nonlinear longitudinal controller implementation and comparison for automated cars

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Xiao-Yun Lue-mail: [email protected]

Han-Shue [email protected]

Steven [email protected]

J. Karl [email protected]

PATH, ITS, U. C. Berkeley,Richmond Field Station Bldg 452,

1357 S. 46th Street,Richmond, CA 94804

Nonlinear Longitudinal ControllerImplementation and Comparisonfor Automated CarsThis paper presents the implementation and comparison of nonlinear controllers inlongitudinal control of cars for platooning. Based on the previous implementationadaptive sliding mode controller, a dynamic sliding mode controller and a dynaback-stepping multiple surface controller have been implemented. These three conthave been compared with the test results. Realtime run shows that each controller hown characteristics. Emphasis has been put on the dynamic back-stepping multiplface controller. @DOI: 10.1115/1.1368115#

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1 IntroductionResearch on automatic longitudinal control of road vehicles

been in progress for several decades~Fenton and Bender@1#, Gar-rard et al.@2#, Shladover@3#!. These prior works mainly concentrate on the control design at the level of using the desiredtance, actual relative distances and relative speeds betwvehicles to generate a desired acceleration or torque commaneach vehicle. To design a controller at this level, one needknow vehicle dynamics. However, to implement such a controlone also needs to generate lower-level commands on each vesuch as throttle control command and brake control commawhich will require inclusion of details of engine and brake dynaics. The works in~Cho and Hedrick@4#, Choi @5#, Gerdes et al.@6#, Maciuca and Hedrick@7#! are crucial steps toward this direction. Based on both vehicle dynamics and engine dynamics slidmode longitudinal controllers have been designed and immented~Rajamani et al.@8#, Hedrick @9#!.

In this paper, the overall vehicle following strategy is the safor all the control design methods, in that each vehicle followsvehicle immediately in front of it. The relative distance and retive speed with respect to the preceding vehicle are measuredfed into the controller. In this formulation, ranging sensor~radar!measurements of the distance between consecutive vehiclebe directly used while absolute position data from coded maginstalled in the pavement~extracted using a magnet distance oserver! is only used as accessory or for software redundancythis latter case, the error accumulation problem in distance emation is avoided, but string stability becomes a prominent prlem. String stability is considered in~Swaroop and Hedrick@10#!.

Previous real-time longitudinal control implementationPATH adopted this vehicle following strategy. It was successfuused in the 1997 NAHSC demonstration using 8 automatedhicles. Recently, this code has been generally updatedrestructured.

Based on the new framework, two controllers have beensigned, implemented in realtime and tested on the track at thC. Berkeley Richmond Field Station~RFS! at a speed of 40 km/hand Crows Landing track at a speed of 72 km/h for two-car ptoons. These two controllers are Dynamic Sliding Mode Contrler and Optimal Dynamic Back-stepping Sliding Surface Contrler. This paper presents the control design strategiesimplementation of these new controllers and compares theresults.

Contributed by the Dynamic Systems and Control Division for publication inJOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscriptreceived by the Dynamic Systems and Control Division June 6, 2000. AssoEditor: Y. Hurmuzlu.

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As in the original code, the system model, including vehicdynamics and engine dynamics, is nonlinear~Cho and Hedrick@4#, Hedrick @9#!. The nonlinear vehicle dynamics is directly usefor control design. The models for engine and brake are repsented by several mappings which are not described here. Prnent differences compared to the previous work are:

1 Overall control design methods are different;2 Data fusion and signal filtering methods are different;3 Some integral filters are used in the calculation of numer

derivative in realtime and used to reduce measurement noisdata filtering, for example, to calculate the vehicle acceleratfrom speed and the derivative of engine manifold pressure.

2 Control Design MethodsThis section presents three nonlinear control design meth

and integral filters which are based on a new viewpoint. The mdifferences among these three methods are in the control delogic and composite reference data. They are, in order of incring complexity:

sliding mode control→ dynamic sliding surface control

→ optimal dynamic back-stepping

sliding surface control

2.1 Integral Filters. Supposef (t) is a continuously mea-sured signal which may not be differentiable analytically, e.g.may contain measurement noise. In controller designd/dt( f t). f (t1Dt)2 f (t)/Dt is required in practice. To avoid noise amplification and propagation in this calculation, integral filters aused to estimate the derivative. These integral filters are basethe solution of an inverse Taylor series problem. These secoorder and third-order integral filters are generalized from the fiorder filter used in~Swaroop et al.@11#!.

2.1.1 Inverse Taylor Series Problem.For an arbitrary timeinstants5t1 , let t5t12Dt with uDtu sufficiently small. In orderto find the value off (s) at s5t without resort to differentiation attime instantt, construct aformal Taylor seriesas follows.

f ~ t1!5 f ~ t1Dt !

5 f ~ t !1 f ~ t !Dt11

2!f ~ t !~Dt !21O~~Dt !3!

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wheref (t), f (t) and f (t) are unknown and are to be found. Matematically, this is aninverse Taylor series problem. In a usualTaylor series, from knownf (t), f (t) and f (t), unknownf (t1) isconstructed whent5t12Dt with uDtu sufficiently small. An im-portant point here is that, to approximatef (t1) with f (t)1 f (t)Dt11/2! f (t)(Dt)2, the error is in the order ofO((Dt)3).

Let t5Dt. The following polynomials with coefficients from aTaylor series

G1~s!511ts

G2~s!515ts11

2!t2s2

G3~s!511ts11

2!t2s21

1

3!t3s3

are all Hurwitz polynomials. This guarantees global stabilitythe filter models to be discussed.

2.1.2 First-Order Filter. Construct the following filter fromthe first-order Taylor expansion

t z52z1 f ~ t !, t.0 (1)

Thus it is expected that

z51

t~2z1 f ~ t !!.

d

dtf ~ t !

If the measured signal is corrupted by a noiseD(t), the actualsignal fed into the filter is

f ~ t !5 f ~ t !1D~ t !

Now the estimation becomes

1

t~2z1 f ~ t !!1

1

tD~ t !

from which one can conclude that

~a! the measurement noise directly propagated to its derivaestimation;

~b! the effort in reducingt in the filter in order to reduce timedelay will cause the noise to be magnified in the derivatestimation.

This is the reason why using first-order filters in nonlinearaltime control is not effective.

2.1.3 Second-Order Filter Model.Similarly, construct thefollowing second-order filter from the second-order Taylor expsion

z15z2

z252

t2 ~2tz22z11 f ~ t1!! (2)

t.0

The following approximation is expected

z1→ f ~ t1!

z11tz2→ f ~ t1! (3)

z2→d

dtf ~ t1!

Based on these filters, one can construct other higher-ordeters ~Lu and Hedrick@12#!.

2.2 Sliding Mode Control. It is generally recognized thasliding mode control is robust with respect to external distbances and is widely used in nonlinear control design~Cavallo

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and Maria@13#, DeCarlo et al.@14#, Fernandez and Hedrick@15#,Gopalswamy and Hedrick@16#, Lu and Spurgeon@17#, Lu andSpurgeon@18#, Slotine and Coetsee@19#, Slotine and Hedrick@20#, Sira-Ramirez@21#, Utkin @22#!. Sliding mode control designstrategy may be divided into two independent procedures whare concerned with the choice of sliding surface and the choicreachability condition to ensure that the sliding surface is reachThis method can be applied to the following nonlinear modwith matched uncertainties~Lu and Spurgeon@23#!.

x~1!5 f ~1!~x~1!,x~2!!

x~2!5 f ~2!~x~1!,x~2!!1g~2!~x~1!,x~2!!u~ t !1D~ t !

where (x(1),x(2))PD,Rn, f 5( f (1)(.), f (2)(.)): D°Rm andg(2)(.,.): D°Rm are piecewise continuous and

f ~0!50

det@g~2!~0!#Þ0.

D(t): @0, )°Rm, which is uncertainty. It is assumed to be Lbesgue measurable.

Proper choice of sliding surface:

~a! Choosings(x)5@s1(x), . . . ,sm(x)#T. This may be consid-ered as a single geometric manifold which may be determineda set of geometric equations~DeCarlo et al.@14#, Utkin @22#, Luand Spurgeon@23#!. This is termed astatic sliding surfacebe-cause it produces static feedback.

~b! Choosing a sliding surface which is a set of differentequations~Fernandez and Hedrick@15#, Lu and Spurgeon@17#, Luand Spurgeon@18#, Sira-Ramirez@21#, Slotine and Coetsee@19#!.This is termed adynamic sliding surfacebecause it usually pro-duces dynamic feedback.

Proper choice of sliding reachability condition:It is required thatsTs,0, if sÞ0. The sliding reachability con-

dition may produce a discontinuous control signal, which is unsirable from the practical point of view. Further work has consered methods to reduce this chattering. There are several effeways to accomplish this task.

~i! Introduce a boundary layer of thickness 0,e!1 aroundthe sliding surface such that, whenisi.e, the controller devel-oped from the sliding mode reachability condition is employand whenisi<e, an alternative continuous control policy is employed ~Slotine and Coetsee@14#, Utkin @22#!.

~ii ! Adopt a dynamic sliding mode feedback where effectifiltering of the control reduces chattering naturally~Cavallo andMaria @13#, Lu and Spurgeon@17#, Lu and Spurgeon@18#, Sira-Ramirez@2#, Slotine and Coetsee@19#!.

~iii ! Adopt a continuous reachability condition~DeCarlo et al.@14#, Hedrick @9#, Lu and Spurgeon@23#!.

There are many such possible reachability conditions whmay be broadly defined by:

DefinitionA general sliding reachability conditionis defined as

s52g~s!(4)

s5@s1 , . . . ,sm#T

whereg(s)5@g1(s), . . . ,gm(s)#T satisfies~1! g(s) is continuous ifsÞ0;~2! ~4! is globally asymptotically stable.Particularly, the following two conditions imply condition~2!.~a!

sTg~s!.0, sÞ0

~b! g(s) is bounded in a neighborhood of 0.Example 1. Consider the following sliding reachability cond

tions of dimension 1.

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~1! A discontinuous sliding reachability condition may be chsen as

g~s!5ks1k0 sign~s!

k.0, k0>0.

~2! Its continuous (C0) round-off is

g~s!5ks1k0 satS s

e Dk.0, k0>0

wheree.0 determines the thickness of the boundary layer.However, sliding mode control can only deal with match

uncertainties appearing additively in systems model. Thuspractical application, some other method needs to be incorporto deal with un-matched uncertainties.

2.3 Dynamic Surface Control. The dynamic sliding sur-face control is one step forward compared to the previous slidmode control. The difference lies in the formulation of the erdynamics. The sliding surface still has first-order dynamicsabove. The sliding reachability condition here is composed ocontinuous part and continuous round-off of a switching law. Tprominent difference is that a third-order integral filter is usedcalculate the acceleration and the derivative of desired enmanifold pressure. This effectively avoids the above primitive nmerical derivative in real-time implementation. Dynamically, this equivalent to inserting a third-order system and a second-osystem in the closed loop. This results naturally in a dynamfeedback.

2.4 Optimal Dynamic Back-Stepping Sliding Surface Con-trol. This method combines all the advantages of the abmethods. Besides, suboptimization is involved. The main featuof this design method can be described as:

1 Sliding mode method: Using sliding surface and slidireachability condition as above.

2 Back-stepping is used in control design logic in formulatierror dynamics. In each step, a proper sliding surface is choThus multiple sliding surfaces naturally result in the end. This ipromising way of dealing with additive unmatched uncertaintin nonlinear models~Swaroop et al.@11#!.

3 Integral filters are used to calculate the derivatives of reence signals at each consecutive step. Thus analytic differentiaof reference signals is avoided.

4 The choice of sliding gain for each sliding surface is innonlinearH` sub-optimization approach~Byrnes et al.@24#, Hilland Moylan@25#, Isidori @26#!.

This design method was first studied in~Hedrick @9#, Swaroopet al. @11#! without sub-optimization. Consider the followinSISO nonlinear uncertain system in strict feedback form withcertainties

x15x21w1~x1!1D1~x1!

x25x31w2~x1 ,x2!1D2~x1 ,x2!

. . . . . . (5)

xn215xn1wn21~x1 ,x2 , . . . ,xn21!1Dn21~x1 ,xn21 , . . . ,xn21!

xn5w~x1 ,x2 , . . . ,xn!1c~x1 ,x2 , . . . ,xn!u1Dn~x1 ,x2 , . . . ,xn!

where

uD i u<r i ix~ i !i1 l i

x~ i !5~x1 , . . . ,xi !

r i>0, l i>0

i 51, . . . ,n.

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Here the only assumption is that

c~x1 ,x2 , . . . ,xn!Þ0

in the region of interest. The control tasks are

~a! to makex1 to track x1d, which may not be differentiablewith ultimately bounded error;

~b! to render the closed loop system robustly ultimatebounded with respect to the uncertainties.

Design a controller with the following procedure with slidinreachability conditionS52kS2k0sat(S).

Step 1Let

S15x12x1d

Formally differentiateS1 and set

S152g1~S1!1S21D1

S25x22x2d

wherex2d is to be determined. Besides,x1d may not be differen-tiable. To avoid this difficulty, use the constructed derivatix1d2dot from an integral filter to replace the formal derivativd/dt(x1d) in S. It is obtained that

x15x1–dot2g1~S1!1S21D1~x1!

5x21w1~x1!1D1~x1!

which leads to

x2d5x1d–dot2g1~S1!2w1~x1!.

Thus, to makex1 trackx1d , it is necessary and sufficient thatx2track x2d .

Step i ( i .1) Let

Si 115xi 112x~ i 11!d

wherex( i 11)d is to be determined. Formally differentiateSi andset

Si52g i~Si !1Si 111D i

Then use the filtered signalxid –dot to replaced/dt(xid) in Si , itis obtained that

xi5xid –dot2g i~Si !1Si 111D i

5xi1w i~x1 ,x2 , . . . ,xi !1D i

which leads to

x~ i 11!d5xid –dot2g i~Si !2w i~x1 ,x2 , . . . ,xi !

This is true fori 51,2, . . . ,n22.Step nÀ1 Formally differentiate

Sn5xn2xnd

xnd5xn21d–dot2gn21~Sn21!2wn21~x1 ,x2 , . . . ,xn21!.

Set

Sn52gn~Sn!1Dn

and replacexnd with the filtered signalxnd–dot constructed froma filter. It is obtained that

xn5xnd–dot2gn~Sn!1Dn

5wn~x1 ,x2 , . . . ,xn!1c~x1 ,x2 , . . . ,xn!u1D i

from which the controller is constructed as

u51

c~x1 ,x2 , . . . ,xn!@xnd–dot2gn~Sn!2wn~x1 ,x2 , . . . ,xn!#

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3 Controller Construction and ImplementationThis section addresses main possible uncertainties and u

level controller constructions using the above three methods.

3.1 Real-Time Uncertainties. Uncertainties in real-timeimplementation may be classified follows:

1 Model mismatch and external disturbances: such as swing on/off of air conditioner, tire inflation differences, differennumber of passengers, vehicle aerodynamic force variationlateral cornering force on the steered wheels, which has a comnent in the longitudinal direction.

Perhaps, the most significant disturbance is caused byshifting. The torque transmitted from engine to drivetrain isduring gear shifting. If the vehicle is in cruising mode or deceration mode, the desired torque would be 0 or a negative vaSuch a loss of torque to drivetrain would not affect vehicle spand distance tracking errors. However, if the vehicle is in acceration mode, desired torque is positive. The loss of torque todrivetrain is equivalent to the loss of control, i.e., the system ding this short period of time is not controllable. This will definitely lead to a great impact on the speed and distance tracerrors. This error increases with the increase of the desired aeration profile.

2 System parameter uncertainty: System parameters, whicclude vehicle parameters, sensor parameters, and actuator peters, are subject to dynamic variation and different road situat

3 Measurement noise on signals: This includes the noise tospeedometer, to radar distance measurement, magnets instaerrors on the ground and the magnetometer measurement ethe error caused by actuators etc.

4 The distribution of the normal forces among the wheewhich determine traction on each wheel, strongly influencewheel’s dynamics. If we take the average of the line speeds othe wheels as the vehicle speed in longitudinal direction, as inpaper, then the distribution of these forces do not affect very mthe vehicle longitudinal speed. Thus the speed difference betwwheels is not taken into consideration for longitudinal controlthis stage.

To improve performance, all the three types of uncertainneed to be compensated in practical control implementation.simplest case is to assume that the uncertainties of type~1! and~2!are approximated by additive uncertainties in the model, whcan be classified as~a! matched and~b! unmatched.

From a dynamics point of view, additive uncertainties are tocompensated in control design although one cannot alwachieve this. The measurement noise fed into the controller, hever, needs to be treated with proper filters. It cannot be comsated by the controller itself in general.

3.2 Sliding Surface Controller. In ~Choi @5#!, the slidingsurface is chosen as follows for each pair of consecutive cars

Si5 e i1a1e i1q1~ xi2 xlead!1q2S xi2xlead1(j 51

i 21

D j Dwherea1 , q1 , q2.0 and

e i5~xi2xi 21!2Di

Di — desired spacing betweeni th and (i 11)th carxi — the position ofi th care i — distance error to be regulated

Then the synthetic acceleration can be solved from the slidreachability condition by using this sliding surface.

It is noted that, in the calculation ofS, one has to face theproblem of differentiatinge i which is essentially the relativespeed between cari and cari 21. This is implemented simply byusing a moving average filter and a numerical derivative

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whereDt is the sampling time period. It is noted that, using thmethod, only the matched uncertaintyD(t), i.e., appearing in thespeed channel, can be expelled by properly choosing slidreachability conditions. However, the uncertainty appearing indistance channel cannot be expelled in this way. The measurenoise is expected to be filtered out by the moving average filbut it brings a prominent delay to the controller at the same timAnother feature is that the sliding reachability condition usedcontinuous which is actually an exponential reaching law.

Measured data used in this controller construction are: speecurrent car, distance measurement between two consecutivepreceding speed and preceding acceleration.

3.3 Dynamic Sliding Surface Controller. Sliding surfaceconstruction is similar as above but the sliding reachability cdition is chosen discontinuous. Data measurement is also the sas previous method. Besides, the following third-order filterused to calculate the derivative of the speed error.

z151

t1~2z11 e i !

z25z3

z352

t22 S 2t2z32z21

1

t1~2z11 e i ! D

wheret1 , t2.0 are integral filter gains. Now using the followinestimate

d

dt~ e i !'z21t2z3

A similar filter has been used in the calculation of the derivatof manifold pressure.

Test results show that throttle command chattering has bgreatly reduced by use of this filter. However, the integral filcan only improve measurement noise reduction as expectedoes not help in dealing with un-matched uncertainties. Thawhy there is not much improvement in distance tracking ercompared to previous work~Choi @5#!.

3.4 Optimal Dynamic Back-Stepping Sliding Surface Con-trol. Measured data used in this controller construction aspeed of current car, distance measurement between two contive cars, and preceding speed, while preceding acceleration isused. This means that data required is one less compared tprevious controllers. This is good from practical viewpoint.

Here the design logic is also different from the previous twmethods. It serves to deal with unmatched uncertainties. Theretwo sliding surfaces for each pair of cars.

Si15~xi2xi 21!2Di

The first sliding reachability condition is chosen as

Si152g1~Si1!1Si2 .

g1~Si1!52ki1Si12ki10satS Si1

e Dwheree satisfies 0,e!1. It controls the thickness of the boundary layer. The second sliding surface is chosen as

Si25 xi2x2d– i

wherex2d– i is

x2d– i5d

dt~Di !2g1~Si1!



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Choose a reachability condition

Si252g~Si2!52ki2Si22ki20satS Si2

e DIn the calculation of synthetic acceleration, the derivative ofx2d– i

is estimated by the following filter

z151

t1~2z11x2d– i !

z25z3

z352

t22 S 2t2z32z21

1

t1~2z11x2d– i ! D

wheret1 , t2.0. This leads to the estimation

d

dt~x2d– i !'z21t2z3

A similar filter has been used in the calculation of the derivatof manifold pressure.

The prominent differences here are that

1 both the uncertainties in the distance channel and that inspeed channel are expelled by proper choice of sliding reaability conditions;

2 the switching gainski10 andki20 are chosen according to aestimated magnitude of the additive uncertainties;

3 the gainski1 and ki2 are chosen such that the closed-losystem is internally stable with the matched uncertaintiestenuated~Lu and Hedrick@27#!.

3.5 Real-Time Implementation. Lower-level control com-mands are generated similarly for all the three controllers aboThe sampling time period is 20 milliseconds in the realtime coThis is also the realtime control command step. Precedingspeed is obtained from the preceding car and from the leadthrough communication; current car line speed is an averagthe four wheel line speeds. The distance to the preceding cobtained by the filtering and fusing of radar range measuremand distance estimation using magnets.

In both simulation and real-time testing, it appeared tsecond- and third-order integral filters are much better than fiorder integral filters in the sense that they provide stronger nreduction and produce better approximations. Compared tmoving average filter plus numerical derivative, one more advtage is that they cause much less time delay. In the real-time cthe integral filters are implemented with the third-order RunKutta method.

4 Real-Time TestThe controllers above have been tested at both high and

speeds. From real-time tests, it can be observed that speed dences have significant impact on longitudinal controllers.

In general, at lower speed, some performance aspects sudistance and speed tracking errors in both transient and ststate and throttle command chattering tend to be worse whileceleration requirements and gear shift disturbances are not prnent. These are based on the following observations:

1 At lower speed, distance measurement by speedometusually not as good as that at higher speed;

2 Throttle command at lower speed is more difficult to manabecause the desired throttle command is close to idle spthrottle.

3 Speed variation is larger relative to the maximum speedsteady state.

4 IC engines tend to have poorer output torque when speelow.



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For passenger’s comfort, the acceleration profile for commcial vehicle can be set relatively low. However, robust stabilitythe closed-loop dynamics at higher speed is extremely importTo test this property, reference trajectory is usually designed sthe vehicle should reach the maximum cruise speed in the shotime. In this case, vehicle dynamics are quite different in tphases: acceleration/deceleration and cruising. High acceleracapability is required and thus transient performance tends topoorer in acceleration phase. Once vehicle speed reaches mmum cruise speed, it tends to be more stable in cruising dularge vehicle inertia. High speed test also means longer runntime, which helps the controller to settle down. Thus betsteady-state is usually achieved at high speed test.

The following test results justify the discussion above. Toduce paper length, only high speed tests are presented. High-stests of the three longitudinal controllers were performed atCrows Landing test track. The maximum speed is 72 km/h. Tcars were tested using three controllers:

1 restructured demo code2 dynamic sliding mode controller combined with integral fi

ters3 suboptimal dynamic back-stepping sliding surface contro

combined with integral filters

1. Restructured Demo CodeThe restructured demo code has a similar performance to

original demo code. Its performance is very difficult to improjust by gain tuning. The parameter tuning seems to be optiwith respect to the control design method. Some disadvantahave been observed.

1 Large time delays due to filtering: It may affect the perfomance and/or stability of the closed-loop system.

2 Large estimation errors due filtering: It significantly affecthe closed-loop dynamics;

3 Noise reduction is not effective and chattering in throtcommand is significant.

The data collected using this controller are shown in Fig. 12. Dynamic Sliding Mode ControllerCompared to the demo code, the performance, particularly

distance tracking error, is not improved as shown in Fig. 2. Thibecause the disturbances in the distance channel cannot bpelled in this design method. However, due to the use to intefilters and slightly different formation of the sliding surface, chatering has been largely reduced in throttle command.

Fig. 1 Reconstructed demo code, 2nd car, following distance4.0 m.

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3. Optimal Dynamic Back-stepping Multiple Surface Con-troller

The performance has been improved in the following aspectshown in Fig. 3:

~a! Transient performance is much better: less distance traing error:20.2;0.3 m compared to that of20.5;0.6 m in democode; Settling time is shorter235 seconds after control startecompared to 60 seconds in demo code;

~b! Distance tracking error in steady state is less than 0.0mcompared to 0.2m in demo code;

~c! Throttle chatter is effectively reduced;~d! It seems more robust with respect to external disturbanc

5 Concluding RemarksA preliminary experimental implementation shows that the O

timal Dynamic Back-stepping Sliding Surface Controller seesuitable for longitudinal control of car platooning, as both steadstate performance and transient performance are better thanfor the baseline~Demo ’97! controller. It has been successfullapplied toautomated vehicle merging announcerwhich is timeand distance critical~Lu et al. @28#!. This may be due to the fol-lowing reasons:

Fig. 2 Dynamic sliding mode controller, 2nd car, following dis-tance 4.0 m

Fig. 3 Dynamic back-stepping sliding surface controller, 2ndcar, following distance 4.0 m

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1 Back-stepping design logic emphasizes speed control wassigning less importance to distance regulation. This may heshape the closed-loop dynamics;

2 The sliding reachability condition when combined with bacstepping design logic effectively expels uncertainties in both dtance and speed channels, which results in a robust closed-system;

3 Integral filters effectively reduce measurement noise, brbetter estimation of the derivative, incur much less time decompared to the moving average plus numerical derivative. Dianalytic differentiation of the reference state is avoided.

4 Optimal sliding gain choice in a nonlinearH` approach fur-ther increases the robust stability margin and improves both tsient and steady-state performance.

Recent fine tuning further improves string stability and perfmance for platooning. These will be addressed in future work

AcknowledgmentThis work was supported by California PATH Program of t

University of California and Caltrans. The contents of this repreflect the views of the authors who are responsible for the faand the accuracy of the data presented herein. The contents dnecessarily reflect the official views or policies of the StateCalifornia. This paper does not constitute a standard, specition, or regulation. Other PATH AVCS group staff, notably, DaEmpey, Wei-Bin Zhang, Paul Kretz, Benedicte Bougler have ctributed in the project management, hardware support, consttive suggestions, providing documents of previous softwaretest driving. They are gratefully acknowledged.

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