Research Article

Robust adaptive cruise control of high speed trains

Mohammadreza Faieghi a,n, Aliakbar Jalali b, Seyed Kamal-e-ddin Mousavi Mashhadi b

a Department of Electrical Engineering, Miyaneh Branch, Islamic Azad University, Miyaneh, Iranb School of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran

a r t i c l e i n f o

Article history:Received 11 July 2013Received in revised form13 November 2013Accepted 3 December 2013Available online 27 December 2013This paper was recommended forpublication by Prof. A.B. Rad

Keywords:Cruise controlHigh speed train (HST)Robust adaptive controlNonlinear systemsNon-minimum phase system

a b s t r a c t

The cruise control problem of high speed trains in the presence of unknown parameters and externaldisturbances is considered. In particular a Lyapunov-based robust adaptive controller is presented toachieve asymptotic tracking and disturbance rejection. The system under consideration is nonlinear,MIMO and non-minimum phase. To deal with the limitations arising from the unstable zero-dynamicswe do an output redefinition such that the zero-dynamics with respect to new outputs becomes stable.Rigorous stability analyses are presented which establish the boundedness of all the internal states andsimultaneously asymptotic stability of the tracking error dynamics. The results are presented for twocommon configurations of high speed trains, i.e. the DD and PPD designs, based on the multi-body modeland are verified by several numerical simulations.

& 2013 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

High speed train offers efficient mobility, green transportationand cost-effective travelling which is known as a feasible alter-native to the aeroplanes for trips under about 650 km. One of thedemanding control problems associated with HSTs is cruise con-trol problem, that is, automatically controlling the train speed tofollow a desired trajectory. The driving forces behind the increas-ing use of automatic control in transportation systems and theinterest in the development of unmanned vehicles require moderntrain control systems to apply new technologies for cruise controlin order to achieve high-precise velocity tracking [1,2].

The methods proposed for cruise control of HST are developedbased on a motion model obtained from Newton's law which can beclassified into two categories. In the first one, which mostly refers toearlier papers, the train consisting of multiple cars is considered as asingle rigid body and its longitudinal motion is characterizedapproximately by a single-point mass Newton equation. Thereforethe dynamics within the train is ignored; see for example [3–6]. Inthe second category a more effective model is considered. As thecouplers between two adjacent cars are not perfectly rigid, theimpacts from the connected cars are taken into account and amulti-body model is obtained; see for example [7–10].

In this paper we consider the multi-body model of train since itprovides more accuracy in characterizing the dynamics of train.

This model is a nonlinear multi-input multi-output (MIMO)representation of the train longitudinal motion and requires morecomplicated stability analysis and design procedure. In addition,defining the cars' velocities as the system outputs yields anunstable zero-dynamics rendering the system is non-minimumphase which challenges the controller design. In the previouspapers linear or simplified nonlinear models are used to designthe controller and thus, dealing with such a complexity is avoided.For example, in [7] the nonlinear model is linearized around anoperating point and a mixed H2=H1 controller is developed.Similarly, in [8] a linearized model is considered and a decouplingcontroller is proposed. Application of nonlinear methods is alsostudied in [9,10]. In these two papers the cars' positions areconsidered as the outputs. Thus, there exists no internal dynamics;the system becomes minimum-phase and position tracking isobtained instead of velocity tracking. In addition, the nonlinearmodel that has been utilized is a simplified version of the multi-body model. More specifically, a second-order differential equa-tion is derived in terms of the first car's position which justpartially describes the train motion. This simplified model is rathersimple in describing the details of train dynamics. To the authors'best knowledge, cruise control design for HST is not addressed yetby using the nonlinear MIMO model, i.e. the original multi-bodymodel and the stability of internal dynamics is not studied as well.

Contemporary HST designs fall into different categories accord-ing to the composition of traction forces. Two important types arePush Pull Driving (PPD) design and Distributed Driving (DD)design. The PPD type has only two motorized cars located at bothends of a train, and the trailers are between the motorized cars.

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ISA Transactions

0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.isatra.2013.12.007

n Corresponding author.E-mail address: mfaieghi@gmail.com (M. Faieghi).

ISA Transactions 53 (2014) 533–541

For the DD type every car has its own motor. In [7] a comparativestudy is provided between these two kinds of designs and it isrevealed that the DD trains are superior in terms of velocity trackingand disturbance rejection. However, the PPD trains have theadvantage of energy saving and low maintenance cost. Velocitytracking problem of PPD trains is more complicated in comparisonwith the DD trains since it has a larger internal dynamics and onehas to show its stability to obtain a meaningful tracking.

In a realistic problem HSTs suffer from unknown parametersand external disturbances. Generally, the weight of passengers andloads vary in each travel and consequently the total weight of trainwill be unknown. This is a considerable source of uncertainty inthe control system. Other parametric uncertainties include themechanical resistance parameters which change according to theenvironmental conditions, the stiffness coefficient of couplers dueto their nonlinear behavior, etc. Wind gust is a major disturbancein HST cruise control which seriously affects stability as well asriding quality. Furthermore, there may be other external distur-bances such as tunnel resistance, ramp resistance, track slope, andcurve resistance. These perturbations are considered in [9,10] androbust controllers are proposed for compensation of their effects.

In this paper we consider the velocity tacking problem in thepresence of both parametric uncertainties and external distur-bances. The proposed controller is based on adaptive controltheory. By the use of adaptive controller there is no need to havea priori information about the bounds on uncertain parametersand the controller is capable of changing itself according to theexisting conditions. These are the main advantages of adaptivemethods in comparison with other robust techniques [11–15].Here, to deal with unknown parameters, an adaptive controller isdesigned by means of Lyapunov direct method and sufficientconditions are obtained to guarantee the stability of closed-loopsystem. Then, in order to deal with external disturbances, weincorporate the adaptive technique with Lyapunov redesign tocome up with a robust adaptive control law which is able toattenuate the effects of unknown parameters and external dis-turbances simultaneously. However, as the system is non-minimum phase, we need to stabilize the internal dynamics toobtain a meaningful tracking [16]. For this purpose, the outputredefinition method is adopted from [17]. The main results of thispaper consist of the following aspects:

� The cruise controller is developed, for the first time, based onthe original multi-body nonlinear MIMO model.

� The stability of internal dynamics is proven for different trainconfigurations and a theoretical justification is provided forstability of all the trailers. Such results are not presented innone of the previous papers [7–10].

� Parametric uncertainties and external disturbances are consid-ered simultaneously and sufficient conditions are given forasymptotic tracking in the presence of such perturbations.

The remainder part of this paper is organized as follows.We introduce the train dynamics and present the problem state-ment in Section 2. The output redefinition approach and stabiliza-tion of the internal dynamics are included in Section 3. Therobust adaptive controller is developed in Section 4. We examinethe performance of the proposed controller via numerical simula-tions in Section 5. An introduction to practical applicabilityof the proposed cruise controller is given in Section 6. Finally,our conclusions appear in Section 7.

Notation: Throughout the paper, unless otherwise mentioned,we will use xi to denote the i-th element of the vector x. When xiitself is a vector, its components will be denoted by xi

j. Vectors andmatrices, if not explicitly stated, are assumed to have appropriatedimensions. J � Jp is used to denote the p-norm of a vector and if

the subscript is dropped, it indicates any p-norm. For a givenmatrix A, λmaxðAÞ and λminðAÞ denote its largest and smallesteigenvalues, respectively. The positive definiteness of A is shownby A40. In shows an n-dimensional identity matrix. In addition,we abuse the notation 0 to denote any zero matrices.

2. Dynamics of HST

2.1. Mathematical modelling

The force diagram of HST is depicted in Fig. 1 where xi, ui andRmi denote position, traction force and mechanical resistance ofthe i-th car, respectively. The aerodynamic drag is indicated by Ra.The behavior of couplers can be described approximately by alinear spring with stiffness coefficient k. Let mi be the mass of i-thcar, then Rmi and Ra are given by

Rmi ¼ ðc0þcv _xiÞmi; ð1Þ

Ra ¼ caM _x21; ð2Þwhere ca, cv and c0 are positive constants and M is the total massof train given by ∑n

i ¼ 1mi. Let Δxi;j ¼ xi�xj, then based on New-ton's equation of motion we obtain

m1 €x1 ¼ �kΔx1;2�Rm1�Raþu1

mi €xi ¼ �kðΔxi;i�1þΔxi;iþ1Þ�Rmiþui; i¼ 2;…;n�1mn €xn ¼ �kΔxn;n�1�Rmnþun:

8><>: ð3Þ

Notice that for the PPD design, we have ui ¼ 0; i¼ 2;…;n�1. Letvi ¼ _xi be the velocity of i-th car, then (3) can be convenientlyrepresented in the following normal form:

_x ¼ v;_v ¼ f ðx; v; tÞþgðtÞu;

(ð4Þ

where x; vARn are position and velocity vectors,

f ðx; v; tÞ ¼

� km1

Δx1;2�c0�cvv1�caMv21m1

� km2

ðΔx2;1þΔx2;3Þ�c0�cvv2

⋮

� kmn

Δxn;n�1�c0�cvvn

0BBBBBBBBBB@

1CCCCCCCCCCA; ð5Þ

and

gðtÞ ¼ ð1=m1;…;1=mnÞT : ð6ÞThe above mathematical modelling is based on [7]. Notice that wehave assumed that the couplers are linear springs which haveconstant stiffness coefficients; however, in practice, the stiffnesscoefficients depend on the displacement nonlinearly. Since such anonlinear behavior is not simply measurable during the trainoperation, we consider any nonlinear deviation of stiffness coeffi-cients as additive parametric uncertainties. In addition, we con-sider nonlinearities arising from track profile such as track slope as

Fig. 1. Force diagram of HST.

M. Faieghi et al. / ISA Transactions 53 (2014) 533–541534

additive external disturbances. Such assumptions are similar to theprevious papers published in the literature [7–10].

2.2. Problem statement

We concentrate on velocity tracking in the presence of unknownparameters and external disturbances. To formulate the controlproblem in case of DD trains, the output vector is defined as

y1 ¼ v; ð7Þwhich implies that the system has relative degree ð1;…;1ÞT . Basedon the above output definition, the system (4) can be represented inthe following input–output form:

_η1 ¼ ξ1_ξ1 ¼A1ðη1;ξ1; tÞþB1ðtÞu1þδ1ðtÞy1 ¼ ξ1

8><>: ð8Þ

where η1 ¼ x indicates the internal states, ξ1 ¼ v is the externalstates vector, u1 ¼ ðu1;…;uvÞT denotes the control input, δ1ðtÞARn

is the additive disturbances, A1ðη; ξ; tÞ ¼ f ðη; ξ; tÞ and B1ðtÞ ¼ gðtÞ. Itis assumed that the vectors A1ðη;ξ; tÞ and B1ðtÞ contain unknownparameters and are linear in terms of them.

Let ξ1 ¼ 0, then the zero-dynamics takes the form _η1 ¼ 0 whichis not stable rendering the system is non-minimum phase. In fact,the velocity tracking problem is to deal with an n-input n-outputsystem having an n-dimensional unstable zero-dynamics. For agiven desired trajectory yd1ARn, let the tracking error bee1 ¼ y1�yd1, then the input–output representation of (8) can berewritten in ðη1; e1Þ coordinate as follows:

_η1 ¼ e1þyd1;_e1 ¼A1ðη1; ξ1; tÞþB1ðtÞu1þδ1ðtÞ� _yd1:

(ð9Þ

The problem is to design an appropriate control law such that inspite of perturbations Je1 J-0 as t-1 while keeping the internalstates η1 bounded.

For PPD trains, the output vector is given by

y2 ¼ ðv1; vnÞT ð10Þand the system relative degree is ð1;1ÞT . In this case the externalstates are expressed by ξ2 ¼ ðv1; vnÞT and the external dynamicscan be regarded as a 2-input 2-output system which refers to themotorized cars locating at the two ends of a train. The internalstates are denoted by η2 ¼ ðx; v2;…; vn�1ÞT which implies thatwe have a 2n�2 dimensional unstable zero-dynamics. The parti-tioned dynamics of PPD trains is described by

_η2 ¼φðη2; ξ2; tÞ_ξ2 ¼A2ðη2;ξ2; tÞþB2ðtÞu2þδ2ðtÞy2 ¼ ξ2

8><>: ð11Þ

where u2 ¼ ðu1;unÞT is the control input, δ2ðtÞAR2 denotesadditive disturbances, φðη2; ξ2; tÞ ¼ ðv; f 2;…; f n�1ÞT , A2ðη2; ξ2; tÞ ¼ðf 1; f nÞT and B2ðtÞ ¼ ðg1; gnÞT . Again, we consider A2ðη2; ξ2; tÞ andB2ðtÞ to have unknown parameters which can be expressed in linearparameter form. Let yd2AR2 be the desired trajectory and define thetracking error e2 ¼ y2�yd2, then the error dynamics becomes

_η2 ¼ e2þyd2;_e2 ¼A2ðη2; ξ2; tÞþB2ðtÞu2þδ2ðtÞ� _yd2:

(ð12Þ

The task of controller is to stabilize the tracking error e2 at the originand makes the zero-dynamics stable despite unknown parametersand external disturbances. Before we start the controller design,some assumptions are presented which will be used in the subse-quent development.

Assumption 1. All the system states are available for feedbackcontrol purpose. These variables can be measured accurately bymeans of highly developed measurement techniques in HST suchas the leaky coaxial synthesized optical cable and the fibre opticgyroscope inertial navigation system [18] or at least they can beaccurately estimated by applying appropriate state observers [19].

Assumption 2. The desired trajectories ydi; i¼ 1;2, are designedsuch that all the entries are positive and ydi; _ydiAL1.

Assumption 3. The additive disturbances δiðtÞ; i¼ 1;2, areassumed to satisfy δiAL1.

3. Stabilization of zero-dynamics

In order to achieve a meaningful velocity tracking, it is requiredto have stable internal dynamics. However, as it is shown inSection 2, both of DD and PPD trains are non-minimum phaseand their internal dynamics are unstable. Therefore the controllerhas a duty to stabilize the internal dynamics and makes itacceptable. To deal with this problem, an output redefinitionapproach is employed in this section. The idea behind the outputredefinition is to redefine the output and subsequently the desiredtrajectory such that asymptotic tracking can be achieved whilesimultaneously the zero-dynamics with respect to new outputs isrendered acceptable. This type of coordinate transformation isadopted from [17]; however, some new analyses are presentedhere (Eqs. (17)–(26)) to have a deep insight into what happensduring the redefinition. We start with the DD trains and thenextend the results for the PPD type.

3.1. DD trains

Consider the internal dynamics of (8) which is given by _η1 ¼ ξ1.It can be seen that there is a direct control of η1 via ξ1, and one canconclude that by suitably adjusting ξ1 the internal dynamics canbe controlled. In fact, the internal dynamics η1 is driven by ξ1 andif we could achieve perfect tracking in finite time, i.e. y1 ¼ yd1 forall tZT for some T40, then the internal dynamics has to satisfy

_η1 ¼ yd1: ð13ÞThis motivates us to do an output redefinition such that theinternal dynamics with respect to new outputs becomes stable.For this purpose, let us consider the following change of variable:

ξ1 ¼ ξ1�K1η1; ð14Þwhere K1ARn�n is a design parameter to be determined. Definethe new output as y1 ¼ ξ1, then the input–output representation(8) can be rewritten in the new coordinate ðη1; ξ1Þ as follows:

_η1 ¼ ξ1þK1η1;_ξ 1 ¼A1ðη1; ξ1; tÞþB1ðtÞu1þδ1ðtÞ�K1ξ1y1 ¼ ξ1

8>><>>: ð15Þ

Now, let ξ1 ¼ 0, then the zero-dynamics becomes _η1 ¼K1η1.Choosing K1 as a Hurwitz matrix yields the internal dynamics of(15) is asymptotically stable which implies that the system isminimum-phase. To get a good insight about the above redefini-tion, we now analyze the system stability in terms of trackingerror. By redefinition of the output, it is required to modify thecommand signal. Let yd1 be the modified desired trajectory anddefine the tracking error as e1 ¼ y1�yd1. Now, by substituting e1and (14) into (15) we obtain

_η1 ¼ e1þK1η1þyd1;

_e 1 ¼A1ðη1; ξ1; tÞþB1ðtÞu1þδ1ðtÞ�K1ξ1� _yd1:

(ð16Þ

M. Faieghi et al. / ISA Transactions 53 (2014) 533–541 535

The zero-dynamics is given by _η1 ¼K1η1þyd1 which can beinterpreted as a nominal plant _η1 ¼K1η1 perturbed by the non-vanishing disturbance yd1. For any Hurwitz choice of K1, thenominal plant will be asymptotically stable and from the converseLyapunov theorem [16] it can be concluded that there exists aLyapunov function Vðη1; tÞ such that for all ðt;η1ÞA ½0;1Þ �Dwhere D¼ fη1ARnjJη1 Jorg, we have

c1 Jη1 J2rV ðη1; tÞrc2 Jη1 J

2 ð17Þ

∂V∂t

þ ∂V∂η1

K1η1r�c3 Jη1 J2 ð18Þ

J∂V=∂η1 Jrc4 Jη1 J ð19Þwhere ci are positive constants. For example, for P1;Q140satisfying KT

1P1þP1K1 ¼ �Q1, the Lyapunov function can betaken as Vðη1; tÞ ¼ ηT1P1η1 and thus we will have c1 ¼ λminðP1Þ,c2 ¼ λmaxðP1Þ, c3 ¼ λmaxðQ1Þ and c4 ¼ λmaxðP1Þ. Taking the deriva-tive of Vðη1; tÞ along the trajectories of _η1 ¼K1η1 and using (18)yield

_V ðη1; tÞr�c3 Jη1 J2þ J∂V=∂η1 J :Jyd1 J : ð20Þ

Let Yd be a positive value such that Jyd1 J1oYd, then by using(19) we obtain

_V ðη1; tÞr�c3 Jη1 J2þc4Yd Jη1 J ; ð21Þ

which results in

_V ðη1; tÞr�c3Vþc4Yd

ffiffiffiffiV

p: ð22Þ

The change of variable W ¼ffiffiffiffiV

pfor Va0 yields the following

linear differential equation:

_Wr�c5Wþc6; ð23Þwhere c5 ¼ c3=2c2 and c6 ¼ c4Ydm=2

ffiffiffiffiffic1

p. By solving the above

inequality and using the comparison lemma [16] we conclude

Wrc6c5ð1�e� c5tÞ: ð24Þ

Making substitution from (17) into (24) results in the followingexpression:

Jη1 Jrc6

c5ffiffiffiffiffic1

p ð1�e� c5tÞr c6c5

ffiffiffiffiffic1

p : ð25Þ

The results will be valid inside the domain D. Thus, we require Yd

to satisfy

Ydr2c1c5rc4

ð26Þ

to obtain Jη1 Jrr. Since _η1 ¼K1η1 is globally asymptoticallystable, we have r-1. Thus, for any bounded Yd the internalstates will be bounded which implies that the zero-dynamics isstable and the system is no longer non-minimum phase. The resultof the above analysis is conceptually important because it estab-lishes stability of zero-dynamics for any bounded yd1. In addition,it presents boundedness of η1. Based on this property, we can useη1 to design a bounded trajectory for yd1. The command signalshould be modified such that when y1-yd1, we also have y-yd.Based on the boundedness of η1 and motivated from (14), wedefine

yd1 ¼ yd1�K1ηd1; ð27Þwhere ηd1 is the solution of (13). We now show that the choice of(27) yields asymptotic tracking. If (13) holds, we will have y1 � yd1for all tZT . Then, from (14) and (27) we obtain

y1 � yd1þK1 ~η1; ð28Þwhere ~η1 ¼ η1�ηd1. Writing the _~η1�equation yields _~η 1 ¼K1 ~η1and since K1 is Hurwitz, ~η1-0 as t-1. It follows from (27) that

y1-yd1. Hence, asymptotic tracking can be attained. We summar-ize the results in the following theorem.

Theorem 1. Consider the input–output representation (8). Supposethat the output is redefined as (14) and let the command signal be(27). Then, asymptotic tracking can be achieved for any Hurwitzmatrix K1.

3.2. PPD trains

The internal dynamics of PPD trains is described by _η2 ¼φðη2; ξ2; tÞ which contains, in contrast with DD trains, both stableand unstable modes. For the purpose of output redefinition, werequire to determine the unstable modes. The zero-dynamicsassociated with x1 and xn is unstable. However, we need to studythe zero-dynamics of all the n�2 trailers to find out whether theyare stable or not. Toward this end, let ζ ¼ ðv2;…; vn�1; x2;…; xn�1ÞTand set ξ2 ¼ 0, then from (11) we obtain

_ζ ¼ AζþB ð29Þwhere

A¼�cvIn�2 A12

In�2 0

!; ð30Þ

A12 ¼

�2k=m2 k=m2 0k=m3 ⋱ ⋱

⋱ k=mn�2

0 k=mn�1 �2k=mn�1

0BBBB@

1CCCCA; ð31Þ

and

B¼ �c0In�2

0

� �: ð32Þ

To examine the stability of (29) we first study the stability of_ζ ¼ Aζ. With this aim consider the block A12 which is a tridiagonalmatrix. For studying the stability properties of this block wepresent a method based on Lyapunov equation. If we could finda solution for the Lyapunov equation

AT12P2þP2A12 ¼ �Q2; ð33Þ

with P2;Q240, then it can be concluded that A12 is Hurwitz. LetP2 ¼ ρ2I2ðn�2Þ where ρ240 is a sufficiently large number. Thisresults in

Q2 ¼

2ρ2k=m2 �ρ2k=m2 0�ρ2k=m3 ⋱ ⋱

⋱ �ρ2k=mn�2

0 �ρ2k=mn�1 2ρ2k=mn�1

0BBBB@

1CCCCA:

ð34ÞIf Q240, then (33) has a solution which implies stability of A12.To show the positive definiteness of Q2, we note that by choosingρ2 sufficiently large all the diagonal entries of (34) will becomelarger than the other entries regardless of the values of k=mi

rendering Q2 is diagonally dominant. From the fact that thenumber of positive eigenvalues of a diagonally dominant matrixis equal to the number of its positive diagonal entries; see [20] forthe details, it can be concluded that all the eigenvalues of Q2 arepositive, in other words Q240. Thus, for a positive definite P2, apositive definite Q2 is obtained which verifies that the Lyapunovequation (33) has a solution and A12 is Hurwitz. Next step is tostudy the stability of the matrix A via evaluation of Lyapunovequation. Let

P ¼P1 P2

P2 P3

!ð35Þ

M. Faieghi et al. / ISA Transactions 53 (2014) 533–541536

with P140 and P3�P2P�11 P240. Note that these two conditions

ensure positive definiteness of P according to the Schur comple-ment [21]. Writing the Lyapunov equation yields the followingexpression:

ATPþPA¼2ð�cvP1þP2Þ P1A12�cvP2þP3

AT12P1�cvP2þP3 AT

12P2þP2A12

!: ð36Þ

Based on Schur complement, the matrix (36) is negative definite ifand only if the following conditions are satisfied:

�cvP1þP2o0 ð37Þ

and

Q2þ2ðP1A12�cvP2þP3Þð�cvP1þP2ÞðAT12P1�cvP2þP3Þ40 ð38Þ

where Q2 is given by (33). If the conditions (37) and (38) aresatisfied, then it can be concluded that the matrix A is Hurwitz andthis results in asymptotic stability of _ζ ¼ Aζ. Notice that it isstraightforward to find a suitable matrix P for the above inequal-ities. For example, if P2 ¼ ρ2I2ðn�2Þ as taken in (34), then a suitablechoice for P1 and P3 could be: P1 ¼ ððρ2þ1Þ=cvÞI2ðn�2Þ andP3 ¼ ðρ2

2cv=ðρ2þ1Þþ1ÞI2ðn�2Þ. It is worth mentioning that in addi-tion to the choice of the matrix P, the conditions (37) and (38) mayimpose some limitations on the parameters of A12. However, in ourexperience on different train parameters the inequalities (37) and(38) are easily satisfied. Next, considering the vector B as non-vanishing disturbance and following the steps (17)–(23) we arriveat _Wr�c5Wþc6 with c5 ¼ c3=2c2 and c6 ¼ c4c0=2

ffiffiffiffiffic1

pwhich

finally gives an upper bound on ζ. Therefore the dynamics of allthe trailers is stable and the unstable modes just refer to x1 and xn.Based on these results, to stabilize the internal dynamics, wedefine

ξ2 ¼ ξ2�K2χ; ð39Þ

where χ ¼ ðx1; xnÞT and K2AR2�2. Defining y2 ¼ ξ2 and lettingy2 ¼ 0 will result in _χ ¼K2χ. Next, consider the re-ordered inter-nal state vector η2 ¼ ðζT ; χT ÞT , then _η2 ¼φðη2; ξ2; tÞ can be con-veniently rewritten as

_η 2 ¼�cvIn�2 A12 0

I 0 00 0 K2

0B@

1CAη2þ

�c0In�2

00

0B@

1CA: ð40Þ

For K2 to be Hurwitz, it can be easily concluded that η2 andconsequently η2 are bounded. It is important to emphasize that bythe output redefinition (39) the desired trajectory should bemodified as

yd2 ¼ yd2�K2ηd2; ð41Þ

to obtain asymptotic tracking. This can be proven similar to thecase of DD trains. Our finding is summarized in the followingtheorem.

Theorem 2. Consider the input–output representation (11). Supposethat the conditions (37) and (38) are satisfied, the output is redefinedas (39) and let the command signal be (41). Then, asymptotic trackingcan be achieved for any Hurwitz matrix K2.

There is one point to be noted. If the trailers undergo anybounded perturbations which can be modelled as additive termsw, then the vector B given by (32) can be redefined as Bnew ¼ Bþwand all the above analysis can be performed for Bnew. Thus,for the case that the trailers suffer from bounded additiveperturbations, boundedness of the internal states is guaranteedand Theorem 2 holds.

4. Robust adaptive controller design

So far it is shown that the internal dynamics can be stabilizedunder certain conditions and asymptotic tracking can be achievedby suitably modifying the command signal. We now proceed todesign a robust adaptive controller to achieve asymptotic trackingin the presence of unknown parameters and external disturbances.To this end a stabilizing state feedback controller is designedfor the external dynamics. Since the design procedure is similarfor either DD or PPD trains, we introduce the following unifiedrepresentation for the external dynamics of both trains:

_e i ¼Aiðηi; ξi; tÞþBiðtÞuiþδiðtÞ�Kiξi� _ydi ð42Þ

where i¼1 indicates the DD type and i¼2 the PPD type. The goal isto make the origin of ei uniformly asymptotically stable, i.e.Jei J-0 as t-1 which results in yi-ydi and consequentlyyi-ydi. To deal with the unknown parameters, first, we designan adaptive controller for the case that δiðtÞ ¼ 0. Then, thecontroller is redesigned to become robust against additive dis-turbances. Notice that the design presented here is based on thestandard feedback linearization techniques for MIMO systemswhich are reported in the literature [22]. As the coupling avoid-ing is inherited in these methods, we are not concerned withthis problem here. To facilitate the design procedure, let M1 ¼diagðm1;…;mnÞ and M2 ¼ diagðm1;mnÞ, then we obtain

Mi_e i ¼Φ0iþΦiθiþui; ð43Þ

where

MiAiðηi; ξi; tÞ�MiðKiξiþ _ydiÞ ¼Φ0iþΦiθi; ð44Þ

in which θiARpi is the vector of unknown parameters, ΦiARni�pi

and Φ0iARni refer to the known terms. We carry on the design byusing the Lyapunov function

Vðei; tÞ ¼ 12 e

Ti Mieiþ1

2~θTi Γ

�1i

~θ i ð45Þ

where Γi is a positive definite diagonal matrix, ~θ i ¼ θi� θ̂ i and θ̂ i

indicates the estimated parameters. Let us calculate the derivativeof Vðei; tÞ along the trajectories of (43). For convenience, we willnot write the argument of the various functions. We have

_V ¼ eTi ðΦ0iþΦiθiþuiÞ�Γ�1i

~θ i_̂θ i ð46Þ

Taking

ui ¼ �Φ0i�Φiθ̂ i�Siei ð47Þ

with Si40 and using the adaptive law

_̂θ i ¼ �ΓiΦ

Ti ei ð48Þ

result in

_V ¼ �eTi Siei: ð49Þ

Since _V is negative semi-definite and _V ¼ 0 ) ei ¼ 0, from LaSal-le's invariance principle [16] it can be concluded that the origin ofei is uniformly asymptotically stable. Let us now consider that thesystem undergoes external disturbances. In this case (43) isrewritten as

M _e i ¼Φ0iþΦiθiþdiþui; ð50Þ

where di ¼Miδi. Applying the following controller:

ui ¼ �Φ0i�Φiθ̂ i�SieiþZi ð51Þ

M. Faieghi et al. / ISA Transactions 53 (2014) 533–541 537

with the adaptation law (48) and using the Lyapunov function (45)yield

_V ¼ �eTi SieiþeTi ðdiþZiÞ: ð52ÞDefine Zi as [23]

Zi ¼ � β2i ei

βi jei j þεie�κi tIni ; ð53Þ

where n1 ¼ n refers to the DD case and n2 ¼ 2 denotes the PPDtrains and βi, εi, κi40 are design parameters. Without loss ofgenerality assume that there exist positive constants Di such that

Jdi J1oDi; ð54Þthen, it can be easily shown that

eTi ðdiþZiÞrniεie�κi t ð55Þfor any βi4Di. Thus, we obtain

_V r�eTi Sieiþniεie�κi t : ð56ÞThen

V�Vð0Þr�Z t

0eTi ðτÞSieiðτÞ dτþ

niεiκi

ð1�e�κi tÞ; ð57Þ

therefore

0rZ t

0eTi ðτÞSieiðτÞ dτrVð0Þþniεi

κið1�e�κi tÞ; ð58Þ

and

limt-1

Z t

0eTi ðτÞSieiðτÞ dτrVð0Þþniεi

κið1�e�κi tÞo1: ð59Þ

Since eTi Siei is a uniformly continuous function, according to theBarbalat lemma limt-1eTi Siei ¼ 0 and consequently ei-0 whichimplies asymptotic tracking. We now give the above results in thefollowing theorem.

Theorem 3. Consider the external dynamics (42) with the unknownparameters θi and the external disturbances δi. Suppose that thecondition (54) is satisfied. Then, the origin of ei is uniformlyasymptotically stable by using the state feedback control law (51)with the adaptation law (48).

5. Numerical simulations

In order to evaluate the performance of the proposed robustadaptive controller, numerical simulations are carried out on aJapan Shinkansen HST which is studied in [7,8]. The train consistsof five cars and its parameter values are listed in Table 1. We limitthe relative displacement between two adjacent cars to 71 m.For the DD configuration, the desired trajectory for all the cars aregiven by the speed profile depicted in Fig. 2. According to theacceleration profile, first, the train will have an acceleration phase0.267 m/s2 in the time interval [0,250]. Then, the acceleration will

be smoothly decreased to zero in the interval [250,350] toachieve the 300 km/h speed. Then, the velocity will be maintainedin [350,700]. We will have a braking phase in [750,950] andthen, the train will be stopped. By stopping the train, the parkingbrakes are employed in practice which is not considered in thesimulations.

To verify the robustness of controller, in all the simulations weapply 20% parametric uncertainty for each parameters in bothdirections randomly based on the provided nominal model. Sinceall the parameters are uncertain, we will have Φ0 ¼ 0AR5,

θ¼ k; c0m1; cvm1; caM;m1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}1st car

; k; k; c0m2; cvm2;m2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}2nd car

;…; k; c0m5; cvm5;m5|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}5th car

0B@

1CA

T

AR24

ð60Þand

Φ¼ ðΦ1 Φ2 Φ3 Φ4 Φ5ÞT AR5�24;

Φ1 ¼ ð�Δx1;2 �1 � _x1 � _x21 �ðK1ξ1þ _yd1Þ 0 ⋯ 0ÞT ;Φ2 ¼ ð0 ⋯ 0 �Δx2;1 �Δx2;3 �1 � _x2 �ðK2ξ2þ _yd2Þ 0 ⋯ 0ÞT ;

Φ3 ¼ ð0 ⋯ 0 �Δx3;2 �Δx3;4 �1 � _x3 �ðK3ξ3þ _yd3Þ 0 ⋯ 0ÞT ;

Φ4 ¼ ð0 ⋯ 0 �Δx4;3 �Δx4;5 �1 � _x4 �ðK4ξ4þ _yd4Þ 0 ⋯ 0ÞT ;

Φ5 ¼ ð0 ⋯ 0 �Δx5;4 �1 � _x5 �ðK5ξ5þ _yd5ÞÞT : ð61ÞThe velocity of all cars is taken as output and we have a5-dimensional tracking error and control input vectors. However,as shown in the block-diagram of overall control system in Fig. 3all the train states are required for feedback and control law.For the acceleration phase, the tracking error and the control inputare demonstrated in Figs. 4 and 5. The amplitude of tracking erroris considerably small and asymptotic tracking, i.e. Je1 J-0, isachieved. In order to reflect the transient state behavior, therelative displacement of all cars is presented in Fig. 6 which issettled at constant values eventually. For the braking phase, simu-lation results are provided in Figs. 7–9 which show that asympto-tic tracking is achieved. Notice that the amplitude of control effortis higher than the acceleration phase since the rate of speedchange is higher in braking phase as shown in Fig. 2.

In order to take into account the effects of external distur-bances, we assume that a wind with resistant force 60 kN hits thetrain directly and remains in all the subsequent time. Since the

Table 1Parameter values of HST used in numerical simulation.

Parameter Value Unit

m1 ;m5 80�103 kgm2 ;m3;m4 40�103 kgc0 0.01176 N/kgcv 0.00077616 N s/m kgca 1.6�10�5 N s2/m2 kgk 80�103 N/m

0 200 400 600 800 1000−200

0

200

400

spee

d pr

ofile

(km

/h)

0 200 400 600 800 1000−1

−0.5

0

0.5

acce

lera

tion

prof

ile (m

/s2 )

time (sec)

Fig. 2. Speed and acceleration profile.

M. Faieghi et al. / ISA Transactions 53 (2014) 533–541538

wind touches the front car only, its effects are observed in thevelocity of the first car. As shown in Fig. 10, the deviation of speedfrom its desired value is compensated after a transient state. Inaddition, we see an increase in sum of the absolute control effortwhich is equal to the amplitude of disturbance. This is reasonablebecause according to Newton's equation of motion, as long as�60 kN wind gust hits the train, the amplitude of control signalmust be 60 kN larger than the normal case to compensate theeffects of external disturbances.

Overall, as shown in the simulation results, the proposed robustadaptive controller is feasible and exhibits good performance interms of tracking and disturbance rejection.

6. Practical application

From practical point of view, the proposed robust adaptivecontroller can be implemented in a computer system of HST as apart of the Automatic Train Operation (ATO) unit. ATO is anoperational safety enhancement device used to help automateoperations of trains and has direct control over traction motorsand brakes. According to International Association of PublicTransport, there are four Grades of Automation (GoA) of trains[24]:

� GoA 1: Corresponds to a fully manual train operation where atrain driver controls not only the starting and stopping of a

Fig. 3. Block diagram of control system.

Fig. 4. Tracking error during acceleration.Fig. 5. Control input during acceleration.

M. Faieghi et al. / ISA Transactions 53 (2014) 533–541 539

train but also the operation of train doors and handling ofemergencies or sudden train diversions.

� GoA 2: Corresponds to a Semi-automatic Train Operation (STO)where the starting and stopping of a train is automated but a

standby train driver remains in the driver's cab to prompt thetrain to start, to control the operation of train doors, tomanually operate the train if needed and to handle emergen-cies. Many ATO systems in the world are of grade GoA 2.

� GoA 3: Corresponds to a Driver-less Train Operation (DTO)where a train can start and stop itself but a train attendant may

Fig. 6. Relative displacement of cars during acceleration.

Fig. 7. Tracking error during braking.

Fig. 8. Control input during braking.

Fig. 9. Relative displacement of cars during braking.

M. Faieghi et al. / ISA Transactions 53 (2014) 533–541540

be present to operate the train doors and to manually drive thetrain in case of emergencies.

� GoA 4: Corresponds to an Unattended Train Operation (UTO)where the starting and stopping of trains, as well as operationof train doors and handling of emergencies are fully automatedwithout any regulatory requirement of staff present in thetrains.

According to the above list, cruise controller is an essential part ofthe last three types of trains. Thus, the proposed robust adaptivecruise controller can be implemented in semi-autonomous andautonomous trains for the purpose of regulating the train velocity.As all the cars' position and velocity can be measured or estimatedaccurately, the control law is easy to implement. The ranges ofdesign parameters are determined theoretically to guarantee thesystem stability and they can be well tuned according to theparticular track profile and train parameters to achieve satisfyingperformance. The controller presented here provides high-precisevelocity tracking according to simulation studies which could havepositive impact on the development of autonomous trains.

7. Conclusion

The aim of this study has been to develop a cruise controller forhigh speed trains based on its nonlinear multi-body model. Earlywork has apparently focused on linear [7,8] or simplified nonlinearmodels [9,10]. The present work considers a nonlinear MIMOnon-minimum phase model of high speed trains which suffersfrom unknown parameters and external disturbances. First, anoutput redefinition is employed to make the zero-dynamics withrespect to new outputs, acceptable. This redefinition together withsuitably modifying the command signal allows us to achieveasymptotic tracking. A Lyapunov-based robust adaptive controlleris proposed which is capable of compensating the effects of

unknown perturbations. Stability analyses are provided whichestablish boundedness of all the internal states and asymptoticstability of the tracking error. The design is performed for eitherDD or PPD configurations and verified by numerical simulations. Inour future work, we will concentrate on the use of simpler controlstructures and dealing with stochastic behavior of external dis-turbances including measurement noise.

References

[1] Shakouri P, Ordys A, Askari MR. Adaptive cruise control with stopgo functionusing the state-dependent nonlinear model predictive control approach. ISATrans 2012;51(5):622–31.

[2] Kiencke U, Nielsen L, Sutton R, Schilling K, Papageorgiou M, Asama H. Theimpact of automatic control on recent developments in transportation andvehicle systems. Annu Rev Control 2006;30(1):81–9.

[3] Howlett P. Optimal strategies for the control of a train. Automatica 1996;32(4):519–32.

[4] Gorgon SP, Lehrer DG. Coordinated train control and energy managementcontrol strategies. In: Proceedings of the ASMELIEEE joint railroad conference,Philadelphia, USA, 1998. p. 165–76.

[5] Liu R, Golovitcher IM. Energy-efficient operation of rail vehicles. Transp ResPart A: Policy Pract 2003;37(10):917–32.

[6] Song Q, Song YD, Cai W. Adaptive backstepping control of train systems withtraction/braking dynamics and uncertain resistive forces. Veh Syst Dyn: Int JVeh Mech Mobil 2011;49(9):1441–54.

[7] Yang C, Sun Y. Mixed H2=H1 cruise controller design for high speed train. Int JControl 2001;74(9):905–20.

[8] Astolfi A, Menini L. Input/output decoupling problem for high speed trains. In:Proceedings of American control conference, 2002. p. 549–54.

[9] Song Q, Song YD, Tang T, Ning B. Computationally inexpensive tracking controlof high-speed trains with traction/braking saturation. IEEE Trans Intell TranspSyst 2011;12(4):1116–25.

[10] Song Q, Song YD. Data-based fault-tolerant control of high-speed trains withtraction/braking notch nonlinearities and actuator failures. IEEE Trans NeuralNetw 2011;22(12):2250–61.

[11] Monda S, Mahanta C. Chattering free adaptive multivariable sliding modecontroller for systems with matched and mismatched uncertainty. ISA Trans2013;52(3):335–41.

[12] Shahriari Kahkeshi M, Sheikholeslam F, Zekri M. Design of adaptive fuzzywavelet neural sliding mode controller for uncertain nonlinear systems. ISATrans 2013;52(3):342–50.

[13] Zheng S, Tang X, Song B, Lu S, Ye B. Stable adaptive PI control for permanentmagnet synchronous motor drive based on improved JITL technique. ISA Trans2013;52(4):539–49.

[14] Zhaoa LD, Fanga JA, Cuia WX, Xua YL, Wang X. Adaptive synchronization andparameter identification of chaotic system with unknown parameters andmixed delays based on a special matrix structure. ISA Trans 2013;52(6):738–43.

[15] Kang HB, Wang JH. Adaptive control of 5 DOF upper-limb exoskeleton robotwith improved safety. ISA Trans 2013;52(6):844–52.

[16] Khalil HK. Nonlinear systems. 3rd ed.Prentice Hall; 2001.[17] Gopalswamy S, Hedrick JK. Tracking nonlinear non-minimum phase systems

using sliding control. Int J Control 1993;57(5):1141–58.[18] Jian S, Yan F, Wang J, Jiang Z. Investigation on real-time trace system for high

speed train. J China Railw Soc 1998;20:71–5.[19] Zhuan X, Xia X. Fault-tolerant control of heavy-haul trains. Veh Syst Dyn: Int J

Veh Mech Mobil 2010;48(6):705–35.[20] Dafchahi F Naeimi. Computing of signs of eigenvalues for diagonally dominant

matrix for inertia problem. Appl Math Sci 2008;2(30):1487–91.[21] Boyd S, El Ghaoui L, Feron E, Balakrishnan V. Linear matrix inequalities in

system and control theory. Philadelphia, PA: SIAM; 1994.[22] Marino R, Tomei P. Nonlinear control design: geometric, adaptive and robust.

Prentice Hall; 1996.[23] Koshkouei AJ, Zinober AS. Adaptive backstepping control of nonlinear systems

with unmatched uncertainty. In: Proceedings of the 39st IEEE conference ondecision and control, vol. 5, 2000. p. 4765–70.

[24] International Association of Public Transport, A global bid for automation:UITP Observatory of Automated Metros confirms sustained growth rates forthe coming years, available online from ⟨http://www.uitp.org/news/pics/pdf/Observatory⟩.

Fig. 10. Tracking error and control effort in the presence of disturbance.

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