II Uni ersity of Modena and Re ggio Emilia ia ignolese 905b 41100 Modena Italy morselliriccardounimoreit Abstract One of the main issue of any contr ol strategy or braking systems is to face the many uncertainties due to the str ong spr ead of the sy ID: 22227 Download Pdf

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II Uni ersity of Modena and Re ggio Emilia ia ignolese 905b 41100 Modena Italy morselliriccardounimoreit Abstract One of the main issue of any contr ol strategy or braking systems is to face the many uncertainties due to the str ong spr ead of the sy

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Self-T uning Contr ol Strategy or Antilock Braking Systems Riccardo Morselli and Roberto Zanasi D.I.I. Uni ersity of Modena and Re ggio Emilia ia ignolese 905/b 41100 Modena, Italy morselli.riccardo@unimore.it Abstract One of the main issue of any contr ol strategy or braking systems is to face the many uncertainties due to the str ong spr ead of the system' parameters: oad conditions, ydraulic actuators, tir beha viour etc. Mor eo er the need or cheap components limits both the number of sensors and the quality of the actuators. This paper pr oposes self-tuning contr ol

strategy or braking systems. The pr oposed contr ol strategy is based on tw light assumptions: 1) the tir longitudinal or ce as function of the tir slip has always unique minimum; 2) the ydraulic actuators can incr ease, decr ease and hold the braking pr essur within limited delay Only the measur of the wheel otational speed and the estimate of the wheel angular acceleration ar equir ed. The contr ol strategy is tested by simulation experiments. Antilock braking systems (ABS) are no commonly installed feature in road ehicles. The are designed to stop ehicles as safely and quickly as possible.

Safety is achie ed by maintaining the steering ef fecti eness and trying to reduce braking distances er the case where the brak es are controlled by the dri er during “panic stop”, see [1]. The ABS control systems are based on the typical tire beha viour described in [2 and briey sho wn in Fig. 1. As demonstrated in [3 ], optimal braking (in terms of minimum tra eled distance) occurs when the longitudinal force operates at its minimum alue along the force-slip curv e. The slip alue corresponding to the minimum longitudinal force depends also on the road conditions, ehicle speed, the

normal force, the tire temperature, the steering angle, etc. In all cases ho we er the shape of the force-slip curv has unique minimum for some alue of the slip The main issue of the ABS control strate gies is to track the optimal slip alue opt corresponding to the minimum longitudinal force using the smallest number of sensors, using the cheapest hardw are and acing the uncertainties due to both the aging of components and the unkno wn orking and en vironmental conditions. Dif ferent control techniques were applied to solv this challenging problem. Man authors ha presented control strate gies

based on the slip control, see [4 ], [5 ], [6 ], [7 and [8 ]. Theoretically the method of slip control is the ideal method. Ho we er tw problems arise: the (unkno wn) optimal slip alue must be identied and the ehicle speed must be measured (as in [5], [6 ]) or estimated in lo cost and reliable ay ercome these problems, either pressure measurement ha been proposed (see [9]) or the braking torque is supposed opt Accelerating Braking opt Accelerating Braking opt Accelerating Braking Fig. 1. Basic tire beha viour: slip ef fects on the longitudinal force. to be kno wn (see [10 ], [11 ], [12

]). These solutions lead to ery good performances, ut do not t the cost requirements. Moreo er man papers gi important theoretical results ut do not deal with the dynamics of the actuators. Currently most commercial ABSs use look-up tab ular approach based on wheel acceleration thresholds, see [1], [13 and [14 ]. These tables are calibrated through iterati laboratory xperiments and engineering eld tests. Therefore, these systems are not adapti and issues such as rob ustness are not addressed. The ork proposed in this paper sho ws that it is possible to track the optimal slip

alue by measuring only the wheel speed and estimating the wheel acceleration. The proposed control strate gy can be seen as minimum seek algorithm based on the phases when the braking pressure (not measured) is ept constant. During these phases the strate gy can infer the control action that will increase the braking force. This control strate gy is rob ust with respect to adhesion ariations, tak es into account the dynamics of the actuators and it is almost hardw are independent. The proposed strate gy is based on the same assumptions and on the same models usually presented in the

literature. Furthermore, dif ferently from the cited papers, the proposed approach tak es into account the dynamics of the alv es and does not require the measure of the slip, of the hydraulic pressure and of the braking torque. The paper is or ganized as follo ws. The dynamic model of standard braking system is described in Section II. Based on this model, the basic operating principle of the proposed control is xplained in Section III. The control strate gy is then described in Section IV and tested by simulations in Section V. Finally some conclusions are dra wn in Section VI.

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yre wheel cylinder br ak disk pump master cylinder built alv damp alv acuum booster br ak pedal eserv oir low mas yre wheel cylinder br ak disk pump master cylinder built alv damp alv acuum booster br ak pedal eserv oir low mas Fig. 2. Schematic of the standard antilock brak system for one wheel. The braking system consists of three subsystems: the tires, the ehicle and the electro-hydraulic actuators. One of the most widely used tire model is based on the acejka' “magic formula”, see [2 ]. This is set of static maps which gi the tire forces (longitudinal force lateral force and

self-aligning torque as function of the longitudinal slip the slip angle the camber angle and the ertical load The static maps are obtained by interpolating xperimental data. The longitudinal slip rate during braking is dened as: (1) where denotes the wheel angular speed, is the rolling radius and is the longitudinal speed of the wheel center in forw ard direction, see Fig. 1. or deceleration with constant slip and camber angles (longitudinal braking), is the speed of the ehicle and qualitati xample of the longitudinal force is sho wn in Fig. 1. The dynamic beha viour of wheel during

braking is described (see [7 ],[14 ], etc) by the dif ferential equation: br (2) where is the oil pressure in the braking system, br denotes the brak gain, br is the braking torque and is the tire longitudinal force. In this ork we consider simplied model of single wheel braking ehicle, the dynamics of this quarter ehicle model is described by: (3) where is the mass of the quarter ehicle and is the aerodynamic drag force. During braking is ne gati e. Since both and are limited, it is possible to nd the minimum car acceleration min (maximum ehicle deceleration) such that min al

ays. The control strate gy (proposed in Section Sec. IV) is almost independent from the hydraulic structure, the only requirement is that the delay of the actuators is limited abo by kno wn alue Ho we er to get the simulation results of Sec. V, the standard (see [9 ], [14 ], [15 ], etc) electro- hydraulic system sho wn in Fig. has been considered. This system is modeled according to [9 with the addition of transient to tak into account the dynamics of the alv es. The duration of the transient is lo wer than kno wn alue The master ylinder the pump and the lo pressure reserv oir are shared among

the wheels, see [1]. Each wheel has tw alv es: uilt alv between the master ylinder and the wheel ylinder and damp alv between the wheel ylinder and the lo pressure reserv oir Both alv es are on/of de vices and, after transient, the can only be in tw positions: closed or open. This hydraulic structure allo ws only three control actions: 1) INCREASE: the uilt alv is open and the damp alv is closed. The braking pressure increases. 2) HOLD: both the uilt alv and the damp alv are closed. The braking pressure at the end of transient, can be assumed to be constant. 3) DECREASE: the uilt alv is closed

and the damp alv is open. The braking pressure decreases. According to [9 ], the dynamics of the braking pressure can be modeled by means of o through the tw alv orices: dP dt mas low (4) the coef cients [0 1] (b uilt alv e) and [0 1] (damp alv e) are when the corresponding alv is closed, when the alv is completely opened. The dynamics of the actuators is described by: if and if and =T else if (1 if (5) where for b; are the control commands which can tak the alues or 1. The abo relations mean that each alv tak es the time to completely open or close. The parameter [0 1)

represents the alv dead zone. If the alv is completely closed or ), it tak es time to be gin to open the alv e. The model of the braking system presented here corre- sponds to the models described in the literature. Furthermore, the dynamics of the alv es is tak en into account with description close to the real hardw are. Optimal braking occurs when the longitudinal force operates at its minimum alue along the force-slip curv e. The proposed ABS control strate gy can be seen as minimum- seek algorithm. Since the force-slip curv has al ays minimum, it is rst necessary to determine

whether the operating point lies in the left or in the right re gion with respect to this minimum. Then the hydraulic actuators are operated to switch from one re gion to the other By this ay “limit ycle around the optimal slip alue arises and

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opt Region P=cost P=cost table points Unstable points Region opt Region P=cost P=cost table points Unstable points Region Fig. 3. Qualitati P- plot. The dotted line denotes the curv ; where The dashed curv ; is within the re gion where the solid curv ; is within the re gion where it guarantees that the

longitudinal force aries around its minimum. Computing the oil pressure in the braking system from equation (2) we obtain: ; br (6) or constant alue of the curv ; has the same shape of the curv Moreo er ; ; consequently an acceleration denes an unique curv ; that does not intersect an other ; curv (see Fig. 3), or gi en and the acceleration is uniquely determined. or an the peak of the curv ; happens for the same alue of opt These properties are sho wn in Fig. 3. This “P- plot is used in the sequel of the paper both to analyze the

dynamic beha viour of the tire during braking and to de elop the proposed control strate gy The torque-spin diagram presented in [14 is similar to the P- plot. This torque-spin plot is introduced in [14 to xplain the design of an ABS controller on the basis of an approximated piece wise tire characteristic. The P- plot presented here is based on equation (6) and then embeds the (unkno wn) true tire characteristic. wealth of information can be found matching the P- plot with the time deri ati of the slip rate From (1) the time deri ati of the slip is: (7) where is the longitudinal acceleration

of the ehicle. Note that when is reaching zero, the slip can ary aster than at high longitudinal speeds. This xplains why the orst performance of the ABS controllers happens usually at lo speed. Pr operty if then Proof: the sign of the slip deri ati is the sign of the term During braking and is limited by the ehicle speed: If then and the slip increases. Pr operty it xists limited angular acceler ation value suc that if then Proof: since the minimum longitudinal acceleration (maximum braking at the best conditions) is limited min and during braking it xists limited angular acceleration alue

such that is ensured. This acceleration alue can be easily found to be min =R indeed: min min min Let be design parameter If then, thanks to Property 1, Let be another design parameter thanks to Property 2, if then Both and are “free parameters that can be tuned to achie the best possible braking performance. Some where between the tw curv es ; (where and ; (where lies the curv ; see Fig. 3. Belo [abo e] the curv ; the slip increases [decreases] for an alue of and If the pressure is ept constant, the points on ; for opt are stable equilibrium points,

while the points on ; for opt are unstable equilibrium points. or an oil pressure if the slip ratio is increasing, if the slip ratio is decreasing. Consequently by measuring the wheel acceleration it is possible to infer some information about the slip ratio. The ne xt step is to nd if an operating point of the tire lies in the stable or in the unstable re gion. Let compute the time deri ati of equation (2) br (8) The follo wing tw properties allo to nd where the operating point of the tire is in some orking conditions: Pr operty if is constant, and then opt Proof:

since from Property follo ws The property can no be deri ed from equation (8) whit Pr operty if is constant, and then opt Proof: since from Property follo ws The property can no be deri ed from equation (8) whit The proposed control strate gy is based on the follo wing assumptions and requirements: A.1) During the HOLD phases the oil pressure and the braking torque br remain constant. This can be considered true at least for short periods. A.2) The wheel angular speed is measured. The wheel angular acceleration is measured or estimated. A.3) Each control action (HOLD, INCREASE and DE- CREASE)

is ensured within limited delay The max- imum delay is kno wn. A.4) The tire characteristic has unique minimum.

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DC HOLD DC DEC INC HOLD none Activation DC DC HOLD HOLD DC DC DEC DEC INC INC HOLD HOLD none none Activation Fig. 4. State chart of the proposed control strate gy Let denote the current sampling instant and let be the sampling period of the controller Properties 3) and 4) of the pre vious section requires the second deri ati of the wheel speed. This is problem in real applications where only the wheel speed is measured. ercome this problem, the acceleration ariation

is measured instead of the second deri ati method to get reliable measure, is to compute by linearly interpolating the acceleration alues for :::; where is design parameter that denotes the number of sampling periods that are needed to get reliable measure of ith small if the acceleration ariation is small the measurement noise will af fect the measure. ith good lo w-noise sensors can be small. The proposed control strate gy is based on state algorithm. The state chart of the algorithm is sho wn in Fig. 4. The basic orking ycle is gi en by the sequence of states (1)-(2)-(3)-(4)-( 5) -(6 )- (1)

see Fig 5. When the control strate gy is acti e, the control commands can only be HOLD, INCREASE or DECREASE. or some states, simple initialization assignement is ecuted once when the algorithm enters the state. The ents of each state are check ed follo wing the gi en sequence. The description of the states is the follo wing: (0) Contr ol command none Oper ations if “emer genc brak e then ne xt state (3). Description The ABS control is not acti e. If “emer genc brak e is detected the ABS control is acti ated. The acti ation mode does not af fect the beha viour of the proposed control and it is

out of the scope of the paper (1) Contr ol command HOLD Initialization := Events if then ne xt state (3). if then ne xt state (2). Description Actuators delay compensation. When the actuators delay has been compensated and the HOLD phase has certainly be gun. opt opt Fig. 5. Basic orking ycle represented on the P- plot. (2) Contr ol command HOLD Initialization := Events if then ne xt state (3). if then ne xt state (5). if and then ne xt state (3). Case (a) of Fig 5. if then ne xt state (6). Case (c) of Fig 5. Description Actuators delay as compensated while in state (1) or (4), the HOLD phase

is established and the pressure can be considered constant. If the measure of can be considered as reliable. The three cases (a), (b) and (c) of Fig are no possible. Case (a) corresponds to property 3). In case (b) the HOLD control command is ept since opt and Case (c) is similar to (b), moreo er it allo ws to re-establish an acceleration lo wer than By this ay sub-c ycle (1)-(2)-(6)-(1) can arise to mak closer to opt The second operation is not necessary if the force-slip curv remains constant. It is helpful in case of abrupt changes of the road conditions. (3) Contr ol command DECREASE

Events if then ne xt state (4). Description The DECREASE control action is established as soon as the operating point is found to be in the unstable re gion or when the wheel is lock ed. By decreasing the brak pressure, the term becomes dominant in equation (2) and the wheel acceleration becomes positi e. (4) Contr ol command HOLD Initialization := Events if then ne xt state (3). if then ne xt state (5). Description Similar to state (1): actuators delay compensation.

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(5) Contr ol command HOLD Initialization := Events if then ne xt state (3). if then ne xt state (2). if and

then ne xt state (6). Case (e) of Fig 5. Description Similar to state (2). Actuators delay as compensated while in state (4) or (1), therefore the HOLD phase is established and the pressure can be considered constant. If the measure of can be considered as reliable. The tw cases (d) and (e) of Fig are possible. Case (e) corresponds to property 4). In case (d) the HOLD control command is ept since opt and The second operation is not necessary if the force-slip curv remains constant. It is helpful in case of abrupt changes of the road conditions. (6) Contr ol command INCREASE Events if then ne

xt state (1). Description The INCREASE control action is established as soon as the operating point is found to be in the stable re gion. By increasing the brak pressure, the term br becomes dominant in equation (2) and the wheel acceleration becomes ne gati e. This section describes the results of tw dif ferent sim- ulations obtained by changing the alv dynamics and the road conditions. The ehicle and the hydraulic frame ork are the same for all the simulations. The sampling period is ms. erify the self-tuning properties, braking on arying road conditions (i.e. dry-wet-dry) ha been consid-

ered. Fig. sho ws the tw force-slip curv es that represent the tire beha viour in the tw dif ferent road conditions. The transition between the tw conditions depends on the tra eled distance The dynamics of the alv es plays an important role for the system performances. Some data about the alv es settling time were found in [17 ]. The system of simulation has erage alv es 20 ms) and sensors 10 ), gradual adhesion ariations dry-wet-dry or the simulation the system has slo alv es up to 50 ms), noisy sensors 20 ), abrupt adhesion ariation dry-wet-dry Simulation results There are oscillations of

the wheel speed and slip due to the alv dynamics, ho we er both the optimal slip and wheel speed are well track ed by the proposed control strate gy as sho wn in Fig. and in Fig. 8. The braking force is around its maximum, as sho wn in Fig. and Fig. 9. The performances decay only at lo speed (less than 8km/h, last 50cm) when the wheel locks- up. As well kno wn, the controllers based on acceleration thresholds induce oscillations on the braking pressure as sho wn in Fig. 10. −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 −4000

−3500 −3000 −2500 −2000 −1500 −1000 PSfrag replacements Slip orce [N] (–), opt ), and (- -) Fig. 6. Simulation 1. ire longitudinal force (–), tire characteristics and (- -), position of the optimal operating points ). 10 15 20 25 30 35 40 45 50 55 10 20 30 40 50 60 70 80 90 100 110 PSfrag replacements Distance [m] Speed [km/h] (- -), (–), opt Fig. 7. Simulation 1. ehicle speed (- -), wheel peripheral speed (–) and optimal wheel speed opt ). Simulation results The amplitude of the wheel speed and slip oscillations increases with the alv es slo wness. Ho we er

these oscillations are still around the optimal slip, as sho wn in Fig. 11 and in Fig. 12. Consequently the braking force is still around its maximum, as sho wn in Fig. 13. self-tuning control strate gy for antilock braking systems has been proposed. The paper has sho wn that it is possible to track the optimal slip alue by measuring only the wheel speed and estimating the wheel acceleration. The proposed control strate gy is rob ust with respect to adhesion ariations, tak es into account the dynamics of the actuators and is al- most hardw are independent. The ef fecti eness of the control

strate gy has been tested by simulation xperiments. The authors wish to thank Dr Nicola Sponghi and Dr Nicola Beschin of the Uni ersity of Modena and Re ggio Emilia for their aluable collaboration.

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0.5 1.5 2.5 3.5 0.4 0.3 0.2 0.1 Time [s] Slip Fig. 8. Simulation 1. racking of the optimal slip opt slip (-) and optimal slip opt ). 0.5 1.5 2.5 3.5 50 60 70 80 90 100 Time [s] [%] Fig. 9. Simulation 1. Performance aluation: ratio between the longitudi- nal force and the maximum achie able force opt (–). Comparison with the same ratio at wheel lock-up (- -). 0.5 1.5 2.5 3.5 10 15 20

25 30 35 Time [s] Pressure [bar] Fig. 10. Simulation 1. Braking pressure (-) and optimal braking pressure opt ). [1] Robert Bosch GmbH, Automoti Handbook”. SAE book, ISBN 0- 7680-0669-4, 2000, pp. 659-673. [2] H.B. acejka, “tire and ehicle Dynamics”. SAE book, ISBN 0-7680- 1126-4, 2002. [3] Tsiotras and C. Canudas de it, “On the Optimal Braking of Wheeled ehicles”, Pr oc. of the American Contr ol Confer ence Chicago, Illinois, June 2000. [4] Chih-Min Lin and Chun-Fei Hsu, “Self-Learning Fuzzy Sliding-Mode Control for Antilock Braking Systems”, IEEE ansactions on Contr ol Systems ec hnolo gy

ol. 11, No. 2, March 2003. [5] Han-Shue an and Masayoshi omizuka, “Discrete-T ime Controller Design for Rob ust ehicle raction”, Contr ol System Ma gazine ol. 10, No. 3, pp.107-113, April 1990. [6] S. Armeni and E. Mosca, ABS with Constrained Minimum Ener gy Control La w”, Pr oc. of the Confer ence on Contr ol Applications 2003 ol.1, pp.19-24, June 2003. [7] Reza Kazemi and Khosro Jaf ari Za viyeh, “De elopment of Ne ABS for assenger Cars Using Dynamic Surf ace Control Method”, Pr oc. of the American Contr ol Confer ence Arlington, A, June 2001. [8] S. Sa aresi, M. anelli, C. Cantoni, D.

Charalambakis, Pre vidi, S. Bittanti, “Slip-Deceleration Control in Anti-Lock Braking Systems”, Pr oc. 16 IF world congr ess Prague, Czech Republic, July 2005. [9] S. Drakuno U. Ozg uner Dix, and B. Ashra ABS Control Using Optimum Search ia Sliding Modes”, Pr oc. of the 33r Confer ence on Decision and Contr ol Lak Buena ista, FL, USA, December 1994. [10] .K. Lennon and K.M. assino, “Intelligent Control for Brak Sys- tems”, IEEE ansactions on Contr ol Systems ec hnolo gy ol. 7, No. 2, March 1999. 0.5 1.5 2.5 3.5 0.4 0.3 0.2 0.1 Time [s] Slip PSfrag replacements ime [s] Slip Slip (–), optimal

slip opt Fig. 11. Simulation 2. racking of the optimal slip opt slip (-) and optimal slip opt ). 0.5 1.5 2.5 3.5 50 60 70 80 90 100 Time [s] [%] PSfrag replacements ime [s] [%] =F opt (–), 1) =F opt (- -) Fig. 12. Simulation 2. Performance aluation: ratio between the longi- tudinal force and the maximum achie able force opt (–). Comparison with the same ratio at wheel lock-up (- -). 10 15 20 25 30 35 40 45 50 55 10 20 30 40 50 60 70 80 90 100 110 PSfrag replacements Distance [m] Speed [km/h] (- -), (–), opt Fig. 13. Simulation 2. ehicle speed (- -), wheel peripheral speed (–) and optimal wheel

speed opt ). [11] Chamaillard, G.L. Gissinger J.M. Perrone and M. Renner An Original Braking Controller ith orque Sensor”, Pr oc. of the Confer ence on Contr ol Applications 1994 ol.1, pp.619-625, August 1994. [12] Cem Unsal and Pushkin Kachroo, “Sliding Mode Measurement Feed- back Control for Antilock Braking Systems”, IEEE ansactions on Contr ol Systems ec hnolo gy ol. 7, No. 2, March 1999. [13] U. Kienck and L.Nielsen, utomotive Contr ol Systems Springer ISBN 3-540-66922-1, 2000. [14] .E. ellstead and N.B.O.L. Pettit, Analysis and redesign of an antilock brak system controller”, IEE Pr

oc.-Contr ol Theory Appl. ol. 144, No. 5, September 1997. [15] M. Maier and K. uller ABS 5.3 The Ne and Compact ABS5 Unit for assengers Cars”, SAE paper n.950757 [16] H. Saito, N. Sasaki, Nakamura, M. ume, H. anaka and M. Nishika a, Acceleration Sensor for ABS”, SAE paper n.920477 [17] Naito, H. ak euchi, H. uromitsu and K. Okamoto, “De elopment of our Solenoid ABS”, SAE paper n.960958

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