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A Lyapunov Function Approach to Longitudinal Control of Vehicle by Tae Soo No

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The Lyapunov second method is used to derive a control law that can be used to control the spacing between vehicles in a platoon. A third-order system is adopted to model the vehicle and powertrain dynamics. In addition, the concept of “expected spacing error” is introduced and used to form a Lyapunov function. Then a control law that always decreases the Lyapunov function is selected. A platoon of four vehicles and various scenarios are used to demonstrate the performance of the proposed control law.
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  116 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 50, NO. 1, JANUARY 2001 A Lyapunov Function Approach to LongitudinalControl of Vehicles in a Platoon Tae Soo No, Kil-To Chong, and Do-Hwan Roh  Abstract— The Lyapunov second method is used to derive a con-trol law that can be used to control the spacing between vehiclesin a platoon. A third-order system is adopted to model the vehicleand powertrain dynamics. In addition, the concept of “expectedspacing error” is introduced and used to form a Lyapunov func-tion. Then a control law that always decreases the Lyapunov func-tionisselected.Aplatoonoffourvehiclesandvariousscenariosareused to demonstrate the performance of the proposed control law.  Index Terms— Intelligent transportation systems, longitudinalcontrol, platoon, vehicle dynamics and control. I. I NTRODUCTION I T HAS been shown that the concept of operating vehiclesin a platoon can significantly improve the efficiency of existing road systems. In order to meet such expectations,it is necessary to develop advanced vehicle control systems(AVCSs), which form a part of intelligent vehicle/highwaysystems (IVHSs) [1]–[4]. As shown in Fig. 1, one of the many functions of AVCSs is longitudinal control, in which thepreassigned spacing between the predecessor and thefollower vehicle in the platoon is to be tightly maintained.Most earlier works assume that the follower vehicle hasaccess tothe velocity and acceleration of theplatoon leadvehicle and can measure the relative distance, velocity, andacceleration between itself and the vehicle in front of it [5]–[8]. There are many strategies for designing longitudinal controllaws. To cite a few: proportional-integral-derivative (PID)-typecontrollers have been proposed by Shladover [5], [6], Sheik- holeslam [7], [8], and Fujioka  et al.  [9]. Hedrick   et al.  [10], [11] have extensively used the sliding mode control methodology.Readers are referred to [1]–[4] for the pertinent literature. In the works cited above, the general structure of the controlinput to vehicle may be written as(1)where denote the absolute position, velocity, and ac-celeration of vehicle , respectively. It has been reported that1morlessspacingcanbemaintainedinaplatoonbyusingsuch Manuscript received October 13, 1999; revised July 26, 2000. This work wassupported by the KoreaScience and Engineering Foundation and by the Mecha-tronics Research Center at Chonbuk National University, Korea.T.S.NoiswiththeDepartmentofAerospaceEngineering,ChonbukNationalUniversity, Chonju, Korea.K.-T. Chong is with the Department of Measurement and Control Engi-neering, Chonbuk National University, Chonju, Korea.D.-H. Roh is with the Department of Electrical Engineering, Chonbuk Na-tional University, Chonju, Korea.Publisher Item Identifier S 0018-9545(01)01133-1. controllers [5]–[9]. Sheikholeslam and Desoer [12] showed that tight spacing control is still possible with no communication of information about the lead vehicle.In this paper, we propose another form of a longitudinal con-trol law(2)that completely eliminates the need for a communication link between the platoon’s leading vehicle and the follower vehiclesand uses the local link between the predecessor and followervehicles, as depicted in Fig. 2. We will show that the procedureused in this paper for designing control laws somewhat miti-gates the ad hoc nature of selecting the variables for feedback orthestructureoftheslidingsurface.Also,iteffectivelyaccom-modates wide variations in vehicle dynamics and is suitable forthe spacing control of mixed vehicles.II. V EHICLE  D YNAMICS  M ODELING Various models for vehicle dynamics have been used in thestudy of longitudinal control of platoons. For a vehicle nom-inally traveling with a constant speed and direction, we haveadopted the following third-order linear model [5], [6]: (3)(4)(5)where , the time constant, is a single parameter that representsthe characteristics of vehicle propulsion system, including theengine, transmission, tires and wheels, and any other internalcontrollers.Fig.3showstheblockdiagramofvehicledynamics.Jerk and acceleration limits may be included if necessary. Re-ferring to Fig. 1, the error in spacing distance between vehiclesand can be written as(6)where and represent the relative velocity and acceleration,respectively.III. L ONGITUDINAL  C ONTROL  L AW  D ESIGN The objective of longitudinal control is to maintain thespacing error below a predetermined level or, if possible, at 0018–9545/01$10.00 © 2001 IEEE  NO  et al. : LYAPUNOV FUNCTION APPROACH TO LONGITUDINAL CONTROL OF VEHICLES 117 Fig. 1. Platoon with global link.Fig. 2. Platoon with local link.Fig. 3. Block diagram of vehicle dynamics. zero. For example, the PID-type controller may be written inits general form as(7)where , etc., are controller gains. With this control law, thecontrolcommandisdeterminedusingtheinformationmeasuredand/or estimated at the current time. In other words, the con-troller is always working, as long as there is a spacing error atthe current time. The only time the controller stops working iswhen the following condition is satisfied:(8)However, if some predictive nature is incorporated into the con-troller, one can make it behave more intelligently. Even if cur-rent is large, if it is expected to be decreasing, no further con-trol action would be necessary. By contrast, some action is re-quired when is expected to increase even though its currentmagnitude is below the predetermined level.With the above reasoning in mind, we introduce the conceptof “expected spacing error” as follows:(9)where is the time-to-go until the future time and may be  118 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 50, NO. 1, JANUARY 2001 Fig. 4. Platoon simulation environment.TABLE IV EHICLE AND  C ONTROLLER  C HARACTERISTICS written as(10)For the sake of brevity, we assume that is zero. As one mayeasily understand, is the expected spacing error at the futuretime if both the predecessor and follower vehicles keep theirrespective current accelerations constant for .TheLyapunovdirectmethodoftenprovidesastraightforwardway of obtaining a control law for many dynamic systems. Forthis purpose, we need to find an appropriate positive definitefunction, which is referred to as a Lyapunov function. Also, acontrol law is designed so that it assures the negative definite-ness of the derivative of the Lyapunov function [13].In this paper, we use the expected spacing error (9) to definea Lyapunov function as follows:(11)IfwedifferentiatetheaboveLyapunovfunction,thenwesimplyobtain(12)Also, differentiating (9) yields(13)Then, we substitute (5) into (13) to get(14)In order to use the Lyapunov direct method, we may selectsuch that(15)where is a positive constant. Then, (12) may be rewritten as(16)which is a negative definite function. This implies that the Lya-punovfunction isastrictlydecreasingfunction,meaningthatLyapunov stability is assured. Finally, the controller that satis-fies the above argument is(17)where and are left undetermined as design parameters.While the proposed control law still needs to measure or es-timate the spacing error and relative speed/acceleration in com-puting the expected spacing error, the vehicle does not receiveany information directly from the platoon lead vehicles otherthanthefirstvehicle.Buttheunidirectionalcommunicationlink is required because each vehicle needs to receive a control com-mand from the preceding vehicle. Another feature of the pro-posed control law is that it systematically accommodates thedynamic characteristics of the vehicles participating in the pla-toon, since the vehicle time constant appears in the controlcommand. This feature will be beneficial if a platoon is com-posed of vehicles of various kinds. Therefore, due to the cas-caded structure of the control laws, a vehicle will have full ac-cess to information on the dynamics and control characteristicsof all the preceding vehicles.  NO  et al. : LYAPUNOV FUNCTION APPROACH TO LONGITUDINAL CONTROL OF VEHICLES 119 (a)(b)Fig. 5. Response to sudden speed change. (a) PID controller. (b) Lyapunov controller. IV. S IMULATION AND  P ERFORMANCE  A NALYSIS As shown in Fig. 4, a four-vehicle platoon is chosen and usedinthesimulationbecauseitmustretaintheessentialcharacteris-ticsofaplatoon.Individualvehiclesareassumedtobeidentica,landthedesignparameters and areselectedaftersomenu-merical experiments. Just for comparison purposes, a PID-typecontroller given by (7) and a simulation scenario were adoptedfrom [6]. The relevant numbers used in the simulation are sum-marizedinTableI.Thespacingdistance issetatzeroforeaseof presenting the simulation results. Then, the control objectiveis that every vehicle line up with the platoon leader vehicle.  A. Response to Abrupt Speed Changes by the Lead Vehicle In this example, the platoon leader vehicle accelerates at 1m/s for 2 s to speed up by 2 m/s, and there are no initialspacing errors between vehicles. As one may see from Fig. 5,the PID-type controller performs just as human drivers do. Intheearly stage,thespacing error betweenthe platoonleader andthe first vehicle is the largest. That is because of the delay inpropulsion or engine dynamics. Although the first vehicle no-tices the acceleration of the lead vehicle earlier than the othervehicles behind it, the difference in acceleration and speed arestill great, producing a large spacing error. But the spacing error
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