Robust grey-box closed-loop stop-and-go control

Transcription

1 Author manuscript, publishe in "7th IEEE Conference on Decision an Control, Cancun : Mexico (8" Robust grey-box close-loop stop-an-go control Jorge Villagra, Brigitte Anréa-Novel, Michel Fliess an Hugues Mounier inria-399, version - 8 Sep 8 Abstract This paper presents a robust stop-an-go control law, especially well aapte to car following scenarios in urban environments. Since many vehicle/roa interaction factors (roa slope, rolling resistance, aeroynamic forces are very poorly known an measurements are quite noisy, a robust strategy is propose within an algebraic framework. On the one han, noisy signals will be processe in orer to obtain accurate erivatives, an thereafter, variable estimates. On the other han, a grey-box close-loop control will be implemente to compensate all kin of unmoele ynamics or parameter uncertainties. A. Generalities I. INTRODUCTION Aaptive cruise control (ACC an stop-an-go control systems have been eeply stuie in recent years [7]. Let us recall that while ACC automatically accelerates or ecelerates the vehicle to keep a quasi-constant target velocity an heaway istance, stop-an-go eals with the vehicle circulating in towns with frequent an sometimes har stops an accelerations. Both situations present completely ifferent comfort an safety constraints, an therefore, in most of the reporte works, ACC an stop-an-go problems are treate separately. Some approaches ([], [9] have trie to reprouce human behavior in orer to achieve a comfort-base control. Unfortunately, this kin of strategy may not necessarily lea to safe operation (see e.g. [7]. Besies, external factors such as roa characteristics, weather conitions, an traffic loa shoul be taken into account in a robust an safe control system. Furthermore, an accepte comfort criteria is to guarantee boune longituinal accelerations an jerks. Using this iea, many authors (e.g. [], [], [6] have moele interistance using ifferent types of time polynomials, whose coefficients are obtaine respecting safety acceleration an jerk constraints. In general, these approaches prouce acceptable results in an ACC scenario. However, uring a suen eceleration of the preceing car, the vehicles present a large transitory relative velocity an the actual inter-istance ecreases abruptly. J. Villagra is with the Departamento e Ingenieria e Sistemas y Automatica, Universia Carlos III, Leganes (Mari, Spain, jvillagr@ing.uc3m.es B. Anréa-Novel is with the Centre e Robotique, École es Mines e Paris, 6, bl. Saint Michel, 77 Paris Ceex, France, brigitte.anrea-novel@ensmp.fr M. Fliess is with INRIA-ALIEN & LIX (CNRS, UMR 76 École polytechnique, 98 Palaiseau, France, Michel.Fliess@polytechnique.eu H. Mounier is with the Institut Électronique Fonamentale (CNRS, UMR 86 Université Paris-Su, 9 Orsay, France, Hugues.Mounier@ief.u-psu.fr Hence, this ynamical scenario woul not be suitably represente by static polynomial moels, but by some kin of inter-istance ynamic moel. In [], the authors propose a nonlinear reference moel taking into account safe an comfort specification in an intuitive way. In aition, the moel is combine with a simple feeback loop use to compensate unmoele ynamics an external isturbances. However, this work makes two assumptions that are never met in real situations: the inter-istance an the velocity of the leaer vehicle are perfectly measure from suitable sensors; the reference acceleration generate by the ynamic inter-istance moel is instantaneously applie to the following vehicle. Our contribution consists in elaborating the engine/brake torque to prouce the expecte reference acceleration of the follower vehicle, that is, when taking into account measurement noises as well as unmoele ynamics, such as roa inclination, aeroynamic forces or rolling resistance. To achieve this task, a unifie approach on estimation an control has been use. An algebraic framework is propose to eal with filtering, estimating erivatives, an finally, moel free control esign.r It is important to point out that these filters, ifferentiators an estimators are not of asymptotic nature, an o not require any statistical knowlege of the corrupting noises. This original way of treating conventional problems can be viewe as a change of paraigm in many control an signal processing aspects (cf. []. Finally, in orer to minimize the loss of performances ue to uncertain roa parameters, a moel-free control philosophy will be use, which will be aapte by incluing specific well-known ynamics, in a kin of grey-box moel control. B. Outline of the paper The general control scheme will be presente in Section II. In the thir Section, the algebraic setting for moel-free control will be introuce. Section IV will be evote to recall the vehicle ynamics an the feeforwar control, where a longituinal acceleration an a consequent torque is generate uner ieal circumstances. This section also shows the motivation for the choice of algebraic techniques introuce in Section III. Noise an parameter robustness will be tackle with a grey-box close-loop control approach in Section IV. Simulation results will show a very goo Examples of aaptive car following controllers ealing with this problem can be foun in works by [], [7] or [].

2 inria-399, version - 8 Sep 8 compromise between performance an robustness. Finally, the conclusion an some future work will be rawn in Section V. II. CONTROL SCHEME Figure graphically summarizes the whole control scheme. The stop-an-go system uses raar information an CAN bus accessible ata to generate, via a ynamic moel an a reliable follower velocity estimation (see e.g. [8], the esire acceleration. This moel provies a safe an comfortable reference inter-istance between the leaer an the following vehicle. A reference longituinal acceleration is then generate as a feeforwar control. Since this moel is base on corrupte measures an not always vali assumptions, a feeback term is introuce. This close-loop will not only behave as a typical PID controller, but it will also estimate linear or non-linear unmoele ynamics (roa slope, win, rolling resistance in orer to anticipate the controller action. The resulting control will provie an acceleration as close as possible to the esire one. ^ Vxl Dynamic interistance moel Leaer velocity estimator r + - Feeforwar control. ^ Vxf Fig.. u r Grey-Box Close-loop Control Follower velocity estimator + + u i Torque generator g Roa slope Win force Rolling resist. Vehicle ( DOF with tire an suspension moels General Stop-an-go control scheme. III. ALGEBRAIC SETTING FOR MODEL-FREE CONTROL A. Numerical ifferentiation Start with a polynomial time function x N (t = N ν= x(ν ( tν ν! R[t], t, of egree N. The usual notations of operational calculus (see, e.g., [] yiel X N (s = N ν= x (ν ( s ν+ Multiply both sies by positive powers of s. The quantities x (ν (, ν =,,...,N, which are linearly ientifiable, satisfy the following triangular system of linear equations: α s N+ X N s α ( N = α s α x (ν (s N ν ν= Vxl Raar ESP x y i α N ( Multiplying both sies of Eq. ( by s N, N > N, permit to get ri of time erivatives, i.e., of s µ ι X N s ι, µ =,...,N, ι N. Consier now an analytic time function, efine by the power series x(t = ν= x(ν ( ν! tν, which is assume to be convergent aroun t =. Approximate x(t by the truncate Controller Area Network Taylor expansion x N (t = N ν= x(ν ( tν ν! of orer N. Goo estimates of the erivatives are obtaine by the same calculations as above. Remark 3.: A most elegant an powerful algorithmic proceure for obtaining a corresponing numerical ifferentiator is provie in []. It will be exploite in the sequel. B. Moel-free control 3 Take a finite-imensional SISO system E(t,y,ẏ,...,y (ι,u, u,...,u (κ = which is linear or not, where E is a sufficiently smooth function of its arguments. Assume that for some integer n, E < n ι, y (n. The implicit function theorem yiels then locally y (n = E(t,y,ẏ,...,y (n,y (n+,...,y (ι,u, u,...,u (κ This equation becomes by setting E = F + αu: where y (n = F + αu ( α R is a non-physical constant parameter, which is chosen by the engineer in such a way that F an αu are of the same magnitue, F is etermine thanks to the knowlege of u, α, an of the estimate of y (n. Remark 3.: A system might only be partially unknown as in Sect. IV-E. It is straightforwar to aapt the previous metho in this case. In all the known examples until toay, n was chosen to be equal to or in Eq. (. If n =, the esire behavior is obtaine via the intelligent PID controller, which is of the form u = e (ÿ F + K P e+k I et + K D (3 α t where y is a reference trajectory e = y y is the tracking error, K P, K I, K D R are suitable gains, the tuning of which is quite straightforwar. IV. HIGH-LEVEL LOOP CONTROL The feeforwar high level control will be briefly recalle to point out its main features an the funamental limitations that have been aresse in the present work: The close-loop control is not at all robust to raar noisy measurements. When suen accelerations/ecelerations are neee, the corresponing open-loop engine/brake torques may be har to compute. After a brief introuction to vehicle longituinal ynamics, the next subsections will etail how each issue has been aresse, uner the algebraic framework presente in Section III. 3 See [3] for more etails, for numerous computer simulations, an for references on alreay existing applications.

3 inria-399, version - 8 Sep 8 A. Vehicle longituinal ynamics A force balance along the vehicle longituinal axis (cf. [6] yiels Mẍ = F x f + F xr F a R x f R xr mgsinθ where F x f, F xr are respectively the front an rear longituinal tire forces, R x f an R xr the front an rear tire forces ue to rolling resistance, θ the angle of inclination of the roa, an F a is the longituinal aeroynamic rag force. The rolling resistance forces are often moele as a timevarying linear function of normal forces on each tire, i.e. R x = kf z, with k the rolling resistance coefficient. The aeroynamic forces can be written (e.g. [9] as F a = ρc A F (V x +V win whith ρ being the mass ensity of air, C the aeroynamic rag coefficient, A F the frontal area of the vehicle (the projecte area of the vehicle in the irection of travel an V x, V win respectively the longituinal vehicle an win velocities. Finally, Pacejka moel [] is use for longituinal tire/roa interaction forces F x. They epen on many factors, but essentially on longituinal slip an normal forces. These normal forces will be compute as realistically as possible within a.o.f vehicle moel (6.o.f. of the vehicle center of gravity an one supplementary.o.f. on each wheel. B. Feeforwar control A reference moel propose by [] will act as a feeforwar term into the longituinal high level control law. The basis of this moel will be sketche in the next lines. The inter-istance reference moel escribes a virtual vehicle ynamics which is positione at a istance r (the reference istance from the leaer vehicle. The reference moel ynamics is given by r = ẍ l ẍ r f ( where ẍ l is the leaer vehicle acceleration an ẍ r f = ur ( r, r ( is a nonlinear function of the inter-istance an of its time erivative. Introucing r in (, where is the safe nominal inter-istance, the control problem is then to fin a suitable control when : u r = u (,, such that all the solutions of the ynamics ( fulfill the following comfort an safety constraints: r c, with c the minimal inter-istance. ẍ r B max, where B max is the maximum attainable longituinal acceleration.... x r J max, with J max a boun on the river esire jerk. The authors of [] propose to use a nonlinear amper/spring moel u = c, which can be introuce in the ynamics equation ( to give: = c ẍ l. The previous equation may be analytically integrate an expresse backwars in terms of r as follows, assuming that ẋ l ( = : r = c ( r + ẋ l (t β, β = ẋ r f (+ c ( r (. (6 From (, the feeforwar control law is then obtaine applying ẍ r f = u r = c r r (7 where the inter-istance evolution comes from the numerical integration of (6. Remark.: In practice, the leaer velocity is not measure an we have to construct an estimator, using for example techniques evelope in section III. C. Close-loop control Some kin of feeback control must be introuce in orer to avoi errors inuce by measurement noises. A stanar PID compensation leas to extremely noisy perturbe results when a erivative term is use, an to instability or important tracking errors when it is not. In orer to avoi this kin of problem, a PD compensator has been implemente, where inter-istance an its timeerivatives are obtaine using Sect. III-A. The signal can be locally approximate by a linear polynomial (N =. Thus, (t = + t, t,, R. In the first case, an estimator for is sought; in the secon one, will be estimate. If we take for instance N = 3 an N =, the estimators can be respectively written as follows : ˆ= ˆ = T T (T 3τ(ττ ˆ = ˆ = 3! T (T τ(ττ (8 T 3 Figure shows 6 the ifference between applying two iscrete PD controller with a ifferent low-pass filter an an algebraic PD controller. Note that the parameter c is an algebraic function of safe an comfort parameters c, V max, B max an J max (cf. []. The integral term is not use in orer to avoi an unstable behavior of the system (see []. 6 Since the inter-istance reference trajectory epens on the close-loop behavior, it is ifficult to exactly obtain the same testbe for the 3 cases presente in Figure.

4 3 interistance (m 3 jerk (ms algebraic estimator to (8 will then be use to compute ˆ ω i. Finally, figure 3 compares inter-istances between openloop generate torque uner no slipping assumption an open-loop torque with our ynamic estimation approach. A remarkable improvement can be obtaine when this new strategy is use in emaning situations. Inee, results shown in figure 3 are obtaine with longituinal accelerations up to. ms, which are rarely foun in an ACC context. inter istance (m jerk (ms 3 3 Torque with w/t: Σ(y yref =3 Torque without w/t: Σ(y yref = interistance (m inria-399, version - 8 Sep 8 interistance (m 3 jerk (ms Fig.. Inter-istance an jerks evolution with ifferent close-loop controllers: a iscrete PD with low-pass filter of cut-off frequency equal to Hz (top, another one with cut-off frequency equal to Hz (mile an an algebraic PD (bottom. The P =.7 an D =. parameters are ientical in all cases. D. torque generation The wheel rotation ynamics can be written as follows I ω = rf x + τ e τ b (9 where I is the rotation inertia moment, ω the wheel angular velocity, r is the tire raius, τ e the applie engine torque, an τ b the brake torque, both of them applie to the wheel center. A commonly use assumption ([],[3],[6] consists in consiering rolling without slipping, i.e. V x = R g rω, where R g is the gear ratio. However, in a stop-an-go context, where fast responses to suen ecelerations are require, this is not an acceptable hypothesis. If a generalize wheel torque τ eb = τ e τ b is consiere, it is straightforwar to see its epenence on tire/roa interaction forces. Therefore, a realistic estimation of this generalize torque from equation (9 turns out to be quite har. The sum of the wheels rotation ynamics equations an of the vehicle ynamic longituinal equation Mγ x = i= F x i yiels 3 Fig. 3. Comparison between torque generation uner no slipping assumption, an with expression (. E. Grey-box feeback control The proceure escribe in Sect. III-B can here be applie in a particular way. Since some specific ynamics are very well known, it is worth to integrate them in our preictive scheme. Thus, the esign parameter α correspons here to well-known quantities. Recall the local input-output moel introuce in Eq. ( an compare it with the reorere torque expression ( ( γ x = τ g I ω i + G(t, ( Mr i= ( G(t = r F a R x f R xr Mgsinθ If rolling without turning is consiere ( V x = γ x, the next equation can then be written: V x = F(t+αu(t+β(t ( where F = G Mr, α = Mr an β = I i= ω i an u = τ g is Mr the control variable. The goal is to obtain an accurate close-loop estimation of F. Following the theoretical ieas escribe in Sect. III, the proceure consists, first of all, in rewriting ( in the operational omain, with the assumption F = F in the estimation time winow, τ g = τ eb = I i= ω i + rmγ x. ( The main inconvenient for such an estimator is that a goo numerical ifferentiator for ω i is neee. An equivalent sv x V = F s + ατ g(s+β(s (3 an then applying the operator with the aim of eliminating the initial conition V s

5 inria-399, version - 8 Sep 8 V x + s V x s = F s + α τ g s + β s Finally, s ν, with ν = is applie in orer to eliminate any non causal term F s = s V x V x s s + α τ g s s + β s s which, expresse backwars in the time omain, yiels F = 3! T T 3 (( T + tv x (t (T tt(ατ g (t+β(tt ( The final close-loop control is then, applying (3 to our case an consiering rolling without turning (i.e. V r x = ẍ r f = u r : τ g = Mr interistance (m Accelerations (ms 3 ( u r F + I Mr ω i + K P e + K D ė, e = r i= ( 3 6 Leaer Follower 6 3 Torque (Nm Velocities (ms Leaer Follower 3 3 Fig.. Inter-istance, velocity, acceleration an generalize torque for highly emaning scenario (up to ms. When the complete strategy is teste on a quite emaning scenario 7, the inter-istance moel reference is pretty well tracke (see figure, the follower acceleration remains uner the comfort constraints, an consequently, the jerk bouns are also guarantee (cf. []. Furthermore, the generalize torque applie to the vehicle seems very robust to noise perturbations. 7 Several heavy accelerations/ecelerations are applie to the vehicle on a flat roa, where neither rolling resistance nor aeroynamic forces are consiere. However, the most important source of uncertainty comes from roa conitions. Thus, if rolling resistance, aeroynamic efforts an a slope roa are introuce, the results are slightly ifferent. Figure a shows that even if the interistance trens are alreay very well respecte, a variable bias cannot be annihilate with the stanar control. The grey-box control strategy has been applie in orer to obtain more robust results. The ashe line in figure a represents the tracking performance when the estimator ˆF of global isturbances is introuce. A consierable improvement (almost % is obtaine when the global effects of isturbances are estimate via equation (. F (ms inter istance (m =9. =39 without F with F F F s F r F a F=F s +F r +F a Fig.. (a Inter-istance evolution with an without F estimation. (b Comparison between real F an its estimate F. Aeroynamic (F a, roa slope (F s an rolling resistance (F r terms are also epicte. It can be appreciate from figure b that roa slope, rolling resistance an aeroynamic forces are pretty well estimate in an overall term ˆF = F. Note that aeroynamic forces are not very significant when compare with the roa slope. However, big win gusts can appear at high spees. In this case, a reliable an fast estimator shoul applie. Figure 6 shows the behavior of control law propose in ( when severe win gusts longituinally knock the car. It can be appreciate that our control is much more robust when F is estimate. Moreover, the time winow estimation size T e can be use as a tuning parameter for safety or comfort. When tracking performance is more important than comfort (suen changes in acceleration, a small winow will be use. If important jerks are not esire, a bigger winow estimation will be more appropriate. V. CONCLUDING REMARKS A grey-box close-loop stop-an-go control for vehicles has been presente. Its main feature is its ability to eal with

6 inria-399, version - 8 Sep 8 usual isturbances (win, roa slope, rolling resistance... which are not easily measurable. It can be seen that, as expecte, our metho leas to a close-loop robust behavior with respect to noises an unmoele ynamics. The next step will be to evelop a low-level control incluing the engine an brake ynamics. An algebraic approach is uner stuy to generate the physical control variables: the throttle angle (see alreay [8] an brake pressure. Furthermore, the whole algorithm is being aapte to real vehicles an will be soon presente. REFERENCES [] M. Brackstone an M. McDonal, Car-following: A historical review, Transportation Research F, Vol., pp. 8-96, [] C. Chien an P. Ioannou, Automatic vehicle-following Proc. Amer. Contr. Conf., pp. 78-7, 99 [3] M. Fliess, C. Join, Commane sans moèle et commane à moèle restreint, e-sta, Vol., 8 (available at [] M. Fliess, C. Join, an H. Sira-Ramírez, Non-linear estimation is easy, Int. J. Moelling Ientification Control, Vol. 3, 8 (available at [] J. K. Herick, D. McMahon, V. Narenran, D. Swaroop, Longituinal Vehicle Controller Design for IVHS Systems, Proc. Amer. Control Conf., pp. 37-3, Boston, 99 [6] T. Hiraoka, T. Kunimatsu, O. Nishihara an H. Kumamoto, Moeling of river following behavior base on minimum-jerk theory, Proc. th Worl Congress ITS, paper 36, San Francisco,. [7] P. Ioannou an Z. Xu, Throttle an Brake Control Systems for Automatic Vehicle Following IVHS Journal, Vol., pp.3-377, 99. [8] C. Join, J. Masse, an M. Fliess, Étue préliminaire une commane sans moèle pour papillon e moteur, J. europ. syst. automat., Vol., pp , 8. [9] U. Kiencke an L. Nielsen, Automotive Control Systems, Springer, [] M. Liubakka, D. Rhoe, J. Winkelman, an P. Kokotovic, Aaptive automotive spee control, IEEE Trans. Automat. Control, Vol. 38, pp. 66, 993 [] J. Martinez an C. Canuas-e-Wit, A Safe Longituinal Control for Aaptive Cruise Control an Stop-an-Go Scenarios, IEEE Trans. Control Systems Technology, Vol., pp. 6-8, 7 [] M. Mboup, C. Join an M. Fliess, A revise look at numerical ifferentiation with an application to nonlinear feeback control, Proc. th Meiterrean Conf. Control Automation, Athens, 6 (available on [3] L. Nouvelière an S. Mammar, Experimental vehicle longituinal control using a secon orer sliing moe technique, Control Eng. Practice, Vol., pp , 7. [] H. Pacejka, E. Baker, The magic formula tyre moel, Proc. st Internat. Coll. Tyre Moels Vehicle System Analysis, pp. -8, 99. [] M. Persson, F. Botling, E. Hesslow an R. Johansson, Stop & Go Controller for Aaptive Cruise Control, Proc. of the IEEE International Conference on Control Applications, pp , Hawaii, 999 [6] R. Rajamani, Vehicle Dynamics An Control, Springer, [7] A. Vahii an A. Eskanarian, Research Avances in Intelligent Collision Avoiance an Aaptive Cruise Control, IEEE Trans. Intelligent Transportation Systems, Vol., pp 3-3, 3. [8] J. Villagra, B. Anréa-Novel, M. Fliess an H. Mounier, Estimation of longituinal an lateral vehicle velocities: an algebraic approach, Proc of 8American Control Conference, Seattle, 8 (available at [9] K. Yi an I. Moon, A Driver-Aaptive Stop-an-Go Cruise Control Strategy, Proc. of IEEE Int. Conf. on Networking. Sensing & Control, pp. 6-6, Taipei,. [] K. Yosia, Operational Calculus: A Theory of Hyperfunctions, Springer, New York, 98 (translate from the Japanese. [] K. Youcef-Toumi, Y. Sasage, Y. Arini, an S. Huang, Application of time elay control to an intelligent cruise control system Proc. American Control Conf., pp , Chicago, 99 Win velocity (ms inter istance (m acceleration (ms F (ms =9.7 =7.8 =87.7 with F (T =T e s with F (T e =3T s real without F with F (T e =T s estimation with F (Te=3T s estimation without F F F (T =T e s F (T e =3T s Fig. 6. (a Win velocity. (b Inter-istance evolution with ifferent sliing winows for F estimation. (c Longituinal acceleration with ifferent estimation time winows. ( F estimation