Active fault-tolerant control of rotation angle sensor in steer-by-wire system based on multi-objective constraint fault estimator

Qinjie Yang; Guozhe Shen; Chao Liu; Zheng Wang; Kai Zheng; Rencheng Zheng

doi:10.1108/jicv-08-2020-0007

Active fault-tolerant control of rotation angle sensor in steer-by-wire system based on multi-objective constraint fault estimator

Yang, Qinjie; Shen, Guozhe; Liu, Chao; Wang, Zheng; Zheng, Kai; Zheng, Rencheng 2021-04-26 00:00:00 Purpose – Steer-by-wire (SBW) system mainly relies on sensors, controllers and motors to replace the traditionally mechanical transmission mechanism to realize steering functions. However, the sensors in the SBW system are particularly vulnerable to external inﬂuences, which can cause systemic faults, leading to poor steering performance and even system instability. Therefore, this paper aims to adopt a fault-tolerant control method to solve the safety problem of the SBW system caused by sensors failure. Design/methodology/approach – This paper proposes an active fault-tolerant control framework to deal with sensors failure in the SBW system by hierarchically introducing fault observer, fault estimator, fault reconstructor. Firstly, the fault observer is used to obtain the observation output of the SBW system and then obtain the residual between the observation output and the SBW system output. And then judge whether the SBW system fails according to the residual. Secondly, dependent on the residual obtained by the fault observer, a fault estimator is designed using bounded real lemma and regional pole conﬁguration to estimate the amplitude and time-varying characteristics of the faulty sensor. Eventually, a fault reconstructor is designed based on the estimation value of sensors fault obtained by the fault estimator and SBW system output to tolerate the faulty sensor. Findings – The numerical analysis shows that the fault observer can be rapidly activated to detect the fault while the sensors fault occurs. Moreover, the estimation accuracy of the fault estimator can reach to 98%, and the fault reconstructor can make the faulty SBW system to retain the steering characteristics, comparing to those of the fault-free SBW system. In addition, it was veriﬁed for the feasibility and effectiveness of the proposed control framework. Research limitations/implications – As the SBW fault diagnosis and fault-tolerant control in this paper only carry out numerical simulation research on sensors faults in matrix and laboratory/Simulink, the subsequent hardware in the loop test is needed for further veriﬁcation. Originality/value – Aiming at the SBW system with parameter perturbation and sensors failure, this paper proposes an active fault-tolerant control framework, which integrates fault observer, fault estimator and fault reconstructor so that the steering performance of SBW system with sensors faults is basically consistent with that of the fault-free SBW system. Keywords Active fault-tolerant control, Fault estimation, Sensors failure, Steer-by-wire Paper type Research paper 1. Introduction The steer-by-wire (SBW) is originated from the ﬂy-by-wire © Qinjie Yang, Guozhe Shen, Chao Liu, Zheng Wang, Kai Zheng and system in the airplane (Waraus, 2009), which is different from Rencheng Zheng. Published in Journal of Intelligent and Connected Vehicles. the traditional steering system. The SBW system uses control Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may signals to replace the traditional mechanical connection between reproduce, distribute, translate and create derivative works of this article the steering wheel and the road wheel, only relying on sensors, (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence maybe seen at http://creativecommons.org/licences/by/4.0/ The current issue and full text archive of this journal is available on Emerald legalcode Insight at: https://www.emerald.com/insight/2399-9802.htm This study was supported in part by the State Key Laboratory of Automotive Safety and Energy under Project No. KF1815, and the National Natural Science Foundation of China (No. 52071047 and No. 51975089). Journal of Intelligent and Connected Vehicles Received 3 August 2020 4/1 (2021) 1–15 Revised 26 October 2020 Emerald Publishing Limited [ISSN 2399-9802] [DOI 10.1108/JICV-08-2020-0007] Accepted 7 November 2020 1 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 motors and controllers to achieve steering. This new technology adaptive observer is proposed for the simultaneous actuator and completely gets rid of the limitations of the traditional steering sensor faults in aircraft engines, and a fault-tolerant control system by brings many signiﬁcant beneﬁts, such as setting a system is designed on this basis of the adaptive observer. In Yang variable transmission ratio to reduce the driving burden, et al. (2015), the problem is studied about sensors faults increasing the safety of the collision and improving vehicle estimation, actuator FD and isolation for a class of uncertain stability and mobility (Mi et al., 2018). The SBW system usually non-linear systems. However, the existing conditions of the fault uses the measurement output of the sensors as a feedback signal estimation observer are not given in these articles, which makes it to control vehicle gestures, but the sensors are vulnerable to difﬁcult to judge whether the fault estimation observer to be unexpected changes in external surroundings, resulting in stuck, designed is suitable for the controlled system. Second, the fault gain, deviation and signal interruption (Gao et al., 2017). This estimation algorithm used in these articles is based on the output will cause the SBW controller to generate wrong control error between the adaptive observer and the system. The commands, so the performance of the attitude control system generated output error includes the fault estimation error and the will be degraded and the steering system will be unstable state estimation error. There is a coupling between the fault resulting in driving safety problems. estimation error and the output error so that the designed fault On the one hand, to ensure the security of the SBW system, a estimation algorithm cannot take into account the accuracy and hardware redundancy can be used as a backup system. In other speed of the fault estimation. Therefore, it is necessary to further words, the SBW system uses conventional mechanical steering suppress the impact of system uncertainty on the fault estimation, linkage, multiple sensors, multiple microprocessors and to improve the transient performance of the fault estimation. In multiple actuators to ensure safety, such as Inﬁniti Q50, addition, previous research studies had few mentioned the General Motors’ Hy-wire, Danfoss original equipment problem of sensors faults estimation in SBW system. Inspired by manufactures and Delphi’s four-wheel steering vehicle. this sensors faults estimation method, and at the same time, to However, the backup system is more expensive, not only overcome the difﬁculties and deﬁciencies in the above designs, increasing the weight of the vehicle but also increasing the this study proposed a multi-objective constrained fault estimator complexity and development cost of the SBW system. (MCFE) based on residual information, so that the fault value On the other hand, a software redundancy can be adopted, from the fault estimator can be identical to the actual fault value, that the fault-tolerant control method can be used to solve the and gives the existing condition of the fault estimator. security problem of the SBW system, which can not only In addition, apart from FD and estimation in the event of a reduce the total number of redundant hardware components component failure in the SBW system, fault reconstruction is of and the cost of system development but also to ensure the great signiﬁcance to make the car run smoothly as soon as overall security and steering performance of the SBW system possible in such a hazardous situation. In Mortazavizadeh et al. (Ito and Yoshikazu, 2013). (2020), a novel FD, isolation and reconstruction control In recent years, many scholars have studied the fault technique was proposed for the failure of voltage and current detection (FD) theory based on the mathematical model of the sensors in the SBW system, in which the problem of SBW system (Anwar and Niu, 2010; Tian et al., 2009; Zhang simultaneous failure of voltage and current sensors could not be and Zhao, 2016; Chengwei et al.,2010; Lu et al.,2017). Its solved. A comprehensive method of reconﬁgurable fault- core idea is to construct residual by using the SBW system and tolerant control system for SBW vehicles is proposed in Wada designed observer, and then use some decision rules to judge et al. (2013). However, the system has actuator redundancy, the occurrence of faults. However, the FD based on the which will increase system development costs. Huang et al. mathematical model adopts an accurate SBW mathematical (2018) adopt the minimax model predictive control (MPC) in model. These FD methods are always inaccurate in the the delta-domain to realize the tracking performance under presence of parameter variations in the SBW system, such as actuator fault, system uncertainties and disturbance. However, variation of the tire cornering stiffness and the system damping the MPC controller is lacking in solving the problem of the coefﬁcient. To resolve this problem, sliding mode control actuator, which makes the system in an unstable state. (Huang et al., 2017; Dhahri et al., 2012) can be used to design In view of the above motivations, this paper proposes an the fault observer or the H_/H1 index (Chen and Patton, active fault-tolerant control framework for the SBW system 2017; Aouaouda et al., 2015; Hou and Patton, 1996; Zhou subject to sensors failure by introducing sensors fault detection, et al., 2017; Yang et al.,2013; Chilali and Gahinet, 1996)can estimation and reconstruction techniques so as to realize be used to design the fault observer, to ensure that the FD is higher-level safety performance. The main contributions of this robust to inﬂuence of interference. However, although the fault paper can be marked as follows: observer designed in these previous studies can detect system A hierarchical fail-safe control framework is presented for failure, it is still difﬁcult to determine faulty components and cooperating detection, estimation, reconstruction, by which identify the fault size and time-varying characteristics. the higher-level safety performance of the SBW system As an indispensable part of fault diagnosis, fault estimation has subject to sensors failure is effectively assured. So the fault attracted more and more attention because of its ability to estimator can not only make up for the shortcomings of the determine the time when the fault occurred, the size and time- fault observer but also the combination can give full play to varying characteristics of the fault and its superiority in reducing their respective advantages. system redundancy. Extensive investigations have been Different from the previous algorithms related to sensors conducted on sensors faults estimation in satellite control systems faults estimation, the designed algorithm in this study and ﬂight control systems (Zhang et al.,2013; Olfa et al.,2015; achieves the decoupling of the fault estimation error and the Liang et al.,2019; Wenhan et al., 2020). In Xiao et al. (2019),an output error between the fault estimator and the SBW 2 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 system, and the application of regional pole conﬁguration can 3. Modeling analysis improve the transient performance of the fault estimation. 3.1 Vehicle model The rest of this paper is mainly organized as: an active fault- In the driving procedure, the lateral stiffness, used to tolerant control problem for SBW system is described in characterize the interaction between the tire and the road Section 2, the SBW model with parameter perturbation and surface, is susceptible to changes in tire inﬂation pressure, road sensors failure is developed in Section 3, the fault tolerance surface and weather conditions and it is often difﬁcult to control framework is proposed based on MCFE in Section 4 determine accurately. This paper assumes that the uncertainty and the numerical analyzes are processed to validate the of the cornering stiffness of the front wheel is DK,and accuracy of fault estimation and effectiveness of the presented establishes a linear two-degree-of-freedom vehicle model with control framework in Section 5. Finally, the research is parameter perturbation, as shown in Figure 2. concluded in Section 6. The equation of motion can be presented as: > ðÞ K 1DK 1 K bK a:ðÞ K 1DK ðÞ K 1DK f f r r f f f f 2. Problem description > b¼ b 1 v v 1 d > r r f < 2 mv mv mv x x As shown in Figure 1, it assumed that the rotation angle sensor, > 2 2 > bK a: K 1DK b K 1 a K 1DK a: K 1DK ðÞ ðÞ ðÞ r f f r f f f f > v _ ¼ b v 1 d : r r f the yaw rate sensor and the lateral acceleration sensor in the I I v I z z x z steering module have a sudden failure due to changes in the (1) external environment. Normally, the SBW system uses the measurement output of the sensor as a feedback signal to where K and K are the cornering stiffness of the front and rear f r control the attitude of the vehicle; thereby, if the sensor fails, it tires, respectively. b is the slip angle of the mass center. v is will cause system instability and even cause trafﬁc accidents. In the yaw angle speed. a and b are the distance from the front and addition, due to component manufacturing and measuring rear axis to mass center, respectively. I is the rotary inertia of errors, it is difﬁcult to obtain accurate SBW parameter values in the vehicle body around z-axis. v is the vehicle longitudinal practice. In addition, it is often difﬁcult to accurately determine speed and m is the full-vehicle mass. the cornering stiffness of a tire during driving. The changes in the above parameters will have a certain impact on the 3.2 Systematic modeling performance of the SBW system. Therefore, this paper also According to Figure 1, the structure of the SBW system mainly considers the perturbation of parameters equivalent to the consists of three parts, namely, the steering wheel module, the damping coefﬁcient of the front wheel and steering mechanism steering module and the electronic control unit. The steering on the steering shaft, the damping coefﬁcient of the motor shaft wheel module mainly transmits the driving intention of drivers and the front wheel deﬂection stiffness. and feeds back the road sense. The main hardware includes a The active fault-tolerant control framework contains three road sense analog motor, rotation angle sensor, current sensor, parts under the problem setting, namely, FD, fault estimation torque sensor, traditional steering wheel and steering column. and fault reconstruction. First, a fault observer is designed to The electronic control unit mainly implements three functions, detect whether the system is faulty in real time, and if a fault namely, controlling the road sense analog motor to generate the occurs, it warns the driver and starts fault tolerance control. road feel, controlling the steering execution motor to generate the Then, a multi-objective fault estimator based on residual steering torque and the fault-tolerant control of the main information obtained by the fault observer is designed to components of the entire system. This research focuses on the estimate the fault size of the sensors. Finally, the fault steering module, as shown in Figure 3, which mainly realizes the estimation value of the sensors and the fault output of the steering of the vehicle. It is based on the steering column assisted sensors are used for active fault-tolerance control. EPS, and the main hardware includes the steering execution motor, rotation angle sensor, current sensor, rack displacement Figure 1 Illustration of the SBW system with sensors failure sensor and traditional gear rack steering. The current articles on SBW system research are mainly based on an accurate SBW model; however, due to the Figure 2 Linear 2-DOF vehicle model with parameter perturbation 3 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 where t is the pneumatic trail. t is the mechanical trail. d is Figure 3 Structure diagram of steering module p m f the steering angle and i is the steering ratio. The SBW system usually uses the measuring output of the sensor as a feedback signal to control the vehicle attitude; however, the rotation angle sensor, the yaw rate sensor and the lateral acceleration sensor will be suddenly failure due to the increase of the use cycle and the inﬂuence of external factors. Assuming that the sensors will continue to output measurement data at this time, but these data are not accurate. These data are speciﬁcally expressed as multiples of the correct data, some ﬁxed value difference from the correct data, constant value and zero value. When the j (j = 1, 2, 3) sensors in the SBW system has a sudden failure, the corresponding measurement output can be expressed as: y ¼ D y 1 a ¼ y 1ðÞ D 1 y 1 a ¼ y 1 f ; (5) jf j j j j j j j j sj existence of component manufacturing and measurement errors, it is difﬁcult to obtain accurate parameter value in where y is the fault output of j (j = 1, 2, 3) sensor. y is the jf j practice. In addition, some parameters of SBW will also change actual output. D is the fault gain and a is the fault constant j j because of environmental changes. Uncertain changes in the deviation or lock value. Specially, D = 0 and a = 0 indicate that j j above parameters will have a certain impact on the performance the sensor signal is interrupt. of the SBW system. Therefore, this paper considers While f =(D 2 1) y 1 a , combining with equation (1) jf j j j the parameter perturbation equivalent to the damping equation (5) and choosing the state vector coefﬁcients of the front wheels and steering mechanism on the _ _ xtðÞ ¼ , control input bv u u I u u r m m m c c steering shaft, and the damping coefﬁcient of the motor shaft, vector (t) = [U] and measurement output vector and their uncertainties are DB and DB , respectively. m c bv a I u ytðÞ ¼ to establish SBW system r y m c Taking the steering motor as the research objective, the dynamic model with parameter perturbation, sensors failure can be equation can be obtained according to newton’s law as follows: expressed as: € _ ðÞ T ¼ J u 1 B 1DB u 1 T ðÞ > m m m m m m a xt _ðÞ ¼ A1DA xtðÞ1 BuðÞ t ; (6) T ¼ KðÞ u G u a m m m e y ðÞ t ¼ CxðÞ t 1 F f ðÞ t f s s (2) > T ¼ K I m t m 1 2 3 > where F ¼ F F F is the fault vector of yaw rate s s s _ _ U ¼ LI 1 RI 1 K u m m e m sensor, lateral acceleration sensor and rotation angle sensor, whose values are separately F ¼ 0 100 0 , where u , J , B and K separately denote the angular s m m m m T T 2 3 position, moment inertia, viscous damping and shaft stiffness of F ¼ 001 00 and F ¼ 00 010 .fs = s s the steering motor. G is the motor speed-reducing device m [f f f ] is the fault values of yaw rate sensor, lateral s1 s2 s3 transmission ratio. T and T are the power motor torque and m a acceleration sensor and rotation angle sensor, respectively. the assist torque acting on the steering gear pinion, respectively. This paper converts the uncertainty magnitude of the system K and K are motor torque constant and counter electromotive t e model into additional interference, and combines it with the force constant, respectively. L, R and I are the inductance, m road information provided to the driver by the road surface to resistance and current of motor armature winding, respectively. form system interference. Equation (6) can be rewritten as: U is the terminal voltage of the power motor. ( xt _ðÞ ¼ AxðÞ t 1 BuðÞ t 1 DdðÞ t Taking rack and front wheel steering components as the ; (7) research object, the dynamic equation is as follows: y t ¼ Cx t 1 F f t ðÞ ðÞ ðÞ f s s € _ J u 1ðÞ B 1DB u ¼GT 1 M ; (3) c c c c c a z where: 2 3 where J is the moment of inertia equivalent to the steering shaft 1 6 7 of the front tire and steering mechanism. B is a viscous friction c mv 6 7 6 7 coefﬁcient equivalent to the steering shaft of the steering 6 7 6 7 mechanism and the front wheel. u is the rotation angle sensor 2 3 6 7 a u 6 7 of the pinion shaft and M is the tire aligning torque. z 6 7 DK b 1 v 00 0 f r 6 7 6 7 v G 6 x 1 7 Assuming that the front wheel slip angle is less than ﬁve 6 7 6 7 D ¼ 6 7d ¼ : 6 7 degrees, the tire aligning torque M can be estimated using the 0 0 _ z 6 7 DB :u 4 m m 5 6 m 7 following formula (Wenhan et al., 2020), 6 7 6 7 M DB :u 00 0 z c e 6 7 av < r 6 7 M ¼ t 1 t F ¼ K b 1 d : t 1 t ðÞ ðÞ z p m yf f f p m 6 00 0 7 v 6 7 : 4 5 d ¼ u =i f c (4) 4 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 Figure 4 Diagram of active fault-tolerant control framework Figure 5 Structure diagram of fault observer Let etðÞ ¼ xtðÞ ^xtðÞ be the state estimation error, it follows from equations (7) and (8) that the error dynamics can be described by: 4. Active fault-tolerance control ðÞ As shown in Figure 4, the sensors fault tolerance control et _ðÞ ¼ A LC etðÞ LF f ðÞ t 1 DdðÞ t s s ; (9) framework of the SBW system is mainly composed of a fault rtðÞ ¼ CeðÞ t 1 F f ðÞ t s s observer, a MCFE, and a fault reconstructor. The fault observer is used to obtain the residual between the observer and the SBW To facilitate the analysis of the residual robustness performance system, to determine whether the SBW system is faulty. The and fault-sensitive performance, according to the superposition multi-objective constraint fault estimator uses the residual theorem of linear systems, equation (9) is decomposed into the obtained by the fault observer to determine the size, time and following two subsystems: time-varying characteristics of the faulty sensors. The ( _ ðÞ e ðÞ t ¼ A LC e ðÞ t 1 DdðÞ t d d reconstructor uses the values of fault estimation and the faulty ; (10) sensor output in the SBW system, to achieve fault-tolerant r ðÞ t ¼ Ce ðÞ t d d control on the faulty sensor. In this way, the performance of SBW with sensors faults can still have the steering characteristics close ðÞ e_ ðÞ t ¼ A LC e ðÞ t LF f ðÞ t f f s s to the fault-free SBW system, thereby achieving fault tolerance. ; (11) r t ¼ Ce t 1 F f t ðÞ ðÞ ðÞ f f s s 4.1 Fault observer design where equation (10) represents the estimation error subsystem In practical applications, SBW systems are often affected by that is only affected by interference, and equation (11) system unmodeled dynamics and parameter changes. represents the estimation error subsystem that is only affected Therefore, when a fault occurs, it is necessary to design a multi- by the fault. The two subsystems satisfy: objective fault observer. On the one hand, the generated residual is sensitive to faults. Usually, the H_ index of the fault- etðÞ ¼ e ðÞ t 1 e ðÞ t d f ; (12) to-residual transfer function G s is used to describe the ðÞ r f rtðÞ ¼ r ðÞ t 1 r ðÞ t d f sensitivity of residual to faults in the worst case. On the other hand, the generated residual is robust to Next, the solution theorem of the multi-objective fault observer disturbance, and the robust performance of residual to gain matrix L is given. disturbance is usually characterized by the H norm of the Theorem 1 According to the bounded and real lemma, for transfer function G ðÞ s . Therefore, the structure of the fault r d the equation (7), given scalarg > 0and r > 0, design the fault observer is shown in Figure 5. Its design is the process of solving observer shown in equation (8), if there are symmetric positive the feedback gain matrix L. After L is obtained, the residual can deﬁnite matrices P and Q and have a suitable dimensional be obtained and the residual can be compared with the set matrix M, N and the following linear matrix inequality (LMI) threshold to judge whether the system has a sensor fault. inequalities are established at the same time, then the error According to the above ﬁgure, the following equation can be dynamic equation (9) is gradually stable, while satisfying: obtained: kG ðÞ s k < g ; kG ðÞ s k > r: r d r f d 1 1 f 2 3 > T ^xtðÞ ¼ A^xtðÞ1 BuðÞ t 1Ly ðÞ t ^ytðÞ > > f T T > ðÞ < A P 1 PA MC MC1 C CPD > 4 5 > < 0 ; (8) > ^ytðÞ ¼ C^xtðÞ > 2 > g I rtðÞ ¼ y ðÞ t ^ytðÞ 2 3 > T T T A Q1 QA ðÞ NC NC1 C C NF C F > s s 4 5 > < 0 ^ ^ where x, y and r denotes the estimated state, estimated output > : T 2 F F 1 r I s s and residual vector, respectively. The matrices L is an observer (13) gain that is to be designed. 5 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 ð ð t t T 2 T Proof: Assume that the SBW system has not failed, and only Because of kG ðÞ s k < g () r r dt < g d ddt,it is r d d d d 1 0 0 consider that the system has unknown interference inputs. Let f (t) = 0 and bring it into equation (9) to get equation (10). to choose Lyapunov function asVe ¼ e Pe > 0, where P is s ðÞ d d a symmetrically positive matrix: T T T T ðÞ ðÞ Ve ¼ e_ Pe 1 e Pe_ ¼ e A LC P 1PA LC e 1 2e PDd ðÞ d d d d d d d d ð ð ð t t t T 2 T T 2 T J ¼ r r dt g d ddt ¼ r r g d d dt d d d d 0 0 0 T 2 T < r r g d d1Ve dt ðÞ d d d () "# t T T T e e ðÞ d A P 1 PA PLC PLC1 C CPD d ¼ < 0 d d g I As V(e ) > 0, it only needs to satisfy: Similarly, if the SBW system fails, there is no unknown "# interference input. Let d (t) = 0 and bring it into equation (9) to T T ðÞ A P 1 PA PLC PLC1 C CPD ð ð get equation (11). t t < 0; T T g I Because of kG ðÞ s k > r () r r dt < r f f dt,it is r f f f s s 0 0 (14) to choose Lyapunov function asVe ¼ e Qe > 0, where Q is ðÞ f f symmetrically positive matrix: T T T T Ve ¼ e_ Qe 1 e Qe_ ¼ e ðÞ A LC Q1QAðÞ LC e 2e QLF f ðÞ f f f f s s f f f f ð ð ð t t t T 2 T T 2 T J ¼ r r dt r f f dt ¼ r r r f f dt f f s s f f s s 0 0 0 T 2 T < r r r f f Ve dt1Ve f f s s f f ðÞ ðÞ 8 9 2 3 "# "# ð T t< T T = e e ðÞ ðÞ f A LC Q QA LC 1ðÞ QLC C C QLF 1 C F f s s 4 5 ¼ 1Ve > 0 ðÞ f : T 2 ; f f s F F r I s s s AsVe > 0, it only needs to satisfy: mathematical model and the actual system, which will ðÞ f 2 3 invalidate the FD result. T T T ðÞ A Q1 QA NC NC1 C C NF C F Therefore, this paper chooses the dynamic threshold J to s s th 4 5 < 0; compare with the residual evaluation function, so as to reduce T 2 F F 1 r I s s the false alarm rate of the fault observer and improve the (15) credibility of the FD result. In this paper, the residual evaluation function when the SBW system contains parameter Let PL = M and QL = N into the equations (14) and (15) to get perturbation, but no sensor failure occurs is taken as the equation (13), which proof the Theorem 1. dynamic threshold, and the FD decision logic is: After generating the residual by the fault observer, it is JtðÞ > J ) AFaultOccurs necessary to select an appropriate residual evaluation method th ; (16) to judge whether the system has a fault. Ideally, if the residual is JtðÞ J ) NOFault th not zero, it indicates that the system has failed. However, in practical applications, the control system is inevitably affected where the residual evaluation function is deﬁned as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ by unmodeled dynamics and parameter changes, so there is a t1 T JtðÞ ¼ krk ¼ r ðÞ t rtðÞdt. RMS T t large deviation between the theoretically obtained 6 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 4.2 Multi-objective constraint fault estimator e_ ðÞ t ¼ f ðÞ t GCe ðÞ t Fe 1ðÞ F GF f ðÞ t ; (19) fs s x fs s s In this section, an MCFE, as shown in Figure 6, based on the 2 3 residual information obtained from the fault observer is 4 5 Let e ¼ and d ¼ f , the augmented error matrix is designed for parameter perturbation and sensor failure in the fs SBW system. In addition, the gain matrix F and G in the obtained as: estimator is obtained by using the bounded real theorem and regional pole assignment lemma, and the size, occurrence time e ¼ Ae 1 Dd and time-varying characteristics of the fault can be determined (20) e ¼ Ce according to the residual obtained in the previous section. fs And the fault estimator is described by the following form: A LC 0 D LF 0 < _ ^ ^ where A ¼ , D ¼ and f ðÞ t ¼ FtðÞf ðÞ t 1 GrðÞ t s s GC F 0 F GF I ; (17) ^ ^ ZtðÞ ¼ f ðÞ t ¼ Ef ðÞ t C ¼ 0 I . s s r If the extended error dynamic equation (20) converges where f is the estimation value of the sensor fault value f . F and asymptotically and stably to zero, it is guaranteed that ^x and f G are the gain matrices with appropriate dimensions to be are accurate estimates of state x and fault f , respectively. The determined later E (t)= I . method of ﬁnding the matrices F, G are given below. Deﬁning state error e ðÞ t ¼ xtðÞ ^xtðÞ and fault estimation Theorem 2 According to the bounded and real lemma, if there error e ðÞ t ¼ f ðÞ t f ðÞ t , by the equations (8) and (17), the fs s is a symmetric matrix T > 0 and the following linear matrix state estimation error is: inequality LMI is satisﬁed, the augmented error equation (20) ðÞ e_ ðÞ t ¼ A LC e ðÞ t LF f ðÞ t 1 DdðÞ t ; (18) x x s s converges to zero asymptotically and steadily. The fault estimator equation (16) can obtain stable state estimation and Fault estimation error is: fault estimation, and the generalized disturbancedtðÞ meets the fault estimation error: kG s k < g ðÞ e d 1 fs 2 3 T T T T T T A1 A T M C C M C M T D M F 00 1 1 1 1 1 s 1 2 6 7 6 7 M M 0 M M F T I 3 3 2 s 2 6 7 6 7 gI 00 0 6 7 < 0; (21) 6 7 6 7 gI 00 6 7 6 7 gI 0 4 5 gI Proof: refer to Section 2.1 for the proof process. only if there is a symmetric positive matrix T satisfying the To further suppress the inﬂuence of parameter variation on following matrix inequality: fault estimation, improve the dynamic characteristics and 2 3 transient performance of fault estimator, and improve the 2h T TA 1 A T 0 4 5 < 0; accuracy of fault estimation, a regional pole assignment lemma TA 1 A T 2h T is introduced in this paper. nn Lemma 1 All the eigenvalues of the state matrix A 2 R of (22) a given system are in the vertical bar area ðÞ ðÞ D l 2 C : h < Re l < h , then the system is D stable and Substitute T = diag (T , T ) into the above formula. vs 1 2 1 2 2 3 T T T T T 2h T T A A T 1 M C1 C M C M 00 1 1 1 1 1 1 2 6 7 6 7 2h T 1 M 1 M 00 1 2 3 6 3 7 < 0; 6 7 T T T T T 6 7 T A1 A T M C C M 2h T C M 1 1 1 2 1 4 1 2 5 M M 2h T 3 2 2 (23) 7 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 Proof: As rank ¼ n, where l is all the Figure 6 Structure diagram of fault estimator i A l I eigenvalues of the system matrix A, (A, C) can be observed. Rewrite equation (19) as: ~ ~ ~ _ ðÞ e ¼ A LC e 1 Dd ; (25) e ¼ Ce fs A 0 L ~ ~ ~ where A ¼ , L ¼ , C ¼ C 0 , 0 F G D LF 0 Theorem 2 and Lemma 1 give the design method of the D ¼ and C ¼ 0 I . 0 F GF I MCFE. Next, we discuss the existence conditions of the According to the linear system theory, we can obtain the MCFE (17). ~ ~ sufﬁcient and necessary condition of equation (23) that A; C Theorem 3 Let rank (A)= n, rank (C)= m and rank (F)= r, is observable, that is: then the sufﬁcient and necessary conditions for the existence of the fault estimator (17) are: A lI n1 r rank ¼ n1 r;8s 2 C; Re s 0 (26) 02 31 ðÞ A lI 0 @4 5A rank 0 F lI ¼ n1 r;8s 2 C; ReðÞ s 0 C 0 A 0 ~ ~ Substitute A ¼ ; C ¼ C 0 into the above 0 F (24) formula: 02 31 02 31 ! "# A lI 0 A lI 0 n n B6 7C B6 7C A lI n1 r B6 7C B6 7C rank ¼ rank 0 F lI ¼ rank C 0 @4 5A @4 5A C 0 0 F lI r (27) ! "# A lI 0 ¼ rank 1 rank 0 F l I C 0 It can be seen from the above that (A, C) is observable, so maintain the driving stability and safety, and the reconstructor is designed as: A l I 0 rank ¼ n and rank (F)= r from the meaning C 0 2 3 010 00 of the question, it is easy to get rank ([0 F l I,]) = r, which is 6 7 proved. ^ 6 7^ y ðÞ t ¼ y ðÞ t F f ðÞ t ¼ y ðÞ t 001 00 f ðÞ t ; ft f s f s s 4 5 000 01 4.3 Fault reconstructor (29) When the sensor fails, it can be seen from Section 4.1 that the fault output of the SBW system satisﬁes: where y , F ,f and y are the fault sensor output, fault switching f s s ft y ¼ y1 f (28) f s matrix, value of fault estimation and fault-tolerant output, respectively. where y, f and y are sensor fault-free output, sensor fault value Keeping the proportion integration differentiation (PID) s f and the fault sensor output, respectively.y can be measured by controller structure unchanged, switching y to the feedback ft the sensor after the failure of the sensor. If the values of f and f are approximately equal, then the value without failure of the Figure 7 Schematic diagram of reconstructor sensor can be obtained through equation (28). Using the above ideas and based on the sensor fault estimated value f obtained by the MCFE designed in Section 4.2 and the fault vector F obtained in Section 3.2, the fault reconstructor shown in Figure 7 is designed for fault sensor reconstruction, making the SBW system can still guarantee the basic steering function in the event of sensors failure, and 8 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 loop of the PID controller can make the SBW system with Table 1 Fault description in details sensor fault still have steering characteristics close to that of the Fault types Fault behavior no-fault SBW system, thus achieving the purpose of fault- Continuous deviation- Rotation angle sensor deviation fault after 6 s tolerant control. gain failure and signal interruption after 9 s Simultaneous gain- The yaw rate sensor has a gain failure of D = 5. Numerical analysis stuck failure 0.3 at t = 9 s, and the rotation angle sensor is To demonstrate the effectiveness of the FD, fault estimation lock and fault-tolerant control strategies are proposed considering Simultaneous Lateral acceleration sensor interrupt fault at t = parameter perturbation and sensors fault in the SBW system. interruption-gain 5 s, and rotation angle sensor gain fault with The simulation environment is set as follows. failure D = 0.4 at the same time It is assumed that the damping coefﬁcient of the front wheels and steering mechanism on the steering shaft, the damping coefﬁcient of the motor shaft and the module, FD module, fault estimation module and fault perturbation amount of the front wheel deﬂection stiffness tolerance control module of the SBW system are parameter are set within 10% (Huang et al., 2017). established, respectively, to analyze the timeliness and This article considers that a single sensor has a mixed accuracy of FD, the accuracy of fault estimation and the failure, and multiple sensors have a simultaneous failure. effect of fault-tolerant control. The failure is set as described in Table 1 for simulation veriﬁcation. 5.1 Results of fault detection Set the vehicle speed v = 15 m/s, given the target steering Using the method in Section 2.1 to design the fault observer, wheel angle as shown in Figure 8. The main parameters in and the mincx command to solve equation (18) can obtain g = the simulation of this paper are shown in Table 2. Under 5.273 10–4, r = 0.4979, as well as the optimal observer the matrix and laboratory (MATLAB)/Simulink environment, the PID control module, fault setting feedback gain matrix L as: 2 3 7 4 5 1199:30 7:98898 10 1:22795 10 150568 1:79442 10 6 5 3 4 7 15152:31:05084 10 1:69501 10 186578 2:48836 10 6 7 6 7 6 10 8 9 7 52:5015 5:02841 10 4:25239 10 10590 7:22559 10 6 7 6 7 10 12 6 7 L ¼ 3:08463 10 0:0184117 316:691 6:13696 10 48:8070 6 7 6 7 9 7 8 6 7 157:260 1:58503 10 1:50850 10 31721 2:53699 10 6 7 6 7 7 4 9 2 6 7 1:09552 10 1:78152 10 0:0727597 1:51664 10 1:14525 10 4 5 10 11 2:17262 10 0:0464384 21:1417 2:34267 10 3:24072 Substituting the matrix L into the FD observer can obtain the that the norm of the residual exceeds the FD threshold at about FD results shown in Figures 9–11. 5 s, indicating that the system is faulty. However, the test results As shown in Figure 9, for the FD of the continuous deviation- also cannot indicate what type of fault occurred in which sensor. From the above simulation results, it can be concluded that gain failure of rotation angle sensor, when there is parameter the designed fault observer can detect a fault, when they occur perturbation and sensor failure in the SBW system, the residual norm exceeds the diagnostic threshold at about 5 s, indicating that a sensor failure has occurred in the system at this time. Figure 8 Angular curve of steering wheel As shown in Figure 10, for the FD results of the simultaneous gain-stuck fault of two sensors, it can be seen that the norm of the residual is lower than the diagnostic threshold between 0 s–9 s and the norm of the residual signal exceeds the FD threshold at about 9.1 s. It indicates that the system has a sensor failure at this time, but it cannot be known, which sensor has failed. The norm of the residual does not exceed the diagnostic threshold at 9 s because there is an error in the ﬁtting of the generated residual curve, which leads to FD false alarms; however, the overall FD performance is satisfactory. As shown in Figure 11, for the FD result of the simultaneous interruption of the two sensors-gain fault, it can be observed 9 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 Table 2 Parameters of SBW model cannot determine the type of fault, nor can it determine, which sensor is faulty. The multi-objective constraint fault estimator Parameters Symbols Values can solve this problem, to further estimate the size and time- Body mass m 1,270 kg varying characteristics of the fault. Body roll moment of inertia Iz 1,537 kg.m Distance from mass center to front a 1.015 m 5.2 Results of fault estimation wheel The method in Section 2.2 was used to design the MCFE, and Distance from mass center to rear b 1.895 m the narrow strip region was selected as 100 <Re (l ) < 10. In wheel MATLAB, mincx command was used to solve equations (26) Front wheel cornering stiffness K 135,000 N/rad Rear wheel cornering stiffness K 175,000 N/rad and (28) to obtaing = 0.0105, and the optimal estimator gain r 2 Pneumatic trail t 0.006 m matrices F and G are: 2 3 Mechanical trail t 0.014 m 6 5 97:79 8:41 10 5:37 10 Armature inductance L 0.003 H 6 7 6 7 Armature resistance R 0.034 X F ¼ 8:43 10 97:63 0:02 ; 4 5 Motor torque coefﬁcient K 0.086 N.m/A t 5 5:37 10 0:02 97:79 Counter electromotive force coefﬁcient K 0.009 V.s/rad Motor stiffness coefﬁcient K 130 N.m/rad 2 3 6 5 Motor rotary inertia J 0.006 Kg/m m 1198:17 97:79 8:60 10 0:18 5:38 10 6 7 Motor damping B 0.1 N.m.s/rad 5 6 m 6 7 G ¼ : 1:13 10 8:45 10 97:41 6:23 0:06 4 5 Steering tube rotary inertia J 0.01 Kg/m c 4 5 1:91 10 5:35 10 0:06 0:70 97:78 Steering tube damping J 0.3 N.m.s/rad Steering tube stiffness coefﬁcient K 0.5 N.m/rad To show that the MCFE can provide better estimation Motor speed-reducing device G 18 performance than the fast-adaptive fault estimation observer, transmission ratio the simulation calculation of the fast-adaptive fault estimation Steering ratio G 15.5 observer (Olfa et al.,2015) design is also given below. The LMI toolbox solves the linear matrix inequality to obtain g = 8.266 10 , the observer gain matrix L and the fault for different time periods of a single sensor and multiple sensors simultaneously appearing different types of fault. However, it estimation gain G are: 2 3 6 5 4 7 5 3:91 10 1:39 10 1:77 10 2:03 10 1:10 10 6 7 6 7 6 5 8 6 7 5:32 10 1:88 10 2:40 10 2:76 10 1:49 10 6 7 6 7 9 8 7 10 8 6 7 7:02 10 2:49 10 3:17 10 3:63 10 1:97 10 6 7 6 7 11 10 9 12 9 6 7 L ¼ 2:97 10 1:05 10 1:34 10 1:83 10 8:32 10 6 7 6 7 6 7 1186:96 41:50 5:27 1:35 10 32:83 6 7 6 7 9 8 7 10 8 6 7 6:30 10 2:23 10 2:84 10 3:27 10 1:77 10 4 5 11 10 9 12 10 4:40 10 1:58 10 2:01 10 2:31 10 1:25 10 2 3 7 4 5 4 5:36 10 1:18 10 2:30 10 7136:16 -9:45 10 6 7 8 7 6 6 7 6 7 G ¼ : 9:19 10 3:42 10 4:14 10 1:46 10 2:71 10 4 5 7 6 5 5 6 9:41 10 5:33 10 8:35 10 1:89 10 4:15 10 Substituting the above-obtained matrices F and G into the constant deviation fault after 5 s, the output of the sensor is MCFE, and L and G into the fast adaptive fault always about 10 degrees higher than the real value. In addition, estimation observer, the fault estimation results can be the gain fault occurs after 7 s, and the fault estimation curve can obtained in Figures 12–21. accurately approximate the real fault from the time when the fault As shown in Figure 12, for the estimation results of continuous occurs. However, the adaptive fault estimation method has deviation-gain fault of rotation angle sensor, it can be seen that obvious overshoot in 4 s 7 s, and obvious oscillation in 7 s the multi-objective constraint fault estimator can accurately 7.5 s. Although the phases of the fault setting curve and the estimate the fault value as 0, when the rotation angle sensor does fault estimation curve are the same in 7 s 12 s, the amplitudes at not fail during 0 s 5 s. When the rotation angle sensor has a the peaks and troughs are signiﬁcantly different. 10 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 Figure 9 Fault detection results of the rotation angle sensor Figure 13 Fault estimation error curve of the rotation angle sensor by estimation method in this paper Figure 10 Results of simultaneous gain-stuck detection of two sensors Figure 14 Fault estimation curve of the yaw rate sensor Figure 11 Results of two sensors interrupted at the same time-gain fault detection Figure 15 Fault estimation error curve of the yaw rate sensor by estimation method in this paper Figure 12 Fault estimation curve of the rotation angle sensor Figure 16 Fault estimation curve of the rotation angle sensor As shown in Figure 13, the error of the fault estimation error is large when the fault occurs. It is because the estimator needs a reaction time to adapt to this sudden fault, and the fault estimation error at all other times is less than 0.7 degrees. 11 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 Figure 17 Fault estimation error curve of the rotation angle sensor by Figure 21 Fault estimation error curve of the rotation angle sensor by estimation method in this paper estimation method in this paper The fault estimation results of the simultaneous gain-stuck fault of two sensors are shown in Figures 14–17. Figure 18 Fault estimation curve of the lateral acceleration sensor As demonstrated from Figures 14 and 16, when the yaw rate sensor and the rotation angle sensor have not failed from 0 s to 9 s, the fault estimator can accurately estimate the fault value as 0. When the yaw rate sensor and rotation angle sensor fail simultaneously after 9 s, the yaw rate sensor output is always 0.3 times greater than the true value and the rotation angle sensor output becomes a ﬁxed value from the moment of the failure, and the fault estimation value curve can accurately approximate the real fault from the moment of the fault. However, the adaptive fault estimation algorithm has obvious oscillations from 4 s to 9 s. Although the phases of the fault setting curve and the fault estimation curve are the same between 9 s and 12 s, the amount of overshoot is large at the peak and valley. As presented in Figures 15 and 17, the fault estimator needs reaction time to adapt to this type of failure, resulting in a spike Figure 19 Fault estimation error curve of the lateral acceleration in the estimation error at 9 s. At other times, the maximum sensor by estimation method in this paper estimation error of the yaw rate and rotation angle are 0.108 deg/s and 0.4 degrees, respectively. The fault estimation results of the simultaneous interruption- gain fault of two sensors are shown in Figures 18–21. As shown in Figures 18 and 20, when the lateral acceleration sensor and rotation angle sensor have not failed from 0 s to 5 s, the fault estimator can accurately estimate the fault value as 0. When the lateral acceleration sensor and the rotation angle sensor fail simultaneously after 5 s, the actual output of the lateral acceleration sensor becomes 0, the output of the rotation angle sensor is always 0.3 times more than the true value, and the fault estimation value curve can accurately approximate the real fault from the moment of the fault. However, the adaptive fault estimation algorithm has different amplitudes at the peaks Figure 20 Fault estimation curve of the rotation angle sensor and valleys, for the fault setting curve and the fault estimation curve from the moment of the fault. As shown in Figures 19 and 21, the fault estimation error is large at 5 s because the estimator needs time to adapt to this type of fault. The peak value of the fault estimation error of the lateral acceleration at the remaining time does not exceed 0.04 m/s . In addition, the peak value of the fault estimation error of the rotation angle of the pinion shaft does not exceed 0.4 degrees at the other moments. From the above fault estimation results, the MCFE designed has strong robustness to system parameter perturbation, and can effectively suppress the inﬂuence of system parameter 12 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 perturbation on sensor fault estimation results. The fault Figure 23 Simultaneous gain-stuck fault tolerant performance before estimation curve obtained by the multi-objective constraint and after comparison curves fault estimator can accurately approximate the real fault from the moment of the fault, with an accuracy rate of up to 98%, which veriﬁes the effectiveness of the fault estimation algorithm designed in this paper. The comparison of the above fault estimation results also reﬂects the limitations of the method designed in (Olfa et al., 2015). Due to the error between the adaptive fault observer and the system output, the coupling effect of fault estimation error, state estimation error and the change of adaptive parameters, the gain matrix of the fault is changed, which causes the oscillation of the fault estimation results and a long convergence time. Comprehensive FD and fault estimation results, when there is parameter perturbation in the SBW system, when a single sensor or multiple sensors have gain, deviation or stuck faults, the fault observer designed in this paper can detect the time when the sensor fails. The fault estimator can accurately estimate the fault amplitude and time-varying characteristics, and the error of fault estimation is generally small, which lays a foundation for the next step of fault-tolerant control. 5.3 Fault-tolerant control results As shown in Figures 22–24, the comparison curves are presented for the actual sensor output, fault-tolerant control output and fault output under continuous deviation-gain failure, simultaneous gain-stuck failure and simultaneous interruption-gain failure. In the ﬁgure, the fault-free output refers to the value when the sensor has not failed, which is Figure 24 Simultaneous interruption-gain fault tolerant performance represented by a solid black line. The fault-tolerant control before and after comparison curves outputs refers to the value when the SBW system sensor starts fault-tolerant control after a fault occurrence at time t, which is represented by a red dot line. The fault output refers to the value after a fault occurrence, which is represented by a blue dotted line. As can be seen from Figure 22, when the rotation angle sensor has no fault from 0 s to 5 s, the actual output, fault- tolerant control output and fault output values are all the same. When a deviation fault occurs after 5 s, the angle of the pinion shaft is always 10 degrees higher than that of the true output. When a gain fault occurs after 7 s, the output of the rotation angle sensor is 0.4 times the normal output, and the peak value of the angle error reaches 32 degrees. When the fault-tolerant Figure 22 Continuous deviation-gain fault tolerant performance before and after comparison curves 13 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 control is started, the output of the fault-tolerant control is regional pole conﬁguration to estimate the amplitude and time- consistent with the output of the fault-free SBW system. varying characteristics of the faulty sensor, and the existence As can be seen from Figure 23,when the yaw rate sensor and condition of the fault estimation observer is given. Third, a the rotation angle sensor have not failed from 0 s to 9 s, the fault-tolerant algorithm is designed based on the sensor fault actual output, fault-tolerant control output and fault output estimate and fault output. Finally, numerical analysis is carried values are all the same. When the yaw rate sensor has a gain out to verify the proposed method. failure after 9 s, the yaw rate error peak value reaches 10 deg/s. The analyzed results validated that the designed fault observer can accurately and timely diagnose the faults of the At the same time, when the rotation angle sensor is stuck, the sensors of the SBW system, when different types of faults occur output of the rotation angle sensor remains at 10 degrees. When in the sensors of the SBW system. The designed MCFE can the fault-tolerant control is started, the output of the fault- accurately estimate the sensor fault size and time-varying tolerant control is consistent with the output of the fault-free characteristics, and is robust to parameter perturbation; and SBW steering system. It can be seen that fault-tolerant control the fault-tolerant control strategy can make the SBW system can signiﬁcantly reduce the impact of faults on the performance with sensors faulty close to the faultless SBW system. of the SBW system, and restore the performance of the SBW Moreover, the steering characteristics of the system can meet system, to be close to that of the fault-free SBW system. the basic requirements of the wire control system, in which the As can be seen from Figure 24, when the lateral acceleration feasibility and effectiveness are fully veriﬁed while applying the sensor and the rotation angle sensor have not failed from 0 s to active fault-tolerant control framework to the SBW system. 5 s, the actual output, fault-tolerant control output and fault output values are all the same. When the signal of the lateral acceleration sensor is interrupted after 5 s, the output of the References lateral acceleration sensor is always 0, and the peak value of the lateral acceleration error reaches 2.3 m/s . At the same time, Anwar, S. and Niu, W.A. (2010), “Nonlinear observer based when the pinion shaft rotation angle sensor has a gain failure, analytical redundancy for predictive fault tolerant control of the peak angle error of the pinion shaft reaches 18 degrees. a steer-by-wire system”, 2010 IEEE International Conference When the fault-tolerant control is started, the output of the on Industrial Technology, Vina del Mar, Chile, pp. 321-334. fault-tolerant control is consistent with the output of the fault- Aouaouda, S., Chadli, M., Shi, P. and Karimi, H.R. (2015), free SBW system. 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(2020), “A state augmentation approach to interval fault Corresponding author estimation for descriptor systems”, European Journal of Rencheng Zheng can be contacted at: rencheng.zheng@tju. Control, Vol. 51, pp. 19-29. edu.cn For instructions on how to order reprints of this article, please visit our website: www.emeraldgrouppublishing.com/licensing/reprints.htm Or contact us for further details: permissions@emeraldinsight.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Intelligent and Connected Vehicles Emerald Publishing http://www.deepdyve.com/lp/emerald-publishing/active-fault-tolerant-control-of-rotation-angle-sensor-in-steer-by-0h9m6Pdb52

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Publisher: Emerald Publishing
Copyright: © Qinjie Yang, Guozhe Shen, Chao Liu, Zheng Wang, Kai Zheng and Rencheng Zheng.
ISSN: 2399-9802
DOI: 10.1108/jicv-08-2020-0007
Publisher site: See Article on Publisher Site

Abstract

Purpose – Steer-by-wire (SBW) system mainly relies on sensors, controllers and motors to replace the traditionally mechanical transmission mechanism to realize steering functions. However, the sensors in the SBW system are particularly vulnerable to external inﬂuences, which can cause systemic faults, leading to poor steering performance and even system instability. Therefore, this paper aims to adopt a fault-tolerant control method to solve the safety problem of the SBW system caused by sensors failure. Design/methodology/approach – This paper proposes an active fault-tolerant control framework to deal with sensors failure in the SBW system by hierarchically introducing fault observer, fault estimator, fault reconstructor. Firstly, the fault observer is used to obtain the observation output of the SBW system and then obtain the residual between the observation output and the SBW system output. And then judge whether the SBW system fails according to the residual. Secondly, dependent on the residual obtained by the fault observer, a fault estimator is designed using bounded real lemma and regional pole conﬁguration to estimate the amplitude and time-varying characteristics of the faulty sensor. Eventually, a fault reconstructor is designed based on the estimation value of sensors fault obtained by the fault estimator and SBW system output to tolerate the faulty sensor. Findings – The numerical analysis shows that the fault observer can be rapidly activated to detect the fault while the sensors fault occurs. Moreover, the estimation accuracy of the fault estimator can reach to 98%, and the fault reconstructor can make the faulty SBW system to retain the steering characteristics, comparing to those of the fault-free SBW system. In addition, it was veriﬁed for the feasibility and effectiveness of the proposed control framework. Research limitations/implications – As the SBW fault diagnosis and fault-tolerant control in this paper only carry out numerical simulation research on sensors faults in matrix and laboratory/Simulink, the subsequent hardware in the loop test is needed for further veriﬁcation. Originality/value – Aiming at the SBW system with parameter perturbation and sensors failure, this paper proposes an active fault-tolerant control framework, which integrates fault observer, fault estimator and fault reconstructor so that the steering performance of SBW system with sensors faults is basically consistent with that of the fault-free SBW system. Keywords Active fault-tolerant control, Fault estimation, Sensors failure, Steer-by-wire Paper type Research paper 1. Introduction The steer-by-wire (SBW) is originated from the ﬂy-by-wire © Qinjie Yang, Guozhe Shen, Chao Liu, Zheng Wang, Kai Zheng and system in the airplane (Waraus, 2009), which is different from Rencheng Zheng. Published in Journal of Intelligent and Connected Vehicles. the traditional steering system. The SBW system uses control Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may signals to replace the traditional mechanical connection between reproduce, distribute, translate and create derivative works of this article the steering wheel and the road wheel, only relying on sensors, (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence maybe seen at http://creativecommons.org/licences/by/4.0/ The current issue and full text archive of this journal is available on Emerald legalcode Insight at: https://www.emerald.com/insight/2399-9802.htm This study was supported in part by the State Key Laboratory of Automotive Safety and Energy under Project No. KF1815, and the National Natural Science Foundation of China (No. 52071047 and No. 51975089). Journal of Intelligent and Connected Vehicles Received 3 August 2020 4/1 (2021) 1–15 Revised 26 October 2020 Emerald Publishing Limited [ISSN 2399-9802] [DOI 10.1108/JICV-08-2020-0007] Accepted 7 November 2020 1 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 motors and controllers to achieve steering. This new technology adaptive observer is proposed for the simultaneous actuator and completely gets rid of the limitations of the traditional steering sensor faults in aircraft engines, and a fault-tolerant control system by brings many signiﬁcant beneﬁts, such as setting a system is designed on this basis of the adaptive observer. In Yang variable transmission ratio to reduce the driving burden, et al. (2015), the problem is studied about sensors faults increasing the safety of the collision and improving vehicle estimation, actuator FD and isolation for a class of uncertain stability and mobility (Mi et al., 2018). The SBW system usually non-linear systems. However, the existing conditions of the fault uses the measurement output of the sensors as a feedback signal estimation observer are not given in these articles, which makes it to control vehicle gestures, but the sensors are vulnerable to difﬁcult to judge whether the fault estimation observer to be unexpected changes in external surroundings, resulting in stuck, designed is suitable for the controlled system. Second, the fault gain, deviation and signal interruption (Gao et al., 2017). This estimation algorithm used in these articles is based on the output will cause the SBW controller to generate wrong control error between the adaptive observer and the system. The commands, so the performance of the attitude control system generated output error includes the fault estimation error and the will be degraded and the steering system will be unstable state estimation error. There is a coupling between the fault resulting in driving safety problems. estimation error and the output error so that the designed fault On the one hand, to ensure the security of the SBW system, a estimation algorithm cannot take into account the accuracy and hardware redundancy can be used as a backup system. In other speed of the fault estimation. Therefore, it is necessary to further words, the SBW system uses conventional mechanical steering suppress the impact of system uncertainty on the fault estimation, linkage, multiple sensors, multiple microprocessors and to improve the transient performance of the fault estimation. In multiple actuators to ensure safety, such as Inﬁniti Q50, addition, previous research studies had few mentioned the General Motors’ Hy-wire, Danfoss original equipment problem of sensors faults estimation in SBW system. Inspired by manufactures and Delphi’s four-wheel steering vehicle. this sensors faults estimation method, and at the same time, to However, the backup system is more expensive, not only overcome the difﬁculties and deﬁciencies in the above designs, increasing the weight of the vehicle but also increasing the this study proposed a multi-objective constrained fault estimator complexity and development cost of the SBW system. (MCFE) based on residual information, so that the fault value On the other hand, a software redundancy can be adopted, from the fault estimator can be identical to the actual fault value, that the fault-tolerant control method can be used to solve the and gives the existing condition of the fault estimator. security problem of the SBW system, which can not only In addition, apart from FD and estimation in the event of a reduce the total number of redundant hardware components component failure in the SBW system, fault reconstruction is of and the cost of system development but also to ensure the great signiﬁcance to make the car run smoothly as soon as overall security and steering performance of the SBW system possible in such a hazardous situation. In Mortazavizadeh et al. (Ito and Yoshikazu, 2013). (2020), a novel FD, isolation and reconstruction control In recent years, many scholars have studied the fault technique was proposed for the failure of voltage and current detection (FD) theory based on the mathematical model of the sensors in the SBW system, in which the problem of SBW system (Anwar and Niu, 2010; Tian et al., 2009; Zhang simultaneous failure of voltage and current sensors could not be and Zhao, 2016; Chengwei et al.,2010; Lu et al.,2017). Its solved. A comprehensive method of reconﬁgurable fault- core idea is to construct residual by using the SBW system and tolerant control system for SBW vehicles is proposed in Wada designed observer, and then use some decision rules to judge et al. (2013). However, the system has actuator redundancy, the occurrence of faults. However, the FD based on the which will increase system development costs. Huang et al. mathematical model adopts an accurate SBW mathematical (2018) adopt the minimax model predictive control (MPC) in model. These FD methods are always inaccurate in the the delta-domain to realize the tracking performance under presence of parameter variations in the SBW system, such as actuator fault, system uncertainties and disturbance. However, variation of the tire cornering stiffness and the system damping the MPC controller is lacking in solving the problem of the coefﬁcient. To resolve this problem, sliding mode control actuator, which makes the system in an unstable state. (Huang et al., 2017; Dhahri et al., 2012) can be used to design In view of the above motivations, this paper proposes an the fault observer or the H_/H1 index (Chen and Patton, active fault-tolerant control framework for the SBW system 2017; Aouaouda et al., 2015; Hou and Patton, 1996; Zhou subject to sensors failure by introducing sensors fault detection, et al., 2017; Yang et al.,2013; Chilali and Gahinet, 1996)can estimation and reconstruction techniques so as to realize be used to design the fault observer, to ensure that the FD is higher-level safety performance. The main contributions of this robust to inﬂuence of interference. However, although the fault paper can be marked as follows: observer designed in these previous studies can detect system A hierarchical fail-safe control framework is presented for failure, it is still difﬁcult to determine faulty components and cooperating detection, estimation, reconstruction, by which identify the fault size and time-varying characteristics. the higher-level safety performance of the SBW system As an indispensable part of fault diagnosis, fault estimation has subject to sensors failure is effectively assured. So the fault attracted more and more attention because of its ability to estimator can not only make up for the shortcomings of the determine the time when the fault occurred, the size and time- fault observer but also the combination can give full play to varying characteristics of the fault and its superiority in reducing their respective advantages. system redundancy. Extensive investigations have been Different from the previous algorithms related to sensors conducted on sensors faults estimation in satellite control systems faults estimation, the designed algorithm in this study and ﬂight control systems (Zhang et al.,2013; Olfa et al.,2015; achieves the decoupling of the fault estimation error and the Liang et al.,2019; Wenhan et al., 2020). In Xiao et al. (2019),an output error between the fault estimator and the SBW 2 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 system, and the application of regional pole conﬁguration can 3. Modeling analysis improve the transient performance of the fault estimation. 3.1 Vehicle model The rest of this paper is mainly organized as: an active fault- In the driving procedure, the lateral stiffness, used to tolerant control problem for SBW system is described in characterize the interaction between the tire and the road Section 2, the SBW model with parameter perturbation and surface, is susceptible to changes in tire inﬂation pressure, road sensors failure is developed in Section 3, the fault tolerance surface and weather conditions and it is often difﬁcult to control framework is proposed based on MCFE in Section 4 determine accurately. This paper assumes that the uncertainty and the numerical analyzes are processed to validate the of the cornering stiffness of the front wheel is DK,and accuracy of fault estimation and effectiveness of the presented establishes a linear two-degree-of-freedom vehicle model with control framework in Section 5. Finally, the research is parameter perturbation, as shown in Figure 2. concluded in Section 6. The equation of motion can be presented as: > ðÞ K 1DK 1 K bK a:ðÞ K 1DK ðÞ K 1DK f f r r f f f f 2. Problem description > b¼ b 1 v v 1 d > r r f < 2 mv mv mv x x As shown in Figure 1, it assumed that the rotation angle sensor, > 2 2 > bK a: K 1DK b K 1 a K 1DK a: K 1DK ðÞ ðÞ ðÞ r f f r f f f f > v _ ¼ b v 1 d : r r f the yaw rate sensor and the lateral acceleration sensor in the I I v I z z x z steering module have a sudden failure due to changes in the (1) external environment. Normally, the SBW system uses the measurement output of the sensor as a feedback signal to where K and K are the cornering stiffness of the front and rear f r control the attitude of the vehicle; thereby, if the sensor fails, it tires, respectively. b is the slip angle of the mass center. v is will cause system instability and even cause trafﬁc accidents. In the yaw angle speed. a and b are the distance from the front and addition, due to component manufacturing and measuring rear axis to mass center, respectively. I is the rotary inertia of errors, it is difﬁcult to obtain accurate SBW parameter values in the vehicle body around z-axis. v is the vehicle longitudinal practice. In addition, it is often difﬁcult to accurately determine speed and m is the full-vehicle mass. the cornering stiffness of a tire during driving. The changes in the above parameters will have a certain impact on the 3.2 Systematic modeling performance of the SBW system. Therefore, this paper also According to Figure 1, the structure of the SBW system mainly considers the perturbation of parameters equivalent to the consists of three parts, namely, the steering wheel module, the damping coefﬁcient of the front wheel and steering mechanism steering module and the electronic control unit. The steering on the steering shaft, the damping coefﬁcient of the motor shaft wheel module mainly transmits the driving intention of drivers and the front wheel deﬂection stiffness. and feeds back the road sense. The main hardware includes a The active fault-tolerant control framework contains three road sense analog motor, rotation angle sensor, current sensor, parts under the problem setting, namely, FD, fault estimation torque sensor, traditional steering wheel and steering column. and fault reconstruction. First, a fault observer is designed to The electronic control unit mainly implements three functions, detect whether the system is faulty in real time, and if a fault namely, controlling the road sense analog motor to generate the occurs, it warns the driver and starts fault tolerance control. road feel, controlling the steering execution motor to generate the Then, a multi-objective fault estimator based on residual steering torque and the fault-tolerant control of the main information obtained by the fault observer is designed to components of the entire system. This research focuses on the estimate the fault size of the sensors. Finally, the fault steering module, as shown in Figure 3, which mainly realizes the estimation value of the sensors and the fault output of the steering of the vehicle. It is based on the steering column assisted sensors are used for active fault-tolerance control. EPS, and the main hardware includes the steering execution motor, rotation angle sensor, current sensor, rack displacement Figure 1 Illustration of the SBW system with sensors failure sensor and traditional gear rack steering. The current articles on SBW system research are mainly based on an accurate SBW model; however, due to the Figure 2 Linear 2-DOF vehicle model with parameter perturbation 3 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 where t is the pneumatic trail. t is the mechanical trail. d is Figure 3 Structure diagram of steering module p m f the steering angle and i is the steering ratio. The SBW system usually uses the measuring output of the sensor as a feedback signal to control the vehicle attitude; however, the rotation angle sensor, the yaw rate sensor and the lateral acceleration sensor will be suddenly failure due to the increase of the use cycle and the inﬂuence of external factors. Assuming that the sensors will continue to output measurement data at this time, but these data are not accurate. These data are speciﬁcally expressed as multiples of the correct data, some ﬁxed value difference from the correct data, constant value and zero value. When the j (j = 1, 2, 3) sensors in the SBW system has a sudden failure, the corresponding measurement output can be expressed as: y ¼ D y 1 a ¼ y 1ðÞ D 1 y 1 a ¼ y 1 f ; (5) jf j j j j j j j j sj existence of component manufacturing and measurement errors, it is difﬁcult to obtain accurate parameter value in where y is the fault output of j (j = 1, 2, 3) sensor. y is the jf j practice. In addition, some parameters of SBW will also change actual output. D is the fault gain and a is the fault constant j j because of environmental changes. Uncertain changes in the deviation or lock value. Specially, D = 0 and a = 0 indicate that j j above parameters will have a certain impact on the performance the sensor signal is interrupt. of the SBW system. Therefore, this paper considers While f =(D 2 1) y 1 a , combining with equation (1) jf j j j the parameter perturbation equivalent to the damping equation (5) and choosing the state vector coefﬁcients of the front wheels and steering mechanism on the _ _ xtðÞ ¼ , control input bv u u I u u r m m m c c steering shaft, and the damping coefﬁcient of the motor shaft, vector (t) = [U] and measurement output vector and their uncertainties are DB and DB , respectively. m c bv a I u ytðÞ ¼ to establish SBW system r y m c Taking the steering motor as the research objective, the dynamic model with parameter perturbation, sensors failure can be equation can be obtained according to newton’s law as follows: expressed as: € _ ðÞ T ¼ J u 1 B 1DB u 1 T ðÞ > m m m m m m a xt _ðÞ ¼ A1DA xtðÞ1 BuðÞ t ; (6) T ¼ KðÞ u G u a m m m e y ðÞ t ¼ CxðÞ t 1 F f ðÞ t f s s (2) > T ¼ K I m t m 1 2 3 > where F ¼ F F F is the fault vector of yaw rate s s s _ _ U ¼ LI 1 RI 1 K u m m e m sensor, lateral acceleration sensor and rotation angle sensor, whose values are separately F ¼ 0 100 0 , where u , J , B and K separately denote the angular s m m m m T T 2 3 position, moment inertia, viscous damping and shaft stiffness of F ¼ 001 00 and F ¼ 00 010 .fs = s s the steering motor. G is the motor speed-reducing device m [f f f ] is the fault values of yaw rate sensor, lateral s1 s2 s3 transmission ratio. T and T are the power motor torque and m a acceleration sensor and rotation angle sensor, respectively. the assist torque acting on the steering gear pinion, respectively. This paper converts the uncertainty magnitude of the system K and K are motor torque constant and counter electromotive t e model into additional interference, and combines it with the force constant, respectively. L, R and I are the inductance, m road information provided to the driver by the road surface to resistance and current of motor armature winding, respectively. form system interference. Equation (6) can be rewritten as: U is the terminal voltage of the power motor. ( xt _ðÞ ¼ AxðÞ t 1 BuðÞ t 1 DdðÞ t Taking rack and front wheel steering components as the ; (7) research object, the dynamic equation is as follows: y t ¼ Cx t 1 F f t ðÞ ðÞ ðÞ f s s € _ J u 1ðÞ B 1DB u ¼GT 1 M ; (3) c c c c c a z where: 2 3 where J is the moment of inertia equivalent to the steering shaft 1 6 7 of the front tire and steering mechanism. B is a viscous friction c mv 6 7 6 7 coefﬁcient equivalent to the steering shaft of the steering 6 7 6 7 mechanism and the front wheel. u is the rotation angle sensor 2 3 6 7 a u 6 7 of the pinion shaft and M is the tire aligning torque. z 6 7 DK b 1 v 00 0 f r 6 7 6 7 v G 6 x 1 7 Assuming that the front wheel slip angle is less than ﬁve 6 7 6 7 D ¼ 6 7d ¼ : 6 7 degrees, the tire aligning torque M can be estimated using the 0 0 _ z 6 7 DB :u 4 m m 5 6 m 7 following formula (Wenhan et al., 2020), 6 7 6 7 M DB :u 00 0 z c e 6 7 av < r 6 7 M ¼ t 1 t F ¼ K b 1 d : t 1 t ðÞ ðÞ z p m yf f f p m 6 00 0 7 v 6 7 : 4 5 d ¼ u =i f c (4) 4 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 Figure 4 Diagram of active fault-tolerant control framework Figure 5 Structure diagram of fault observer Let etðÞ ¼ xtðÞ ^xtðÞ be the state estimation error, it follows from equations (7) and (8) that the error dynamics can be described by: 4. Active fault-tolerance control ðÞ As shown in Figure 4, the sensors fault tolerance control et _ðÞ ¼ A LC etðÞ LF f ðÞ t 1 DdðÞ t s s ; (9) framework of the SBW system is mainly composed of a fault rtðÞ ¼ CeðÞ t 1 F f ðÞ t s s observer, a MCFE, and a fault reconstructor. The fault observer is used to obtain the residual between the observer and the SBW To facilitate the analysis of the residual robustness performance system, to determine whether the SBW system is faulty. The and fault-sensitive performance, according to the superposition multi-objective constraint fault estimator uses the residual theorem of linear systems, equation (9) is decomposed into the obtained by the fault observer to determine the size, time and following two subsystems: time-varying characteristics of the faulty sensors. The ( _ ðÞ e ðÞ t ¼ A LC e ðÞ t 1 DdðÞ t d d reconstructor uses the values of fault estimation and the faulty ; (10) sensor output in the SBW system, to achieve fault-tolerant r ðÞ t ¼ Ce ðÞ t d d control on the faulty sensor. In this way, the performance of SBW with sensors faults can still have the steering characteristics close ðÞ e_ ðÞ t ¼ A LC e ðÞ t LF f ðÞ t f f s s to the fault-free SBW system, thereby achieving fault tolerance. ; (11) r t ¼ Ce t 1 F f t ðÞ ðÞ ðÞ f f s s 4.1 Fault observer design where equation (10) represents the estimation error subsystem In practical applications, SBW systems are often affected by that is only affected by interference, and equation (11) system unmodeled dynamics and parameter changes. represents the estimation error subsystem that is only affected Therefore, when a fault occurs, it is necessary to design a multi- by the fault. The two subsystems satisfy: objective fault observer. On the one hand, the generated residual is sensitive to faults. Usually, the H_ index of the fault- etðÞ ¼ e ðÞ t 1 e ðÞ t d f ; (12) to-residual transfer function G s is used to describe the ðÞ r f rtðÞ ¼ r ðÞ t 1 r ðÞ t d f sensitivity of residual to faults in the worst case. On the other hand, the generated residual is robust to Next, the solution theorem of the multi-objective fault observer disturbance, and the robust performance of residual to gain matrix L is given. disturbance is usually characterized by the H norm of the Theorem 1 According to the bounded and real lemma, for transfer function G ðÞ s . Therefore, the structure of the fault r d the equation (7), given scalarg > 0and r > 0, design the fault observer is shown in Figure 5. Its design is the process of solving observer shown in equation (8), if there are symmetric positive the feedback gain matrix L. After L is obtained, the residual can deﬁnite matrices P and Q and have a suitable dimensional be obtained and the residual can be compared with the set matrix M, N and the following linear matrix inequality (LMI) threshold to judge whether the system has a sensor fault. inequalities are established at the same time, then the error According to the above ﬁgure, the following equation can be dynamic equation (9) is gradually stable, while satisfying: obtained: kG ðÞ s k < g ; kG ðÞ s k > r: r d r f d 1 1 f 2 3 > T ^xtðÞ ¼ A^xtðÞ1 BuðÞ t 1Ly ðÞ t ^ytðÞ > > f T T > ðÞ < A P 1 PA MC MC1 C CPD > 4 5 > < 0 ; (8) > ^ytðÞ ¼ C^xtðÞ > 2 > g I rtðÞ ¼ y ðÞ t ^ytðÞ 2 3 > T T T A Q1 QA ðÞ NC NC1 C C NF C F > s s 4 5 > < 0 ^ ^ where x, y and r denotes the estimated state, estimated output > : T 2 F F 1 r I s s and residual vector, respectively. The matrices L is an observer (13) gain that is to be designed. 5 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 ð ð t t T 2 T Proof: Assume that the SBW system has not failed, and only Because of kG ðÞ s k < g () r r dt < g d ddt,it is r d d d d 1 0 0 consider that the system has unknown interference inputs. Let f (t) = 0 and bring it into equation (9) to get equation (10). to choose Lyapunov function asVe ¼ e Pe > 0, where P is s ðÞ d d a symmetrically positive matrix: T T T T ðÞ ðÞ Ve ¼ e_ Pe 1 e Pe_ ¼ e A LC P 1PA LC e 1 2e PDd ðÞ d d d d d d d d ð ð ð t t t T 2 T T 2 T J ¼ r r dt g d ddt ¼ r r g d d dt d d d d 0 0 0 T 2 T < r r g d d1Ve dt ðÞ d d d () "# t T T T e e ðÞ d A P 1 PA PLC PLC1 C CPD d ¼ < 0 d d g I As V(e ) > 0, it only needs to satisfy: Similarly, if the SBW system fails, there is no unknown "# interference input. Let d (t) = 0 and bring it into equation (9) to T T ðÞ A P 1 PA PLC PLC1 C CPD ð ð get equation (11). t t < 0; T T g I Because of kG ðÞ s k > r () r r dt < r f f dt,it is r f f f s s 0 0 (14) to choose Lyapunov function asVe ¼ e Qe > 0, where Q is ðÞ f f symmetrically positive matrix: T T T T Ve ¼ e_ Qe 1 e Qe_ ¼ e ðÞ A LC Q1QAðÞ LC e 2e QLF f ðÞ f f f f s s f f f f ð ð ð t t t T 2 T T 2 T J ¼ r r dt r f f dt ¼ r r r f f dt f f s s f f s s 0 0 0 T 2 T < r r r f f Ve dt1Ve f f s s f f ðÞ ðÞ 8 9 2 3 "# "# ð T t< T T = e e ðÞ ðÞ f A LC Q QA LC 1ðÞ QLC C C QLF 1 C F f s s 4 5 ¼ 1Ve > 0 ðÞ f : T 2 ; f f s F F r I s s s AsVe > 0, it only needs to satisfy: mathematical model and the actual system, which will ðÞ f 2 3 invalidate the FD result. T T T ðÞ A Q1 QA NC NC1 C C NF C F Therefore, this paper chooses the dynamic threshold J to s s th 4 5 < 0; compare with the residual evaluation function, so as to reduce T 2 F F 1 r I s s the false alarm rate of the fault observer and improve the (15) credibility of the FD result. In this paper, the residual evaluation function when the SBW system contains parameter Let PL = M and QL = N into the equations (14) and (15) to get perturbation, but no sensor failure occurs is taken as the equation (13), which proof the Theorem 1. dynamic threshold, and the FD decision logic is: After generating the residual by the fault observer, it is JtðÞ > J ) AFaultOccurs necessary to select an appropriate residual evaluation method th ; (16) to judge whether the system has a fault. Ideally, if the residual is JtðÞ J ) NOFault th not zero, it indicates that the system has failed. However, in practical applications, the control system is inevitably affected where the residual evaluation function is deﬁned as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ by unmodeled dynamics and parameter changes, so there is a t1 T JtðÞ ¼ krk ¼ r ðÞ t rtðÞdt. RMS T t large deviation between the theoretically obtained 6 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 4.2 Multi-objective constraint fault estimator e_ ðÞ t ¼ f ðÞ t GCe ðÞ t Fe 1ðÞ F GF f ðÞ t ; (19) fs s x fs s s In this section, an MCFE, as shown in Figure 6, based on the 2 3 residual information obtained from the fault observer is 4 5 Let e ¼ and d ¼ f , the augmented error matrix is designed for parameter perturbation and sensor failure in the fs SBW system. In addition, the gain matrix F and G in the obtained as: estimator is obtained by using the bounded real theorem and regional pole assignment lemma, and the size, occurrence time e ¼ Ae 1 Dd and time-varying characteristics of the fault can be determined (20) e ¼ Ce according to the residual obtained in the previous section. fs And the fault estimator is described by the following form: A LC 0 D LF 0 < _ ^ ^ where A ¼ , D ¼ and f ðÞ t ¼ FtðÞf ðÞ t 1 GrðÞ t s s GC F 0 F GF I ; (17) ^ ^ ZtðÞ ¼ f ðÞ t ¼ Ef ðÞ t C ¼ 0 I . s s r If the extended error dynamic equation (20) converges where f is the estimation value of the sensor fault value f . F and asymptotically and stably to zero, it is guaranteed that ^x and f G are the gain matrices with appropriate dimensions to be are accurate estimates of state x and fault f , respectively. The determined later E (t)= I . method of ﬁnding the matrices F, G are given below. Deﬁning state error e ðÞ t ¼ xtðÞ ^xtðÞ and fault estimation Theorem 2 According to the bounded and real lemma, if there error e ðÞ t ¼ f ðÞ t f ðÞ t , by the equations (8) and (17), the fs s is a symmetric matrix T > 0 and the following linear matrix state estimation error is: inequality LMI is satisﬁed, the augmented error equation (20) ðÞ e_ ðÞ t ¼ A LC e ðÞ t LF f ðÞ t 1 DdðÞ t ; (18) x x s s converges to zero asymptotically and steadily. The fault estimator equation (16) can obtain stable state estimation and Fault estimation error is: fault estimation, and the generalized disturbancedtðÞ meets the fault estimation error: kG s k < g ðÞ e d 1 fs 2 3 T T T T T T A1 A T M C C M C M T D M F 00 1 1 1 1 1 s 1 2 6 7 6 7 M M 0 M M F T I 3 3 2 s 2 6 7 6 7 gI 00 0 6 7 < 0; (21) 6 7 6 7 gI 00 6 7 6 7 gI 0 4 5 gI Proof: refer to Section 2.1 for the proof process. only if there is a symmetric positive matrix T satisfying the To further suppress the inﬂuence of parameter variation on following matrix inequality: fault estimation, improve the dynamic characteristics and 2 3 transient performance of fault estimator, and improve the 2h T TA 1 A T 0 4 5 < 0; accuracy of fault estimation, a regional pole assignment lemma TA 1 A T 2h T is introduced in this paper. nn Lemma 1 All the eigenvalues of the state matrix A 2 R of (22) a given system are in the vertical bar area ðÞ ðÞ D l 2 C : h < Re l < h , then the system is D stable and Substitute T = diag (T , T ) into the above formula. vs 1 2 1 2 2 3 T T T T T 2h T T A A T 1 M C1 C M C M 00 1 1 1 1 1 1 2 6 7 6 7 2h T 1 M 1 M 00 1 2 3 6 3 7 < 0; 6 7 T T T T T 6 7 T A1 A T M C C M 2h T C M 1 1 1 2 1 4 1 2 5 M M 2h T 3 2 2 (23) 7 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 Proof: As rank ¼ n, where l is all the Figure 6 Structure diagram of fault estimator i A l I eigenvalues of the system matrix A, (A, C) can be observed. Rewrite equation (19) as: ~ ~ ~ _ ðÞ e ¼ A LC e 1 Dd ; (25) e ¼ Ce fs A 0 L ~ ~ ~ where A ¼ , L ¼ , C ¼ C 0 , 0 F G D LF 0 Theorem 2 and Lemma 1 give the design method of the D ¼ and C ¼ 0 I . 0 F GF I MCFE. Next, we discuss the existence conditions of the According to the linear system theory, we can obtain the MCFE (17). ~ ~ sufﬁcient and necessary condition of equation (23) that A; C Theorem 3 Let rank (A)= n, rank (C)= m and rank (F)= r, is observable, that is: then the sufﬁcient and necessary conditions for the existence of the fault estimator (17) are: A lI n1 r rank ¼ n1 r;8s 2 C; Re s 0 (26) 02 31 ðÞ A lI 0 @4 5A rank 0 F lI ¼ n1 r;8s 2 C; ReðÞ s 0 C 0 A 0 ~ ~ Substitute A ¼ ; C ¼ C 0 into the above 0 F (24) formula: 02 31 02 31 ! "# A lI 0 A lI 0 n n B6 7C B6 7C A lI n1 r B6 7C B6 7C rank ¼ rank 0 F lI ¼ rank C 0 @4 5A @4 5A C 0 0 F lI r (27) ! "# A lI 0 ¼ rank 1 rank 0 F l I C 0 It can be seen from the above that (A, C) is observable, so maintain the driving stability and safety, and the reconstructor is designed as: A l I 0 rank ¼ n and rank (F)= r from the meaning C 0 2 3 010 00 of the question, it is easy to get rank ([0 F l I,]) = r, which is 6 7 proved. ^ 6 7^ y ðÞ t ¼ y ðÞ t F f ðÞ t ¼ y ðÞ t 001 00 f ðÞ t ; ft f s f s s 4 5 000 01 4.3 Fault reconstructor (29) When the sensor fails, it can be seen from Section 4.1 that the fault output of the SBW system satisﬁes: where y , F ,f and y are the fault sensor output, fault switching f s s ft y ¼ y1 f (28) f s matrix, value of fault estimation and fault-tolerant output, respectively. where y, f and y are sensor fault-free output, sensor fault value Keeping the proportion integration differentiation (PID) s f and the fault sensor output, respectively.y can be measured by controller structure unchanged, switching y to the feedback ft the sensor after the failure of the sensor. If the values of f and f are approximately equal, then the value without failure of the Figure 7 Schematic diagram of reconstructor sensor can be obtained through equation (28). Using the above ideas and based on the sensor fault estimated value f obtained by the MCFE designed in Section 4.2 and the fault vector F obtained in Section 3.2, the fault reconstructor shown in Figure 7 is designed for fault sensor reconstruction, making the SBW system can still guarantee the basic steering function in the event of sensors failure, and 8 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 loop of the PID controller can make the SBW system with Table 1 Fault description in details sensor fault still have steering characteristics close to that of the Fault types Fault behavior no-fault SBW system, thus achieving the purpose of fault- Continuous deviation- Rotation angle sensor deviation fault after 6 s tolerant control. gain failure and signal interruption after 9 s Simultaneous gain- The yaw rate sensor has a gain failure of D = 5. Numerical analysis stuck failure 0.3 at t = 9 s, and the rotation angle sensor is To demonstrate the effectiveness of the FD, fault estimation lock and fault-tolerant control strategies are proposed considering Simultaneous Lateral acceleration sensor interrupt fault at t = parameter perturbation and sensors fault in the SBW system. interruption-gain 5 s, and rotation angle sensor gain fault with The simulation environment is set as follows. failure D = 0.4 at the same time It is assumed that the damping coefﬁcient of the front wheels and steering mechanism on the steering shaft, the damping coefﬁcient of the motor shaft and the module, FD module, fault estimation module and fault perturbation amount of the front wheel deﬂection stiffness tolerance control module of the SBW system are parameter are set within 10% (Huang et al., 2017). established, respectively, to analyze the timeliness and This article considers that a single sensor has a mixed accuracy of FD, the accuracy of fault estimation and the failure, and multiple sensors have a simultaneous failure. effect of fault-tolerant control. The failure is set as described in Table 1 for simulation veriﬁcation. 5.1 Results of fault detection Set the vehicle speed v = 15 m/s, given the target steering Using the method in Section 2.1 to design the fault observer, wheel angle as shown in Figure 8. The main parameters in and the mincx command to solve equation (18) can obtain g = the simulation of this paper are shown in Table 2. Under 5.273 10–4, r = 0.4979, as well as the optimal observer the matrix and laboratory (MATLAB)/Simulink environment, the PID control module, fault setting feedback gain matrix L as: 2 3 7 4 5 1199:30 7:98898 10 1:22795 10 150568 1:79442 10 6 5 3 4 7 15152:31:05084 10 1:69501 10 186578 2:48836 10 6 7 6 7 6 10 8 9 7 52:5015 5:02841 10 4:25239 10 10590 7:22559 10 6 7 6 7 10 12 6 7 L ¼ 3:08463 10 0:0184117 316:691 6:13696 10 48:8070 6 7 6 7 9 7 8 6 7 157:260 1:58503 10 1:50850 10 31721 2:53699 10 6 7 6 7 7 4 9 2 6 7 1:09552 10 1:78152 10 0:0727597 1:51664 10 1:14525 10 4 5 10 11 2:17262 10 0:0464384 21:1417 2:34267 10 3:24072 Substituting the matrix L into the FD observer can obtain the that the norm of the residual exceeds the FD threshold at about FD results shown in Figures 9–11. 5 s, indicating that the system is faulty. However, the test results As shown in Figure 9, for the FD of the continuous deviation- also cannot indicate what type of fault occurred in which sensor. From the above simulation results, it can be concluded that gain failure of rotation angle sensor, when there is parameter the designed fault observer can detect a fault, when they occur perturbation and sensor failure in the SBW system, the residual norm exceeds the diagnostic threshold at about 5 s, indicating that a sensor failure has occurred in the system at this time. Figure 8 Angular curve of steering wheel As shown in Figure 10, for the FD results of the simultaneous gain-stuck fault of two sensors, it can be seen that the norm of the residual is lower than the diagnostic threshold between 0 s–9 s and the norm of the residual signal exceeds the FD threshold at about 9.1 s. It indicates that the system has a sensor failure at this time, but it cannot be known, which sensor has failed. The norm of the residual does not exceed the diagnostic threshold at 9 s because there is an error in the ﬁtting of the generated residual curve, which leads to FD false alarms; however, the overall FD performance is satisfactory. As shown in Figure 11, for the FD result of the simultaneous interruption of the two sensors-gain fault, it can be observed 9 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 Table 2 Parameters of SBW model cannot determine the type of fault, nor can it determine, which sensor is faulty. The multi-objective constraint fault estimator Parameters Symbols Values can solve this problem, to further estimate the size and time- Body mass m 1,270 kg varying characteristics of the fault. Body roll moment of inertia Iz 1,537 kg.m Distance from mass center to front a 1.015 m 5.2 Results of fault estimation wheel The method in Section 2.2 was used to design the MCFE, and Distance from mass center to rear b 1.895 m the narrow strip region was selected as 100 <Re (l ) < 10. In wheel MATLAB, mincx command was used to solve equations (26) Front wheel cornering stiffness K 135,000 N/rad Rear wheel cornering stiffness K 175,000 N/rad and (28) to obtaing = 0.0105, and the optimal estimator gain r 2 Pneumatic trail t 0.006 m matrices F and G are: 2 3 Mechanical trail t 0.014 m 6 5 97:79 8:41 10 5:37 10 Armature inductance L 0.003 H 6 7 6 7 Armature resistance R 0.034 X F ¼ 8:43 10 97:63 0:02 ; 4 5 Motor torque coefﬁcient K 0.086 N.m/A t 5 5:37 10 0:02 97:79 Counter electromotive force coefﬁcient K 0.009 V.s/rad Motor stiffness coefﬁcient K 130 N.m/rad 2 3 6 5 Motor rotary inertia J 0.006 Kg/m m 1198:17 97:79 8:60 10 0:18 5:38 10 6 7 Motor damping B 0.1 N.m.s/rad 5 6 m 6 7 G ¼ : 1:13 10 8:45 10 97:41 6:23 0:06 4 5 Steering tube rotary inertia J 0.01 Kg/m c 4 5 1:91 10 5:35 10 0:06 0:70 97:78 Steering tube damping J 0.3 N.m.s/rad Steering tube stiffness coefﬁcient K 0.5 N.m/rad To show that the MCFE can provide better estimation Motor speed-reducing device G 18 performance than the fast-adaptive fault estimation observer, transmission ratio the simulation calculation of the fast-adaptive fault estimation Steering ratio G 15.5 observer (Olfa et al.,2015) design is also given below. The LMI toolbox solves the linear matrix inequality to obtain g = 8.266 10 , the observer gain matrix L and the fault for different time periods of a single sensor and multiple sensors simultaneously appearing different types of fault. However, it estimation gain G are: 2 3 6 5 4 7 5 3:91 10 1:39 10 1:77 10 2:03 10 1:10 10 6 7 6 7 6 5 8 6 7 5:32 10 1:88 10 2:40 10 2:76 10 1:49 10 6 7 6 7 9 8 7 10 8 6 7 7:02 10 2:49 10 3:17 10 3:63 10 1:97 10 6 7 6 7 11 10 9 12 9 6 7 L ¼ 2:97 10 1:05 10 1:34 10 1:83 10 8:32 10 6 7 6 7 6 7 1186:96 41:50 5:27 1:35 10 32:83 6 7 6 7 9 8 7 10 8 6 7 6:30 10 2:23 10 2:84 10 3:27 10 1:77 10 4 5 11 10 9 12 10 4:40 10 1:58 10 2:01 10 2:31 10 1:25 10 2 3 7 4 5 4 5:36 10 1:18 10 2:30 10 7136:16 -9:45 10 6 7 8 7 6 6 7 6 7 G ¼ : 9:19 10 3:42 10 4:14 10 1:46 10 2:71 10 4 5 7 6 5 5 6 9:41 10 5:33 10 8:35 10 1:89 10 4:15 10 Substituting the above-obtained matrices F and G into the constant deviation fault after 5 s, the output of the sensor is MCFE, and L and G into the fast adaptive fault always about 10 degrees higher than the real value. In addition, estimation observer, the fault estimation results can be the gain fault occurs after 7 s, and the fault estimation curve can obtained in Figures 12–21. accurately approximate the real fault from the time when the fault As shown in Figure 12, for the estimation results of continuous occurs. However, the adaptive fault estimation method has deviation-gain fault of rotation angle sensor, it can be seen that obvious overshoot in 4 s 7 s, and obvious oscillation in 7 s the multi-objective constraint fault estimator can accurately 7.5 s. Although the phases of the fault setting curve and the estimate the fault value as 0, when the rotation angle sensor does fault estimation curve are the same in 7 s 12 s, the amplitudes at not fail during 0 s 5 s. When the rotation angle sensor has a the peaks and troughs are signiﬁcantly different. 10 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 Figure 9 Fault detection results of the rotation angle sensor Figure 13 Fault estimation error curve of the rotation angle sensor by estimation method in this paper Figure 10 Results of simultaneous gain-stuck detection of two sensors Figure 14 Fault estimation curve of the yaw rate sensor Figure 11 Results of two sensors interrupted at the same time-gain fault detection Figure 15 Fault estimation error curve of the yaw rate sensor by estimation method in this paper Figure 12 Fault estimation curve of the rotation angle sensor Figure 16 Fault estimation curve of the rotation angle sensor As shown in Figure 13, the error of the fault estimation error is large when the fault occurs. It is because the estimator needs a reaction time to adapt to this sudden fault, and the fault estimation error at all other times is less than 0.7 degrees. 11 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 Figure 17 Fault estimation error curve of the rotation angle sensor by Figure 21 Fault estimation error curve of the rotation angle sensor by estimation method in this paper estimation method in this paper The fault estimation results of the simultaneous gain-stuck fault of two sensors are shown in Figures 14–17. Figure 18 Fault estimation curve of the lateral acceleration sensor As demonstrated from Figures 14 and 16, when the yaw rate sensor and the rotation angle sensor have not failed from 0 s to 9 s, the fault estimator can accurately estimate the fault value as 0. When the yaw rate sensor and rotation angle sensor fail simultaneously after 9 s, the yaw rate sensor output is always 0.3 times greater than the true value and the rotation angle sensor output becomes a ﬁxed value from the moment of the failure, and the fault estimation value curve can accurately approximate the real fault from the moment of the fault. However, the adaptive fault estimation algorithm has obvious oscillations from 4 s to 9 s. Although the phases of the fault setting curve and the fault estimation curve are the same between 9 s and 12 s, the amount of overshoot is large at the peak and valley. As presented in Figures 15 and 17, the fault estimator needs reaction time to adapt to this type of failure, resulting in a spike Figure 19 Fault estimation error curve of the lateral acceleration in the estimation error at 9 s. At other times, the maximum sensor by estimation method in this paper estimation error of the yaw rate and rotation angle are 0.108 deg/s and 0.4 degrees, respectively. The fault estimation results of the simultaneous interruption- gain fault of two sensors are shown in Figures 18–21. As shown in Figures 18 and 20, when the lateral acceleration sensor and rotation angle sensor have not failed from 0 s to 5 s, the fault estimator can accurately estimate the fault value as 0. When the lateral acceleration sensor and the rotation angle sensor fail simultaneously after 5 s, the actual output of the lateral acceleration sensor becomes 0, the output of the rotation angle sensor is always 0.3 times more than the true value, and the fault estimation value curve can accurately approximate the real fault from the moment of the fault. However, the adaptive fault estimation algorithm has different amplitudes at the peaks Figure 20 Fault estimation curve of the rotation angle sensor and valleys, for the fault setting curve and the fault estimation curve from the moment of the fault. As shown in Figures 19 and 21, the fault estimation error is large at 5 s because the estimator needs time to adapt to this type of fault. The peak value of the fault estimation error of the lateral acceleration at the remaining time does not exceed 0.04 m/s . In addition, the peak value of the fault estimation error of the rotation angle of the pinion shaft does not exceed 0.4 degrees at the other moments. From the above fault estimation results, the MCFE designed has strong robustness to system parameter perturbation, and can effectively suppress the inﬂuence of system parameter 12 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 perturbation on sensor fault estimation results. The fault Figure 23 Simultaneous gain-stuck fault tolerant performance before estimation curve obtained by the multi-objective constraint and after comparison curves fault estimator can accurately approximate the real fault from the moment of the fault, with an accuracy rate of up to 98%, which veriﬁes the effectiveness of the fault estimation algorithm designed in this paper. The comparison of the above fault estimation results also reﬂects the limitations of the method designed in (Olfa et al., 2015). Due to the error between the adaptive fault observer and the system output, the coupling effect of fault estimation error, state estimation error and the change of adaptive parameters, the gain matrix of the fault is changed, which causes the oscillation of the fault estimation results and a long convergence time. Comprehensive FD and fault estimation results, when there is parameter perturbation in the SBW system, when a single sensor or multiple sensors have gain, deviation or stuck faults, the fault observer designed in this paper can detect the time when the sensor fails. The fault estimator can accurately estimate the fault amplitude and time-varying characteristics, and the error of fault estimation is generally small, which lays a foundation for the next step of fault-tolerant control. 5.3 Fault-tolerant control results As shown in Figures 22–24, the comparison curves are presented for the actual sensor output, fault-tolerant control output and fault output under continuous deviation-gain failure, simultaneous gain-stuck failure and simultaneous interruption-gain failure. In the ﬁgure, the fault-free output refers to the value when the sensor has not failed, which is Figure 24 Simultaneous interruption-gain fault tolerant performance represented by a solid black line. The fault-tolerant control before and after comparison curves outputs refers to the value when the SBW system sensor starts fault-tolerant control after a fault occurrence at time t, which is represented by a red dot line. The fault output refers to the value after a fault occurrence, which is represented by a blue dotted line. As can be seen from Figure 22, when the rotation angle sensor has no fault from 0 s to 5 s, the actual output, fault- tolerant control output and fault output values are all the same. When a deviation fault occurs after 5 s, the angle of the pinion shaft is always 10 degrees higher than that of the true output. When a gain fault occurs after 7 s, the output of the rotation angle sensor is 0.4 times the normal output, and the peak value of the angle error reaches 32 degrees. When the fault-tolerant Figure 22 Continuous deviation-gain fault tolerant performance before and after comparison curves 13 Rotation angle sensor Journal of Intelligent and Connected Vehicles Qinjie Yang et al. Volume 4 · Number 1 · 2021 · 1–15 control is started, the output of the fault-tolerant control is regional pole conﬁguration to estimate the amplitude and time- consistent with the output of the fault-free SBW system. varying characteristics of the faulty sensor, and the existence As can be seen from Figure 23,when the yaw rate sensor and condition of the fault estimation observer is given. Third, a the rotation angle sensor have not failed from 0 s to 9 s, the fault-tolerant algorithm is designed based on the sensor fault actual output, fault-tolerant control output and fault output estimate and fault output. Finally, numerical analysis is carried values are all the same. When the yaw rate sensor has a gain out to verify the proposed method. failure after 9 s, the yaw rate error peak value reaches 10 deg/s. The analyzed results validated that the designed fault observer can accurately and timely diagnose the faults of the At the same time, when the rotation angle sensor is stuck, the sensors of the SBW system, when different types of faults occur output of the rotation angle sensor remains at 10 degrees. When in the sensors of the SBW system. The designed MCFE can the fault-tolerant control is started, the output of the fault- accurately estimate the sensor fault size and time-varying tolerant control is consistent with the output of the fault-free characteristics, and is robust to parameter perturbation; and SBW steering system. It can be seen that fault-tolerant control the fault-tolerant control strategy can make the SBW system can signiﬁcantly reduce the impact of faults on the performance with sensors faulty close to the faultless SBW system. of the SBW system, and restore the performance of the SBW Moreover, the steering characteristics of the system can meet system, to be close to that of the fault-free SBW system. the basic requirements of the wire control system, in which the As can be seen from Figure 24, when the lateral acceleration feasibility and effectiveness are fully veriﬁed while applying the sensor and the rotation angle sensor have not failed from 0 s to active fault-tolerant control framework to the SBW system. 5 s, the actual output, fault-tolerant control output and fault output values are all the same. When the signal of the lateral acceleration sensor is interrupted after 5 s, the output of the References lateral acceleration sensor is always 0, and the peak value of the lateral acceleration error reaches 2.3 m/s . At the same time, Anwar, S. and Niu, W.A. (2010), “Nonlinear observer based when the pinion shaft rotation angle sensor has a gain failure, analytical redundancy for predictive fault tolerant control of the peak angle error of the pinion shaft reaches 18 degrees. a steer-by-wire system”, 2010 IEEE International Conference When the fault-tolerant control is started, the output of the on Industrial Technology, Vina del Mar, Chile, pp. 321-334. fault-tolerant control is consistent with the output of the fault- Aouaouda, S., Chadli, M., Shi, P. and Karimi, H.R. (2015), free SBW system. It can be seen that fault-tolerant control can “Discrete-time H_/H1 sensor fault detection observer signiﬁcantly reduce the impact of faults on the performance of design for nonlinear systems with parameter uncertainty”, the SBW steering system, and restore the performance of the International Journal of Robust and Nonlinear Control, Vol. 25 SBW system to be close to that of the fault-free SBW system. No. 3, pp. 339-361. The above simulation results show that when a single sensor Chen, L. and Patton, R.J. (2017), “A time-domain LPV H_/ H1 fault detection ﬁlter”, IFAC-Papers OnLine,Vol. 50 of the SBW system has different faults at different times and No. 1, pp. 8600-8605. multiple sensors fail at the same time, the steering performance Chengwei, T., Changfu, Z., Xiang, W. and Dong, Y. (2010), of the SBW system will be affected to varying degrees. The fault “Fault-tolerant control of sensors in steer-by-wire vehicle”, observer designed in this paper can detect the fault of the sensor Journal of Jilin University (Engineering Science),Vol.40 No. 1, more accurately. The multi-objective constraint fault estimator pp. 6-12 (in Chinese). can estimate the fault amplitude of the sensor more accurately Chilali, M. and Gahinet, P. (1996), “H1 design with pole when the sensor fails, and the overall error of the fault placement constraints: an LMI approach”, IEEE Transactions estimation is smaller. The fault-tolerant control algorithm on Automatic Control,Vol.41No.3,pp.358-367. designed based on the fault estimation information can restore Dhahri, S., Sellami, A.B. and Hmida, F. (2012), “Robust the steering performance of the SBW system to be close to the sensor fault detection and isolation for a steer-by-wire system performance of the fault-free SBW system when the sensor fails. based on sliding mode observer”, 16th IEEE Mediterranean Therefore, it is veriﬁed about the effectiveness and feasibility for Electrotechnical Conference, Yasmine Hammamet, Tunisia, the fault-tolerant control strategy proposed in this paper. pp. 450-454. Gao, Z., Yang, L., Wang, H. and Li, X. (2017), “Active fault 6. Conclusions tolerant control of electric power steering system with sensor This paper proposes a design method of MCFE, to deal with fault”, Proceedings of the 36th Chinese Control Conference, sensor faults of SBW systems under parameter uncertainties. Dalian, China, pp. 9630-9636. The residual can be correspondingly obtained by the fault Hou, M. and Patton, R.J. (1996), “An LMI approach to H_/ observer to be used as the control input. Therefore, an active H1 fault detection observers”, UKACC International fault-tolerant control framework that integrates the fault Conference on Control, Exeter, pp. 305-310. observer, fault estimator and fault reconstructor is designed to Huang, C., Naghdy, F. and Du, H. (2018), “Observer-based process the rotation angle sensor fault of the SBW system under fault-tolerant controller for uncertain steer-by-wire systems parameter disturbance. using the delta operator”, IEEE/ASME Transactions on First, an SBW system model with parameter perturbation Mechatronics, Vol. 23 No. 6, pp. 2587-2598. and sensor failure is established, and a multi-objective H_/H1 Huang, C., Naghdy, F. and Du, H. 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(2017), “H_/H1 tolerant control method for steer-by-wire system”, 2009 fault detection observer design in ﬁnite-frequency domain International Conference on Mechatronics and Automation, for lipschitz non-linear systems”, IET Control Theory & Changchun, China, pp. 291-295. Applications, Vol. 11 No. 14, pp. 2361-2369. Wada, N., Fujii, K. and Saeki, M. (2013), “Reconﬁgurable fault-tolerant controller synthesis for a steer-by-wire vehicle using independently driven wheels”, Vehicle System Further reading Dynamics, Vol. 51 No. 9, pp. 1438-1465. Waraus, D. (2009), “Steer-by-wire system based on FlexRay Balachandran, A. and Gerdes, J.C. (2013), “Artiﬁcial steering protocol”, IEEE Transactions on Applied Electronics, Pilsen, feel design for steer-by-wire vehicles”, IFAC Proceedings Czech Republic, pp. 269-272. Volumes, Vol. 46 No. 21, pp. 404-409. Wenhan, Z., Zhenhua, W., Tarek, R., Ye, W. and Yi, S. (2020), “A state augmentation approach to interval fault Corresponding author estimation for descriptor systems”, European Journal of Rencheng Zheng can be contacted at: rencheng.zheng@tju. Control, Vol. 51, pp. 19-29. edu.cn For instructions on how to order reprints of this article, please visit our website: www.emeraldgrouppublishing.com/licensing/reprints.htm Or contact us for further details: permissions@emeraldinsight.com

Journal

Journal of Intelligent and Connected Vehicles – Emerald Publishing

Published: Apr 26, 2021

Keywords: Active fault-tolerant control; Fault estimation; Sensors failure; Steer-by-wire

References

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Anwar, S.; Niu, W.A.
Discrete‐time H_/H∞ sensor fault detection observer design for nonlinear systems with parameter uncertainty

Aouaouda, S.; Chadli, M.; Shi, P.; Karimi, H.R.
A time-domain LPV H_/H∞ fault detection filter

Chen, L.; Patton, R.J.
Fault-tolerant control of sensors in steer-by-wire vehicle

Chengwei, T.; Changfu, Z.; Xiang, W.; Dong, Y.
H∞ design with pole placement constraints: an LMI approach

Chilali, M.; Gahinet, P.
Robust sensor fault detection and isolation for a steer-by-wire system based on sliding mode observer

Dhahri, S.; Sellami, A.B.; Hmida, F.
Active fault tolerant control of electric power steering system with sensor fault

Gao, Z.; Yang, L.; Wang, H.; Li, X.
An LMI approach to H_/H∞ fault detection observers

Hou, M.; Patton, R.J.
Observer-based fault-tolerant controller for uncertain steer-by-wire systems using the delta operator

Huang, C.; Naghdy, F.; Du, H.
Fault tolerant sliding mode predictive control for uncertain steer-by-wire system

Huang, C.; Naghdy, F.; Du, H.
Design of fault tolerant control system for electric vehicles with steer-by-wire and in-wheel motors

Ito, A.; Yoshikazu, H.
Observer-based H∞ fault-tolerant attitude control for satellite with actuator and sensor faults

Liang, X.; Wang, Q.; Hu, C.; Dong, C.
Fault diagnosis of steer-by-wire system for unmanned vehicle

Lu, X.; Zhiqiang, F.; Zengliang, L.; Renxie, Z.
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Mortazavizadeh, S.A.; Ebrahimi, M.; Ghaderi, A.; Hajian, M.
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Olfa, H.; Boumedyen, B.; Ahmed, Z.
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Tian, C.; Zong, C.; He, L.; Dong, Y.
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Wada, N.; Fujii, K.; Saeki, M.
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Waraus, D.
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Wenhan, Z.; Zhenhua, W.; Tarek, R.; Ye, W.; Yi, S.
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Xiao, L.; Meng, Z.; Huang, X.; Ma, L.
Robust sliding-mode observer-based sensor fault estimation, actuator fault detection and isolation for uncertain nonlinear systems

Yang, J.; Zhu, F.; Wang, X.; Bu, X.
Mixed H_/H∞ fault detection observer design for multi model systems via nonsmooth optimization approach

Yang, J.; Hamelin, F.; Apkarian, P.; Sauter, D.
Robust sensor fault estimation scheme for satellite attitude control systems

Zhang, J.; Swain, A.K.; Nguang, S.K.
Two-way H∞ control method with a fault-tolerant module for steer-by-wire system

Zhang, H.; Zhao, W.
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Zhou, M.; Wang, Z.; Shen, Y.; Shen, M.
Artificial steering feel design for steer-by-wire vehicles

Balachandran, A.; Gerdes, J.C.

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Active fault-tolerant control of rotation angle sensor in steer-by-wire system based on multi-objective constraint fault estimator

Active fault-tolerant control of rotation angle sensor in steer-by-wire system based on multi-objective constraint fault estimator

Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

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Active fault-tolerant control of rotation angle sensor in steer-by-wire system based on multi-objective constraint fault estimator

Active fault-tolerant control of rotation angle sensor in steer-by-wire system based on multi-objective constraint fault estimator

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