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Fractional Steepest Ascent Method for TCU Fault Detection
10th IFAC Symposium on Fault Detection, 10th IFAC on Fault Detection, 10th IFAC Symposium Symposium Detection, Supervision and Safetyon forFault Technical Processes Supervision and Safety for Technical Processes Supervision and Safety for Technical Processesonline at www.sciencedirect.com Warsaw, Poland, August 29-31, 2018 Available 10th IFAC Symposium on Fault Detection, Warsaw, Poland, August 29-31, 2018 Warsaw, Poland, August 29-31, 2018 10th IFAC Symposium Detection, Supervision and Safetyon forFault Technical Processes Supervision and Safety Technical Warsaw, Poland, Augustfor 29-31, 2018 Processes Warsaw, Poland, August 29-31, 2018
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IFAC PapersOnLine 51-24 (2018) 1336–1342 Fractional Fractional Steepest Steepest Ascent Ascent Method Method for for TCU TCU Fault Fault Detection Detection Fractional Steepest for Detection Hongqiu Zhu, Zhiliang Wu,Ascent Chunhua Method Yang, Tao Peng, ZhiwenFault Chen, Xiaoyue Yang Fractional Steepest Ascent Method for TCU TCU Fault Detection
Hongqiu Zhu, Zhu, Zhiliang Zhiliang Wu, Wu, Chunhua Chunhua Yang, Yang, Tao Tao Peng, Peng, Zhiwen Zhiwen Chen, Chen, Xiaoyue Xiaoyue Yang Yang Hongqiu Hongqiu Zhu, Zhiliang Wu, Chunhua Yang, Tao Peng, Zhiwen Chen, Xiaoyue Yang Hongqiu Zhu, Zhiliang Wu, Chunhua Yang, Tao Peng, Zhiwen Chen, Xiaoyue Yang School of Information Science and Engineering ， Central South University， School School of of Information Information Science Science and and Engineering Engineering ， ，Central Central South South University University， ， School of Information Science and Engineering ， Central South University ， Changsha ， China (Tel: 086-0731-88836876; e-mail: [email protected]). School of410083 Information Science and Engineering ，Central South University， Changsha 410083 ， (Tel: e-mail: [email protected]). Changsha 410083 ，China China (Tel: 086-0731-88836876; 086-0731-88836876; e-mail: [email protected]). Changsha 410083，China (Tel: 086-0731-88836876; e-mail: [email protected]). Changsha 410083a， China (Tel: 086-0731-88836876; e-mail: [email protected]). Abstract: This paper proposes Fractional Steepest Ascent Method (FSAM) for detecting detecting faults in in Traction Abstract: This Abstract: This paper paper proposes proposes aa Fractional Fractional Steepest Steepest Ascent Ascent Method Method (FSAM) (FSAM) for for detecting faults faults in Traction Traction Control Unit (TCU), which aims at detecting the transient fault caused by electromagnetic radiation, Control Unit (TCU), which aims at detecting the transient fault caused by electromagnetic radiation, Control Unit (TCU), which aims at detecting the transient fault caused by electromagnetic radiation, Abstract: Thisviolent paper vibration, proposes aetc. Fractional Steepest Ascent Method forquadratic detectingenergy faults infunctional Traction overheating, In FSAM, FSAM, the maximum maximum value of of(FSAM) the signal Abstract: Thisviolent paper vibration, proposes aetc. Fractional Steepest Ascent Method forquadratic detectingenergy faults infunctional Traction overheating, In the value the overheating, vibration, etc. In thethe maximum value of(FSAM) the signal signal energy functional Control Unitinviolent (TCU), whichby aims at FSAM, detecting transient fault caused by quadratic electromagnetic radiation, is obtained time domain means of Grü mwald Letniko (G-L) fractional order technique. For fault Control Unit (TCU), which aims at detecting the transient fault caused by electromagnetic radiation, is obtained in time domain by means of Grü mwald Letniko (G-L) fractional order technique. For is obtained inviolent time domain byetc. means of Grü mwald Letniko (G-L) fractional order technique. For fault fault overheating, vibration, In FSAM, the maximum value of the signal quadratic energy functional detection purpose, the detection threshold is determined with the Root Mean Square (RMS) of the velocity overheating, violentthe vibration, etc. In FSAM, the maximum of theMean signal quadratic energy functional detection purpose, the detection threshold is determined withvalue the Root Mean Square (RMS) of the the velocity detection purpose, detection threshold is determined with the Root Square (RMS) of velocity is obtained in time domain by means of Grü mwald Letniko (G-L) fractional order technique. For fault residual absolute value and the fractional order is optimized optimized by(G-L) searching the maximum maximum ratio of of RMS. The is obtained in time domain by fractional means of order Grümwald Letniko fractional order technique. For fault residual absolute value and the the fractional order is by searching the ratio RMS. The residual absolute value and is optimized by searching the maximum ratio ofthe RMS. The detection purpose, the detection threshold is determined with the Root Mean Square (RMS) of velocity simulation results verify that the method proposed is not only sensitive to the TCU fault signals, but also detection purpose, the detection threshold is determined with the Root Mean Square (RMS) of the velocity simulation results verify that the method proposed is not only sensitive to the TCU fault signals, but also simulation results verify that the method proposed is not only sensitive to the TCU fault signals, but also residual absolute value and the fractional order is optimized by searching the maximum ratio of RMS. The can effectively suppress the noise to extent. residual absolute value and the fractional is optimized by searching the maximum ratio of RMS. The can effectively suppress the noise to some someorder extent. can effectively suppress to some extent. simulation results verify the thatnoise the method proposed is not only sensitive to the TCU fault signals, but also simulation results verify that the method proposed is not only sensitive to the TCU fault signals, but also © 2018, IFAC (International of Automatic can effectively suppress the Federation noise to some extent. Control) Hosting by Elsevier Ltd. All rights reserved. can effectively suppress the noise to some extent. Keywords: TCU, fault detection, detection, quadratic energy energy functional, FSAM. FSAM. Keywords: TCU, fault Keywords: TCU, fault detection, quadratic quadratic energy functional, functional, FSAM. Keywords: TCU, fault detection, quadratic energy functional, FSAM. Keywords: TCU, fault detection, quadratic energy functional, domainFSAM. for rolling rolling element bearing bearing weak fault. fault. Chen et et al. domain 1. INTRODUCTION domain for for rolling element element bearing weak weak fault. Chen Chen et al. al. 1. INTRODUCTION (2014) have used the fractional-order self-synchronization 1. INTRODUCTION (2014) have used the fractional-order self-synchronization (2014) have used the fractional-order self-synchronization for Fuzzy rollingPetri element bearing fault. Chen al. error-based nets to to detect weak nontechnical losseset and Traction Control Control Unit1.(TCU) (TCU) as one one of the the important electronic electronic domain INTRODUCTION domain for Fuzzy rollingPetri element bearing fault. Chen al. detect nontechnical losses Traction as error-based nets to detect weak nontechnical lossesetand and Traction Control Unit Unit1.(TCU) as one of of the important important electronic error-based (2014) haveFuzzy used Petri the nets fractional-order self-synchronization INTRODUCTION outage fault events. Zhou et al. (2016) have introduced a novel devices in traction drive control system (TDCS) of the high(2014) have used the fractional-order self-synchronization outage fault events. Zhou et al. (2016) have introduced a novel devices in traction drive control system of fault Fuzzy events.Petri Zhounets et al. have introduced a novel devices in traction drive control system (TDCS) of the the highhigh- outage error-based to (2016) detect nontechnical and Traction Control Unit (TCU) asisone the(TDCS) important adaptive variable time fractional-order anisotropiclosses diffusion speed train, its main main function to of control traction electronic motor for for adaptive error-based Fuzzy time Petri fractional-order nets to detect nontechnical losses and variable anisotropic diffusion Traction Control Unitfunction (TCU) asis one of the important electronic speed train, its to control traction motor adaptive variable time fractional-order anisotropic diffusion speed train, its main function is to control traction motor for outage fault events. Zhou et al. (2016) have introduced novel devices in the traction drive controlofsystem (TDCS) ofhigh-speed the high- filtering for seismic data noise attenuation. Long et al. a(2017) achieving normal operation the TDCS. The outage fault events. Zhou et al. (2016) have introduced a novel filtering for seismic data noise attenuation. Long et al. (2017) devices in traction drive control system (TDCS) of the highachieving the normal operation of the TDCS. The high-speed filtering for seismictime datafractional-order noise attenuation. Long et al. (2017) achieving theitsnormal operationisoftothe TDCS. The high-speed variable speed main function control traction for adaptive have presented time frequency filtering anisotropic algorithm for fordiffusion bearing train istrain, usually running at above above 200km/h under motor the harsh harsh adaptive variable timefrequency fractional-order diffusion aaa time filtering algorithm bearing speed its main function is to 200km/h control traction motor for have train usually running at under the have presented presented time frequency filtering anisotropic algorithm for bearing train is istrain, usually running at above 200km/h under the harsh filtering for seismic data noise attenuation. Long et al. (2017) achieving the normal operation of the TDCS. The high-speed fault detection detection baseddata on the the fractional lower Long order et S transform. transform. complex and changeable environment, forThe example, the fault for seismic noise attenuation. al. (2017) based on fractional lower order achieving the normal operation of the TDCS. high-speed complex and changeable environment, for example, the fault detection based on the fractional lower order S Sfor transform. complex and changeable environment, forunder example, the filtering have presented a time frequency filtering algorithm bearing train is usually running at above 200km/h the harsh Pineda-Sanchez et al. al.frequency (2010) filtering have utilized utilized the for fractional electromagnetic radiationatand and input power variation, have presented a time algorithm bearing et (2010) have the fractional train is usually running above under theviolent harsh Pineda-Sanchez electromagnetic radiation input200km/h powerfor variation, violent Pineda-Sanchez et al. (2010) have lower utilized theS transform. fractional electromagnetic radiation and input power variation, violent fault detection based onits the fractional order complex and changeable environment, example, the Fourier tranform and optimization method to detect the vibration, bad weather, and so on, which probably cause the fault detection based onits theoptimization fractional lower orderto S transform. Fourier tranform and method detect the complex and changeable environment, for example, the vibration, bad weather, and so on, which probably cause Fourier tranform and its optimization method to detect the vibration, bad weather, and so on, which probably cause the Pineda-Sanchez et al. for (2010) have utilized the fractional electromagnetic radiation and inputorpower variation, violent harmonic components characterizing each type of fault. processor module, register module interface module faults Pineda-Sanchez et al. (2010) have utilized the fractional harmonic components for characterizing each type of fault. electromagnetic radiation and inputor power variation, violent processor module, register module or interface module faults harmonic components for characterizing each type of fault. processor module, register module interface module faults andhave its optimization method to detect the vibration, bad weather, and so on, which cause the Fourier Pisano ettranform al. (2014) proposed aa discontinuous discontinuous dynamic in et 2017). These faults probably are non-permanent non-permanent Fourier andhave its optimization method to detect the (2014) proposed dynamic et al. vibration, bad weather, and so on, which probably cause the Pisano in TCU TCU (Yang (Yang et al., al., 2017). These faults are Pisano ettranform al. (2014) have proposed a discontinuous dynamic in TCU (Yang et al., 2017). These faults are non-permanent harmonic components for characterizing each type order of fault. processor module, register module or interface module faults system fault detection method based on fractional for and submerged by noise in time domain, which may induce the harmonic components for characterizing each type of fault. system fault detection method based on fractional order for processor module, register module or interface module faults and submerged by noise in time domain, which may induce the system et fault method baseda discontinuous on fractional order for and submerged noise in time domain, which may induce the Pisano al. detection (2014) have proposed dynamic in TCU (Yang by ettimely al., 2017). These faults are non-permanent fault simulation of hydraulic plants. Kuo et al. (2017) have failure without detection (Maniatakos et al., 2011, Pisano et al. (2014) have proposed a Kuo discontinuous dynamic fault of plants. et have in TCUwithout (Yang ettimely al., 2017). These(Maniatakos faults are non-permanent et al., failure detection fault simulation simulation of hydraulic hydraulic plants. Kuo et al. al. (2017) (2017) have failure without timely detection (Maniatakos et induce al., 2011, 2011, system fault detection method based on fractional order for and submerged by noise in time domain, which may the proposed detection method based on on fractional fractional order dynamic Kelkar et al., al., 2014). 2014). system fault detection method method based for aaa detection dynamic and submerged by noise in time domain, which may induce the proposed Kelkar et proposed detection method based Kuo on fractional fractional dynamic Kelkar et al., 2014). simulation of hydraulic plants. et al. (2017) have failure without timely detection (Maniatakos et al., 2011, fault deviation for solar solarofPV PV panel failure. failure. However, most of the the have fault fault simulation hydraulic plants. Kuo et most al. (2017) deviation for panel However, of fault failure without timely detection (Maniatakos et al., 2011, deviation for solar PV panel failure. However, most of the fault Moreover, for safety reasons, TCU redundancy is often used proposed a detection method based on fractional dynamic Kelkar et al., Moreover, for safety detection methods based on fractional fractional order mentioned above Moreover, for2014). safety reasons, reasons, TCU TCU redundancy redundancy is is often often used used detection proposed methods a detection method based on fractional dynamic based on order mentioned Kelkar al., 2014). methods based on failure. fractional order mentioned above to avoidetproblems problems (Ronanki et et al., al., 2017), 2017), besides, besides, TCU TCU fault fault detection deviation for solar PV panel However, most of theabove fault to avoid (Ronanki are usually based on known fault models or obvious fault to avoid problems (Ronanki et al., 2017), besides, TCU fault deviation forbased solar PV panel failure. However, most of the fault Moreover,generally for safetyhave reasons, TCUduration redundancy is often used are usually on known fault models or obvious are usually based on known fault models or obvious fault signals short and no fixed detection methods based on fractional order mentioned above Moreover, for safety reasons, TCU redundancy is often used signals generally have short duration and no fixed features, which are not suitable for TCU fault detection signals generally (Ronanki have short duration and TCU no fixed detection methods based on fractional order mentioned above to avoid problems et al., 2017), besides, fault features, which are not suitable for TCU fault detection features, which are not suitable for TCU fault detection characteristic frequency, which not only cause wasting of usually or Besides, obvious when fault to avoid problems (Ronanki et al.,not 2017), TCU fault characteristic frequency, which only cause of without faultbased modelon orknown obviousfault faultmodels features. characteristic frequency, which only besides, cause wasting wasting of are are usually based onor known fault models or Besides, obvious when fault signals generally short not duration no fixed fault model obvious fault features. without fault model ornot obvious faultfor features. Besides, when equipment resourceshave and lacking lacking of enough enoughand concerning on without features, which are suitable TCU fault detection signals generally have short duration and no fixed equipment resources and of concerning on too much complicated signal processing needed, it would lead equipment resources and which lackingnot of only enough concerning on features, which are not suitable for TCU fault detection characteristic frequency, cause wasting of too much complicated signal processing needed, it would lead too muchfault complicated signal processing needed,Besides, it wouldwhen lead TCU fault, but butfrequency, also bring bring challenges challenges for thecause traditional signal model orburden obvious faultetfeatures. characteristic which not for only wasting of without TCU fault, also the traditional signal to high computation (Yang al., 2018) and cannot TCU fault, but also bring challenges for the traditional signal without fault model orburden obvious faultet Besides, when equipment resourcestoand lacking offault. enough concerning on to high computation (Yang etfeatures. al., 2018) and cannot to high computation burden (Yang al., 2018) and cannot processing methods detect TCU However, in recent too much complicated signal processing needed, it would lead equipment resourcesto lacking enough concerning on achieve the timely TCU fault detection. processing methods detect TCU However, in processing methods toand detect TCUoffault. fault. However, in recent recent too muchthe complicated signal processing needed, it would lead TCU fault, but also bring challenges for the traditional signal achieve timely TCU fault detection. achieve the timely TCU fault detection. years, as the development of fractional theory and application, to high computation burden (Yang et al., 2018) and cannot TCU fault, but also bring of challenges for the traditional signal to high computation burden (Yang et al., 2018) and cannot years, as development fractional theory and years, as the themethods development of fractional theory and application, application, processing to detect TCU However, recent Different from most of the the methods mentioned above, above, in this this with its unique unique characteristics, such fault. as strong strong global in memory achieve the timely TCU fault detection. from most of methods mentioned processing methods to detect TCU fault. However, in recent Different with its characteristics, such as global memory Different from most of the methods mentioned above, in in this with its unique characteristics, such as strong global memory achieveinspired the timely TCU fault detection. years, as the (Podlubny, development1999, of fractional theory and2010), application, paper, by (Pu, 2015), we propose a Fractional Steepest correlation Monje et al., non(Pu, we aa Fractional Steepest years, as the (Podlubny, development1999, of fractional theory and2010), application, correlation Monje et nonpaper, inspired inspired by (Pu,of2015), 2015), we propose propose Fractional Steepest correlation (Podlubny, 1999, Monje et al., al., 2010), non- paper, Different from by most the mentioned above, this with its unique such strong global memory Ascent Method (FSAM) formethods the purpose purpose of detecting detecting thein TCU localized (whichcharacteristics, can be be processed processed inas time domain, frequency Different from most of the methods mentioned above, inTCU this Ascent Method (FSAM) for the of the with its unique characteristics, such as strong global memory localized (which can in time domain, frequency Ascentinspired Methodby (FSAM) for the purpose of detecting the TCU localized (which can be processed in time domain, frequency paper, (Pu, 2015), we propose a Fractional Steepest correlation (Podlubny, 1999, Monje et al., 2010), nonfault quickly. FSAM is2015), to use usewe thepropose maximum quadraticSteepest energy domain, s domain domain and zz domain), domain), weak singularity singularity (Pu, 2016), paper, inspired by (Pu, a Fractional fault quickly. FSAM is to the maximum quadratic energy correlation (Podlubny, 1999, Monje et al., 2010), non(Pu, 2016), domain, s and weak fault quickly. is to use maximum quadratictheenergy domain, s (which domaincan andbe z domain), weak singularity (Pu, 2016), Ascent MethodFSAM (FSAM) thethe purpose TCU localized processed in time frequency of the velocity velocity residual for absolute value of asdetecting the optimization optimization fractional order has has been widely used used in domain, signal processing, Ascent Method (FSAM) for the purpose the TCU the residual absolute value the localized can been be processed in time frequency of fractional order widely in signal processing, of thequickly. velocity residual absolute value ofas asdetecting the optimization fractionals (which order has been widelyweak used in domain, signal processing, fault FSAM is to use the maximum quadratic energy domain, domain and z domain), singularity (Pu, 2016), target, and performs performs the G-L fractional order processing processing on the the especially in nonlinear, unsteady fault quickly. FSAMthe is to usefractional the maximum quadratic energy G-L order on domain, s domain andwith z domain), weaknon-Gaussian, singularity (Pu, 2016), target, especially in dealing dealing with nonlinear, non-Gaussian, unsteady target, and performs the G-L fractional order processing on the especially in dealing with nonlinear, non-Gaussian, unsteady of the and velocity residual absolute value astime the domain. optimization fractional order hasetbeen widely used in signal processing, velocity residual absolute value directly in The signals ( Sabatier al., 2007, Machado et al., 2011). of the velocity residual absolute value in astime the domain. optimization residual absolute value directly fractional order haset widely in signal velocity residual absolute value directly in time domain. The al., 2007, used Machado et al., 2011). velocity signals (Sabatier signals Sabatier etbeen al., nonlinear, 2007, Machado et processing, al.,unsteady 2011). target, and performs the G-L fractional order processing onThe the especially inindealing with Root Mean Square (RMS) (RMS) of the velocity velocity residual absolute absolute Meanwhile, fault detection, detection, many non-Gaussian, practical fault fault detection detection target, and performs the G-L of fractional order residual processing on the Root Mean Square the especially in dealing with nonlinear, non-Gaussian, unsteady Root Mean Square (RMS) of the velocity residual absolute Meanwhile, in fault many practical velocity residual absolute value directly in time domain. The Meanwhile, in faultetdetection, manyMachado practical et faultal.,detection signals ( Sabatier al., 2007, 2011). value and the maximum ratio of RMS are used as the detection methods based on the the principle of fractional fractional order have2011). been value velocity absolute value directly time The RMS are used as the and the maximum ratio signals Sabatier et principle al., 2007, Machado order et al., andresidual theSquare maximum ratio ofthe RMS are in used as domain. the detection detection methods on of have been Root Mean (RMS) ofof velocity residual absolute methods (based based on the principle of fractional order have been value threshold setting and the fractional order optimization, Meanwhile, in fault detection, many practical fault detection put forward for different objects. For example, Wang et al. Root Mean Square (RMS) of the velocity residual absolute threshold setting and the fractional order optimization, threshold setting and ratio the of fractional orderas optimization, Meanwhile, in detection, many practical fault detection put forward for objects. example, Wang et al. and the maximum RMS used the detection put forward forfault different objects. For example, Wang et al. value respectively. The effectiveness effectiveness of are FSAM for TCU fault methods based ondifferent the principle of For fractional order havebased been (2017) have proposed fault feature feature extraction method value and the maximum ratio of RMS used for as the detection The of FSAM TCU fault respectively. The and effectiveness of are FSAM for TCU fault methods based on the aaaprinciple of fractional order havebased been respectively. (2017) have proposed fault extraction method threshold setting the fractional order optimization, (2017) have proposed fault feature extraction method based detection is verified by the results of simulation. put forward for different objects. For example, Wang et al. on the fractional Hilbertobjects. transform defined inWang frequency threshold and theresults fractional order optimization, verified by of detection is issetting verified by the the results of simulation. simulation. put forward for different For defined example,in et al. detection on the fractional Hilbert transform frequency respectively. The effectiveness of FSAM for TCU on the fractional Hilbert transform defined in frequency (2017) have proposed a fault feature extraction method based respectively. The effectiveness of FSAM for TCU fault fault (2017) have proposed a fault feature extraction method based detection is verified by the results of simulation. on the fractional Hilbert transform defined in frequency detection is verified by the results of simulation. on the fractional Hilbert transform defined in frequency Copyright © 2018 IFAC 1336 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 1336Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 1336 Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2018.09.561 Copyright © 2018 IFAC 1336 Copyright © 2018 IFAC 1336
IFAC SAFEPROCESS 2018 Warsaw, Poland, August 29-31, 2018
Hongqiu Zhu et al. / IFAC PapersOnLine 51-24 (2018) 1336–1342
2. BASICS OF FSAM When any fault in the processor module, register module or interface module occurs, the velocity residual absolute value will be perturbed on magnitude. The duration of such a perturbation is short, which has no fixed characteristic frequency, and difficult to detect the fault signal through the traditional frequency domain method. However, such a short perturbation brings energy fluctuations to the signal, in other words, the disturbance of the fault will usually cause the energy of the signal to contain the faulty mutation information. Therefore, in order to be able to deal with the fault signal directly in time domain, FSAM is proposed to detect TCU fault by searching the fault signal pulses with the maximum quadratic energy. Refer to (Pu, 2015), we define the quadratic energy functional as 1 𝐸𝐸 = 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 − 𝜂𝜂(𝑠𝑠1∗ − 𝑠𝑠)2 , (𝜂𝜂 > 0)
(1)
Where 𝐸𝐸 denotes the quadratic energy of the signal 𝑠𝑠, 𝜂𝜂 is a positive number that controls the degree of quadratic energy 1 is the first-order extreme of functional bump convexity, 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 1∗ 𝐸𝐸, 𝑠𝑠 is the first-order optimal argument value corresponding 1 . to 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚
1 ) is the From Fig. 1, it can be seen that the point A(𝑠𝑠1∗ , 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣 𝑣𝑣∗ first-order extreme of 𝐸𝐸, the point B(𝑠𝑠 , 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 ) is the 𝑣𝑣-order extremum of 𝐸𝐸, 𝑠𝑠0 is the initial value of a randomly selected iteration for searching.
At present, the commonly used fractional order is defined as Grümwald.Letniko (G-L) fractional order, or RiemannLiouville (R-L) fractional order, or Caputo fractional order. The G-L fractional order derivative is approximated by its effective discrete-time model based on Laguerre filters (Stanisławski et al., 2016). So, in this paper, we use the definition of G-L fractional order, that is 𝐺𝐺𝐺𝐺 𝑣𝑣 𝑎𝑎𝐷𝐷𝑡𝑡 𝑠𝑠(𝑡𝑡)
(2)
s.t. 0 The iteration form of 𝑠𝑠 is defined as 𝑠𝑠𝑘𝑘+1|𝑘𝑘 = 𝑠𝑠𝑘𝑘|𝑘𝑘−1 + 𝜇𝜇 (−
𝑑𝑑 𝑣𝑣 𝐸𝐸
| ) 𝑑𝑑𝑠𝑠 𝑣𝑣 𝑠𝑠=𝑠𝑠𝑘𝑘
(3)
In (3), 𝑘𝑘 is the number of iterations, 𝑣𝑣 is a real number, 𝑠𝑠𝑘𝑘|𝑘𝑘−1 is the iteration adjustment value of 𝑠𝑠 at the moment 𝑘𝑘, 𝑠𝑠𝑘𝑘+1|𝑘𝑘 is the iteration adjustment value of 𝑠𝑠 at the moment 𝑘𝑘 + 1, 𝑑𝑑 𝑣𝑣 𝐸𝐸
| is the 𝑣𝑣-order fractional gradient value of 𝐸𝐸 at 𝑠𝑠 = 𝑑𝑑𝑠𝑠 𝑣𝑣 𝑠𝑠=𝑠𝑠𝑘𝑘 𝑠𝑠𝑘𝑘 , 𝜇𝜇 is the coefficient that controls the stability and convergence rate of FSAM. From (1) and (3), it can be seen that FSAM is a parabola 𝐸𝐸 along an opening and incremental search in the positive direction of the 𝑣𝑣 -order fractional gradient. The iterative search process of FSAM is shown in Fig. 1.
∙
(4)
𝑘𝑘!(𝑣𝑣−𝑘𝑘)!
𝑑𝑑 𝑣𝑣 𝐸𝐸
𝑣𝑣 |𝑠𝑠=𝑠𝑠𝑘𝑘 =
2
1 [𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 −𝜂𝜂(𝑠𝑠 1∗ ) ]𝑠𝑠𝑘𝑘−𝑣𝑣
𝛤𝛤(1−𝑣𝑣)
2𝜂𝜂𝑠𝑠 1∗ 𝑠𝑠𝑘𝑘1−𝑣𝑣
+
𝛤𝛤(2−𝑣𝑣)
−
2𝜂𝜂𝑠𝑠𝑘𝑘2−𝑣𝑣
𝛤𝛤(3−𝑣𝑣)
(5)
In (5), 𝑣𝑣 > 0, which means that the quadratic energy of the signal 𝑠𝑠 is fractional differential; at the same time, 𝑠𝑠𝑘𝑘−𝑣𝑣 , 𝑠𝑠𝑘𝑘1−𝑣𝑣 , and 𝑠𝑠𝑘𝑘2−𝑣𝑣 are the power function of the signal 𝑠𝑠 .In fact, according to the fractional order properties (Mohammad, 2016), we let 0 < 𝑣𝑣 < 1 .
When the iterative search process of the FSAM from 𝑠𝑠0 to the stable point B, there are 𝐷𝐷𝑠𝑠𝑣𝑣𝑘𝑘 = 0, 𝑠𝑠𝑘𝑘+1 = 𝑠𝑠𝑘𝑘 = 𝑠𝑠 𝑣𝑣∗ , and 𝐸𝐸 = 𝑣𝑣 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 , then i)
𝑣𝑣 1 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 − 𝜂𝜂(𝑠𝑠1∗ )2 + 2𝜂𝜂𝑠𝑠1∗ 𝑠𝑠 𝑣𝑣∗ − 𝜂𝜂(𝑠𝑠 𝑣𝑣∗ )2
(6)
ii) 2
1 [𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 −𝜂𝜂(𝑠𝑠 1∗ ) ]
(𝑠𝑠 𝑣𝑣∗ )−𝑣𝑣 {
Γ(1−𝑣𝑣)
+
2𝜂𝜂𝑠𝑠 1∗
∙ 𝑠𝑠 𝑣𝑣∗ −
Γ(2−𝑣𝑣)
Since (𝑠𝑠 𝑣𝑣∗ )−𝑣𝑣 ≠ 0, we have 2
1 [𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 −𝜂𝜂(𝑠𝑠 1∗ ) ]
+
2𝜂𝜂𝑠𝑠 1∗
𝛤𝛤(2−𝑣𝑣)
∙ 𝑠𝑠 𝑣𝑣∗ −
2𝜂𝜂
∙ (𝑠𝑠 𝑣𝑣∗ )2 } = 0
2𝜂𝜂
∙ (𝑠𝑠 𝑣𝑣∗ )2 = 0 (7)
Γ(3−𝑣𝑣)
𝛤𝛤(3−𝑣𝑣)
when 𝑣𝑣 ≠ 1, 2, 3, with 𝑠𝑠 𝑣𝑣∗ as an independent variable, the root of equation (7) can be solved as
A
𝑣𝑣 𝐸𝐸𝑚𝑚𝑎𝑎𝑥𝑥
− 𝑘𝑘ℎ) , 𝑘𝑘 ∈ 𝑍𝑍
From (1) and (4), the fractional 𝑣𝑣-order of 𝐸𝐸 for 𝑠𝑠(𝑡𝑡) at 𝑡𝑡 = 𝑘𝑘 can be obtained
𝛤𝛤(1−𝑣𝑣)
E 1 𝐸𝐸𝑚𝑚𝑎𝑎𝑥𝑥
1
ℎ→0 ℎ𝑣𝑣 (−1)𝑘𝑘 𝛤𝛤(𝑣𝑣+1) ∑∞ 𝑘𝑘=0 𝛤𝛤(𝑘𝑘+1)𝛤𝛤(𝑣𝑣−𝑘𝑘+1) 𝑠𝑠(𝑡𝑡
𝛤𝛤(𝑘𝑘+1)𝛤𝛤(𝑣𝑣−𝑘𝑘+1)
𝑑𝑑𝑠𝑠
s
= 𝑙𝑙𝑙𝑙𝑙𝑙
In (4), 𝐺𝐺𝐺𝐺𝑎𝑎𝐷𝐷𝑡𝑡𝑣𝑣 𝑠𝑠(𝑡𝑡) represents the operator of the fractional 𝑣𝑣order of the signal 𝑠𝑠(𝑡𝑡) at the interval [𝑎𝑎, 𝑡𝑡], ℎ represents the calculation step size, 𝛤𝛤 represents the Euler’s Gamma function, 𝛤𝛤(𝑣𝑣+1) 𝑣𝑣! = . and there is
The objective function of FSAM is 1 J FSAM max Emax (s1* s)2
1337
B
𝑣𝑣∗ = 𝑠𝑠1,2
Γ(3−𝑣𝑣) 2𝜂𝜂
×{
𝜂𝜂𝑠𝑠 1∗
Γ(2−𝑣𝑣)
𝜂𝜂 2 (𝑠𝑠 1∗ )2
±√
Γ2 (2−𝑣𝑣)
+
1 2𝜂𝜂[𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 −𝜂𝜂(𝑠𝑠 1∗ )2 ]
Γ(1−𝑣𝑣)Γ(3−𝑣𝑣)
}
(8)
To ensure the convergence of the algorithm, the two roots in (8) should be equal, then we can obtain
0
𝑠𝑠
1∗
𝑠𝑠 𝑣𝑣∗ ⋯ 𝑠𝑠0
Γ(1−𝑣𝑣)Γ(3−𝑣𝑣) Γ2 (2−𝑣𝑣)
s
=−
𝑠𝑠 𝑣𝑣∗ =
Fig. 1. An iterative search process of FSAM
1337
1 2𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚
𝜂𝜂(𝑠𝑠 1∗ )2
Γ(3−𝑣𝑣)𝑠𝑠 1∗ 2Γ(2−𝑣𝑣)
+2
(9) (10)
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In (9), it shows that if the 𝑠𝑠1∗ and 𝜂𝜂 have been given, the 1 of the signal s. optimization of 𝑣𝑣 is actually to obtain the 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 By substituting (10) into (1), the following equation is given 𝑣𝑣 𝐸𝐸 = 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 + 𝜂𝜂 [
−4𝛤𝛤(2−𝑣𝑣)+𝛤𝛤(3−𝑣𝑣) 𝛤𝛤(3−𝑣𝑣)
(𝑠𝑠 𝑣𝑣∗ )2 +
4𝛤𝛤(2−𝑣𝑣) 𝑣𝑣∗ 𝑠𝑠 𝑠𝑠 𝛤𝛤(3−𝑣𝑣)
− 𝑠𝑠 2 ]
2
1 𝜇𝜇[𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 −𝜂𝜂(𝑠𝑠 1∗ ) ]𝑠𝑠𝑘𝑘−𝜈𝜈
−
𝛤𝛤(1−𝜈𝜈)
2𝜇𝜇𝜇𝜇𝑠𝑠 1∗ 𝑠𝑠𝑘𝑘1−𝜈𝜈 𝛤𝛤(2−𝜈𝜈)
+
𝑒𝑒(𝑡𝑡) = |𝑉𝑉𝐹𝐹 (𝑡𝑡) − 𝑉𝑉𝑁𝑁 (𝑡𝑡)|
(11)
By substituting (5) into (3), we can obtain 𝑠𝑠𝑘𝑘+1|k = 𝑠𝑠𝑘𝑘|𝑘𝑘−1 −
velocity measurements 𝑉𝑉𝑁𝑁 (𝑡𝑡) and 𝑉𝑉𝐹𝐹 (𝑡𝑡) at the 𝑡𝑡 moment under the normal state and the TCU faulty state by the velocity sensors, respectively, then the velocity residual absolute value 𝑒𝑒(𝑡𝑡) and its discretization velocity residual absolute value 𝑒𝑒(𝑘𝑘) are respectively as
2𝜇𝜇𝜇𝜇𝑠𝑠𝑘𝑘2−𝜈𝜈 𝛤𝛤(3−𝜈𝜈)
(12)
𝑒𝑒(𝑘𝑘) = |𝑉𝑉𝐹𝐹 (𝑘𝑘) − 𝑉𝑉𝑁𝑁 (𝑘𝑘)|
2𝜇𝜇𝜇𝜇
[−𝑠𝑠𝑘𝑘2 +
𝛤𝛤(3−𝜈𝜈) 𝛤𝛤(3−𝜈𝜈) 1∗ 𝑣𝑣∗ 𝑠𝑠 𝑠𝑠 ] 𝑠𝑠𝑘𝑘−𝑣𝑣 𝛤𝛤(2−𝜈𝜈)
𝛤𝛤(3−𝜈𝜈) 1∗ 𝑠𝑠 𝑠𝑠𝑘𝑘 𝛤𝛤(2−𝜈𝜈)
𝑁𝑁
+ (𝑠𝑠 𝑣𝑣∗ )2 − (13)
2𝜇𝜇𝜇𝜇
𝛤𝛤(3−𝜈𝜈)
(14)
And 𝑠𝑠 𝑣𝑣 about (𝑠𝑠 − 𝑠𝑠 𝑣𝑣∗ ) can be expressed as 𝑠𝑠 𝑣𝑣 = ∑∞ 𝑛𝑛=0
𝛤𝛤(1+𝑣𝑣)(𝑠𝑠 𝑣𝑣∗ )𝑣𝑣−𝑛𝑛 (𝑠𝑠−𝑠𝑠 𝑣𝑣∗ )𝑛𝑛 𝛤𝛤(𝑛𝑛+1)𝛤𝛤(1−𝑛𝑛+𝑣𝑣)
, 𝑣𝑣 ≤ 𝑛𝑛
When 𝑛𝑛 = 0 (it doesn’t satisfy the condition of 𝑣𝑣 ≤ 𝑛𝑛 ) and 𝑛𝑛 = 1, from (15), it can be obtained the approximation of 𝑠𝑠 𝑣𝑣 as 𝑠𝑠 𝑣𝑣 ≈ 𝑣𝑣(𝑠𝑠 𝑣𝑣∗ )𝑣𝑣−1 (𝑠𝑠 − 𝑠𝑠 𝑣𝑣∗ )
(16)
(17)
Combine (14), (16) and (17), we have 𝐷𝐷𝑡𝑡1 𝑠𝑠(𝑡𝑡|𝑡𝑡
2𝜇𝜇𝜇𝜇
𝛤𝛤(3−𝜈𝜈)𝑣𝑣(𝑠𝑠 𝑣𝑣∗ )𝑣𝑣−1
‖𝑒𝑒 ∗ ‖𝑅𝑅𝑅𝑅𝑅𝑅
[𝑠𝑠(𝑡𝑡) − 𝑠𝑠 𝑣𝑣∗ ]
2𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇
𝛤𝛤(3−𝜈𝜈)𝑣𝑣(𝑠𝑠 𝑣𝑣∗ )𝑣𝑣−1
In order to let the fault signal and the noise can be better distinguished, the maximum of 𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 is used as the fractional order optimal objective function, that is 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = max(𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 )
Let
2𝜇𝜇𝜇𝜇
𝛤𝛤(3−𝜈𝜈)𝑣𝑣(𝑠𝑠 𝑣𝑣∗ )𝑣𝑣−1
𝑙𝑙𝑙𝑙𝑙𝑙
2𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇
{
𝑒𝑒(𝑘𝑘 + 1|𝑘𝑘) ≥ ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 ⟹ 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑒𝑒(𝑘𝑘 + 1|𝑘𝑘) < ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 ⟹ 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑙𝑙
(28)
3.2 Algorithm overview
Start
(18)
e(𝑡𝑡)
)
e(𝑘𝑘)
FSAM
(19)
𝑒𝑒(𝑘𝑘 + 1|𝑘𝑘)
= −∞
‖𝑒𝑒‖𝑅𝑅𝑀𝑀𝑆𝑆
(20)
= 𝜒𝜒, then from (20), 𝜒𝜒 should satisfy 𝜒𝜒 < 0
(27)
Finally, the TCU fault detection logic can be written as
In order to ensure the stability of the iteration, the second term of (19) should be satisfied that 𝑘𝑘→∞ 𝛤𝛤(3−𝜈𝜈)𝑣𝑣(𝑠𝑠 𝑣𝑣∗ )𝑣𝑣−1
(25)
(26)
‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅
(18) is a first-order ordinary differential equation with variables that are separable. By solving (18), the solution 𝑠𝑠(𝑡𝑡) can be obtained, then its discretization 𝑠𝑠𝑘𝑘 with a sampling period 𝜏𝜏 can be given as 𝑠𝑠𝑘𝑘 = 𝑠𝑠 𝑣𝑣∗ + 𝑒𝑒𝑒𝑒𝑒𝑒 (
1 2
+ 1|𝑘𝑘)) e(𝑘𝑘 + 1|𝑘𝑘))) 𝑇𝑇
A flow chart of the TCU fault detection method based on FSAM is shown in Fig. 2.
− ℎ) ≈ 𝑠𝑠𝑘𝑘+1|k − 𝑠𝑠𝑘𝑘|𝑘𝑘−1 ≈
( (∑𝑁𝑁−1 𝑘𝑘=0 (𝑒𝑒(𝑘𝑘 𝑁𝑁
Let 𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 be the ratio of ‖𝑒𝑒 ∗ ‖𝑅𝑅𝑅𝑅𝑅𝑅 to ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 , that is
The first order difference is used as the approximation of the first order differential𝐷𝐷𝑡𝑡1 𝑠𝑠(𝑡𝑡|𝑡𝑡 − ℎ), which is 𝐷𝐷𝑡𝑡1 𝑠𝑠(𝑡𝑡|𝑡𝑡 − ℎ) ≈ 𝑠𝑠𝑘𝑘+1|𝑘𝑘 − 𝑠𝑠𝑘𝑘|𝑘𝑘−1
=
𝑅𝑅𝑅𝑅𝑅𝑅
1
𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 =
(15)
(24)
RMS of FSAM output 𝑒𝑒(𝑘𝑘 + 1|𝑘𝑘) is calculated as ‖𝑒𝑒 ∗ ‖
(𝑠𝑠𝑘𝑘 − 𝑠𝑠 𝑣𝑣∗ )2 𝑠𝑠𝑘𝑘−𝑣𝑣
1 2
1
𝑇𝑇 ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 = ( (∑𝑁𝑁−1 𝑘𝑘=0 𝑒𝑒 (𝑘𝑘) 𝑒𝑒(𝑘𝑘)))
Furthermore, (13) can be rewritten as 𝑠𝑠𝑘𝑘+1|𝑘𝑘 = 𝑠𝑠𝑘𝑘|𝑘𝑘−1 +
(23)
Then, 𝑒𝑒(𝑘𝑘) is used as the input of FSAM, that is 𝑠𝑠(𝑘𝑘) = 𝑒𝑒(𝑘𝑘), the output of FSAM is 𝑒𝑒(𝑘𝑘 + 1|𝑘𝑘) . RMS of the velocity residual absolute value 𝑒𝑒(𝑘𝑘) is calculated as:
By substituting (7) and (10) into (12), we have 𝑠𝑠𝑘𝑘+1|𝑘𝑘 = 𝑠𝑠𝑘𝑘|𝑘𝑘−1 −
(22)
Normal
N
𝑒𝑒(𝑘𝑘 + 1|𝑘𝑘) ≥ ‖𝑒𝑒‖𝑅𝑅𝑀𝑀𝑆𝑆 Y
(21)
Fault
3. FSAM-BASED TCU FAULT DTECTION METHOD End
3.1 TCU fault detection method Firstly, by setting the fault free condition and TCU fault injection on the simulation platform, then we can collect the
Fig. 2. A flow chart of the TCU fault detection method based on FSAM
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The steps of the proposed TCU fault detection method and the parameters setting guide are as follows： 1. Obtain the measurements and carry on the velocity residual value to take the absolute value processing and the discretization processing; 2. The RMS of the velocity residual absolute value and the ratios of RMS are calculated according to (24), (25), and (26), respectively； 3. Randomly select the initial value of 𝑠𝑠0 , and set the 𝑠𝑠1∗ to a very small positive number for the TCU weak fault signal (recommend 0 < 𝑠𝑠1∗ ≤ 0.001), then by (11) can obtain the 𝑠𝑠 𝑣𝑣∗ , as for the 𝜒𝜒 , we recommend −2 < 𝜒𝜒 < −1 to ensure the underdamped oscillation, once the 𝜂𝜂 and 𝜒𝜒 have been set, we 2𝜇𝜇𝜇𝜇 can use 𝑣𝑣∗ )𝑣𝑣−1 = 𝜒𝜒 to obtain the 𝜇𝜇, then use (14) for 𝛤𝛤(3−𝜈𝜈)𝑣𝑣(𝑠𝑠
iterative processing;
4. Set the range of fractional order 𝑣𝑣 as (0,1) with the step 0.01,the optimal order is where the maximum of 𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 can be obtained, that is 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = max(𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 );
The experimental data used in this paper is collected from the traction drive control system fault injection benchmark (TDCS-FIB) of the CRH2 traction drive control system, the version number is 1.3. For downloading the benchmark and detailed information, please visit http://gfist.csu.edu.cn/. The simulation time in this paper is set to be 4s, TCU fault occurs after 1.2s, the data sampling frequency is set to be 100 kHz. As shown in Fig. 3(a), the velocity of high-speed train in normal operation is a non-linear, non-Gaussian, unsteady wave signal due to the interference of system noise and environmental noise (Chen et al., 2017), especially the weak fault signal of the interface module fault in TCU is usually submerged by noise in time domain (see Fig. 3(b)). The initial value of FSAM is set to: 𝑠𝑠0 = 0, 𝑠𝑠1∗ = 0.001, 𝜂𝜂 = 1.25, 𝜒𝜒 = −1.5, the 0 ~ 4s of ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 is used as the threshold for judging whether the TCU fault occurs (indicated by a red dotted line). First, we use FSAM to detect the processor module fault and register module fault in TCU, the results are shown in Fig. 4 and Fig.5 respectively. It can be seen from Fig. 4 that both failures occurred at about 2.5s, and the maximum of magnitude have been magnified more than 7 times at least.
250
250
200
200
150 100 50 0 0
0.5
1
1.5
196.102 196.1 196.098 196.096 196.094 196.092 2.45
2 t(s)
2.5
3
3.5
150 100 50 0 0
4
0.5
1
1.5
2 t(s)
2.5
3
3.5
4
196.11 VF（km/h）
VN（km/h）
4. SIMULATION
VF（km/h）
VN（km/h）
5. Use the ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 as the threshold to determine whether the TCU fault occurs.
1339
2.5 t(s)
196.1 196.09 196.08
2.55
2.45
2.5 t(s)
(a)
2.55
(b)
Fig. 3. (a) was the speed curve of high-speed train under normal operation, (b) was the speed curve of high-speed train under the interface module fault in TCU. 0.06
0.4 e(k)
e(k+1|k) ||eRMS||
0.35
0.05
0.3 0.25 e(k+1|k)
e(k)
0.04 0.03
0.2 0.15
0.02
0.1 0.01
0.05 0 2.4
2.6
2.8
3
3.2 t(s)
3.4
3.6
3.8
0 2.4
4
(a)
2.6
2.8
3
3.2 t(s)
3.4
3.6
3.8
4
(b)
Fig. 4. The effect of FSAM to detect the processor module fault in TCU, the fault occurred at 2.503s, the order was 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = 0.30, the threshold was ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 = 0.0144 , (a) was the input signal, (b) was the output signal. 1339
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However, the interface module fault in TCU is much more difficult to detect than the processor module fault and register module fault in TCU. Then, in order to quantitatively describe the severity of the interface module fault in TCU and verify the robustness of FRAM, the severity of the interface module fault in TCU is set to Level 1, Level 2 and Level 3, respectively,
depending on the parameters named A, P and Q on the TDCSFIB V1.3. The specific set values of A, P and Q for Level 1, Level 2 and Level 3 are shown in Table 1.The results of the comparison of the processing are shown in Fig. 6, Fig. 7, and Fig. 8, respectively, and 𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 with different severity of the interface module fault in TCU is shown in Table 2. 10
5
10
10
e(k+1|k) ||eRMS||
e(k)
5
10
0
lg(e(k+1|k))
lg(e(k))
10
-5
10
0
10
-5
10
-10
10
0
0.5
1
1.5
2 t(s)
2.5
3
3.5
0
4
0.5
1
1.5
(a)
2 t(s)
2.5
3
3.5
4
(b)
Fig. 5. The effect of FSAM to detect the register module fault in TCU, the fault occurred at 2.502s, the order was 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = 0.09, the threshold was ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 = 77.1 , (a) was the input signal, (b) was the output signal. 0.02
0.06 e(k)
0.018
e(k+1|k) ||eRMS||
0.05
0.016 0.014
0.04 e(k+1|k)
e(k)
0.012 0.01 0.008
0.03 0.02
0.006 0.004
0.01
0.002 0 2.44
2.46
2.48
2.5
2.52
0 2.44
2.54
t(s)
2.46
2.48
2.5
2.52
2.54
t(s)
(a)
(b)
Fig. 6. The effect of FSAM to detect the interface module fault in TCU, the fault occurred at 2.502s, the severity level was Level 1, the order was 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = 0.31, the threshold was ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 = 0.0047 , (a) was the input signal, (b) was the output signal. 0.025
0.08
e(k)
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0.07
0.02 0.06 0.05
e(k)
e(k+1|k)
0.015
0.01
0.04 0.03 0.02
0.005
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0 2.98
3
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3.06
0 2.98
3.08
t(s)
3
3.02
3.04
3.06
3.08
t(s)
(a)
(b)
Fig. 7. The effect of FSAM to detect the interface module fault in TCU, the fault occurred at 3.001s, the severity level was Level 2, the order was 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = 0.31, the threshold was ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 = 0.0045 , (a) was the input signal, (b) was the output signal. 1340
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Hongqiu Zhu et al. / IFAC PapersOnLine 51-24 (2018) 1336–1342
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0.08
0.025
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e(k)
0.07 0.02
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0 2.44
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Fig. 8. The effect of FSAM to detect the interface module fault in TCU, the fault occurred at 2.501s, the severity level was Level 3, the order was 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = 0.31, the threshold was ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 = 0.0048 , (a) was the input signal, (b) was the output signal. Table 1. The severity with the specific set values of A, P and Q for the interface module fault in TCU Fault severity rating Level 1
A
P
Q
25
-100
-200
Level 2
25
-200
-500
Level 3
25
-500
-1000
Table 2. 𝑱𝑱𝑹𝑹𝑹𝑹𝑹𝑹 with different severity for the interface module fault in TCU Severity rating Level 1 Level 2 Level 3
Status Before failure 0.5218 0.5394 0.5302
Failure 2.3427 2.6081 2.7328
It can be seen from Fig. 6 that when the TCU fault occurs at 2.502s, and the fault severity level is Level 1, the velocity of the residual is very small, the fault signal and the noise barely can be resolved; but after FSAM, it can be seen that RMS of the TCU fault signal is enhanced about 2.34 times, while the non-linear characteristic of the fault signal can be well preserved, and the noise is also well suppressed. Moreover, it can be seen from Fig. 7 and Fig. 8 that when the TCU fault occurs at 3.001s and 2.501s, and the fault severity is Level 2 and Level 3, respectively, the residual of velocity absolute value is smaller, and the fault signal is also more gradually buried in the noise, it is more difficult to be distinguished; but after processing by FSAM, it can be seen from Table 2 that 𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 are about 2.61 times and 2.73 times enhancement, respectively, but also can retain the output response of the nonlinear features well, and the noise has also been a very good inhibition. 5. CONCLUSION Aiming at the problem that the TCU fault is difficult to detect when in the case of non-linear, non-Gaussian, unsteady, this paper proposes a method based on FSAM to search the maximum quadratic energy of the velocity residual absolute
After failure 0.5212 0.3290 0.4457
value, the results show that it cannot only separate the fault signal from the strong noise, but also suppress the noise interference at the non-fault time, which can realize the purpose of detecting the TCU fault. The simulation results show that FSAM is practical and effective, and it has some guiding significance for the practical engineering application. What we should note is that FSAM is not suitable for the fault detection that the features of the fault signals are very similar to the noise, about this issue, we will develop FSAM in our future work. ACKNOWLEDGEMENTS This work was supported by the Major Program of the National Natural Science Foundation of China (Grant No. 61490702), the National Natural Science Foundation of China (Grant No. 61773407) and the Fundamental Research Funds for the Central Universities of Central South University (Grant No. 2018zzts603). REFERENCES Chen, S., Zhan, T., Huang, C., Chen, J. and Lin, C. (2015). Nontechnical Loss and Outage Detection Using FractionalOrder Self-Synchronization Error-Based Fuzzy Petri Nets in
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IFAC PapersOnLine 51-24 (2018) 1336–1342 Fractional Fractional Steepest Steepest Ascent Ascent Method Method for for TCU TCU Fault Fault Detection Detection Fractional Steepest for Detection Hongqiu Zhu, Zhiliang Wu,Ascent Chunhua Method Yang, Tao Peng, ZhiwenFault Chen, Xiaoyue Yang Fractional Steepest Ascent Method for TCU TCU Fault Detection
Hongqiu Zhu, Zhu, Zhiliang Zhiliang Wu, Wu, Chunhua Chunhua Yang, Yang, Tao Tao Peng, Peng, Zhiwen Zhiwen Chen, Chen, Xiaoyue Xiaoyue Yang Yang Hongqiu Hongqiu Zhu, Zhiliang Wu, Chunhua Yang, Tao Peng, Zhiwen Chen, Xiaoyue Yang Hongqiu Zhu, Zhiliang Wu, Chunhua Yang, Tao Peng, Zhiwen Chen, Xiaoyue Yang School of Information Science and Engineering ， Central South University， School School of of Information Information Science Science and and Engineering Engineering ， ，Central Central South South University University， ， School of Information Science and Engineering ， Central South University ， Changsha ， China (Tel: 086-0731-88836876; e-mail: [email protected]). School of410083 Information Science and Engineering ，Central South University， Changsha 410083 ， (Tel: e-mail: [email protected]). Changsha 410083 ，China China (Tel: 086-0731-88836876; 086-0731-88836876; e-mail: [email protected]). Changsha 410083，China (Tel: 086-0731-88836876; e-mail: [email protected]). Changsha 410083a， China (Tel: 086-0731-88836876; e-mail: [email protected]). Abstract: This paper proposes Fractional Steepest Ascent Method (FSAM) for detecting detecting faults in in Traction Abstract: This Abstract: This paper paper proposes proposes aa Fractional Fractional Steepest Steepest Ascent Ascent Method Method (FSAM) (FSAM) for for detecting faults faults in Traction Traction Control Unit (TCU), which aims at detecting the transient fault caused by electromagnetic radiation, Control Unit (TCU), which aims at detecting the transient fault caused by electromagnetic radiation, Control Unit (TCU), which aims at detecting the transient fault caused by electromagnetic radiation, Abstract: Thisviolent paper vibration, proposes aetc. Fractional Steepest Ascent Method forquadratic detectingenergy faults infunctional Traction overheating, In FSAM, FSAM, the maximum maximum value of of(FSAM) the signal Abstract: Thisviolent paper vibration, proposes aetc. Fractional Steepest Ascent Method forquadratic detectingenergy faults infunctional Traction overheating, In the value the overheating, vibration, etc. In thethe maximum value of(FSAM) the signal signal energy functional Control Unitinviolent (TCU), whichby aims at FSAM, detecting transient fault caused by quadratic electromagnetic radiation, is obtained time domain means of Grü mwald Letniko (G-L) fractional order technique. For fault Control Unit (TCU), which aims at detecting the transient fault caused by electromagnetic radiation, is obtained in time domain by means of Grü mwald Letniko (G-L) fractional order technique. For is obtained inviolent time domain byetc. means of Grü mwald Letniko (G-L) fractional order technique. For fault fault overheating, vibration, In FSAM, the maximum value of the signal quadratic energy functional detection purpose, the detection threshold is determined with the Root Mean Square (RMS) of the velocity overheating, violentthe vibration, etc. In FSAM, the maximum of theMean signal quadratic energy functional detection purpose, the detection threshold is determined withvalue the Root Mean Square (RMS) of the the velocity detection purpose, detection threshold is determined with the Root Square (RMS) of velocity is obtained in time domain by means of Grü mwald Letniko (G-L) fractional order technique. For fault residual absolute value and the fractional order is optimized optimized by(G-L) searching the maximum maximum ratio of of RMS. The is obtained in time domain by fractional means of order Grümwald Letniko fractional order technique. For fault residual absolute value and the the fractional order is by searching the ratio RMS. The residual absolute value and is optimized by searching the maximum ratio ofthe RMS. The detection purpose, the detection threshold is determined with the Root Mean Square (RMS) of velocity simulation results verify that the method proposed is not only sensitive to the TCU fault signals, but also detection purpose, the detection threshold is determined with the Root Mean Square (RMS) of the velocity simulation results verify that the method proposed is not only sensitive to the TCU fault signals, but also simulation results verify that the method proposed is not only sensitive to the TCU fault signals, but also residual absolute value and the fractional order is optimized by searching the maximum ratio of RMS. The can effectively suppress the noise to extent. residual absolute value and the fractional is optimized by searching the maximum ratio of RMS. The can effectively suppress the noise to some someorder extent. can effectively suppress to some extent. simulation results verify the thatnoise the method proposed is not only sensitive to the TCU fault signals, but also simulation results verify that the method proposed is not only sensitive to the TCU fault signals, but also © 2018, IFAC (International of Automatic can effectively suppress the Federation noise to some extent. Control) Hosting by Elsevier Ltd. All rights reserved. can effectively suppress the noise to some extent. Keywords: TCU, fault detection, detection, quadratic energy energy functional, FSAM. FSAM. Keywords: TCU, fault Keywords: TCU, fault detection, quadratic quadratic energy functional, functional, FSAM. Keywords: TCU, fault detection, quadratic energy functional, FSAM. Keywords: TCU, fault detection, quadratic energy functional, domainFSAM. for rolling rolling element bearing bearing weak fault. fault. Chen et et al. domain 1. INTRODUCTION domain for for rolling element element bearing weak weak fault. Chen Chen et al. al. 1. INTRODUCTION (2014) have used the fractional-order self-synchronization 1. INTRODUCTION (2014) have used the fractional-order self-synchronization (2014) have used the fractional-order self-synchronization for Fuzzy rollingPetri element bearing fault. Chen al. error-based nets to to detect weak nontechnical losseset and Traction Control Control Unit1.(TCU) (TCU) as one one of the the important electronic electronic domain INTRODUCTION domain for Fuzzy rollingPetri element bearing fault. Chen al. detect nontechnical losses Traction as error-based nets to detect weak nontechnical lossesetand and Traction Control Unit Unit1.(TCU) as one of of the important important electronic error-based (2014) haveFuzzy used Petri the nets fractional-order self-synchronization INTRODUCTION outage fault events. Zhou et al. (2016) have introduced a novel devices in traction drive control system (TDCS) of the high(2014) have used the fractional-order self-synchronization outage fault events. Zhou et al. (2016) have introduced a novel devices in traction drive control system of fault Fuzzy events.Petri Zhounets et al. have introduced a novel devices in traction drive control system (TDCS) of the the highhigh- outage error-based to (2016) detect nontechnical and Traction Control Unit (TCU) asisone the(TDCS) important adaptive variable time fractional-order anisotropiclosses diffusion speed train, its main main function to of control traction electronic motor for for adaptive error-based Fuzzy time Petri fractional-order nets to detect nontechnical losses and variable anisotropic diffusion Traction Control Unitfunction (TCU) asis one of the important electronic speed train, its to control traction motor adaptive variable time fractional-order anisotropic diffusion speed train, its main function is to control traction motor for outage fault events. Zhou et al. (2016) have introduced novel devices in the traction drive controlofsystem (TDCS) ofhigh-speed the high- filtering for seismic data noise attenuation. Long et al. a(2017) achieving normal operation the TDCS. The outage fault events. Zhou et al. (2016) have introduced a novel filtering for seismic data noise attenuation. Long et al. (2017) devices in traction drive control system (TDCS) of the highachieving the normal operation of the TDCS. The high-speed filtering for seismictime datafractional-order noise attenuation. Long et al. (2017) achieving theitsnormal operationisoftothe TDCS. The high-speed variable speed main function control traction for adaptive have presented time frequency filtering anisotropic algorithm for fordiffusion bearing train istrain, usually running at above above 200km/h under motor the harsh harsh adaptive variable timefrequency fractional-order diffusion aaa time filtering algorithm bearing speed its main function is to 200km/h control traction motor for have train usually running at under the have presented presented time frequency filtering anisotropic algorithm for bearing train is istrain, usually running at above 200km/h under the harsh filtering for seismic data noise attenuation. Long et al. (2017) achieving the normal operation of the TDCS. The high-speed fault detection detection baseddata on the the fractional lower Long order et S transform. transform. complex and changeable environment, forThe example, the fault for seismic noise attenuation. al. (2017) based on fractional lower order achieving the normal operation of the TDCS. high-speed complex and changeable environment, for example, the fault detection based on the fractional lower order S Sfor transform. complex and changeable environment, forunder example, the filtering have presented a time frequency filtering algorithm bearing train is usually running at above 200km/h the harsh Pineda-Sanchez et al. al.frequency (2010) filtering have utilized utilized the for fractional electromagnetic radiationatand and input power variation, have presented a time algorithm bearing et (2010) have the fractional train is usually running above under theviolent harsh Pineda-Sanchez electromagnetic radiation input200km/h powerfor variation, violent Pineda-Sanchez et al. (2010) have lower utilized theS transform. fractional electromagnetic radiation and input power variation, violent fault detection based onits the fractional order complex and changeable environment, example, the Fourier tranform and optimization method to detect the vibration, bad weather, and so on, which probably cause the fault detection based onits theoptimization fractional lower orderto S transform. Fourier tranform and method detect the complex and changeable environment, for example, the vibration, bad weather, and so on, which probably cause Fourier tranform and its optimization method to detect the vibration, bad weather, and so on, which probably cause the Pineda-Sanchez et al. for (2010) have utilized the fractional electromagnetic radiation and inputorpower variation, violent harmonic components characterizing each type of fault. processor module, register module interface module faults Pineda-Sanchez et al. (2010) have utilized the fractional harmonic components for characterizing each type of fault. electromagnetic radiation and inputor power variation, violent processor module, register module or interface module faults harmonic components for characterizing each type of fault. processor module, register module interface module faults andhave its optimization method to detect the vibration, bad weather, and so on, which cause the Fourier Pisano ettranform al. (2014) proposed aa discontinuous discontinuous dynamic in et 2017). These faults probably are non-permanent non-permanent Fourier andhave its optimization method to detect the (2014) proposed dynamic et al. vibration, bad weather, and so on, which probably cause the Pisano in TCU TCU (Yang (Yang et al., al., 2017). These faults are Pisano ettranform al. (2014) have proposed a discontinuous dynamic in TCU (Yang et al., 2017). These faults are non-permanent harmonic components for characterizing each type order of fault. processor module, register module or interface module faults system fault detection method based on fractional for and submerged by noise in time domain, which may induce the harmonic components for characterizing each type of fault. system fault detection method based on fractional order for processor module, register module or interface module faults and submerged by noise in time domain, which may induce the system et fault method baseda discontinuous on fractional order for and submerged noise in time domain, which may induce the Pisano al. detection (2014) have proposed dynamic in TCU (Yang by ettimely al., 2017). These faults are non-permanent fault simulation of hydraulic plants. Kuo et al. (2017) have failure without detection (Maniatakos et al., 2011, Pisano et al. (2014) have proposed a Kuo discontinuous dynamic fault of plants. et have in TCUwithout (Yang ettimely al., 2017). These(Maniatakos faults are non-permanent et al., failure detection fault simulation simulation of hydraulic hydraulic plants. Kuo et al. al. (2017) (2017) have failure without timely detection (Maniatakos et induce al., 2011, 2011, system fault detection method based on fractional order for and submerged by noise in time domain, which may the proposed detection method based on on fractional fractional order dynamic Kelkar et al., al., 2014). 2014). system fault detection method method based for aaa detection dynamic and submerged by noise in time domain, which may induce the proposed Kelkar et proposed detection method based Kuo on fractional fractional dynamic Kelkar et al., 2014). simulation of hydraulic plants. et al. (2017) have failure without timely detection (Maniatakos et al., 2011, fault deviation for solar solarofPV PV panel failure. failure. However, most of the the have fault fault simulation hydraulic plants. Kuo et most al. (2017) deviation for panel However, of fault failure without timely detection (Maniatakos et al., 2011, deviation for solar PV panel failure. However, most of the fault Moreover, for safety reasons, TCU redundancy is often used proposed a detection method based on fractional dynamic Kelkar et al., Moreover, for safety detection methods based on fractional fractional order mentioned above Moreover, for2014). safety reasons, reasons, TCU TCU redundancy redundancy is is often often used used detection proposed methods a detection method based on fractional dynamic based on order mentioned Kelkar al., 2014). methods based on failure. fractional order mentioned above to avoidetproblems problems (Ronanki et et al., al., 2017), 2017), besides, besides, TCU TCU fault fault detection deviation for solar PV panel However, most of theabove fault to avoid (Ronanki are usually based on known fault models or obvious fault to avoid problems (Ronanki et al., 2017), besides, TCU fault deviation forbased solar PV panel failure. However, most of the fault Moreover,generally for safetyhave reasons, TCUduration redundancy is often used are usually on known fault models or obvious are usually based on known fault models or obvious fault signals short and no fixed detection methods based on fractional order mentioned above Moreover, for safety reasons, TCU redundancy is often used signals generally have short duration and no fixed features, which are not suitable for TCU fault detection signals generally (Ronanki have short duration and TCU no fixed detection methods based on fractional order mentioned above to avoid problems et al., 2017), besides, fault features, which are not suitable for TCU fault detection features, which are not suitable for TCU fault detection characteristic frequency, which not only cause wasting of usually or Besides, obvious when fault to avoid problems (Ronanki et al.,not 2017), TCU fault characteristic frequency, which only cause of without faultbased modelon orknown obviousfault faultmodels features. characteristic frequency, which only besides, cause wasting wasting of are are usually based onor known fault models or Besides, obvious when fault signals generally short not duration no fixed fault model obvious fault features. without fault model ornot obvious faultfor features. Besides, when equipment resourceshave and lacking lacking of enough enoughand concerning on without features, which are suitable TCU fault detection signals generally have short duration and no fixed equipment resources and of concerning on too much complicated signal processing needed, it would lead equipment resources and which lackingnot of only enough concerning on features, which are not suitable for TCU fault detection characteristic frequency, cause wasting of too much complicated signal processing needed, it would lead too muchfault complicated signal processing needed,Besides, it wouldwhen lead TCU fault, but butfrequency, also bring bring challenges challenges for thecause traditional signal model orburden obvious faultetfeatures. characteristic which not for only wasting of without TCU fault, also the traditional signal to high computation (Yang al., 2018) and cannot TCU fault, but also bring challenges for the traditional signal without fault model orburden obvious faultet Besides, when equipment resourcestoand lacking offault. enough concerning on to high computation (Yang etfeatures. al., 2018) and cannot to high computation burden (Yang al., 2018) and cannot processing methods detect TCU However, in recent too much complicated signal processing needed, it would lead equipment resourcesto lacking enough concerning on achieve the timely TCU fault detection. processing methods detect TCU However, in processing methods toand detect TCUoffault. fault. However, in recent recent too muchthe complicated signal processing needed, it would lead TCU fault, but also bring challenges for the traditional signal achieve timely TCU fault detection. achieve the timely TCU fault detection. years, as the development of fractional theory and application, to high computation burden (Yang et al., 2018) and cannot TCU fault, but also bring of challenges for the traditional signal to high computation burden (Yang et al., 2018) and cannot years, as development fractional theory and years, as the themethods development of fractional theory and application, application, processing to detect TCU However, recent Different from most of the the methods mentioned above, above, in this this with its unique unique characteristics, such fault. as strong strong global in memory achieve the timely TCU fault detection. from most of methods mentioned processing methods to detect TCU fault. However, in recent Different with its characteristics, such as global memory Different from most of the methods mentioned above, in in this with its unique characteristics, such as strong global memory achieveinspired the timely TCU fault detection. years, as the (Podlubny, development1999, of fractional theory and2010), application, paper, by (Pu, 2015), we propose a Fractional Steepest correlation Monje et al., non(Pu, we aa Fractional Steepest years, as the (Podlubny, development1999, of fractional theory and2010), application, correlation Monje et nonpaper, inspired inspired by (Pu,of2015), 2015), we propose propose Fractional Steepest correlation (Podlubny, 1999, Monje et al., al., 2010), non- paper, Different from by most the mentioned above, this with its unique such strong global memory Ascent Method (FSAM) formethods the purpose purpose of detecting detecting thein TCU localized (whichcharacteristics, can be be processed processed inas time domain, frequency Different from most of the methods mentioned above, inTCU this Ascent Method (FSAM) for the of the with its unique characteristics, such as strong global memory localized (which can in time domain, frequency Ascentinspired Methodby (FSAM) for the purpose of detecting the TCU localized (which can be processed in time domain, frequency paper, (Pu, 2015), we propose a Fractional Steepest correlation (Podlubny, 1999, Monje et al., 2010), nonfault quickly. FSAM is2015), to use usewe thepropose maximum quadraticSteepest energy domain, s domain domain and zz domain), domain), weak singularity singularity (Pu, 2016), paper, inspired by (Pu, a Fractional fault quickly. FSAM is to the maximum quadratic energy correlation (Podlubny, 1999, Monje et al., 2010), non(Pu, 2016), domain, s and weak fault quickly. is to use maximum quadratictheenergy domain, s (which domaincan andbe z domain), weak singularity (Pu, 2016), Ascent MethodFSAM (FSAM) thethe purpose TCU localized processed in time frequency of the velocity velocity residual for absolute value of asdetecting the optimization optimization fractional order has has been widely used used in domain, signal processing, Ascent Method (FSAM) for the purpose the TCU the residual absolute value the localized can been be processed in time frequency of fractional order widely in signal processing, of thequickly. velocity residual absolute value ofas asdetecting the optimization fractionals (which order has been widelyweak used in domain, signal processing, fault FSAM is to use the maximum quadratic energy domain, domain and z domain), singularity (Pu, 2016), target, and performs performs the G-L fractional order processing processing on the the especially in nonlinear, unsteady fault quickly. FSAMthe is to usefractional the maximum quadratic energy G-L order on domain, s domain andwith z domain), weaknon-Gaussian, singularity (Pu, 2016), target, especially in dealing dealing with nonlinear, non-Gaussian, unsteady target, and performs the G-L fractional order processing on the especially in dealing with nonlinear, non-Gaussian, unsteady of the and velocity residual absolute value astime the domain. optimization fractional order hasetbeen widely used in signal processing, velocity residual absolute value directly in The signals ( Sabatier al., 2007, Machado et al., 2011). of the velocity residual absolute value in astime the domain. optimization residual absolute value directly fractional order haset widely in signal velocity residual absolute value directly in time domain. The al., 2007, used Machado et al., 2011). velocity signals (Sabatier signals Sabatier etbeen al., nonlinear, 2007, Machado et processing, al.,unsteady 2011). target, and performs the G-L fractional order processing onThe the especially inindealing with Root Mean Square (RMS) (RMS) of the velocity velocity residual absolute absolute Meanwhile, fault detection, detection, many non-Gaussian, practical fault fault detection detection target, and performs the G-L of fractional order residual processing on the Root Mean Square the especially in dealing with nonlinear, non-Gaussian, unsteady Root Mean Square (RMS) of the velocity residual absolute Meanwhile, in fault many practical velocity residual absolute value directly in time domain. The Meanwhile, in faultetdetection, manyMachado practical et faultal.,detection signals ( Sabatier al., 2007, 2011). value and the maximum ratio of RMS are used as the detection methods based on the the principle of fractional fractional order have2011). been value velocity absolute value directly time The RMS are used as the and the maximum ratio signals Sabatier et principle al., 2007, Machado order et al., andresidual theSquare maximum ratio ofthe RMS are in used as domain. the detection detection methods on of have been Root Mean (RMS) ofof velocity residual absolute methods (based based on the principle of fractional order have been value threshold setting and the fractional order optimization, Meanwhile, in fault detection, many practical fault detection put forward for different objects. For example, Wang et al. Root Mean Square (RMS) of the velocity residual absolute threshold setting and the fractional order optimization, threshold setting and ratio the of fractional orderas optimization, Meanwhile, in detection, many practical fault detection put forward for objects. example, Wang et al. and the maximum RMS used the detection put forward forfault different objects. For example, Wang et al. value respectively. The effectiveness effectiveness of are FSAM for TCU fault methods based ondifferent the principle of For fractional order havebased been (2017) have proposed fault feature feature extraction method value and the maximum ratio of RMS used for as the detection The of FSAM TCU fault respectively. The and effectiveness of are FSAM for TCU fault methods based on the aaaprinciple of fractional order havebased been respectively. (2017) have proposed fault extraction method threshold setting the fractional order optimization, (2017) have proposed fault feature extraction method based detection is verified by the results of simulation. put forward for different objects. For example, Wang et al. on the fractional Hilbertobjects. transform defined inWang frequency threshold and theresults fractional order optimization, verified by of detection is issetting verified by the the results of simulation. simulation. put forward for different For defined example,in et al. detection on the fractional Hilbert transform frequency respectively. The effectiveness of FSAM for TCU on the fractional Hilbert transform defined in frequency (2017) have proposed a fault feature extraction method based respectively. The effectiveness of FSAM for TCU fault fault (2017) have proposed a fault feature extraction method based detection is verified by the results of simulation. on the fractional Hilbert transform defined in frequency detection is verified by the results of simulation. on the fractional Hilbert transform defined in frequency Copyright © 2018 IFAC 1336 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 1336Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 1336 Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2018.09.561 Copyright © 2018 IFAC 1336 Copyright © 2018 IFAC 1336
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2. BASICS OF FSAM When any fault in the processor module, register module or interface module occurs, the velocity residual absolute value will be perturbed on magnitude. The duration of such a perturbation is short, which has no fixed characteristic frequency, and difficult to detect the fault signal through the traditional frequency domain method. However, such a short perturbation brings energy fluctuations to the signal, in other words, the disturbance of the fault will usually cause the energy of the signal to contain the faulty mutation information. Therefore, in order to be able to deal with the fault signal directly in time domain, FSAM is proposed to detect TCU fault by searching the fault signal pulses with the maximum quadratic energy. Refer to (Pu, 2015), we define the quadratic energy functional as 1 𝐸𝐸 = 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 − 𝜂𝜂(𝑠𝑠1∗ − 𝑠𝑠)2 , (𝜂𝜂 > 0)
(1)
Where 𝐸𝐸 denotes the quadratic energy of the signal 𝑠𝑠, 𝜂𝜂 is a positive number that controls the degree of quadratic energy 1 is the first-order extreme of functional bump convexity, 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 1∗ 𝐸𝐸, 𝑠𝑠 is the first-order optimal argument value corresponding 1 . to 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚
1 ) is the From Fig. 1, it can be seen that the point A(𝑠𝑠1∗ , 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 𝑣𝑣 𝑣𝑣∗ first-order extreme of 𝐸𝐸, the point B(𝑠𝑠 , 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 ) is the 𝑣𝑣-order extremum of 𝐸𝐸, 𝑠𝑠0 is the initial value of a randomly selected iteration for searching.
At present, the commonly used fractional order is defined as Grümwald.Letniko (G-L) fractional order, or RiemannLiouville (R-L) fractional order, or Caputo fractional order. The G-L fractional order derivative is approximated by its effective discrete-time model based on Laguerre filters (Stanisławski et al., 2016). So, in this paper, we use the definition of G-L fractional order, that is 𝐺𝐺𝐺𝐺 𝑣𝑣 𝑎𝑎𝐷𝐷𝑡𝑡 𝑠𝑠(𝑡𝑡)
(2)
s.t. 0 The iteration form of 𝑠𝑠 is defined as 𝑠𝑠𝑘𝑘+1|𝑘𝑘 = 𝑠𝑠𝑘𝑘|𝑘𝑘−1 + 𝜇𝜇 (−
𝑑𝑑 𝑣𝑣 𝐸𝐸
| ) 𝑑𝑑𝑠𝑠 𝑣𝑣 𝑠𝑠=𝑠𝑠𝑘𝑘
(3)
In (3), 𝑘𝑘 is the number of iterations, 𝑣𝑣 is a real number, 𝑠𝑠𝑘𝑘|𝑘𝑘−1 is the iteration adjustment value of 𝑠𝑠 at the moment 𝑘𝑘, 𝑠𝑠𝑘𝑘+1|𝑘𝑘 is the iteration adjustment value of 𝑠𝑠 at the moment 𝑘𝑘 + 1, 𝑑𝑑 𝑣𝑣 𝐸𝐸
| is the 𝑣𝑣-order fractional gradient value of 𝐸𝐸 at 𝑠𝑠 = 𝑑𝑑𝑠𝑠 𝑣𝑣 𝑠𝑠=𝑠𝑠𝑘𝑘 𝑠𝑠𝑘𝑘 , 𝜇𝜇 is the coefficient that controls the stability and convergence rate of FSAM. From (1) and (3), it can be seen that FSAM is a parabola 𝐸𝐸 along an opening and incremental search in the positive direction of the 𝑣𝑣 -order fractional gradient. The iterative search process of FSAM is shown in Fig. 1.
∙
(4)
𝑘𝑘!(𝑣𝑣−𝑘𝑘)!
𝑑𝑑 𝑣𝑣 𝐸𝐸
𝑣𝑣 |𝑠𝑠=𝑠𝑠𝑘𝑘 =
2
1 [𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 −𝜂𝜂(𝑠𝑠 1∗ ) ]𝑠𝑠𝑘𝑘−𝑣𝑣
𝛤𝛤(1−𝑣𝑣)
2𝜂𝜂𝑠𝑠 1∗ 𝑠𝑠𝑘𝑘1−𝑣𝑣
+
𝛤𝛤(2−𝑣𝑣)
−
2𝜂𝜂𝑠𝑠𝑘𝑘2−𝑣𝑣
𝛤𝛤(3−𝑣𝑣)
(5)
In (5), 𝑣𝑣 > 0, which means that the quadratic energy of the signal 𝑠𝑠 is fractional differential; at the same time, 𝑠𝑠𝑘𝑘−𝑣𝑣 , 𝑠𝑠𝑘𝑘1−𝑣𝑣 , and 𝑠𝑠𝑘𝑘2−𝑣𝑣 are the power function of the signal 𝑠𝑠 .In fact, according to the fractional order properties (Mohammad, 2016), we let 0 < 𝑣𝑣 < 1 .
When the iterative search process of the FSAM from 𝑠𝑠0 to the stable point B, there are 𝐷𝐷𝑠𝑠𝑣𝑣𝑘𝑘 = 0, 𝑠𝑠𝑘𝑘+1 = 𝑠𝑠𝑘𝑘 = 𝑠𝑠 𝑣𝑣∗ , and 𝐸𝐸 = 𝑣𝑣 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 , then i)
𝑣𝑣 1 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 − 𝜂𝜂(𝑠𝑠1∗ )2 + 2𝜂𝜂𝑠𝑠1∗ 𝑠𝑠 𝑣𝑣∗ − 𝜂𝜂(𝑠𝑠 𝑣𝑣∗ )2
(6)
ii) 2
1 [𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 −𝜂𝜂(𝑠𝑠 1∗ ) ]
(𝑠𝑠 𝑣𝑣∗ )−𝑣𝑣 {
Γ(1−𝑣𝑣)
+
2𝜂𝜂𝑠𝑠 1∗
∙ 𝑠𝑠 𝑣𝑣∗ −
Γ(2−𝑣𝑣)
Since (𝑠𝑠 𝑣𝑣∗ )−𝑣𝑣 ≠ 0, we have 2
1 [𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 −𝜂𝜂(𝑠𝑠 1∗ ) ]
+
2𝜂𝜂𝑠𝑠 1∗
𝛤𝛤(2−𝑣𝑣)
∙ 𝑠𝑠 𝑣𝑣∗ −
2𝜂𝜂
∙ (𝑠𝑠 𝑣𝑣∗ )2 } = 0
2𝜂𝜂
∙ (𝑠𝑠 𝑣𝑣∗ )2 = 0 (7)
Γ(3−𝑣𝑣)
𝛤𝛤(3−𝑣𝑣)
when 𝑣𝑣 ≠ 1, 2, 3, with 𝑠𝑠 𝑣𝑣∗ as an independent variable, the root of equation (7) can be solved as
A
𝑣𝑣 𝐸𝐸𝑚𝑚𝑎𝑎𝑥𝑥
− 𝑘𝑘ℎ) , 𝑘𝑘 ∈ 𝑍𝑍
From (1) and (4), the fractional 𝑣𝑣-order of 𝐸𝐸 for 𝑠𝑠(𝑡𝑡) at 𝑡𝑡 = 𝑘𝑘 can be obtained
𝛤𝛤(1−𝑣𝑣)
E 1 𝐸𝐸𝑚𝑚𝑎𝑎𝑥𝑥
1
ℎ→0 ℎ𝑣𝑣 (−1)𝑘𝑘 𝛤𝛤(𝑣𝑣+1) ∑∞ 𝑘𝑘=0 𝛤𝛤(𝑘𝑘+1)𝛤𝛤(𝑣𝑣−𝑘𝑘+1) 𝑠𝑠(𝑡𝑡
𝛤𝛤(𝑘𝑘+1)𝛤𝛤(𝑣𝑣−𝑘𝑘+1)
𝑑𝑑𝑠𝑠
s
= 𝑙𝑙𝑙𝑙𝑙𝑙
In (4), 𝐺𝐺𝐺𝐺𝑎𝑎𝐷𝐷𝑡𝑡𝑣𝑣 𝑠𝑠(𝑡𝑡) represents the operator of the fractional 𝑣𝑣order of the signal 𝑠𝑠(𝑡𝑡) at the interval [𝑎𝑎, 𝑡𝑡], ℎ represents the calculation step size, 𝛤𝛤 represents the Euler’s Gamma function, 𝛤𝛤(𝑣𝑣+1) 𝑣𝑣! = . and there is
The objective function of FSAM is 1 J FSAM max Emax (s1* s)2
1337
B
𝑣𝑣∗ = 𝑠𝑠1,2
Γ(3−𝑣𝑣) 2𝜂𝜂
×{
𝜂𝜂𝑠𝑠 1∗
Γ(2−𝑣𝑣)
𝜂𝜂 2 (𝑠𝑠 1∗ )2
±√
Γ2 (2−𝑣𝑣)
+
1 2𝜂𝜂[𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 −𝜂𝜂(𝑠𝑠 1∗ )2 ]
Γ(1−𝑣𝑣)Γ(3−𝑣𝑣)
}
(8)
To ensure the convergence of the algorithm, the two roots in (8) should be equal, then we can obtain
0
𝑠𝑠
1∗
𝑠𝑠 𝑣𝑣∗ ⋯ 𝑠𝑠0
Γ(1−𝑣𝑣)Γ(3−𝑣𝑣) Γ2 (2−𝑣𝑣)
s
=−
𝑠𝑠 𝑣𝑣∗ =
Fig. 1. An iterative search process of FSAM
1337
1 2𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚
𝜂𝜂(𝑠𝑠 1∗ )2
Γ(3−𝑣𝑣)𝑠𝑠 1∗ 2Γ(2−𝑣𝑣)
+2
(9) (10)
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In (9), it shows that if the 𝑠𝑠1∗ and 𝜂𝜂 have been given, the 1 of the signal s. optimization of 𝑣𝑣 is actually to obtain the 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 By substituting (10) into (1), the following equation is given 𝑣𝑣 𝐸𝐸 = 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 + 𝜂𝜂 [
−4𝛤𝛤(2−𝑣𝑣)+𝛤𝛤(3−𝑣𝑣) 𝛤𝛤(3−𝑣𝑣)
(𝑠𝑠 𝑣𝑣∗ )2 +
4𝛤𝛤(2−𝑣𝑣) 𝑣𝑣∗ 𝑠𝑠 𝑠𝑠 𝛤𝛤(3−𝑣𝑣)
− 𝑠𝑠 2 ]
2
1 𝜇𝜇[𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 −𝜂𝜂(𝑠𝑠 1∗ ) ]𝑠𝑠𝑘𝑘−𝜈𝜈
−
𝛤𝛤(1−𝜈𝜈)
2𝜇𝜇𝜇𝜇𝑠𝑠 1∗ 𝑠𝑠𝑘𝑘1−𝜈𝜈 𝛤𝛤(2−𝜈𝜈)
+
𝑒𝑒(𝑡𝑡) = |𝑉𝑉𝐹𝐹 (𝑡𝑡) − 𝑉𝑉𝑁𝑁 (𝑡𝑡)|
(11)
By substituting (5) into (3), we can obtain 𝑠𝑠𝑘𝑘+1|k = 𝑠𝑠𝑘𝑘|𝑘𝑘−1 −
velocity measurements 𝑉𝑉𝑁𝑁 (𝑡𝑡) and 𝑉𝑉𝐹𝐹 (𝑡𝑡) at the 𝑡𝑡 moment under the normal state and the TCU faulty state by the velocity sensors, respectively, then the velocity residual absolute value 𝑒𝑒(𝑡𝑡) and its discretization velocity residual absolute value 𝑒𝑒(𝑘𝑘) are respectively as
2𝜇𝜇𝜇𝜇𝑠𝑠𝑘𝑘2−𝜈𝜈 𝛤𝛤(3−𝜈𝜈)
(12)
𝑒𝑒(𝑘𝑘) = |𝑉𝑉𝐹𝐹 (𝑘𝑘) − 𝑉𝑉𝑁𝑁 (𝑘𝑘)|
2𝜇𝜇𝜇𝜇
[−𝑠𝑠𝑘𝑘2 +
𝛤𝛤(3−𝜈𝜈) 𝛤𝛤(3−𝜈𝜈) 1∗ 𝑣𝑣∗ 𝑠𝑠 𝑠𝑠 ] 𝑠𝑠𝑘𝑘−𝑣𝑣 𝛤𝛤(2−𝜈𝜈)
𝛤𝛤(3−𝜈𝜈) 1∗ 𝑠𝑠 𝑠𝑠𝑘𝑘 𝛤𝛤(2−𝜈𝜈)
𝑁𝑁
+ (𝑠𝑠 𝑣𝑣∗ )2 − (13)
2𝜇𝜇𝜇𝜇
𝛤𝛤(3−𝜈𝜈)
(14)
And 𝑠𝑠 𝑣𝑣 about (𝑠𝑠 − 𝑠𝑠 𝑣𝑣∗ ) can be expressed as 𝑠𝑠 𝑣𝑣 = ∑∞ 𝑛𝑛=0
𝛤𝛤(1+𝑣𝑣)(𝑠𝑠 𝑣𝑣∗ )𝑣𝑣−𝑛𝑛 (𝑠𝑠−𝑠𝑠 𝑣𝑣∗ )𝑛𝑛 𝛤𝛤(𝑛𝑛+1)𝛤𝛤(1−𝑛𝑛+𝑣𝑣)
, 𝑣𝑣 ≤ 𝑛𝑛
When 𝑛𝑛 = 0 (it doesn’t satisfy the condition of 𝑣𝑣 ≤ 𝑛𝑛 ) and 𝑛𝑛 = 1, from (15), it can be obtained the approximation of 𝑠𝑠 𝑣𝑣 as 𝑠𝑠 𝑣𝑣 ≈ 𝑣𝑣(𝑠𝑠 𝑣𝑣∗ )𝑣𝑣−1 (𝑠𝑠 − 𝑠𝑠 𝑣𝑣∗ )
(16)
(17)
Combine (14), (16) and (17), we have 𝐷𝐷𝑡𝑡1 𝑠𝑠(𝑡𝑡|𝑡𝑡
2𝜇𝜇𝜇𝜇
𝛤𝛤(3−𝜈𝜈)𝑣𝑣(𝑠𝑠 𝑣𝑣∗ )𝑣𝑣−1
‖𝑒𝑒 ∗ ‖𝑅𝑅𝑅𝑅𝑅𝑅
[𝑠𝑠(𝑡𝑡) − 𝑠𝑠 𝑣𝑣∗ ]
2𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇
𝛤𝛤(3−𝜈𝜈)𝑣𝑣(𝑠𝑠 𝑣𝑣∗ )𝑣𝑣−1
In order to let the fault signal and the noise can be better distinguished, the maximum of 𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 is used as the fractional order optimal objective function, that is 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = max(𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 )
Let
2𝜇𝜇𝜇𝜇
𝛤𝛤(3−𝜈𝜈)𝑣𝑣(𝑠𝑠 𝑣𝑣∗ )𝑣𝑣−1
𝑙𝑙𝑙𝑙𝑙𝑙
2𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇
{
𝑒𝑒(𝑘𝑘 + 1|𝑘𝑘) ≥ ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 ⟹ 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑒𝑒(𝑘𝑘 + 1|𝑘𝑘) < ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 ⟹ 𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑙𝑙
(28)
3.2 Algorithm overview
Start
(18)
e(𝑡𝑡)
)
e(𝑘𝑘)
FSAM
(19)
𝑒𝑒(𝑘𝑘 + 1|𝑘𝑘)
= −∞
‖𝑒𝑒‖𝑅𝑅𝑀𝑀𝑆𝑆
(20)
= 𝜒𝜒, then from (20), 𝜒𝜒 should satisfy 𝜒𝜒 < 0
(27)
Finally, the TCU fault detection logic can be written as
In order to ensure the stability of the iteration, the second term of (19) should be satisfied that 𝑘𝑘→∞ 𝛤𝛤(3−𝜈𝜈)𝑣𝑣(𝑠𝑠 𝑣𝑣∗ )𝑣𝑣−1
(25)
(26)
‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅
(18) is a first-order ordinary differential equation with variables that are separable. By solving (18), the solution 𝑠𝑠(𝑡𝑡) can be obtained, then its discretization 𝑠𝑠𝑘𝑘 with a sampling period 𝜏𝜏 can be given as 𝑠𝑠𝑘𝑘 = 𝑠𝑠 𝑣𝑣∗ + 𝑒𝑒𝑒𝑒𝑒𝑒 (
1 2
+ 1|𝑘𝑘)) e(𝑘𝑘 + 1|𝑘𝑘))) 𝑇𝑇
A flow chart of the TCU fault detection method based on FSAM is shown in Fig. 2.
− ℎ) ≈ 𝑠𝑠𝑘𝑘+1|k − 𝑠𝑠𝑘𝑘|𝑘𝑘−1 ≈
( (∑𝑁𝑁−1 𝑘𝑘=0 (𝑒𝑒(𝑘𝑘 𝑁𝑁
Let 𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 be the ratio of ‖𝑒𝑒 ∗ ‖𝑅𝑅𝑅𝑅𝑅𝑅 to ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 , that is
The first order difference is used as the approximation of the first order differential𝐷𝐷𝑡𝑡1 𝑠𝑠(𝑡𝑡|𝑡𝑡 − ℎ), which is 𝐷𝐷𝑡𝑡1 𝑠𝑠(𝑡𝑡|𝑡𝑡 − ℎ) ≈ 𝑠𝑠𝑘𝑘+1|𝑘𝑘 − 𝑠𝑠𝑘𝑘|𝑘𝑘−1
=
𝑅𝑅𝑅𝑅𝑅𝑅
1
𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 =
(15)
(24)
RMS of FSAM output 𝑒𝑒(𝑘𝑘 + 1|𝑘𝑘) is calculated as ‖𝑒𝑒 ∗ ‖
(𝑠𝑠𝑘𝑘 − 𝑠𝑠 𝑣𝑣∗ )2 𝑠𝑠𝑘𝑘−𝑣𝑣
1 2
1
𝑇𝑇 ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 = ( (∑𝑁𝑁−1 𝑘𝑘=0 𝑒𝑒 (𝑘𝑘) 𝑒𝑒(𝑘𝑘)))
Furthermore, (13) can be rewritten as 𝑠𝑠𝑘𝑘+1|𝑘𝑘 = 𝑠𝑠𝑘𝑘|𝑘𝑘−1 +
(23)
Then, 𝑒𝑒(𝑘𝑘) is used as the input of FSAM, that is 𝑠𝑠(𝑘𝑘) = 𝑒𝑒(𝑘𝑘), the output of FSAM is 𝑒𝑒(𝑘𝑘 + 1|𝑘𝑘) . RMS of the velocity residual absolute value 𝑒𝑒(𝑘𝑘) is calculated as:
By substituting (7) and (10) into (12), we have 𝑠𝑠𝑘𝑘+1|𝑘𝑘 = 𝑠𝑠𝑘𝑘|𝑘𝑘−1 −
(22)
Normal
N
𝑒𝑒(𝑘𝑘 + 1|𝑘𝑘) ≥ ‖𝑒𝑒‖𝑅𝑅𝑀𝑀𝑆𝑆 Y
(21)
Fault
3. FSAM-BASED TCU FAULT DTECTION METHOD End
3.1 TCU fault detection method Firstly, by setting the fault free condition and TCU fault injection on the simulation platform, then we can collect the
Fig. 2. A flow chart of the TCU fault detection method based on FSAM
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The steps of the proposed TCU fault detection method and the parameters setting guide are as follows： 1. Obtain the measurements and carry on the velocity residual value to take the absolute value processing and the discretization processing; 2. The RMS of the velocity residual absolute value and the ratios of RMS are calculated according to (24), (25), and (26), respectively； 3. Randomly select the initial value of 𝑠𝑠0 , and set the 𝑠𝑠1∗ to a very small positive number for the TCU weak fault signal (recommend 0 < 𝑠𝑠1∗ ≤ 0.001), then by (11) can obtain the 𝑠𝑠 𝑣𝑣∗ , as for the 𝜒𝜒 , we recommend −2 < 𝜒𝜒 < −1 to ensure the underdamped oscillation, once the 𝜂𝜂 and 𝜒𝜒 have been set, we 2𝜇𝜇𝜇𝜇 can use 𝑣𝑣∗ )𝑣𝑣−1 = 𝜒𝜒 to obtain the 𝜇𝜇, then use (14) for 𝛤𝛤(3−𝜈𝜈)𝑣𝑣(𝑠𝑠
iterative processing;
4. Set the range of fractional order 𝑣𝑣 as (0,1) with the step 0.01,the optimal order is where the maximum of 𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 can be obtained, that is 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = max(𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 );
The experimental data used in this paper is collected from the traction drive control system fault injection benchmark (TDCS-FIB) of the CRH2 traction drive control system, the version number is 1.3. For downloading the benchmark and detailed information, please visit http://gfist.csu.edu.cn/. The simulation time in this paper is set to be 4s, TCU fault occurs after 1.2s, the data sampling frequency is set to be 100 kHz. As shown in Fig. 3(a), the velocity of high-speed train in normal operation is a non-linear, non-Gaussian, unsteady wave signal due to the interference of system noise and environmental noise (Chen et al., 2017), especially the weak fault signal of the interface module fault in TCU is usually submerged by noise in time domain (see Fig. 3(b)). The initial value of FSAM is set to: 𝑠𝑠0 = 0, 𝑠𝑠1∗ = 0.001, 𝜂𝜂 = 1.25, 𝜒𝜒 = −1.5, the 0 ~ 4s of ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 is used as the threshold for judging whether the TCU fault occurs (indicated by a red dotted line). First, we use FSAM to detect the processor module fault and register module fault in TCU, the results are shown in Fig. 4 and Fig.5 respectively. It can be seen from Fig. 4 that both failures occurred at about 2.5s, and the maximum of magnitude have been magnified more than 7 times at least.
250
250
200
200
150 100 50 0 0
0.5
1
1.5
196.102 196.1 196.098 196.096 196.094 196.092 2.45
2 t(s)
2.5
3
3.5
150 100 50 0 0
4
0.5
1
1.5
2 t(s)
2.5
3
3.5
4
196.11 VF（km/h）
VN（km/h）
4. SIMULATION
VF（km/h）
VN（km/h）
5. Use the ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 as the threshold to determine whether the TCU fault occurs.
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2.5 t(s)
196.1 196.09 196.08
2.55
2.45
2.5 t(s)
(a)
2.55
(b)
Fig. 3. (a) was the speed curve of high-speed train under normal operation, (b) was the speed curve of high-speed train under the interface module fault in TCU. 0.06
0.4 e(k)
e(k+1|k) ||eRMS||
0.35
0.05
0.3 0.25 e(k+1|k)
e(k)
0.04 0.03
0.2 0.15
0.02
0.1 0.01
0.05 0 2.4
2.6
2.8
3
3.2 t(s)
3.4
3.6
3.8
0 2.4
4
(a)
2.6
2.8
3
3.2 t(s)
3.4
3.6
3.8
4
(b)
Fig. 4. The effect of FSAM to detect the processor module fault in TCU, the fault occurred at 2.503s, the order was 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = 0.30, the threshold was ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 = 0.0144 , (a) was the input signal, (b) was the output signal. 1339
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However, the interface module fault in TCU is much more difficult to detect than the processor module fault and register module fault in TCU. Then, in order to quantitatively describe the severity of the interface module fault in TCU and verify the robustness of FRAM, the severity of the interface module fault in TCU is set to Level 1, Level 2 and Level 3, respectively,
depending on the parameters named A, P and Q on the TDCSFIB V1.3. The specific set values of A, P and Q for Level 1, Level 2 and Level 3 are shown in Table 1.The results of the comparison of the processing are shown in Fig. 6, Fig. 7, and Fig. 8, respectively, and 𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 with different severity of the interface module fault in TCU is shown in Table 2. 10
5
10
10
e(k+1|k) ||eRMS||
e(k)
5
10
0
lg(e(k+1|k))
lg(e(k))
10
-5
10
0
10
-5
10
-10
10
0
0.5
1
1.5
2 t(s)
2.5
3
3.5
0
4
0.5
1
1.5
(a)
2 t(s)
2.5
3
3.5
4
(b)
Fig. 5. The effect of FSAM to detect the register module fault in TCU, the fault occurred at 2.502s, the order was 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = 0.09, the threshold was ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 = 77.1 , (a) was the input signal, (b) was the output signal. 0.02
0.06 e(k)
0.018
e(k+1|k) ||eRMS||
0.05
0.016 0.014
0.04 e(k+1|k)
e(k)
0.012 0.01 0.008
0.03 0.02
0.006 0.004
0.01
0.002 0 2.44
2.46
2.48
2.5
2.52
0 2.44
2.54
t(s)
2.46
2.48
2.5
2.52
2.54
t(s)
(a)
(b)
Fig. 6. The effect of FSAM to detect the interface module fault in TCU, the fault occurred at 2.502s, the severity level was Level 1, the order was 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = 0.31, the threshold was ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 = 0.0047 , (a) was the input signal, (b) was the output signal. 0.025
0.08
e(k)
e(k+1|k) ||eRMS||
0.07
0.02 0.06 0.05
e(k)
e(k+1|k)
0.015
0.01
0.04 0.03 0.02
0.005
0.01
0 2.98
3
3.02
3.04
3.06
0 2.98
3.08
t(s)
3
3.02
3.04
3.06
3.08
t(s)
(a)
(b)
Fig. 7. The effect of FSAM to detect the interface module fault in TCU, the fault occurred at 3.001s, the severity level was Level 2, the order was 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = 0.31, the threshold was ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 = 0.0045 , (a) was the input signal, (b) was the output signal. 1340
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0.08
0.025
e(k+1|k) ||eRMS||
e(k)
0.07 0.02
0.06 0.05
e(k)
e(k+1|k)
0.015
0.01
0.04 0.03 0.02
0.005
0.01 0 2.44
2.46
2.48
2.5
2.52
0 2.44
2.54
2.46
2.48
2.5
t(s)
t(s)
(a)
(b)
2.52
2.54
Fig. 8. The effect of FSAM to detect the interface module fault in TCU, the fault occurred at 2.501s, the severity level was Level 3, the order was 𝑣𝑣𝑜𝑜𝑜𝑜𝑜𝑜 = 0.31, the threshold was ‖𝑒𝑒‖𝑅𝑅𝑅𝑅𝑅𝑅 = 0.0048 , (a) was the input signal, (b) was the output signal. Table 1. The severity with the specific set values of A, P and Q for the interface module fault in TCU Fault severity rating Level 1
A
P
Q
25
-100
-200
Level 2
25
-200
-500
Level 3
25
-500
-1000
Table 2. 𝑱𝑱𝑹𝑹𝑹𝑹𝑹𝑹 with different severity for the interface module fault in TCU Severity rating Level 1 Level 2 Level 3
Status Before failure 0.5218 0.5394 0.5302
Failure 2.3427 2.6081 2.7328
It can be seen from Fig. 6 that when the TCU fault occurs at 2.502s, and the fault severity level is Level 1, the velocity of the residual is very small, the fault signal and the noise barely can be resolved; but after FSAM, it can be seen that RMS of the TCU fault signal is enhanced about 2.34 times, while the non-linear characteristic of the fault signal can be well preserved, and the noise is also well suppressed. Moreover, it can be seen from Fig. 7 and Fig. 8 that when the TCU fault occurs at 3.001s and 2.501s, and the fault severity is Level 2 and Level 3, respectively, the residual of velocity absolute value is smaller, and the fault signal is also more gradually buried in the noise, it is more difficult to be distinguished; but after processing by FSAM, it can be seen from Table 2 that 𝐽𝐽𝑅𝑅𝑅𝑅𝑅𝑅 are about 2.61 times and 2.73 times enhancement, respectively, but also can retain the output response of the nonlinear features well, and the noise has also been a very good inhibition. 5. CONCLUSION Aiming at the problem that the TCU fault is difficult to detect when in the case of non-linear, non-Gaussian, unsteady, this paper proposes a method based on FSAM to search the maximum quadratic energy of the velocity residual absolute
After failure 0.5212 0.3290 0.4457
value, the results show that it cannot only separate the fault signal from the strong noise, but also suppress the noise interference at the non-fault time, which can realize the purpose of detecting the TCU fault. The simulation results show that FSAM is practical and effective, and it has some guiding significance for the practical engineering application. What we should note is that FSAM is not suitable for the fault detection that the features of the fault signals are very similar to the noise, about this issue, we will develop FSAM in our future work. ACKNOWLEDGEMENTS This work was supported by the Major Program of the National Natural Science Foundation of China (Grant No. 61490702), the National Natural Science Foundation of China (Grant No. 61773407) and the Fundamental Research Funds for the Central Universities of Central South University (Grant No. 2018zzts603). REFERENCES Chen, S., Zhan, T., Huang, C., Chen, J. and Lin, C. (2015). Nontechnical Loss and Outage Detection Using FractionalOrder Self-Synchronization Error-Based Fuzzy Petri Nets in
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