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Parameter Identification of Train Basic Resistance Using Multi-Innovation Theory⁎
Proceedings, 10th IFAC International Symposium on Proceedings, 10th IFAC International Symposium on Advanced Control Chemical Processes Proceedings, 10th of IFAC International Symposium on at www.sciencedirect.com Available online Advanced Control of Chemical Processes Proceedings, 10th of IFAC International Symposium on Shenyang,Control Liaoning, China, July 25-27, 2018 Advanced Chemical Processes Shenyang, Liaoning, China, July 25-27, 2018 Advanced of China, Chemical Processes Shenyang,Control Liaoning, July 25-27, 2018 Shenyang, Liaoning, China, July 25-27, 2018
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IFAC PapersOnLine 51-18 (2018) 637–642
Parameter Identification of Train Basic Parameter Identification of Train Basic Parameter Identification of Train Basic Resistance Using Multi-Innovation Theory Parameter Identification of Train Basic Resistance Using Multi-Innovation Theory Resistance Using Multi-Innovation Theory Resistance Using Multi-Innovation Theory Xiaoyu Liu ∗∗ Bin Ning ∗∗ Jing Xun ∗∗ Cheng Wang ∗∗ ∗∗
Xiaoyu Liu ∗∗ Bin Ning ∗∗ ∗∗∗ Jing Xun ∗∗ Cheng Wang ∗∗ ∗∗ ∗ Xiao Xiao Tong Xiaoyu Liu ∗ Bin Ning Jing XunLiu Cheng Wang ∗∗ ∗ ∗ ∗∗∗ ∗ Xiao Xiao Tong Liu ∗∗∗ ∗ ∗ Xiaoyu Liu Bin Jing XunLiuCheng Wang XiaoNing Xiao ∗∗∗ Tong Xiao Xiao ∗∗∗ Tong Liu ∗ ∗ ∗ State Key Laboratory of Rail Traffic Control and Safety, Beijing Key Laboratory of Rail Traffic Control and Safety, Beijing ∗ ∗ StateJiaotong University, Beijing, 100044, (e-mail: StateJiaotong Key Laboratory of Rail Traffic ControlChina and Safety, ∗ University, Beijing, 100044, (e-mail:Beijing StateJiaotong Key Laboratory of Rail Traffic ControlChina and Safety, [email protected]) University, Beijing, 100044, China (e-mail:Beijing [email protected]) ∗∗ Jiaotong University, Beijing, 100044, China (e-mail: [email protected]) ∗∗ School of Internet of Things Engineering, Jiangnan University, of Internet of Things Engineering, Jiangnan ∗∗ ∗∗ School [email protected]) School of 214122, InternetChina of Things Engineering, Jiangnan University, University, Wuxi, (e-mail: [email protected]) ∗∗ Wuxi, 214122, China (e-mail: [email protected]) ∗∗∗ School of Internet of Things Engineering, Jiangnan University, Traffic Control Technology CO., Ltd. Beijing Research Wuxi, 214122, China (e-mail: [email protected]) ∗∗∗ Control Technology CO., Ltd. Beijing Research Institute, Institute, ∗∗∗ Traffic ∗∗∗ Wuxi, 214122, China (e-mail: [email protected]) Beijing, 100070, China (e-mail: [email protected]) Traffic Control Technology CO., Ltd. Beijing Research Institute, ∗∗∗ Beijing, 100070, China (e-mail: [email protected]) Traffic Control Technology CO., Ltd. Beijing Research Institute, Beijing, 100070, China (e-mail: [email protected]) Beijing, 100070, China (e-mail: [email protected]) Abstract: Train Train basic basic resistance resistance is is important for for the design of the automatic train operation, Abstract: the of automatic operation, Abstract: Trainthe basic resistance is important important forprecision, the design design of the theconsumption, automatic train train operation, which influences efficiency, punctuality, stop energy and the which influences the efficiency, punctuality, stop precision, energy consumption, and the safety safety Abstract: Train basic resistance istheory important forprecision, the design of theconsumption, automatic train operation, of the train. The multi-innovation is a novel concept which can improve the accuracy of which influences the efficiency, punctuality, stop energy and the safety of the train. The the multi-innovation theory is stop a novel conceptenergy which consumption, can improve the accuracy of which influences efficiency, punctuality, precision, and the safety parameter estimation and be used to modify the traditional recursive least squares algorithm. of the train. The multi-innovation theory is a novel concept which can improve the accuracy of parameter estimation and be used theory to modify traditional recursive least squares algorithm. of The isform a the novel concept which can improve the accuracy of In the thistrain. paper, we multi-innovation derive theberegularization regularization of the multi-innovation least squares algorithm parameter estimation and used to modify the traditional recursive least squares algorithm. In this paper, we derive the form of the multi-innovation least squares algorithm parameter estimation and be used to modify the traditional recursive least squares algorithm. In this paper, we derive the regularization form of the multi-innovation least squares algorithm and apply apply it it to to the the train train basic basic resistance estimation. The results based and resistance parameter parameter The simulation simulation results based In this paper, wethe derive the regularization form ofthat, theestimation. multi-innovation least squares algorithm on the Yizhuang Line of Beijing Subway indicate compared with traditional least squares and apply it to train basic resistance parameter estimation. The simulation results based on the Yizhuang Linetrain of Beijing Subway indicate that,estimation. compared with traditional results least squares and apply it to the basic resistance parameter The simulation based algorithm, the multi-innovation least squares algorithm can provide higher estimation accuracy on the Yizhuang Line of Beijing Subway indicate that, compared with traditional least squares algorithm, the multi-innovation least squares can provide higher estimation algorithm accuracy on the Yizhuang Line of Beijing Subway indicate that, compared with traditional least squares and robustness, and can be usedleast for online identification. algorithm, the multi-innovation squares algorithm can provide higher estimation accuracy and and for identification. algorithm, the multi-innovation squares algorithm can provide higher estimation accuracy and robustness, robustness, and can can be be used usedleast for online online identification. © 2018, IFAC (International Federation Automatic Control) Hosting by Elsevier Ltd. All rights reserved. and robustness, and can be used forofonline identification. Keywords: Train Train basic basic resistance, resistance, Multi-innovation identification, Recursive identification, identification, Keywords: Multi-innovation identification, Keywords: Train basic resistance, Multi-innovation identification, Recursive Recursive identification, Parameter estimation, Urban rail transit Parameter estimation, Urban transit Keywords: basic resistance, identification, Recursive identification, Parameter Train estimation, Urban rail rail Multi-innovation transit Parameter estimation, Urban rail transit 1. INTRODUCTION constitution 1. constitution of of basic basic resistance, resistance, aa large large number number of of experexper1. INTRODUCTION INTRODUCTION imental data are integrated according to the formation constitution of basic resistance, a large number of experimental data are integrated according to the formation INTRODUCTION ofare basic resistance, a large number of experimental dataand integrated according to the formation mechanism, the classical formula was proposed Urban rail rail transit transit1. has has many advantages, advantages, such such as as large large constitution mechanism, and the classical Davis Davis formula wasformation proposed Urban many imental data are integrated according to the (see Davis (1926), Bernsteen et al. (1980), and Huang Urban rail transit has many advantages, such as large mechanism, and the classical Davis formula was proposed capacity, high high efficiency, efficiency, punctuality, punctuality, reliability reliability and and safety, safety, (see Davis (1926), Bernsteen et al. (1980), and Huang et et al. al. capacity, and the classicaletDavis formula was proposed Urban rail transit has punctuality, many ofadvantages, such assafety, large mechanism, (2000), and Yuan (2015)). capacity, high efficiency, reliability and (see Davis (1926), Bernsteen al. (1980), and Huang et al. so it becomes the backbone urban public transport. (2000), and Yuan (2015)). so it the backbone of public transport. Davis capacity, high efficiency, punctuality, reliability and safety, (2000), and(1926), Yuan Bernsteen (2015)). et al. (1980), and Huang et al. so it becomes becomes theoperation backbone of urban urban public transport. Automatic train (ATO) plays a key key role in (see Currently, aaYuan domestic enterprise Automatic train operation (ATO) plays a role in (2000), and (2015)). so it becomes the backbone of urban public transport. Currently, domestic enterprise needs needs one one experimental experimental Automatic operation (ATO) plays aofkey rolerail in ensuring the thetrain safety and efficient efficient operation urban procedure called coast-down test to determine the paramensuring safety and operation of urban rail Currently, a domestic enterprise needs one experimental Automatic operation (ATO) plays aofdensity key role(see in procedure called coast-down test to determine the paramensuring thetrain safety andhigh efficient operation urban rail transit operating with speed and high Currently, a resistance domestic enterprise needs one experimental procedure called coast-down test to determine the parameter of basic at the expense of time, labor, and transit operating with high speed and high density (see ensuring the safety andIn efficient operation urban (see rail eter of basic resistance at the expense of time, labor, and transit operating with high speed and with highof density Yu and Chen (2011)). recent years, development procedure called coast-down test to determine the parameter of basic resistance at the expense of time, labor, and material, so it is necessary to find a method that does Yu and Chen (2011)). In recent years, with development transit operating with high speed and high density (see material, so resistance it is necessary toexpense find a of method that does Yu and Chen (2011)). Incontrol recent technology, years, with the development of communication and platform eter of basic at the time, labor, and not need coast-down test (see Bernsteen et al. (1980) and of communication and control technology, the platform material, so it is necessary to find a method that does Yu anddoors Chen (PSD) (2011)). Incontrol recentwidely years, used. with the development not need coast-down test (seetoBernsteen et al. (1980) and of communication and technology, platform screen has been In order to material, so it is necessary find a method that does Yuan (2015)). The train resistance depends strongly screen doors (PSD) has been widely used. In order to need coast-down testbasic (see Bernsteen et al. (1980) and of communication control technology, the Yuan (2015)). The train basic resistance depends strongly screen doors (PSD)and hasare been used. In platform order to not ensure that passengers passengers not widely disturbed when they get get not need coast-down test (see Bernsteen et al. (1980) and Yuan (2015)). The train basic resistance depends strongly on weather and track conditions, so many control decisions ensure that are not disturbed when they screen (PSD) hasarebeen used. In order to on weather andThe track conditions, so manydepends control decisions ensure that not widely disturbed when they get on and and doors off on onpassengers the platform platform with PSD, the the more accurate Yuan (2015)). train basic resistance strongly of ATO can be improved if the basic resistance parameters on off the with PSD, more accurate on weather and track conditions, so many control decisions ensure that are not disturbed when they get on of ATO canand be improved if the basic resistance parameters on off onpassengers the platform PSD, the moreand accurate stopand precision is required (seewith Wang et al. al. (2013) Chen weather track conditions, so many control decisions can be online. of is stop precision is required (see Wang et (2013) and Chen of ATO can be improved if thecondition basic resistance on and off on the platform with PSD, the more accurate can be estimated estimated online. The The condition of track trackparameters is complex complex stop precision is required (see Wang et al. (2013) and Chen et al. (2013)). of ATO can be improved if the basic resistance parameters and unique, and it is inevitable that the data is et al. (2013)). be estimated online. The condition of collected track is complex stop and unique, and it is inevitable that the collected data is et al.precision (2013)). is required (see Wang et al. (2013) and Chen can can be estimated online. The condition of collected track isthe complex and unique, and it isand inevitable thatbecause the data is often discontinuous abnormal of unexThe basic resistance is one important factor of stop preciet al. (2013)). often discontinuous and abnormal because of the unexThe basic resistance is one important factor of stop preciand unique, and it is inevitable that the collected data is pected interruptions measurement errors. All of these The basic resistance is one important factor of stop precioften discontinuous and abnormal because of the unexsion, which which is is crucial to to the the design design of of ATO. ATO. Many Many factors factors pected interruptions and measurement errors. All of these sion, often discontinuous and abnormal because ofAll theof unexThe basic resistance istoone of stopfactors preciwill increase the difficulty of parameter identification, ession, whichthe is crucial crucial theimportant design ofasfactor ATO. Many pected interruptions and measurement errors. these will effect basic resistance, such the wheel-rail couwill increase the difficulty of parametererrors. identification, eswill effect the basic resistance, such the wheel-rail couinterruptions and measurement All of these sion, which is crucial to the between design ATO. Many factors pecially online identification. will the basic resistance, suchofas as the wheel-rail couwill increase the difficulty of parameter identification, esplingeffect (including the friction friction axle and bearing, and pected pecially online identification. pling (including the between axle and bearing, and will increase the difficulty of parameter identification, eswill effect the basic resistance, such as the wheel-rail coupecially online identification. pling (including the friction between axle and bearing, the friction between wheel and rail), the train’s outer and The of the friction between wheel and rail), the outer and online identification. pling (including the friction between axle and bearing, The development development of related related fields fields such such as as system system identifiidentifithe friction between wheeleffect), and rail), the train’s train’s outer and pecially air coupling coupling (aerodynamic pantograph and network cation, machine learning, and neural networks contributes air (aerodynamic effect), pantograph and network The development of related fields such as system identifithe friction between wheel and rail), the train’s outer and cation, machine learning, and neural networks contributes air coupling (aerodynamic effect), pantograph and network coupling with electric traction, etc. Due to the complex The development of related such as system identification, machine learning, andfields neural networks contributes to the parameter estimation of the mathematical model. coupling with electric traction, etc. Due to the complex air coupling (aerodynamic effect), pantograph and network the parameter estimation of the mathematical model. coupling with electric traction, etc. Due to the complex to cation, machine learning, andalgorithm neural networks contributes This work is supported by the Beijing Laboratory of Urban Yuan (2015) applied genetic to the basic resisto the parameter estimation of the mathematical coupling with electric traction, etc. Due to the complex Yuan (2015) applied genetic algorithm to the basicmodel. resisThis work is supported by the Beijing Laboratory of Urban to the parameter estimation of the mathematical model. tance parameter estimation, but the algorithm has poor Yuan (2015) applied genetic algorithm to the basic Rail Transit, Beijing Key Urban Rail This work the is supported by Laboratory the Beijing ofLaboratory of Transit Urban tance parameter estimation, but the algorithm has resispoor Rail Transit, the Beijing Key Laboratory of Urban Rail Transit Yuan (2015) applied genetic algorithm to the basic resisThis work is supported by the Beijing Laboratory of Urban Automation and Control, the Laboratory research funds of National Natureal-time performance for online identification because of tance parameter estimation, but the algorithm has poor Rail Transit, the Beijing Key of Urban Rail Transit Automation and Control, the research funds of National Natureal-time performance for online identification because of tance parameter estimation, but identification the algorithm has algopoor Rail Transit, the Control, Beijing of Urban Rail Transit ral Science Foundation of Key China under Grant (No. 61790570, No. real-time performance for online because of Automation and the Laboratory research funds of National Natutremendous number of iterations. The least squares ral Science Foundation of China under Grant (No. 61790570, No. tremendous number of iterations. The least squares algoAutomation and Control, the research funds of National Natureal-time performance for online identification because of 61790573), Jiangsu Province Industry University Prospective Joint ral Science Foundation of China under Grant (No. 61790570, No. rithm is widely used, which can realize off-line identifitremendous number of iterations. The least squares algo61790573), Jiangsu Province Industry University Prospective Joint rithm is widely used, which can The realize off-line identifiral Science Foundation of China under Grant (No. 61790570, (No. No. Research Project (BY2015019-29), Beijing Jiaotong University tremendous number ofregression iterations. least squares algo61790573), Jiangsu Province Industry University Prospective cation of both linear model and pseudo-linear rithm is widely used, which can realize off-line identifiResearch Project (BY2015019-29), Beijing Jiaotong University Joint (No. cation of both linear regression and pseudo-linear 61790573), Jiangsu Province Industry University Prospective Joint 2015JBZ007, No. (BY2015019-29), 2017JBM076), and the State Key Laboratory of Research Project Beijing Jiaotong (No. rithm is used, whichevery canmodel realize off-line identification of widely both linear regression model and pseudo-linear regression model. However, parameter of the basic 2015JBZ007, No. 2017JBM076), and the State KeyUniversity Laboratory of Research Project (BY2015019-29), Beijing Jiaotong University (No. regression model. However, every parameter of the basic Rail Traffic Control and Safety (Beijing Jiaotong University) 2015JBZ007, No. 2017JBM076), and the State Key Laboratory of cation of both linear regression model and pseudo-linear regression model. However, every parameter of the basic Rail Traffic Control and Safety (Beijing Jiaotong University) (No. resistance has its own meaning, and the least squares algo2015JBZ007, No.No. 2017JBM076), and and the Jiaotong State Key Laboratory of RCS2016ZJ004, RCS2017ZT013, No. RCS2018ZT012). Rail Traffic Control and Safety (Beijing University) (No. resistance has its own meaning, and the least squares algoregression model. However, every parameter of the basic RCS2016ZJ004, No. RCS2017ZT013, and No. RCS2018ZT012). resistance has its own meaning, and the least squares algoRail Traffic Control and Safety (Beijing University) (No. RCS2016ZJ004, No. RCS2017ZT013, and Jiaotong No. RCS2018ZT012). resistance has its own meaning, and the least squares algoRCS2016ZJ004, No. RCS2017ZT013, and No. RCS2018ZT012).
2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright 631 Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2018 IFAC 631 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 631 10.1016/j.ifacol.2018.09.352 Copyright © 2018 IFAC 631
2018 IFAC ADCHEM 638 Shenyang, Liaoning, China, July 25-27, 2018Xiaoyu Liu et al. / IFAC PapersOnLine 51-18 (2018) 637–642
rithm can not be used for constrained estimation directly, the regularization form of least squares algorithm was introduced to solve this problem (see Haykin (2009)). Furthermore, the recursive form of the least squares algorithm can be used for online identification (see Goodwin and Sin (1984)). Ding et al. (2010) proposed the multi-innovation identification theory and applied it to the recursive least squares (RLS) algorithm. The multi-innovation identification algorithm has good performance in the case of missing data and anomalous data (see Ding and Chen (2007)). Based on the data of ATO, we preprocess the data of the train and use the regularized least squares algorithm to estimate the train basic resistance parameter. We also apply the recursive least squares (RLS) algorithm, multiinnovation least squares (MILS) algorithm, and intervalvarying multi-innovation least squares (V-MILS) algorithm to the basic resistance parameter estimation. These algorithms can not only provide high estimation accuracy but also performs well in the online identification, which will reduce the expense of time, labor, and material for organizing field test. 2. MODELING AND DATA PREPROCESSING 2.1 Modeling The factors that affect the basic resistance are very complex. An early comprehensive study of train resistance was conducted by Davis (1926). Based on the formation mechanism of basic resistance and empirical data, the aerodynamic resistance generated by aerodynamic effect is regarded as the square function of velocity, and the resistance generated by mechanical resistance (such as wheel and rail coupling, pantograph and network coupling) is considered to be a linear function of velocity. The classical Davis formula was described as DX 2 B TR = A + + Cv + v , (W/n) W where the quantity TR is the basic resistance, W and v are the weight and velocity of the train, respectively, n is the number of axles, and X is an effective frontal cross section. The constants A, B, C, and D are empirically adjusted to fit the particular type of train considered (see Davis (1926) and Bernsteen et al. (1980)).
Thus, the Davis formula is rewritten as follows w0 (t) = a(t) + b(t)v(t) + c(t)v 2 (t) + ε(t), (1) where v(t) is the velocity, a(t), b(t), and c(t) are the parameters to be identified, ε(t) is the noise term. Rewrite (1) in the form of an identification model y(t) = ϕT (t)θ(t) + ε(t), (2) where y(t) = w0 (t) is the basic resistance, ϕT (t) = T 1, v(t), v 2 (t) is the information vector consisting of the system input-output data, ε(t) is a stochastic noise with T zero mean, and θ(t) = [a(t), b(t), c(t)] is the parameter vector to be identified. 2.2 Data preprocessing With the basic resistance of the train modelled, according to the model in (2), we need the basic resistance and velocity information for parameter identification. The velocity is recorded as a function of time by ATO during the train’s operation, so the decelaration is obtained by differentiation, and the retarding force is obtained by applying Newton’s second law, when the train is allowed to coast on the track. Train resistance (w) is divided into basic resistance (w0 ) and additional resistance (including gradient resistance (wi ), curve resistance (wr ), and tunnel resistance (ws )), so w = w0 + w i + w r + w s , (3) we can obtain the velocity and acceleration, then the resistance (w) can be computed. Furthermore, when the train is coasting on the track with no tunnel and no curve, we can get the equation (4), ws = 0, wr = 0, and wi = 1000 sin θ, (4) in urban rail transit, the gradient is expressed in tanθ = i0/00 which is very small, so wi ≈ i. Finally, we can get the basic resistance by w0 = −100α − i, (5) 2 where α is acceleration (m/s ). The Yizhuang Line of Beijing Subway is a typical urban rail transit track contains both ground and underground conditions where the train basic resistance will change significantly. The actual data we used in this paper was collected from the Yizhuang Line at the early morning of October, 2016 and January, 2017.
For the convenience of parameter identification, we usually simplify the Davis formula as follows w0 = a + bv + cv 2 , where w0 is unit basic resistance (N/kN), parameter a comprises resistances which are considered independent of speed, but variable with axle load, parameter b contains resistances which are proportional to the first power of the velocity and originates from losses caused by mechanical resistance, and parameter c comprises resistances which are proportional to the square of the velocity and originates from losses caused by aerodynamic resistance.
The data is exported directly from the ATO, containing the target distance, the speed limit, the velocity, the train’s weight, the slope, and so on. The information is recorded in every 200ms. Then, we compute the acceleration by (6). vt+1 − vt , (6) α= T where T = 200ms, vt and vt+1 are velocity at the sampling point t and t + 1, respectively.
However, parameters a, b, and c are slow time-varying parameters due to weather, route, and train conditions. Furthermore, the aerodynamic resistance and mechanical resistance are both a source of resistance and a class of noise source, and the train is disturbed by lots of noise in the process of operation.
It shows in Fig. 1 that the coasting acceleration of the train can be negative, positive or zero, which is abnormal. The environment is complex, the interval time 200ms is too short to obtain accurate data, because of measurement error. In order to solve this problem, the segmented processing method is used for data preprocessing.
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2018 IFAC ADCHEM Shenyang, Liaoning, China, July 25-27, 2018Xiaoyu Liu et al. / IFAC PapersOnLine 51-18 (2018) 637–642
Fig. 1. Unpreprocessed data from ATO system
where
N
t=1
Fig. 2. Method of segmenting data As shown in Fig. 2, the segmentation interval is 0.5km/h, and the velocity change is 0.5km/h at the end of the segmentation, then the second section starts from the middle of the first section and extends 0.5km/h. The velocity data of each segment is obtained, and the acceleration of each section is computed by the least squares estimation method. The basic resistance is calculated by the (5), then we can identify the parameters.
2
(y(t) − ϕT (t)θ) is the criterion of traditional
least squares erstimation, θ 2 is the L2 norm of θ, and N is the length of data, λ is restraint constant, so we obtain the regularized least squares (LS) algorithm N −1 N T ˆ θ= ϕ (t) ϕ (t) +λI ϕ (t) y (t) . (7) t=1
t=1
We use this method to enhance the generalization ability of the algorithm, and obtain the following results:
3. PARAMETER IDENTIFICATION
Least squares (LS) algorithm is effective for linear and nonlinear parametric systems identification. We can get the LS algorithm by minimizing the Euclidean distance. L −1 L T ˆ θ= ϕ (t) y (t) . ϕ (t) ϕ (t) t=1
We use this algorithm to identify the basic resistance parameter, and obtain the results a = 23.74, b = −6.99 × 10−1 , c = 7.23 × 10−3 . As the estimation result and the Fig. 3 show, the parameter b is negative, but the basic resistance can not be negative. Every parameter of the basic resistance has its own meaning, and the LS algorithm can not be used for 633
Rewrite the identification model in section 2 as y(t) = ϕT (t)θ(t) + ε(t), where y(t) = w0 (t) is the basic resistance, ϕT (t) = T 1, v(t), v 2 (t) is the information vector consisting of the system input-output data, ε(t) is a stochastic noise with T zero mean, and θ(t) = [a(t), b(t), c(t)] is the parameter vector to be identified. This model is a linear parameter model.
t=1
constrained identification directly. The regularization form of LS algorithm can be used to solve this problem (see Haykin (2009)), the new criterion function is N λ 2 2 J(θ) = (y(t) − ϕT (t)θ) + θ 2 , 2 t=1
Fig. 3. Traditional least squares algorithm
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Fig. 4. Regularized least squares algorithm w0 = 5.15×10−1 + 8.597 × 10−2 v + 7.512 × 10−4 v 2 . (8) From the Fig. 4 and the equation (8), it can be seen that the identification results are more consistent with the actual situation. The least squares algorithm performs well in the off-line estimation, but the basic resistance changes with the line conditions, the weather, and the train’s own conditions.
2018 IFAC ADCHEM 640 Shenyang, Liaoning, China, July 25-27, 2018Xiaoyu Liu et al. / IFAC PapersOnLine 51-18 (2018) 637–642
Therefore, it is necessary to estimate the train basic resistance parameter online so that the ATO system can adjust its control with more precise information.
The recursive least squares (RLS) algorithm can be used for online identification (see Goodwin and Sin (1984)). Compared with the traditional least squares algorithm, it is more efficient computationally if we update the estimates as new data becomes available online. The recursive least squares algorithm is given by θˆ (t) = θˆ (t − 1) + L (t) y (t) − ϕT (t) θˆ (t − 1) , (9) −1 L(t) = P (t − 1) ϕ(t) 1 + ϕT (t) P (t − 1) ϕ(t) , (10)
T
P (t) = [I − L(t)ϕ (t)]P (t − 1), P (0) = p0 I, (11) ˆ is the estimate of θˆ at time t, θ(0) ˆ where θ(t) = 13 /p0 , 13 is a 3-dimensional column vector whose elements are 1, P (t) ∈ Rn×n is the covariance matrix, p0 is taken to be a large positive number (e.g. p0 = 106 ), I is identity matrix of suitable order (here, the order of I is 3), and the scalar value e(t) = y (t) − ϕT (t) θˆ (t − 1) ∈ R1 was defined as innovation. Equation (9) updates the estimates at each step using the innvation e(t).
Based on (7) and the derivation process of RLS algorithm, in this paper, we derive the regularization form of RLS algorithm (equation (12)-(16)). ˆ − 1), ˆ = θ(t ˆ − 1) + L(t)e(t) − λ P (t)θ(t (12) θ(t) N −1 (13) L(t) = Q(t)ϕ(t) 1 + ϕT (t)Q(t)ϕ(t) , −1 N N I + P (t − 1) , (14) Q(t) = P (t − 1) λ λ P (t) = [I − L(t)ϕT (t)]Q(t), P (0) = p0 I. e(t) = y (t) − ϕT (t) θˆ (t − 1) .
(15)
(16) The flowchart of the algorithm is shown in Fig. 5. We apply this algorithm to the parameter identification of the train basic resistance. Ding et al. (2010) extended the RLS algorithm from the view point of innovation modification and proposed the multi-innovation least squares (MILS) algorithm. The essential idea is to expand the scalar innovation e(t) to an innovation matrix − 1) y(t) − ϕT (t)θ(t − 1) y(t − 1) − ϕT (t − 1)θ(t . ∈ Rp , E(p, t) = . . − 1) y(t − p + 1) − ϕT (t − p + 1)θ(t where the length of the innovation is p, the input and output are also changed to the vector form Φ(p, t) = [ϕ(t), ϕ(t − 1), ..., ϕ(t − p + 1)] ∈ Rn×p , T
Y (p, t) = [y(t), y(t − 1), ..., y(t − p + 1)] ∈ Rp . Then, − 1). E(p, t) = Y (p, t) − ΦT (p, t)θ(t
The MILS algorithm with the innovation length p was given as follows ˆ = θ(t ˆ − 1) + L(t) Y (p, t) − ΦT (p, t)θ(t ˆ − 1) , θ(t) 634
Fig. 5. The flowchart of the regularized RLS algorithm. L(t) = P (t−1)Φ(p, t) Ip + ΦT (p, t)P (t−1)Φ(p, t) −1, P (t) = P (t − 1) − L(t)ΦT (p, t)P (t − 1), Φ(p, t) = [ϕ(t), ϕ(t − 1), ..., ϕ(t − p + 1)] , T
Y (p, t) = [y(t), y(t − 1), ..., y(t − p + 1)] , where Ip denotes an identity matrix of order p. When p = 1, the MILS algorithm reduces to RLS algorithm. We derive the regularization form of MILS algorithm for train basic resistance parameter estimation. ˆ − 1), ˆ = θ(t ˆ − 1) + L(t)E(p, t) − λ P (t)θ(t θ(t) N −1 L(t) = Q(t)Φ(p, t) Ip + ΦT (p, t)Q(t)Φ(p, t) , −1 N N I + P (t − 1) , Q(t) = P (t − 1) λ λ P (t) = [I − L(t)ΦT (p, t)]Q(t), − 1), E(p, t) = Y (p, t) − ΦT (p, t)θ(t
Φ(p, t) = [ϕ(t), ϕ(t − 1), ..., ϕ(t − p + 1)] , T
Y (p, t) = [y(t), y(t − 1), ..., y(t − p + 1)] . The MILS algorithm extends the length of the innovation. Because the MILS algorithm use not only the current data but also the past data at each recursive step, parameter estimation accuracy can be improved, and the algorithm can also be used for online estimation. We can obtain the results of the RLS algorithm (or the MILS algorithm with p = 1) and the MILS algorithm (with p = 2, p = 4, and p = 6) from Fig. 6. For comparison,
2018 IFAC ADCHEM Shenyang, Liaoning, China, July 25-27, 2018Xiaoyu Liu et al. / IFAC PapersOnLine 51-18 (2018) 637–642
LS RLS p=2 MILS p = 4 p=6
b (10−2 ) 8.597 8.511 5.749 3.591 1.941
c (10−3 ) 0.751 0.758 0.981 1.422 1.262
(10−1 ) 5.632 5.632 5.623 5.618 5.613
Table 1. Parameter identification results a (100 ) 0.515 0.783 0.932 1.638 2.194
Fig. 7. Comparison of LS and MILS algorithms.
table 1 lists the differences between the results of LS algorithm, RLS algorithm and the MILS algorithm. σ2
Fig. 6. MILS algorithm with p = 1, p = 2, p = 4, p = 6.
Algorithms
641
Online Yes/No No Yes Yes Yes Yes
D
E
W
In table 1, a, b, and c are basic resistance parameters, and σ 2 represents the variance. The given results show that with the increase of the length of the innovation, the variance of the identification results is decreasing, so the MILS algorithm can achieve better performance than RLS algorithm. Both RLS and MILS algorithms can be used for online identification. 4. ANALYSIS AND VERIFICATION 4.1 Verification of results We use a set of data for validation. The basic resistance is calculated using the identification results obtained by the LS and MILS algorithms, and then we use the basic resistance to calculate the acceleration and the velocity by (5) and (6). We compare the velocities calculated from LS algorithm and MILS algorithm with the actual velocity and obtain the Fig. 7 and table 2. Table 2. Comparison of different algorithms Algorithms LS MILS
Variances (σ 2 = 10−2 ) 0.99 0.88
F
Fig. 8. Parameter online identification. significantly, the stabilization of parameters demonstrate the real-time characteristic and validity of the algorithm. Generally, the least squares algorithm and RLS algorithm have fast convergence properties and high precision, and the usage of the data is efficient. Thus, sometimes we use the MILS algorithm, the improvement of the parameter estimation accuracy is limited, and we can use LS algorithm for off-line identification and RLS algorithm for online identification. However, the condition of the train and the track is complex, so it is inevitable that the collected data is often discontinuous and abnormal because of the unexpected interruption and measurement error. The multi-innovation theory based identification algorithm has strong robustness, and can improve the accuracy of identification greatly when we encounter this abnormal situation. 4.2 Analysis of robustness
The simulation results shows that the estimation results become more accurate by applying the MILS algorithm to the basic resistance parameter estimation. Parameter online identification results are depicted in Fig. 8, it can be observed from this figure that after about 70 recursions, the value of a, b, and c no longer change 635
There are strict restrictions on the data used for parameter estimation. The results of parameter identification can be more accurate only when the train coasts on the level tangent track. Therefore, the data used for basic resistance parameter estimation collected by ATO usually discontinuous. The train operating on track with complex
2018 IFAC ADCHEM 642 Shenyang, Liaoning, China, July 25-27, 2018Xiaoyu Liu et al. / IFAC PapersOnLine 51-18 (2018) 637–642
conditions, ATO often collects discontinuous data and anomalous data because of the unexpected interruption and measurement error. These phenomena are inevitable and will increase the difficulty of parameter identification especially online identification. The multi-innovation theory based identification algorithm also has good performance in the case of anomalous data and missing-data and can provide fast convergence and high parameter estimation accuracy, the interval-varying multi-innovation least squares (V-MILS) algorithm was given by Ding et al. (2010), ˆ s−1 ) + L(ts ) Y (p, ts ) − ΦT (p, ts ) θ(t ˆ s−1 ) , ˆ s ) = θ(t θ(t L (ts ) =P (ts−1 ) Φ (p, ts ) −1 × Ip + ΦT (p, ts ) P (ts−1 ) Φ (p, ts ) ,
P (ts ) = P (ts−1 ) − L (ts ) ΦT (p, ts ) P (ts−1 ) , Φ(p, ts ) = [ϕ(ts ), ϕ(ts − 1), ..., ϕ(ts − p + 1)] , T
Y (p, ts ) = [y(ts ), y(ts − 1), ..., y(ts − p + 1)] , where 0 = t0 < t1 < t2 < ..., and 1 ≤ t∗s = ts − ts−1 . The parameter estimate θˆ (t) is updated only at instant t = ts , and so is the convariance matrix P . The V-MILS algorithm computes the parameter estimates using the interval-varying iteration, so it can overcome the affect of bad data on the parameter estimates. We also derive the regularization form of V-MILS algorithm for the train basic resistance parameter estimation ˆ s ) = θ(t ˆ s−1 ) + L(ts )E(p, ts ) − λ P (ts )θ(t ˆ s−1 ), θ(t N −1 L(ts ) = Q(ts )Φ(p, ts ) Ip + ΦT (p, ts )Q(ts )Φ(p, ts ) , −1 N N Q(ts ) = P (ts−1 ) I + P (ts−1 ) , λ λ P (ts ) = [I − L(ts )ΦT (p, ts )]Q(ts ), ˆ s−1 ), E(p, ts ) = Y (p, ts ) − ΦT (p, ts )θ(t
Φ(p, ts ) = [ϕ(ts ), ϕ(ts − 1), ..., ϕ(ts − p + 1)] , T
Y (p, ts ) = [y(ts ), y(ts − 1), ..., y(ts − p + 1)] . We chose a set of data that contain anomalous data collected from the Yizhaung Line for verification.
Fig. 9. Comparison of LS and V-MILS algorithms. 636
The given results in Fig. 9 show that the performance of traditional LS algorithm is poor in the case anomalous data, but the V-MILS algorithm can skip the anomalous data so that we can get more reliable parameters. 5. CONCLUSION The multi-innovation theory based identification algorithm can improve the parameter estimation accuracy, we applied it to the basic resistance parameter estimation using the data collected from Yizhaung Line of Beijing Subway. By comparing the MILS algorithm and V-MILS algorithm with traditional least squares algorithm, we conclude that the multi-innovation least squares algorithm can not only provide high estimation accuracy but also perform well in the case of missing data and anomalous data. The MILS algorithm and V-MILS algorithm can also be used for online identification, which will make the train basic resistance parameter estimation simpler and more accurate, and will reduce the expense of time, labor, and material for organizing field test. REFERENCES Bernsteen, S.A., Uher, R.A., and Romualdi, J.P. (1980). The interpretation of train rolling resistance from fundamental mechanics. Industry Applications IEEE Transactions on, IA-19(5), 802-817. Chen, D.W., Chen, R., Li, Y.D., and Tang, T. (2013). Online learning algorithms for train automatic stop control using precise location data of balises. IEEE Transactions on Intelligent Transportation Systems, 14(3), 15261535. Davis, W.J. (1926). The tractive resistance of electric locomotives and cars. General Electr. Rev., 29(10), 685708. Ding, F. and Chen, T. (2007). Performance analysis of multi-innovation gradient type identification methods. Automatica, 43(1), 1-14. Ding, F., Liu, P.X., and Liu, G. (2010). Multiinnovation least-squares identification for system modeling. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 40(3), 767-778. Goodwin, G.C. and Sin, K.S. (1984). Adaptive Filtering, Prediction and Control. Englewood Cliffs, NJ: PrenticeHall. Haykin, S.S. (2009). Neural networks and learning machines. Pearson Schweiz Ag. Huang, W.Y., Yang, N.Q., and Huang, M. (2000). Ponderation on Railway Train Basic Resistance. China Railwayence, 21(3), 44-57. Wang, C., Tang, T., and Luo, R.S. (2013). Study on iterative learning control in automatic train operation. Journal of the China Railway Society, 35(3), 48-52. Yu, Z.Y. and Chen, D.W. (2011). Modeling and system identification of the braking system of urban rail vehicles. Journal of the China Railway Society, 33(10), 37-40. Yuan, L. (2015). Train Basic Resistance Identification and Its Online Update Algorithm. Journal of Information & Computational Science, 12(11), 4161-4171.
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IFAC PapersOnLine 51-18 (2018) 637–642
Parameter Identification of Train Basic Parameter Identification of Train Basic Parameter Identification of Train Basic Resistance Using Multi-Innovation Theory Parameter Identification of Train Basic Resistance Using Multi-Innovation Theory Resistance Using Multi-Innovation Theory Resistance Using Multi-Innovation Theory Xiaoyu Liu ∗∗ Bin Ning ∗∗ Jing Xun ∗∗ Cheng Wang ∗∗ ∗∗
Xiaoyu Liu ∗∗ Bin Ning ∗∗ ∗∗∗ Jing Xun ∗∗ Cheng Wang ∗∗ ∗∗ ∗ Xiao Xiao Tong Xiaoyu Liu ∗ Bin Ning Jing XunLiu Cheng Wang ∗∗ ∗ ∗ ∗∗∗ ∗ Xiao Xiao Tong Liu ∗∗∗ ∗ ∗ Xiaoyu Liu Bin Jing XunLiuCheng Wang XiaoNing Xiao ∗∗∗ Tong Xiao Xiao ∗∗∗ Tong Liu ∗ ∗ ∗ State Key Laboratory of Rail Traffic Control and Safety, Beijing Key Laboratory of Rail Traffic Control and Safety, Beijing ∗ ∗ StateJiaotong University, Beijing, 100044, (e-mail: StateJiaotong Key Laboratory of Rail Traffic ControlChina and Safety, ∗ University, Beijing, 100044, (e-mail:Beijing StateJiaotong Key Laboratory of Rail Traffic ControlChina and Safety, [email protected]) University, Beijing, 100044, China (e-mail:Beijing [email protected]) ∗∗ Jiaotong University, Beijing, 100044, China (e-mail: [email protected]) ∗∗ School of Internet of Things Engineering, Jiangnan University, of Internet of Things Engineering, Jiangnan ∗∗ ∗∗ School [email protected]) School of 214122, InternetChina of Things Engineering, Jiangnan University, University, Wuxi, (e-mail: [email protected]) ∗∗ Wuxi, 214122, China (e-mail: [email protected]) ∗∗∗ School of Internet of Things Engineering, Jiangnan University, Traffic Control Technology CO., Ltd. Beijing Research Wuxi, 214122, China (e-mail: [email protected]) ∗∗∗ Control Technology CO., Ltd. Beijing Research Institute, Institute, ∗∗∗ Traffic ∗∗∗ Wuxi, 214122, China (e-mail: [email protected]) Beijing, 100070, China (e-mail: [email protected]) Traffic Control Technology CO., Ltd. Beijing Research Institute, ∗∗∗ Beijing, 100070, China (e-mail: [email protected]) Traffic Control Technology CO., Ltd. Beijing Research Institute, Beijing, 100070, China (e-mail: [email protected]) Beijing, 100070, China (e-mail: [email protected]) Abstract: Train Train basic basic resistance resistance is is important for for the design of the automatic train operation, Abstract: the of automatic operation, Abstract: Trainthe basic resistance is important important forprecision, the design design of the theconsumption, automatic train train operation, which influences efficiency, punctuality, stop energy and the which influences the efficiency, punctuality, stop precision, energy consumption, and the safety safety Abstract: Train basic resistance istheory important forprecision, the design of theconsumption, automatic train operation, of the train. The multi-innovation is a novel concept which can improve the accuracy of which influences the efficiency, punctuality, stop energy and the safety of the train. The the multi-innovation theory is stop a novel conceptenergy which consumption, can improve the accuracy of which influences efficiency, punctuality, precision, and the safety parameter estimation and be used to modify the traditional recursive least squares algorithm. of the train. The multi-innovation theory is a novel concept which can improve the accuracy of parameter estimation and be used theory to modify traditional recursive least squares algorithm. of The isform a the novel concept which can improve the accuracy of In the thistrain. paper, we multi-innovation derive theberegularization regularization of the multi-innovation least squares algorithm parameter estimation and used to modify the traditional recursive least squares algorithm. In this paper, we derive the form of the multi-innovation least squares algorithm parameter estimation and be used to modify the traditional recursive least squares algorithm. In this paper, we derive the regularization form of the multi-innovation least squares algorithm and apply apply it it to to the the train train basic basic resistance estimation. The results based and resistance parameter parameter The simulation simulation results based In this paper, wethe derive the regularization form ofthat, theestimation. multi-innovation least squares algorithm on the Yizhuang Line of Beijing Subway indicate compared with traditional least squares and apply it to train basic resistance parameter estimation. The simulation results based on the Yizhuang Linetrain of Beijing Subway indicate that,estimation. compared with traditional results least squares and apply it to the basic resistance parameter The simulation based algorithm, the multi-innovation least squares algorithm can provide higher estimation accuracy on the Yizhuang Line of Beijing Subway indicate that, compared with traditional least squares algorithm, the multi-innovation least squares can provide higher estimation algorithm accuracy on the Yizhuang Line of Beijing Subway indicate that, compared with traditional least squares and robustness, and can be usedleast for online identification. algorithm, the multi-innovation squares algorithm can provide higher estimation accuracy and and for identification. algorithm, the multi-innovation squares algorithm can provide higher estimation accuracy and robustness, robustness, and can can be be used usedleast for online online identification. © 2018, IFAC (International Federation Automatic Control) Hosting by Elsevier Ltd. All rights reserved. and robustness, and can be used forofonline identification. Keywords: Train Train basic basic resistance, resistance, Multi-innovation identification, Recursive identification, identification, Keywords: Multi-innovation identification, Keywords: Train basic resistance, Multi-innovation identification, Recursive Recursive identification, Parameter estimation, Urban rail transit Parameter estimation, Urban transit Keywords: basic resistance, identification, Recursive identification, Parameter Train estimation, Urban rail rail Multi-innovation transit Parameter estimation, Urban rail transit 1. INTRODUCTION constitution 1. constitution of of basic basic resistance, resistance, aa large large number number of of experexper1. INTRODUCTION INTRODUCTION imental data are integrated according to the formation constitution of basic resistance, a large number of experimental data are integrated according to the formation INTRODUCTION ofare basic resistance, a large number of experimental dataand integrated according to the formation mechanism, the classical formula was proposed Urban rail rail transit transit1. has has many advantages, advantages, such such as as large large constitution mechanism, and the classical Davis Davis formula wasformation proposed Urban many imental data are integrated according to the (see Davis (1926), Bernsteen et al. (1980), and Huang Urban rail transit has many advantages, such as large mechanism, and the classical Davis formula was proposed capacity, high high efficiency, efficiency, punctuality, punctuality, reliability reliability and and safety, safety, (see Davis (1926), Bernsteen et al. (1980), and Huang et et al. al. capacity, and the classicaletDavis formula was proposed Urban rail transit has punctuality, many ofadvantages, such assafety, large mechanism, (2000), and Yuan (2015)). capacity, high efficiency, reliability and (see Davis (1926), Bernsteen al. (1980), and Huang et al. so it becomes the backbone urban public transport. (2000), and Yuan (2015)). so it the backbone of public transport. Davis capacity, high efficiency, punctuality, reliability and safety, (2000), and(1926), Yuan Bernsteen (2015)). et al. (1980), and Huang et al. so it becomes becomes theoperation backbone of urban urban public transport. Automatic train (ATO) plays a key key role in (see Currently, aaYuan domestic enterprise Automatic train operation (ATO) plays a role in (2000), and (2015)). so it becomes the backbone of urban public transport. Currently, domestic enterprise needs needs one one experimental experimental Automatic operation (ATO) plays aofkey rolerail in ensuring the thetrain safety and efficient efficient operation urban procedure called coast-down test to determine the paramensuring safety and operation of urban rail Currently, a domestic enterprise needs one experimental Automatic operation (ATO) plays aofdensity key role(see in procedure called coast-down test to determine the paramensuring thetrain safety andhigh efficient operation urban rail transit operating with speed and high Currently, a resistance domestic enterprise needs one experimental procedure called coast-down test to determine the parameter of basic at the expense of time, labor, and transit operating with high speed and high density (see ensuring the safety andIn efficient operation urban (see rail eter of basic resistance at the expense of time, labor, and transit operating with high speed and with highof density Yu and Chen (2011)). recent years, development procedure called coast-down test to determine the parameter of basic resistance at the expense of time, labor, and material, so it is necessary to find a method that does Yu and Chen (2011)). In recent years, with development transit operating with high speed and high density (see material, so resistance it is necessary toexpense find a of method that does Yu and Chen (2011)). Incontrol recent technology, years, with the development of communication and platform eter of basic at the time, labor, and not need coast-down test (see Bernsteen et al. (1980) and of communication and control technology, the platform material, so it is necessary to find a method that does Yu anddoors Chen (PSD) (2011)). Incontrol recentwidely years, used. with the development not need coast-down test (seetoBernsteen et al. (1980) and of communication and technology, platform screen has been In order to material, so it is necessary find a method that does Yuan (2015)). The train resistance depends strongly screen doors (PSD) has been widely used. In order to need coast-down testbasic (see Bernsteen et al. (1980) and of communication control technology, the Yuan (2015)). The train basic resistance depends strongly screen doors (PSD)and hasare been used. In platform order to not ensure that passengers passengers not widely disturbed when they get get not need coast-down test (see Bernsteen et al. (1980) and Yuan (2015)). The train basic resistance depends strongly on weather and track conditions, so many control decisions ensure that are not disturbed when they screen (PSD) hasarebeen used. In order to on weather andThe track conditions, so manydepends control decisions ensure that not widely disturbed when they get on and and doors off on onpassengers the platform platform with PSD, the the more accurate Yuan (2015)). train basic resistance strongly of ATO can be improved if the basic resistance parameters on off the with PSD, more accurate on weather and track conditions, so many control decisions ensure that are not disturbed when they get on of ATO canand be improved if the basic resistance parameters on off onpassengers the platform PSD, the moreand accurate stopand precision is required (seewith Wang et al. al. (2013) Chen weather track conditions, so many control decisions can be online. of is stop precision is required (see Wang et (2013) and Chen of ATO can be improved if thecondition basic resistance on and off on the platform with PSD, the more accurate can be estimated estimated online. The The condition of track trackparameters is complex complex stop precision is required (see Wang et al. (2013) and Chen et al. (2013)). of ATO can be improved if the basic resistance parameters and unique, and it is inevitable that the data is et al. (2013)). be estimated online. The condition of collected track is complex stop and unique, and it is inevitable that the collected data is et al.precision (2013)). is required (see Wang et al. (2013) and Chen can can be estimated online. The condition of collected track isthe complex and unique, and it isand inevitable thatbecause the data is often discontinuous abnormal of unexThe basic resistance is one important factor of stop preciet al. (2013)). often discontinuous and abnormal because of the unexThe basic resistance is one important factor of stop preciand unique, and it is inevitable that the collected data is pected interruptions measurement errors. All of these The basic resistance is one important factor of stop precioften discontinuous and abnormal because of the unexsion, which which is is crucial to to the the design design of of ATO. ATO. Many Many factors factors pected interruptions and measurement errors. All of these sion, often discontinuous and abnormal because ofAll theof unexThe basic resistance istoone of stopfactors preciwill increase the difficulty of parameter identification, ession, whichthe is crucial crucial theimportant design ofasfactor ATO. Many pected interruptions and measurement errors. these will effect basic resistance, such the wheel-rail couwill increase the difficulty of parametererrors. identification, eswill effect the basic resistance, such the wheel-rail couinterruptions and measurement All of these sion, which is crucial to the between design ATO. Many factors pecially online identification. will the basic resistance, suchofas as the wheel-rail couwill increase the difficulty of parameter identification, esplingeffect (including the friction friction axle and bearing, and pected pecially online identification. pling (including the between axle and bearing, and will increase the difficulty of parameter identification, eswill effect the basic resistance, such as the wheel-rail coupecially online identification. pling (including the friction between axle and bearing, the friction between wheel and rail), the train’s outer and The of the friction between wheel and rail), the outer and online identification. pling (including the friction between axle and bearing, The development development of related related fields fields such such as as system system identifiidentifithe friction between wheeleffect), and rail), the train’s train’s outer and pecially air coupling coupling (aerodynamic pantograph and network cation, machine learning, and neural networks contributes air (aerodynamic effect), pantograph and network The development of related fields such as system identifithe friction between wheel and rail), the train’s outer and cation, machine learning, and neural networks contributes air coupling (aerodynamic effect), pantograph and network coupling with electric traction, etc. Due to the complex The development of related such as system identification, machine learning, andfields neural networks contributes to the parameter estimation of the mathematical model. coupling with electric traction, etc. Due to the complex air coupling (aerodynamic effect), pantograph and network the parameter estimation of the mathematical model. coupling with electric traction, etc. Due to the complex to cation, machine learning, andalgorithm neural networks contributes This work is supported by the Beijing Laboratory of Urban Yuan (2015) applied genetic to the basic resisto the parameter estimation of the mathematical coupling with electric traction, etc. Due to the complex Yuan (2015) applied genetic algorithm to the basicmodel. resisThis work is supported by the Beijing Laboratory of Urban to the parameter estimation of the mathematical model. tance parameter estimation, but the algorithm has poor Yuan (2015) applied genetic algorithm to the basic Rail Transit, Beijing Key Urban Rail This work the is supported by Laboratory the Beijing ofLaboratory of Transit Urban tance parameter estimation, but the algorithm has resispoor Rail Transit, the Beijing Key Laboratory of Urban Rail Transit Yuan (2015) applied genetic algorithm to the basic resisThis work is supported by the Beijing Laboratory of Urban Automation and Control, the Laboratory research funds of National Natureal-time performance for online identification because of tance parameter estimation, but the algorithm has poor Rail Transit, the Beijing Key of Urban Rail Transit Automation and Control, the research funds of National Natureal-time performance for online identification because of tance parameter estimation, but identification the algorithm has algopoor Rail Transit, the Control, Beijing of Urban Rail Transit ral Science Foundation of Key China under Grant (No. 61790570, No. real-time performance for online because of Automation and the Laboratory research funds of National Natutremendous number of iterations. The least squares ral Science Foundation of China under Grant (No. 61790570, No. tremendous number of iterations. The least squares algoAutomation and Control, the research funds of National Natureal-time performance for online identification because of 61790573), Jiangsu Province Industry University Prospective Joint ral Science Foundation of China under Grant (No. 61790570, No. rithm is widely used, which can realize off-line identifitremendous number of iterations. The least squares algo61790573), Jiangsu Province Industry University Prospective Joint rithm is widely used, which can The realize off-line identifiral Science Foundation of China under Grant (No. 61790570, (No. No. Research Project (BY2015019-29), Beijing Jiaotong University tremendous number ofregression iterations. least squares algo61790573), Jiangsu Province Industry University Prospective cation of both linear model and pseudo-linear rithm is widely used, which can realize off-line identifiResearch Project (BY2015019-29), Beijing Jiaotong University Joint (No. cation of both linear regression and pseudo-linear 61790573), Jiangsu Province Industry University Prospective Joint 2015JBZ007, No. (BY2015019-29), 2017JBM076), and the State Key Laboratory of Research Project Beijing Jiaotong (No. rithm is used, whichevery canmodel realize off-line identification of widely both linear regression model and pseudo-linear regression model. However, parameter of the basic 2015JBZ007, No. 2017JBM076), and the State KeyUniversity Laboratory of Research Project (BY2015019-29), Beijing Jiaotong University (No. regression model. However, every parameter of the basic Rail Traffic Control and Safety (Beijing Jiaotong University) 2015JBZ007, No. 2017JBM076), and the State Key Laboratory of cation of both linear regression model and pseudo-linear regression model. However, every parameter of the basic Rail Traffic Control and Safety (Beijing Jiaotong University) (No. resistance has its own meaning, and the least squares algo2015JBZ007, No.No. 2017JBM076), and and the Jiaotong State Key Laboratory of RCS2016ZJ004, RCS2017ZT013, No. RCS2018ZT012). Rail Traffic Control and Safety (Beijing University) (No. resistance has its own meaning, and the least squares algoregression model. However, every parameter of the basic RCS2016ZJ004, No. RCS2017ZT013, and No. RCS2018ZT012). resistance has its own meaning, and the least squares algoRail Traffic Control and Safety (Beijing University) (No. RCS2016ZJ004, No. RCS2017ZT013, and Jiaotong No. RCS2018ZT012). resistance has its own meaning, and the least squares algoRCS2016ZJ004, No. RCS2017ZT013, and No. RCS2018ZT012).
2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright 631 Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2018 IFAC 631 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 631 10.1016/j.ifacol.2018.09.352 Copyright © 2018 IFAC 631
2018 IFAC ADCHEM 638 Shenyang, Liaoning, China, July 25-27, 2018Xiaoyu Liu et al. / IFAC PapersOnLine 51-18 (2018) 637–642
rithm can not be used for constrained estimation directly, the regularization form of least squares algorithm was introduced to solve this problem (see Haykin (2009)). Furthermore, the recursive form of the least squares algorithm can be used for online identification (see Goodwin and Sin (1984)). Ding et al. (2010) proposed the multi-innovation identification theory and applied it to the recursive least squares (RLS) algorithm. The multi-innovation identification algorithm has good performance in the case of missing data and anomalous data (see Ding and Chen (2007)). Based on the data of ATO, we preprocess the data of the train and use the regularized least squares algorithm to estimate the train basic resistance parameter. We also apply the recursive least squares (RLS) algorithm, multiinnovation least squares (MILS) algorithm, and intervalvarying multi-innovation least squares (V-MILS) algorithm to the basic resistance parameter estimation. These algorithms can not only provide high estimation accuracy but also performs well in the online identification, which will reduce the expense of time, labor, and material for organizing field test. 2. MODELING AND DATA PREPROCESSING 2.1 Modeling The factors that affect the basic resistance are very complex. An early comprehensive study of train resistance was conducted by Davis (1926). Based on the formation mechanism of basic resistance and empirical data, the aerodynamic resistance generated by aerodynamic effect is regarded as the square function of velocity, and the resistance generated by mechanical resistance (such as wheel and rail coupling, pantograph and network coupling) is considered to be a linear function of velocity. The classical Davis formula was described as DX 2 B TR = A + + Cv + v , (W/n) W where the quantity TR is the basic resistance, W and v are the weight and velocity of the train, respectively, n is the number of axles, and X is an effective frontal cross section. The constants A, B, C, and D are empirically adjusted to fit the particular type of train considered (see Davis (1926) and Bernsteen et al. (1980)).
Thus, the Davis formula is rewritten as follows w0 (t) = a(t) + b(t)v(t) + c(t)v 2 (t) + ε(t), (1) where v(t) is the velocity, a(t), b(t), and c(t) are the parameters to be identified, ε(t) is the noise term. Rewrite (1) in the form of an identification model y(t) = ϕT (t)θ(t) + ε(t), (2) where y(t) = w0 (t) is the basic resistance, ϕT (t) = T 1, v(t), v 2 (t) is the information vector consisting of the system input-output data, ε(t) is a stochastic noise with T zero mean, and θ(t) = [a(t), b(t), c(t)] is the parameter vector to be identified. 2.2 Data preprocessing With the basic resistance of the train modelled, according to the model in (2), we need the basic resistance and velocity information for parameter identification. The velocity is recorded as a function of time by ATO during the train’s operation, so the decelaration is obtained by differentiation, and the retarding force is obtained by applying Newton’s second law, when the train is allowed to coast on the track. Train resistance (w) is divided into basic resistance (w0 ) and additional resistance (including gradient resistance (wi ), curve resistance (wr ), and tunnel resistance (ws )), so w = w0 + w i + w r + w s , (3) we can obtain the velocity and acceleration, then the resistance (w) can be computed. Furthermore, when the train is coasting on the track with no tunnel and no curve, we can get the equation (4), ws = 0, wr = 0, and wi = 1000 sin θ, (4) in urban rail transit, the gradient is expressed in tanθ = i0/00 which is very small, so wi ≈ i. Finally, we can get the basic resistance by w0 = −100α − i, (5) 2 where α is acceleration (m/s ). The Yizhuang Line of Beijing Subway is a typical urban rail transit track contains both ground and underground conditions where the train basic resistance will change significantly. The actual data we used in this paper was collected from the Yizhuang Line at the early morning of October, 2016 and January, 2017.
For the convenience of parameter identification, we usually simplify the Davis formula as follows w0 = a + bv + cv 2 , where w0 is unit basic resistance (N/kN), parameter a comprises resistances which are considered independent of speed, but variable with axle load, parameter b contains resistances which are proportional to the first power of the velocity and originates from losses caused by mechanical resistance, and parameter c comprises resistances which are proportional to the square of the velocity and originates from losses caused by aerodynamic resistance.
The data is exported directly from the ATO, containing the target distance, the speed limit, the velocity, the train’s weight, the slope, and so on. The information is recorded in every 200ms. Then, we compute the acceleration by (6). vt+1 − vt , (6) α= T where T = 200ms, vt and vt+1 are velocity at the sampling point t and t + 1, respectively.
However, parameters a, b, and c are slow time-varying parameters due to weather, route, and train conditions. Furthermore, the aerodynamic resistance and mechanical resistance are both a source of resistance and a class of noise source, and the train is disturbed by lots of noise in the process of operation.
It shows in Fig. 1 that the coasting acceleration of the train can be negative, positive or zero, which is abnormal. The environment is complex, the interval time 200ms is too short to obtain accurate data, because of measurement error. In order to solve this problem, the segmented processing method is used for data preprocessing.
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2018 IFAC ADCHEM Shenyang, Liaoning, China, July 25-27, 2018Xiaoyu Liu et al. / IFAC PapersOnLine 51-18 (2018) 637–642
Fig. 1. Unpreprocessed data from ATO system
where
N
t=1
Fig. 2. Method of segmenting data As shown in Fig. 2, the segmentation interval is 0.5km/h, and the velocity change is 0.5km/h at the end of the segmentation, then the second section starts from the middle of the first section and extends 0.5km/h. The velocity data of each segment is obtained, and the acceleration of each section is computed by the least squares estimation method. The basic resistance is calculated by the (5), then we can identify the parameters.
2
(y(t) − ϕT (t)θ) is the criterion of traditional
least squares erstimation, θ 2 is the L2 norm of θ, and N is the length of data, λ is restraint constant, so we obtain the regularized least squares (LS) algorithm N −1 N T ˆ θ= ϕ (t) ϕ (t) +λI ϕ (t) y (t) . (7) t=1
t=1
We use this method to enhance the generalization ability of the algorithm, and obtain the following results:
3. PARAMETER IDENTIFICATION
Least squares (LS) algorithm is effective for linear and nonlinear parametric systems identification. We can get the LS algorithm by minimizing the Euclidean distance. L −1 L T ˆ θ= ϕ (t) y (t) . ϕ (t) ϕ (t) t=1
We use this algorithm to identify the basic resistance parameter, and obtain the results a = 23.74, b = −6.99 × 10−1 , c = 7.23 × 10−3 . As the estimation result and the Fig. 3 show, the parameter b is negative, but the basic resistance can not be negative. Every parameter of the basic resistance has its own meaning, and the LS algorithm can not be used for 633
Rewrite the identification model in section 2 as y(t) = ϕT (t)θ(t) + ε(t), where y(t) = w0 (t) is the basic resistance, ϕT (t) = T 1, v(t), v 2 (t) is the information vector consisting of the system input-output data, ε(t) is a stochastic noise with T zero mean, and θ(t) = [a(t), b(t), c(t)] is the parameter vector to be identified. This model is a linear parameter model.
t=1
constrained identification directly. The regularization form of LS algorithm can be used to solve this problem (see Haykin (2009)), the new criterion function is N λ 2 2 J(θ) = (y(t) − ϕT (t)θ) + θ 2 , 2 t=1
Fig. 3. Traditional least squares algorithm
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Fig. 4. Regularized least squares algorithm w0 = 5.15×10−1 + 8.597 × 10−2 v + 7.512 × 10−4 v 2 . (8) From the Fig. 4 and the equation (8), it can be seen that the identification results are more consistent with the actual situation. The least squares algorithm performs well in the off-line estimation, but the basic resistance changes with the line conditions, the weather, and the train’s own conditions.
2018 IFAC ADCHEM 640 Shenyang, Liaoning, China, July 25-27, 2018Xiaoyu Liu et al. / IFAC PapersOnLine 51-18 (2018) 637–642
Therefore, it is necessary to estimate the train basic resistance parameter online so that the ATO system can adjust its control with more precise information.
The recursive least squares (RLS) algorithm can be used for online identification (see Goodwin and Sin (1984)). Compared with the traditional least squares algorithm, it is more efficient computationally if we update the estimates as new data becomes available online. The recursive least squares algorithm is given by θˆ (t) = θˆ (t − 1) + L (t) y (t) − ϕT (t) θˆ (t − 1) , (9) −1 L(t) = P (t − 1) ϕ(t) 1 + ϕT (t) P (t − 1) ϕ(t) , (10)
T
P (t) = [I − L(t)ϕ (t)]P (t − 1), P (0) = p0 I, (11) ˆ is the estimate of θˆ at time t, θ(0) ˆ where θ(t) = 13 /p0 , 13 is a 3-dimensional column vector whose elements are 1, P (t) ∈ Rn×n is the covariance matrix, p0 is taken to be a large positive number (e.g. p0 = 106 ), I is identity matrix of suitable order (here, the order of I is 3), and the scalar value e(t) = y (t) − ϕT (t) θˆ (t − 1) ∈ R1 was defined as innovation. Equation (9) updates the estimates at each step using the innvation e(t).
Based on (7) and the derivation process of RLS algorithm, in this paper, we derive the regularization form of RLS algorithm (equation (12)-(16)). ˆ − 1), ˆ = θ(t ˆ − 1) + L(t)e(t) − λ P (t)θ(t (12) θ(t) N −1 (13) L(t) = Q(t)ϕ(t) 1 + ϕT (t)Q(t)ϕ(t) , −1 N N I + P (t − 1) , (14) Q(t) = P (t − 1) λ λ P (t) = [I − L(t)ϕT (t)]Q(t), P (0) = p0 I. e(t) = y (t) − ϕT (t) θˆ (t − 1) .
(15)
(16) The flowchart of the algorithm is shown in Fig. 5. We apply this algorithm to the parameter identification of the train basic resistance. Ding et al. (2010) extended the RLS algorithm from the view point of innovation modification and proposed the multi-innovation least squares (MILS) algorithm. The essential idea is to expand the scalar innovation e(t) to an innovation matrix − 1) y(t) − ϕT (t)θ(t − 1) y(t − 1) − ϕT (t − 1)θ(t . ∈ Rp , E(p, t) = . . − 1) y(t − p + 1) − ϕT (t − p + 1)θ(t where the length of the innovation is p, the input and output are also changed to the vector form Φ(p, t) = [ϕ(t), ϕ(t − 1), ..., ϕ(t − p + 1)] ∈ Rn×p , T
Y (p, t) = [y(t), y(t − 1), ..., y(t − p + 1)] ∈ Rp . Then, − 1). E(p, t) = Y (p, t) − ΦT (p, t)θ(t
The MILS algorithm with the innovation length p was given as follows ˆ = θ(t ˆ − 1) + L(t) Y (p, t) − ΦT (p, t)θ(t ˆ − 1) , θ(t) 634
Fig. 5. The flowchart of the regularized RLS algorithm. L(t) = P (t−1)Φ(p, t) Ip + ΦT (p, t)P (t−1)Φ(p, t) −1, P (t) = P (t − 1) − L(t)ΦT (p, t)P (t − 1), Φ(p, t) = [ϕ(t), ϕ(t − 1), ..., ϕ(t − p + 1)] , T
Y (p, t) = [y(t), y(t − 1), ..., y(t − p + 1)] , where Ip denotes an identity matrix of order p. When p = 1, the MILS algorithm reduces to RLS algorithm. We derive the regularization form of MILS algorithm for train basic resistance parameter estimation. ˆ − 1), ˆ = θ(t ˆ − 1) + L(t)E(p, t) − λ P (t)θ(t θ(t) N −1 L(t) = Q(t)Φ(p, t) Ip + ΦT (p, t)Q(t)Φ(p, t) , −1 N N I + P (t − 1) , Q(t) = P (t − 1) λ λ P (t) = [I − L(t)ΦT (p, t)]Q(t), − 1), E(p, t) = Y (p, t) − ΦT (p, t)θ(t
Φ(p, t) = [ϕ(t), ϕ(t − 1), ..., ϕ(t − p + 1)] , T
Y (p, t) = [y(t), y(t − 1), ..., y(t − p + 1)] . The MILS algorithm extends the length of the innovation. Because the MILS algorithm use not only the current data but also the past data at each recursive step, parameter estimation accuracy can be improved, and the algorithm can also be used for online estimation. We can obtain the results of the RLS algorithm (or the MILS algorithm with p = 1) and the MILS algorithm (with p = 2, p = 4, and p = 6) from Fig. 6. For comparison,
2018 IFAC ADCHEM Shenyang, Liaoning, China, July 25-27, 2018Xiaoyu Liu et al. / IFAC PapersOnLine 51-18 (2018) 637–642
LS RLS p=2 MILS p = 4 p=6
b (10−2 ) 8.597 8.511 5.749 3.591 1.941
c (10−3 ) 0.751 0.758 0.981 1.422 1.262
(10−1 ) 5.632 5.632 5.623 5.618 5.613
Table 1. Parameter identification results a (100 ) 0.515 0.783 0.932 1.638 2.194
Fig. 7. Comparison of LS and MILS algorithms.
table 1 lists the differences between the results of LS algorithm, RLS algorithm and the MILS algorithm. σ2
Fig. 6. MILS algorithm with p = 1, p = 2, p = 4, p = 6.
Algorithms
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Online Yes/No No Yes Yes Yes Yes
D
E
W
In table 1, a, b, and c are basic resistance parameters, and σ 2 represents the variance. The given results show that with the increase of the length of the innovation, the variance of the identification results is decreasing, so the MILS algorithm can achieve better performance than RLS algorithm. Both RLS and MILS algorithms can be used for online identification. 4. ANALYSIS AND VERIFICATION 4.1 Verification of results We use a set of data for validation. The basic resistance is calculated using the identification results obtained by the LS and MILS algorithms, and then we use the basic resistance to calculate the acceleration and the velocity by (5) and (6). We compare the velocities calculated from LS algorithm and MILS algorithm with the actual velocity and obtain the Fig. 7 and table 2. Table 2. Comparison of different algorithms Algorithms LS MILS
Variances (σ 2 = 10−2 ) 0.99 0.88
F
Fig. 8. Parameter online identification. significantly, the stabilization of parameters demonstrate the real-time characteristic and validity of the algorithm. Generally, the least squares algorithm and RLS algorithm have fast convergence properties and high precision, and the usage of the data is efficient. Thus, sometimes we use the MILS algorithm, the improvement of the parameter estimation accuracy is limited, and we can use LS algorithm for off-line identification and RLS algorithm for online identification. However, the condition of the train and the track is complex, so it is inevitable that the collected data is often discontinuous and abnormal because of the unexpected interruption and measurement error. The multi-innovation theory based identification algorithm has strong robustness, and can improve the accuracy of identification greatly when we encounter this abnormal situation. 4.2 Analysis of robustness
The simulation results shows that the estimation results become more accurate by applying the MILS algorithm to the basic resistance parameter estimation. Parameter online identification results are depicted in Fig. 8, it can be observed from this figure that after about 70 recursions, the value of a, b, and c no longer change 635
There are strict restrictions on the data used for parameter estimation. The results of parameter identification can be more accurate only when the train coasts on the level tangent track. Therefore, the data used for basic resistance parameter estimation collected by ATO usually discontinuous. The train operating on track with complex
2018 IFAC ADCHEM 642 Shenyang, Liaoning, China, July 25-27, 2018Xiaoyu Liu et al. / IFAC PapersOnLine 51-18 (2018) 637–642
conditions, ATO often collects discontinuous data and anomalous data because of the unexpected interruption and measurement error. These phenomena are inevitable and will increase the difficulty of parameter identification especially online identification. The multi-innovation theory based identification algorithm also has good performance in the case of anomalous data and missing-data and can provide fast convergence and high parameter estimation accuracy, the interval-varying multi-innovation least squares (V-MILS) algorithm was given by Ding et al. (2010), ˆ s−1 ) + L(ts ) Y (p, ts ) − ΦT (p, ts ) θ(t ˆ s−1 ) , ˆ s ) = θ(t θ(t L (ts ) =P (ts−1 ) Φ (p, ts ) −1 × Ip + ΦT (p, ts ) P (ts−1 ) Φ (p, ts ) ,
P (ts ) = P (ts−1 ) − L (ts ) ΦT (p, ts ) P (ts−1 ) , Φ(p, ts ) = [ϕ(ts ), ϕ(ts − 1), ..., ϕ(ts − p + 1)] , T
Y (p, ts ) = [y(ts ), y(ts − 1), ..., y(ts − p + 1)] , where 0 = t0 < t1 < t2 < ..., and 1 ≤ t∗s = ts − ts−1 . The parameter estimate θˆ (t) is updated only at instant t = ts , and so is the convariance matrix P . The V-MILS algorithm computes the parameter estimates using the interval-varying iteration, so it can overcome the affect of bad data on the parameter estimates. We also derive the regularization form of V-MILS algorithm for the train basic resistance parameter estimation ˆ s ) = θ(t ˆ s−1 ) + L(ts )E(p, ts ) − λ P (ts )θ(t ˆ s−1 ), θ(t N −1 L(ts ) = Q(ts )Φ(p, ts ) Ip + ΦT (p, ts )Q(ts )Φ(p, ts ) , −1 N N Q(ts ) = P (ts−1 ) I + P (ts−1 ) , λ λ P (ts ) = [I − L(ts )ΦT (p, ts )]Q(ts ), ˆ s−1 ), E(p, ts ) = Y (p, ts ) − ΦT (p, ts )θ(t
Φ(p, ts ) = [ϕ(ts ), ϕ(ts − 1), ..., ϕ(ts − p + 1)] , T
Y (p, ts ) = [y(ts ), y(ts − 1), ..., y(ts − p + 1)] . We chose a set of data that contain anomalous data collected from the Yizhaung Line for verification.
Fig. 9. Comparison of LS and V-MILS algorithms. 636
The given results in Fig. 9 show that the performance of traditional LS algorithm is poor in the case anomalous data, but the V-MILS algorithm can skip the anomalous data so that we can get more reliable parameters. 5. CONCLUSION The multi-innovation theory based identification algorithm can improve the parameter estimation accuracy, we applied it to the basic resistance parameter estimation using the data collected from Yizhaung Line of Beijing Subway. By comparing the MILS algorithm and V-MILS algorithm with traditional least squares algorithm, we conclude that the multi-innovation least squares algorithm can not only provide high estimation accuracy but also perform well in the case of missing data and anomalous data. The MILS algorithm and V-MILS algorithm can also be used for online identification, which will make the train basic resistance parameter estimation simpler and more accurate, and will reduce the expense of time, labor, and material for organizing field test. REFERENCES Bernsteen, S.A., Uher, R.A., and Romualdi, J.P. (1980). The interpretation of train rolling resistance from fundamental mechanics. Industry Applications IEEE Transactions on, IA-19(5), 802-817. Chen, D.W., Chen, R., Li, Y.D., and Tang, T. (2013). Online learning algorithms for train automatic stop control using precise location data of balises. IEEE Transactions on Intelligent Transportation Systems, 14(3), 15261535. Davis, W.J. (1926). The tractive resistance of electric locomotives and cars. General Electr. Rev., 29(10), 685708. Ding, F. and Chen, T. (2007). Performance analysis of multi-innovation gradient type identification methods. Automatica, 43(1), 1-14. Ding, F., Liu, P.X., and Liu, G. (2010). Multiinnovation least-squares identification for system modeling. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 40(3), 767-778. Goodwin, G.C. and Sin, K.S. (1984). Adaptive Filtering, Prediction and Control. Englewood Cliffs, NJ: PrenticeHall. Haykin, S.S. (2009). Neural networks and learning machines. Pearson Schweiz Ag. Huang, W.Y., Yang, N.Q., and Huang, M. (2000). Ponderation on Railway Train Basic Resistance. China Railwayence, 21(3), 44-57. Wang, C., Tang, T., and Luo, R.S. (2013). Study on iterative learning control in automatic train operation. Journal of the China Railway Society, 35(3), 48-52. Yu, Z.Y. and Chen, D.W. (2011). Modeling and system identification of the braking system of urban rail vehicles. Journal of the China Railway Society, 33(10), 37-40. Yuan, L. (2015). Train Basic Resistance Identification and Its Online Update Algorithm. Journal of Information & Computational Science, 12(11), 4161-4171.