Robust adaptive cruise control of high speed trains

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<ul><li><p>Research Article</p><p>Robust adaptive cruise control of high speed trains</p><p>Mohammadreza Faieghi a,n, Aliakbar Jalali b, Seyed Kamal-e-ddin Mousavi Mashhadi b</p><p>a Department of Electrical Engineering, Miyaneh Branch, Islamic Azad University, Miyaneh, Iranb School of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran</p><p>a r t i c l e i n f o</p><p>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</p><p>Keywords:Cruise controlHigh speed train (HST)Robust adaptive controlNonlinear systemsNon-minimum phase system</p><p>a b s t r a c t</p><p>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.</p><p>&amp; 2013 ISA. Published by Elsevier Ltd. All rights reserved.</p><p>1. Introduction</p><p>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].</p><p>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 [36]. 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 [710].</p><p>In this paper we consider the multi-body model of train since itprovides more accuracy in characterizing the dynamics of train.</p><p>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.</p><p>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.</p><p>Contents lists available at ScienceDirect</p><p>journal homepage:</p><p>ISA Transactions</p><p>0019-0578/$ - see front matter &amp; 2013 ISA. Published by Elsevier Ltd. All rights reserved.</p><p>n Corresponding author.E-mail address: (M. Faieghi).</p><p>ISA Transactions 53 (2014) 533541</p>;domain=pdf;domain=pdf;domain=pdfmailto:mfaieghi@gmail.com</li><li><p>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.</p><p>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.</p><p>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 [1115].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:</p><p> The cruise controller is developed, for the first time, based onthe original multi-body nonlinear MIMO model.</p><p> 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 [710].</p><p> Parametric uncertainties and external disturbances are consid-ered simultaneously and sufficient conditions are given forasymptotic tracking in the presence of such perturbations.</p><p>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.</p><p>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 xij. 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</p><p>the subscript is dropped, it indicates any p-norm. For a givenmatrix A, maxA and minA 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.</p><p>2. Dynamics of HST</p><p>2.1. Mathematical modelling</p><p>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</p><p>Rmi c0cv _ximi; 1</p><p>Ra caM _x21; 2where ca, cv and c0 are positive constants and M is the total massof train given by ni 1mi. Let xi;j xixj, then based on New-ton's equation of motion we obtain</p><p>m1 x1 kx1;2Rm1Rau1mi xi kxi;i1xi;i1Rmiui; i 2;;n1mn xn kxn;n1Rmnun:</p><p>8&gt;: 3Notice that for the PPD design, we have ui 0; i 2;;n1. Letvi _xi be the velocity of i-th car, then (3) can be convenientlyrepresented in the following normal form:</p><p>_x v;_v f x; v; tgtu;</p><p>(4</p><p>where x; vARn are position and velocity vectors,</p><p>f x; v; t </p><p> km1</p><p>x1;2c0cvv1caMv</p><p>21</p><p>m1</p><p> km2</p><p>x2;1x2;3c0cvv2</p><p> kmn</p><p>xn;n1c0cvvn</p><p>0BBBBBBBBBB@</p><p>1CCCCCCCCCCA; 5</p><p>and</p><p>gt 1=m1;;1=mnT : 6The 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</p><p>Fig. 1. Force diagram of HST.</p><p>M. Faieghi et al. / ISA Transactions 53 (2014) 533541534</p></li><li><p>additive external disturbances. Such assumptions are similar to theprevious papers published in the literature [710].</p><p>2.2. Problem statement</p><p>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</p><p>y1 v; 7which implies that the system has relative degree 1;;1T . Basedon the above output definition, the system (4) can be represented inthe following inputoutput form:</p><p>_1 1_1 A11;1; tB1tu11ty1 1</p><p>8&gt;: 8where 1 x indicates the internal states, 1 v is the externalstates vector, u1 u1;;uvT denotes the control input, 1tARnis the additive disturbances, A1; ; t f ; ; t and B1t gt. Itis assumed that the vectors A1;; t and B1t contain unknownparameters and are linear in terms of them.</p><p>Let 1 0, then the zero-dynamics takes the form _1 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 y1yd1, then the inputoutput representation of (8) can berewritten in 1; e1 coordinate as follows:</p><p>_1 e1yd1;_e1 A11; 1; tB1tu11t _yd1:</p><p>(9</p><p>The problem is to design an appropriate control law such that inspite of perturbations Je1 J-0 as t-1 while keeping the internalstates 1 bounded.</p><p>For PPD trains, the output vector is given by</p><p>y2 v1; vnT 10and the system relative degree is 1;1T . In this case the externalstates are expressed by 2 v1; vnT 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 2 x; v2;; vn1T which implies thatwe have a 2n2 dimensional unstable zero-dynamics. The parti-tioned dynamics of PPD trains is described by</p><p>_2 2; 2; t_2 A22;2; tB2tu22ty2 2</p><p>8&gt;: 11where u2 u1;unT is the control input, 2tAR2 denotesadditive disturbances, 2; 2; t v; f 2;; f n1T , A22; 2; t f 1; f nT and B2t g1; gnT . Again, we consider A22; 2; t andB2t to have unknown parameters which can be expressed in linearparameter form. Let yd2AR2 be the desired trajectory and define thetracking error e2 y2yd2, then the error dynamics becomes</p><p>_2 e2yd2;_e2 A22; 2; tB2tu22t _yd2:</p><p>(12</p><p>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.</p><p>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].</p><p>Assumption 2. The desired trajectories ydi; i 1;2, are designedsuch that all the entries are positive and ydi; _ydiAL1.</p><p>Assumption 3. The additive disturbances it; i 1;2, areassumed to satisfy iAL1.</p><p>3. Stabilization of zero-dynamics</p><p>In order to achieve a meaningful velocity tracking, it is requiredto have stable...</p></li></ul>