Nonlinear Dyn (2014) 76:125–138DOI10.1007/s11071-013-1115-1翻译！Robustcontrolofwheelslipinanti-lockbrakesystem ofautomobilesTohidSardarmehni·HosseinRahmani·Mohammad BagherMenhajReceived:3November2012/Accepted:9October2013/Publishedonline:6November2013© Springer Science+Business Media Dordrecht2013Abstract In this paper, performances of two model- free control systems including Fuzzy Logic Control (FLC) and Neural Predictive Control (NPC) on track- ing performance of wheel-slip in Anti-lock Braking System (ABS) are compared. As an accurate andcon- trol oriented model, a half vehicle model is devel- opedtogenerateextensivesimulationdataofthebrak- ing system. Brake system identification is preformed through a Perceptron neural networks model of brake system which is trained with offline data by Gradi- ent Descent Back Propagation (GDBP) algorithm. In order to reduce the time cost of the calculations and improving the robustness of closed loop control sys- tem, an online Perceptron neural network adaptively generatestheoptimumcontrolactions.Byacompara- tive simulation analysis it is shown that the NPC sys- tem has a better tracking performance, shorter stop- ping time and distance than the FLC controllers. The robustness of the proposed control systems are eval- uated under ±25 % uncertainty. It is shown that the NPC system is more robust against both exogenous disturbances and modeling uncertainties than theFLC system.T. Sardarmehni(B) ·H.Rahmani FacultyofMechanicalEngineering,UniversityofTabriz, Tabriz,Irane-mail:t.sardarmehni88@ms.tabrizu.ac.irM.B.MenhajDepartment of Electrical Engineering,AmirkabirUniversity of Technology, Tehran,Iran Keywords ABS ·NPC ·FLCAbbreviationsFLC FuzzyLogicControlNPCNeuralPredictiveControlABS Anti-lock BrakingSystemGDBP Gradient Descent Back PropagationECU Electronic ControlUnitSMC SlidingModecontrolMPCModelPredictiveControlMLP Multi-LayerPerceptronMAE MeanAbsoluteErrorMATE Mean Absolute TrackingErrorVehicle brake systemparametersV Vehicle velocity[m/s]ωWheel angular velocity[rad/s]g Gravity acceleration[m/s2]R Radius of tire[m]J f Moment of inertia of front wheel [kgm2]J r Moment of inertia of rear wheel [kgm2]a Distance from center of gravity to frontaxle[m]b Distance from center of gravity to rearaxle[m]h f Height of front unsprung mass[m]h s Height of the sprung mass[m]h r Height of rear unsprung mass[m]m f Front unsprung mass[kg]m s Sprung mass of the vehicle[kg]m r Rear unsprung mass[kg]126T. Sardarmehni etal. Fig. 1 Friction coefficientversus wheel slip[2]m tot Total mass of the vehicle[kg]P i Hydraulic pressure[kPa]P p Constantpumppressure[kPa]P low Constantreservoirpressure[kPa]C d1 BuildvalvecoefficientC d2Dumpvalvecoefficient A wc Wheelcylinderarea[m 2]hMechanicalefficiencyB F Brakefactorr r Effective radius of braking disk[m]C f CoefficientofflowK br Brake displacement proportionalityconstant1 IntroductionReducingrequiredstoppingtimeanddistanceinbrak- inghasalwaysbeenoneofthemostimportantcontrol goalsindesigningthebrakingsystemsofautomobiles. Without any control on angular velocity of wheels,an ordinary braking system exerts dissident fixed brak- ing torque on wheels. This fixed torque causes abrupt decrease in wheel angular velocity with a greater rate than the vehicle speed which results in wheel lock- ups during braking. Through locking-up, friction co- efficient and road adhesion become smaller which re- ducestheactive-appliedbrakingtorqueontires.Asthe result, the stopping time increases and vehicle stops afteralongerdistance.Furthermore,directionalstabil- ity of vehicle would considerably degrade. Toprevent wheel lock-ups and their probable catastrophicconse- quences during severe braking, ABS has been intro- duced to automotive industry. The main idea in ABS is controlling the active brake torque so that prevents wheel lock-ups. This procedure results in generationofmaximumnegativeaccelerationofvehiclewhiletheRobustcontrolofwheelslipinanti-lockbrakesystemofautomobiles 127 directional stability and steering ability of vehicle isguaranteed[1].Inordertogeneratethemaximumnegativeacceler-ation of vehicle, the longitudinal force should bekept around its peak value which requires thefrictionforce to be at its peak amount [2].Researches show that thevalueofthefrictioncoefficientgreatlydependsonroad conditions. As shown in Fig. 1, the peakvalueoffrictioncoefficientvariesfrom0.02to0.43indifferentroad conditions[2].In general, two different strategies are used tocon- trol the braking torque in ABS for preventingwheel lock-ups. The first method is based on wheeldeceler- ation. In this method no information fromvehicle ve-locitysensorsisrequired.However,theoperatingloadfrom the road cannot be used completely in wheelde-celerationmethod.Thesecondmethoduseswheelslipratio, defined by the ratio of the difference betweenwheel linear velocity and vehicle speed over vehiclespeed. The second method uses the oil pressure inanactuatortoregulatethebrakingtorque.Thisregulation ispreformedasthewheelslipratiotracksapredefinedvalue regarding the friction coefficient peak valuein different road conditions [3,4].PI-controllers, which are commonly used in ABSof automobiles, require long time calibrationprocess and would not function effectively in thepresence of exogenous noises and disturbances.Furthermore, the performance of a PI controller isnot satisfactory robust to the vehicle brake modelsevere nonlineari-ties,structuredorunstructureduncertainties,andtime-varying dynamics due to variation of roadandvehicle conditions. Nowadays, regarding theadvancementsin microcomputer industry, theintelligent and adaptive controltechniquescouldbeimplementedinElectronicControl Units (ECU) of vehicles. Besides thesignif-128T. Sardarmehni etal. icant transient and steady-state performances, these controllers are adjusted simply and also are robust to the uncertainties[5].As the state of the art in the designing of ABSs, FLC and neural networks have been used. Regarding satisfactory transient and steady-state performance of the FLC in nonlinear time-varying systems, signifi- cantresearchhasbeenperformedonthesecontrolsys- tems.MauerdesignedaFLCsystemappliedonaquar- ter vehicle model which could identify different road conditions. The proposed FLC system could generate action brake signals through considering current and past values of the brake pressure and slip ratio [6]. Zhang et al. developed a FLC system by considering data to estimate vehicle speed. The estimated vehi- cle speed was used for predicting the amount of slip. Theproposedcontrollercalculatedthemodifiedbrake torque signal based on the predicted the amount of wheelslip[11].Jacquetetal.proposedaMPCsystem togeneratetheoptimalbraketorque.Theonlinerecon- structionwasdonebaseduponestimationofthebrake adhesion torque and estimation of the wheel speed. Furthermore, the necessity of wheel speed sensors for identifying the tire/road characteristics was avoided by using a torque sensor located in the wheel. The simulation results showed that the proposedcontroller had a fast and stable response under rapid changes of theroadconditions.Moreover,thedesignedcontroller dμ dμdλand ( dλ)/dt as inputs of the controller. The pro- posed FLC system led to better performance in pedal pushingfeeling[3].Sharkawydevelopedaself-tuning PID controller which was accompanied by Fuzzy and Geneticalgorithmtoobtaintheoptimalmodulesofthe designed FLC. The results showed that the proposed controller had a fast response with low overshoot and short stopping distance[7].As a model-based nonlinear control system, the Sliding Mode control (SMC) systems are preferred duetotheirfastandtheeaseoftheirapplicationonthenonlinear models. Besides the possibility of fast im- plementation, the robustness and the stability of SMC could be guaranteed in most applications [8]. There has been much research work on application of SMC systems in ABS. Harifi et al. designed a SMC sys- tem for ABS which used integral switching surface for reducing the chattering effects [9]. Half vehicle model was used so the designed controller provided two separated brake torques for front and rear wheel. Furthermore, the results of this controller were com- paredwiththoseofaFuzzy,aselflearningFuzzyslid- ing mode and a neural network hybrid controller. It wasstatedthatthedesignedcontrollerhadtheshortest stoppingdistance.However,theneuralnetworkhybrid controller had the least amount of wheel slip tracking error regarding to other controllers[9].Model Predictive Control (MPC) is a control tech- nique that tries to minimize the tracking errorthrough predictions of the future outputs of a plant which de- finesthepredictionhorizon[10].Duetotherobustand stable performance of this control system, it has al- ways attracted the attention of control engineers. An- war and Ashrafi presented a MPC system for ABS. The presented method required wheel speed sensors could satisfactorily shorten the stopping distance[1]. In this paper, half vehicle model is used for gen- erating the simulation data of a braking system. Two differentmodel-freecontroltechniquesincludingFLC andneuralnetworkMPCsystemsaredeveloped.Fora better assessment, the overall performance of the pro- posed FLC and MPC systems are compared. A neu- ral network-based algorithm is developed in the MPC system to generate the brake torque through track- ing a modified desired slip trajectory. In the MPC system, identification of the half vehicle brake sys- tem model is performed by a Multi-Layer Perceptron (MLP) neural network for the front wheel. In order to improve the identification error and cope with the se- vere time-varying dynamics of the vehicle brake sys- temmodel,asmalldatacollectionsampletimeisused for collecting the training patterns of the neural net- works.Inordertoincreasethecalculationspeedofthe MPC system, the brake system identification is per- formed by an offline MLP model. Training process of the MLP neural network model of the brake sys- tem is performed with offline data by Gradient De- scent Back-Propagation (GDBP) algorithm. In order toacceleratethecomputationsintheMPCsystem,the prediction horizon is shortened. To accelerate the op- timization and improve the robustness of the control system,adaptivestrategyisusedintheoptimizationof the MPC system. In the NPC system, an online MLP neural network is adaptively designed to generate the optimum values of brake torque such that predicted amount of slip in the NPC tracks its modified desired value. Here, GDBP algorithm is used for the online training of the MLP system in the controller. At the end,robustnessofthedesignedFLCandNPCsystems against modeling uncertainties and inputdisturbancesRobustcontrolofwheelslipinanti-lockbrakesystemofautomobiles 129 F = (m h + m h + m h) ⎪⎪ Fig. 2 Free-body diagramof the halfvehiclemodel[12]are evaluated by imposing 25 % Gaussian noises. Be- tweentheproposedmodel-freecontrolsystems,itwas seenthattheNPCsystemismorerobustthantheFLC system.Thesimulationresultsshowedthatthecontrol action input is a smooth signal in the designed NPC. The normal force can be considered as the difference between the vehicle mass distribution and mass trans- fer of the vehicle during acceleration or deceleration. The normal force due to vehicle mass distribution can be formulated as[12] Besides,thestoppingtimeanddistanceisshorterin ⎧ b the NPC system than the FLC controlsystem.The entire program is running in MATLAB/simu- linksoftware.⎪⎨F zf 1=a +b (m tot g) a ⎪⎩F zr 1=a +b (m tot g) (4)2 SystemdynamicsThe second part of the normal force on front wheel is obtainedfromusingtheconversionlawofmomenton rear wheel as[12] 2.1 A half vehicle dynamicmodelComprehensive vehicle models including all parame- ters are not control oriented models due to their com- plexity and highly nonlinearity. Therefore, a simpli- fied dynamical vehicle model that possesses all the (a +b)F zf 2=(m f h f +m s h s +m r h r )V ˙ Dividing both sides to (a + b ), onehas V ˙ zf 2 f f s s r r (a +b) (5) (6) main characteristics of vehicle systems isconsidered.A free-body diagram of the half vehicle model is shown in Fig.2.Applying the same procedure on the front wheels,the normal force of rear tire can be definedas V ˙ As shown in Fig. 2, steering effects and drag force are not considered in modeling to avoid complexity. The total traction force can be presented as[12] F tot = F xf +F xr (1) F zr 2=−(m f h f +m s h s +m r h r )(a +b) The normal forces of tires are defined as[12] ⎧F zf =F zf 1−F zf 2 ⎪⎪ (7) In(1),F xf and F xr are,respectively,thefrontandrear longitudinaltire-roadcontactforcesandaredefined ⎪⎪⎪ ⎨ b = a +b V ˙ m t g −(m f h f +m s h s +m r h r )(a +b) (8) as[12] ⎪⎪F zr =F zr 1−F zr 2 ⎪ F xf =μ(λf )F zf (2) ⎪⎪ a V ⎪⎪ ˙ F xr =μ(λr )F zr (3) where F zf and F zr arethenormalforcesactingonthe frontandrearwheel.μ(λf )and μ(λr )arefrictioncoef- ficientsbetweenroadandfront/reartires,respectively. ⎩ =a +b m t g +(m f h f +m s h s +m r h r )(a +b) Through some substitutions, the total traction force is obtained as[12] F tot =μ(λf )(m 1g −m 3V ˙)+μ(λr )(m 2g +m 3V ˙)(9)130T. Sardarmehni etal. ⎪ ⎩ ⎨ ⎪ 2J f ⎪⎪ ⎪ − Table 1 Constant values for different road condition [2] Road condition C 1 C 2 C 3where ⎧m 1=b m totand F zr in (14) and (15), angular acceleration for front wheel and rear wheel would be definedas 1 ⎪⎪ . . ⎪⎪ ⎨ m 2= ⎪⎪ a +b a a +btot(10)ω˙f =2J 1 −T bf +μ(λf )m 1gR −μ(λf )m 3R V ˙+T e (16) ⎪⎪⎪m 3= m f h f + m s h s +m r h ra +bω˙r = 2J r .T br +μ(λr )m 2gR +μ(λr )m 3R V ˙. (17) In (10), λ is the wheel slip which can be formulated for the front and rear wheel as[12]Since the main goal is controlling the wheel slip, the defined state variables in this paper are vehicle veloc- ity, the front and rear wheel slip (x 1 = V and x 2 =λf ⎧⎪λf = ⎪⎩λr = V −ωf RV V −ωr R V(11)and x 3 = λr ). Consequently, the state-space equations could be defined as[12] ⎧x ˙1=V ˙=f 1(x 2,x 3) ⎪ Wheel slip changes from 0 to 1 which indicatescom- pletelyrollingtireontheroadwithoutanyslipand⎪⎪⎪x ˙2 ⎪ =λ˙f bf wheel lock-up, respectively. Mostly, wheel lock-ups happen during deceleration while the angular veloc- ⎪⎪⎨ ⎪⎪ f 1(x 2,x 3)(1−x 2)−Rf 2(x 2,x 3)+RT = V (18) ity of wheels becomes zero. Burckhardt tire friction model is used to describe the relationship betweenthe ⎪⎪⎪x ˙3=λ˙r ⎪ RT br friction coefficient and wheel slip[2]:⎪⎪⎩ f 1(x 2,x 3)(1−x 3)−Rf 3(x 2,x 3)+ = V 2J r μ(λ) = C 1.1 − e −C 2λ.−C 3λ (12)where C 1, C 2 and C 3 are constants which depend on where ⎧ ⎪⎪f 1(x 2,x 3)=−g μ(x 2)m 1 +μ(x 3)m 2 roadcondition.Table 1showsthedifferentvaluefor ⎪⎪⎪ m tot 1 −μ(x 2)m 3 +μ(x 3)m 3 various roadconditions.By assuming the front wheel to be the driver one, the engine torque only acts on the front wheel. Free- body diagram of the Front-wheel is shown in Fig. 3. Applying Newton’s second law along the horizontal ⎨f 2(x 2,x 3)= ⎪⎪⎪ ⎪⎩f 3(x 2,x 3)= 2J f 1 2J r .μ(x 2)m 1Rg − μ(x 2)m 3Rf 1.(19) .μ(x 3)m 2Rg −μ(x 3)m 3Rf 1. direction one has[12]m tot V ˙=−F tot (13) 2J f ω˙f =−T bf +μ(λf )F zf R +T e (14) 2J r ω˙r =−T br +μ(λr )F zr R (15) In(14),T e representstheenginetorquewhichwould becomezeroduringdeceleration.Bysubstituting F zf 2.2 Dynamics of the hydraulic brakemodelThe structure of a standard hydraulic brake actuatoris shown in Fig. 4. There are two solenoid valves which can only be open or close. The amount of braking pressure is regulated through the opening conditionof each valve which is simply specified by coefficients C d1 and C d2. For instance, when C d1 = 1 and C d2 =0 f Dryasphalt 1.2801 23.99 0.52 Drycobblestones 1.3713 6.4565 0.6691 Dryconcrete 1.1973 25.168 0.5373 Wetasphalt 0.857 33.822 0.347 Wet cobblestones 0.4004 33.708 0.1204 Snow 0.1946 94.129 0.0646 Ice 0.05 306.39 0 Fig. 3 Front-wheel free-body diagram[12]Robustcontrolofwheelslipinanti-lockbrakesystemofautomobiles 131− = − Fig. 4 Structure of astandard hydraulicmodel[13]Fig. 5a Membershipfunction ofvelocitythe build valve is open and the dump valve is closed (see Fig. 4) which increases the braking pressure and vice versa. The hydraulic system dynamic model can be represented as[13]in this section. The parallel structure of FLC systems enables the control system to activate all rules simul- taneously. This characteristic of the FLC systems im- proves the required time for calculations[6]. In general, FLC systems perform four mainstages C dp i f dt ,2 = A 1C d1 ρ (P p −P i ) which are Fuzzification, rule base, inference mecha- nism and Defuzzification. Fuzzification andDefuzzi- −A 2C d2 ,2 (P i P low ) (20) ρ ficationstagestransformanynumericaldatatoitscor- responding fuzzy value and vice versa. The inferencemechanismdeterminesthematchingdegreeofthecur- Throughsomesimplifications,thebraketorquecanbe defined as[13] T b =P i A wc ηB F r r (21) It is assumed that the brake torque defined by (21), only acts on the driver wheel which is the front one. Theamountoftherearwheelbraketorqueisdepended on that of the front wheel and could be definedas T br =K br T bf (22) The value for K br is often selected to be less than 1 in order to prevent rear wheel lock-ups at the situations which front wheel locks-up[12]. 3 An ABS fuzzy controllerdesign Due to an acceptable performance of the FLC in non- linearsystems,thistypeofcontrolsystemisdevelopedrent fuzzy input with respect to each rule and decides which rules should be fired. At the end, the firedrules are combined to form the control action[14].Mamdani method is applied for designing a fuzzy controller for the front wheel. The input vector of the proposed controller is consisted of the vehicle speed, wheel slip error and oil pressure differences. The wheel slip error is the difference between the current wheel slip and the modified desired wheel slip which varies with the condition of the road. The wheel-slip errors of the front and rear wheel are definedas .e f λd λf (23) e r = λd −λrAs shown in Figs. 5a , 5b , 5c and 5d , six Gaussian membership functions are used for the vehicle speed. Moreover, seven Gaussian membership functions are usedforthewheelsliperror;fortheoilpressurediffer-encetwoS-shapemembershipfunctionsareused.De-130 T. Sardarmehni etal. Fig. 5b Membershipfunction of LambdaerrorFig. 5c Membershipfunction ofΔPFig. 5d Membershipfunction of C dTable 2a Positive pressure difference (ΔP >0) Table 2b Negative pressure difference (ΔP <0)Vλerror VλerrorNVB NB NS ZO PS PB PVB NVB NB NS ZO PS PB PVBVS DEC DEC DEC DEC INC INC INC VS DEC DEC DEC DEC INC INC INC S DEC DEC DEC DEC INC INC INC S DEC DEC DEC DEC INC INC INC M DEC DEC DEC DEC INC INC INC M DEC DEC DEC DEC INC INC INC B DEC DEC DEC DEC INC INC INC B DEC DEC DEC INC INC INC INC VB DEC DEC DEC DEC INC INC INC VB DEC DEC DEC INC INC INC INC VVB DEC DEC DEC DEC INC INC INC VVB DEC DEC DEC INC INC INC INCfuzzification is performed through two Z-shapemem- bershipfunctions.After defining fuzzy sets for inputs and output, fuzzy control rules are designed as described in Ta- bles2a and 2b. Fuzzy control rules are developed for two different positive and negative oil pressurediffer- ence situations. In Tables 2a and 2b, INC represents the command that opens the build valve and closes the dump valve and DEC expresses the commandthat closesthebuildvalveandopensthedumpvalve. 4MultiLayerPerceptron(MLP)modelingofthe brakesystem4.1Structure of the MLP neuralnetworkThe structure of a MLP neural network is shown in Fig. 6. The proposed neural network model of the brake system is consisted of two layers comprising hidden layer and output layer. The input data are re-ceivedanddistributedtothenodesofthehiddenlayer.132T. Sardarmehni etal. 2 2 × f W (i,1) W 1 W (j,1) Fig. 6 Block diagram ofMLP neuralnetworksThe nonlinear activation function, f 1 in everyneuronofthehiddenlayerreceivestheweightedsumofallin-putdataplusthebiastermofeachcorrespondingnode.Intheoutputlayer,theweightedoutputsofthehidden layerarelinearlysummeduptotheoutputlayer’sbiaswhere Q is the total number of training patterns and l defines the in-process training pattern. Based on the back propagation algorithm, the bias term of (27) is updatedas ∂E(k) term and form the output of thenetwork.b 2(k +1)=b 2(k)−η ∂b 2(30) Assuming the number of inputs as p , the numberof neurons in the hidden layer as q and the dimen-sion of the network’s output as 1, the attributes ofthewhere η is a small positive number called thelearning rate. More precisely, it can beshown Q neural network model become as W 1 R q ×p ,b 1R q ×1,2 2 .∂E(k) ∂y(l) net 1 R q ×1, a 1 R q ×1, W 2 R q ×1 and b 2 R 1×1.Theactivationfunction f 1isasigmoidfunctiondefinedas 2b (k + 1) = b (k) −η = b (k) −η l =1 Q . ∂y(l)∂b 2(k) e l (l) (31) f 1(x)=1 +exp (−βx) − 1 (24)l =1 The same procedure can be performed for updating where 0 < β <1 is a tuning parameter and could beadjusted by experience or trial and error. Now, theweightedinputvector,P isgatheredwiththeneuron’sW 1, W 2, and b 1. So onehas W 2(k +1) Q bias term to obtain net 1as= W (k) −η .∂E(k) ∂y(l) 2 net 1 = W 1P +b 1 (25)l =1 Q ∂y(l) ∂W (k) Therefore, the output of the hidden layer, a 1 isob-2 . 1. 1 1. tainedasa 1 =f .net 1. (26)= b (k) −η b 1 l =1 e l (l)f W P (l) +b (32)The network’s overall output can now be obtained asfollows [10, 15,16]:(i,1)(k +1) =b 1 Q ..l (1,i) y = W 2a 1 +b 2 (27)(i,1)(k) −η l =1 e (l) ×W 2 (k) r .1 (i,j) (k)P (i,1)(l)+b 1 (k).. (33) 4.2 Training algorithm of the MLPnetworkThe parameters of MLP model, including b 1, b 2,W 1And similarly, onehas (j,i)(k +1) Q . and W 2,areupdatedbytheGDBPalgorithmbasedonthefollowingerror,e l :e l = y −y d (28)= W 1 (k) (j,i) .p −η.e l (l)W 2f r j l =1 .where y and y d are the MLP’s and desired outputs,respectively.Theindexofperformancecanbedefined..1 × (j,i) i =1 . (k)P (i,1)(l)+b 1 (k). as×P (j,1)(l) 1 Q (34) .. .E =2l =1y(l)−y d (l) (29)Comprehensive details and formulation of GDBP al- gorithm can be found in [10, 15,16].Robustcontrolofwheelslipinanti-lockbrakesystemofautomobiles133Fig. 7a Illustration ofnormalized brake torqueused fortrainingFig. 7b Illustration ofnormalized slip ratio usedfortraining4.3 The MLP model training and validationdataFor the front wheel, an MLP neural network istrainedby the GDBP algorithm. Considering λ(t) as the out-put slip of each MLP network at the current sampletime, t , the input vector of the training process is con-sistedofthebraketorqueandtheslipat(t −1)and(t ) as P = .T b (t), T b (t − 1), λ(t − 1), λ(t −2).T (35)ThroughwidespreadsimulationdataoftheMLPbrakesystem model, four neurons are chosen in the hid-den layer and thereby the neural network parame-ters are specified as W 1 R 4×4, b 1 R 4×1, W 2 R 4×1and b 2 R 1×1. A 4000-tuple random signal of braketorque, which was bounded to the interval 400 to5000 (Newton-meter), is imposed to the brakesystemmodel. To consider both the transient and the steady-state performance of the brake system, one half ofthe training patterns are generated with pulse widthof0.1 second and the other half has a wide pulse width,1.5 second. By imposing the input training data onthe brake system, 4000 output values for slip are gen-erated, which are used in the training process of the neural network systems. As Figs. 7a and 7b show, all trainingpatternsarenormalizedintotheinterval [0,1]. A fixed 0.01 second sample time is used in the gener- ation of the training data. β in (24) is considered2. In training with GDBP, the learning rate is set to 0.001. The initial values for weights and biases were taken as random numbers. The training process was finished when all errors (29) of the training patterns reached 0.01. The training process with GDBP was performedin458.25seconds.Aftertraining,theaccu- racy of the trained MLP model is evaluated by a new set of bounded random samples for the brake torque. Theresulted0.0026MeanAbsoluteError(MAE)cor- responding to the evaluation data in Fig. 8is signifi- cantlysmall. TheconvergedweightandbiasesoftheMLPneural network models for the front wheel are presented in Table 3. 5 MPC The schematic of the proposed NPC system is shownin Fig. 9. In each sample time, the NPC generates N u134T. Sardarmehni etal.⎢ ⎢ ⎢ ⎥⎥Fig. 8 Illustration ofvalidationdataMAE =0.0026Fig. 9 Schematic of NPCsystemTable 3 Final parameters of the MLPmodel⎡0.1791 ⎤ T ⎡0.4025 ⎤ ⎡ 0.2504 −0.1039 0.08790.4792 ⎤ ⎢0.6155 ⎥⎢0.6992 ⎥ ⎢ 0.1193 −0.1287 0.74980.4613 ⎥ b 1=⎢− ⎥ b 2 =−0.0847 W 2 =⎢ ⎥ W 1 =⎢ ⎥ 0.4755 ⎥ −0.2207⎥ ⎢0.6112 0.5432 0.7971 0.5464⎥ ⎣ ⎦1.1208 ⎣ ⎦ 0.6099 ⎣ ⎦ 0.4545 0.3086 0.8599 0.8654number of future constrained control inputs such thatthe predicted amount of system output tracks a modi-fied desired trajectory throughout the prediction hori-zon. N u defines the predictionhorizon.InFig.9,T b (t 0)and λ(t 0)showthebraketorqueand the slip at time t 0, respectively. Therefore, in thetime interval t 0 through t 0+Δt , all the components ofthe following vector, P i are known for i =0: P i =.T b (t 0 +iΔt),T b .t 0 +(i −1)Δt .,λ.t 0+(i −1)Δt .,λ.t 0+(i −2)Δt ..Ti =0,..., N u (36)In the NPC system, the vector P i is given to the MLPmodel of the brake system for predicting the slip.Therefore,intheinterval,t 0to t 0+Δt ,NPCrequiresthe predicted values of T b and λ at time samples t 0 + iΔt (i = 0,..., N u ). The values of λ(t 0 + iΔt) for each P i in Fig. 9is predicted through the trained MLP model of the brake system using P i −1 for i = 1,..., N u . The optimal values of T b (t 0 + iΔt) are obtained through the optimization process fully discussed in Sect.5.1. Theonlineoptimizationresultsinoptimizedvalues of all U i s in each sample time. As shown in Fig. 9, only the first component of P 1, which is shown as U 1,is used as the control input. Similarly, the remain- ing U i s are considered as the initial values for the fu- ture time steps, recursively. This usage of the preced- ing optimized values as the initial values of the cur- renttime results in fast convergence of the NPC sys- tem.Robustcontrolofwheelslipinanti-lockbrakesystemofautomobiles135 2 = − (j,1) = − + = − = − 1 + = 1 − 1 W W ∂Ψ (k) 2 W (j,1) (i,1) . f 5.1 OptimizationForreducingthecalculation’stimecostandimproving therobustnessoftheMPCsystem,optimizationunitis constructedbasedonMLPneuralnetworks.Inthepro- and ∂U 1(k) ∂Φ2(k) =1 (44) Similarly, ψ 2 is updatedas posedoptimizationmethod,anMLPneuralnetworkis adaptivelytrainedtogeneratetheonlineoptimumval- ues for control actions. These optimal control actions aregeneratedbasedonthetrackingperformanceofthe slip in the MPCsystem.As a result, an MLP neural network with thesame Ψ2(k +1)=Ψ2(k)−η∂E(k) ∂Ψ(k) From chain rule, (45) simplifiesto Ψ 2(k +1) Ψ 2(k) η∂E(k) ∂e l (k) ∂λ1(k)∂U 1(k) ∂e l (k) ∂λ1(k) ∂U 1(k) ∂Ψ2(k) (45) (46) structuregiveninSect.4.1isusedforanonlinegener- ation of front wheel brake torque signal. Considering ψ 1 and ψ2 as the first and second layer weightmatri- ces, and ϕ1 and ϕ2 as the first and second layer bias terms, the output can be definedaswhere: ∂U 1(k) ∂Ψ2(k) 4 ..4 .. U 1=Ψ2g .Ψ1X 0+Φ1.+Φ2 (37) = g i =1 . j =1 1 (i,j) X 0(j,1)+Φ1 (47) where g is the activation function defined by (24) for β = 2. For N u = 1, the error terms can be definedas According the GDBP algorithm, Φ1 could be updated as 1 1 ∂E(k) e l = λ(1)−λd (1) (38) where λ(1) is definedasλ(1) = W 2f 1.W 1X 1 + b 1.+b 2 (39)Φ (k + 1) = Φ (k)−η∂Φ1(k) (48) We apply the chain rule andwrite Φ1(k +1) The error term is definedas Φ1(k) η∂E(k) ∂e l (k) ∂λ1(k)∂U 1(k) ∂e l (k)∂λ1(k)∂U 1(k)∂Φ1(k) (49) 1 2with E = 2e l (40) ForonlineupdatingoftheparametersoftheMLPnet- work, back propagation algorithm is used. Hence,one ∂U 1(k) ∂Φ1(k) 4 . .4 .. has = .Ψ 2 r .(i,j)X 0(j,1)+Φ(j,1) Φ2(k 1) Φ2(k) η∂E(k)∂Φ2(k)(41)i =1 (i,1)g Ψ1 j =1 1 (50) We use the chain rule to rewrite (40)as Φ2(k +1) Φ2(k) η∂E(k) ∂e l (k) ∂λ1(k)∂U 1(k)∂e l (k)∂λ1(k)∂U 1(k)∂Φ2(k)(42)By the same procedure ψ1 can be updatedas ∂E(k) Ψ (k 1) Ψ (k) η ∂Ψ1(k) We use the chain rule toobtain Ψ 1(k +1) (51) The first two partial derivative terms on the rightside of(42)areeasilydefinedandthethirdtermcanbe =Ψ1(k)−η∂E(k)∂e l (k)∂λ1(k)∂U 1(k) (52) formulatedas4 .where ∂e l (k) ∂λ1(k) ∂U 1(k)∂Ψ (k) ∂λ1(k) ∂U 1(k)= .i =1 2 (1,i) 1 r(i,1) ∂U 1(k ) 1 (i,j) = Ψ (i,1)X 0(i,1)g r .4. × j =11 (i,j) ..X 1(j,1)+b 1 (43).4 . × j =1 1 (i,j) . X 0(j,1)+Φ1 (53) Ψ Ψ。
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