+ΔFTΘ+Bnβdχ̇1F=ρx2+θ1Tf1(κχ1)xˆ̇i=xˆi+1+θˆiTfi(κˆχ1,xˆ̄i)+kie1,i=2,,n1xˆ̇n=βu+θˆnTfn(κˆχ1,xˆ)+kne1

Referring to the standard backstepping recursive

Simulation example

In this section, a mass–spring–dampersystem is simulated to demonstrate the effectiveness of our main results. By Khalil (2002), the state–space equations of the mass–spring–damper system, depicted in Fig. 2, can be derived as ẋ1=x2ẋ2=b1mx1b1b32mx13b2mx2+1mu+dwhere the physical meaning of the parameters m, bi,i=1,2,3 of the system (43) can be found in Khalil (2002). bi are unknown constant parameters satisfying |bi|12. We assume that the mass of car is known, that is, m=14 kg. Due to

Conclusions

This paper has studied the problem of adaptive sensor failure compensation for a class of nonlinear parametric strict-feedback systems. A novel adaptive state observer has been designed to estimate the unavailable states. For the sake of compensating for the effects of sensor failures, a new error signal related to an adaptive compensation coefficient has been introduced into the backstepping procedure. We have shown that the designed controller can guarantee all the signals in the closed-loop

Ding Zhai received his Ph.D. degrees in Automation from Northeastern University, China, in 2004. He visited the Nanyang Technological University in 2004 as a Research Fellow. He is now a Professor at Northeastern University. His research interests focus on fault detection, adaptive control. Dr. Zhai is an Associate Editor for the International Journal of Control, Automation and Systems and IEEE Access.

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      In Li (2019), a bound estimation approach was proposed to remove the restrictions that the total number of failures is allowed to be infinite. Furthermore, an adaptive output feedback compensation control scheme was proposed for PSFSs with sensor failures (Zhai et al., 2018). Wang, Wen, Zhang et al. (2020) proposed an output-feedback adaptive control method for nonlinear MIMO systems subject to actuator faults and sensor faults.

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    Ding Zhai received his Ph.D. degrees in Automation from Northeastern University, China, in 2004. He visited the Nanyang Technological University in 2004 as a Research Fellow. He is now a Professor at Northeastern University. His research interests focus on fault detection, adaptive control. Dr. Zhai is an Associate Editor for the International Journal of Control, Automation and Systems and IEEE Access.

    Liwei An received the B.S. and M.S. degrees in mathematics from Northeastern University, Shenyang, China, in 2014 and 2016, respectively, where he is currently pursuing the Ph.D. degree in navigation, guidance, and control. His current research interests include adaptive control and security of cyber physical systems.

    Jiuxiang Dong received the Ph.D. degree in navigation guidance and control from Northeastern University, China, in 2009. He is currently a Professor at the College of Information Science and Engineering, Northeastern University. His research interests include fuzzy control, robust control and reliable control.

    Qingling Zhang received the Ph.D. degree from the Automatic Control Department of Northeastern University, Shenyang, China, in 1995. Dr. Zhang is now a Professor at Northeastern University. His research interests include descriptor systems, fuzzy control and robust control.

    The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Gang Tao under the direction of Editor Miroslav Krstic.

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