BULEINUL INSIUULUI POLIEHNIC DIN IAŞI Publicat de Univeritatea ehnică Gheorghe Aachi din Iaşi omul LVII (LXI), Fac. 4, 2011 SecŃia AUOMAICĂ şi CALCULAOARE PREDICIVE CONROL SRAEGY IN DELA DOMAIN FOR DAMPING OSCILLAIONS IN DRIVELINE SYSEM BY CRISINA BUDACIU and CORNELIU LAZĂR Gheorghe Aachi echnical Univerity of Iaşi, Faculty of Automatic Control and Computer Engineering Received: September 3, 2011 Accepted for publication: November 11, 2011 Abtract. he paper propoe a predictive control trategy in dicrete time -repreentation in order to reduce the ocillation in the driveline. hee ocillation appear in modern dieel engine when a high engine torque i generated at low engine peed. he main object of thi paper i to preent a model baed control concept in domain which eem to improve both the control performance and control law implementation uing fixed point repreentation in the context of fat ampling. Several imulation were performed in comparion with claical GPC trategy and dicrete PID controller. he reult obtained with the propoed controller demontrate better control performance. Key word: fat ampling, delta operator, predictive control, ocillation. 2000 Mathematic Subject Claification: 34K35, 94A20, 39A21. 1. Introduction Nowaday, the dieel engine with direct fuel injection generate a high engine torque already at very low engine peed. hi feature lead to the increae of drivetrain train more than other type of engine do. An iue with Correponding author; e-mail: chalauca@ac.tuiai.ro
34 Critina Budaciu and Corneliu Lazăr thi high engine torque conit in the torion of the drivetrain which caue ocillation combined with jerking of the vehicle. hee ocillation are viible during the tip in and tip out maneuver. In the paper (Baumann et al., 2006), the author illutrate the behavior of the car when an abrupt tep on the accelerator pedal i performed and a o-called anti-jerk control baed on a predictive tate pace model i propoed. Mot traditional control algorithm are baed on hift operator decription, but thee trategie are ill conditioned when applied to data taken at the ampling rate that are high relative to the dynamic of the underlying continuou time procee. Moreover, many algorithm are better conditioned numerically uing dicrete delta () operator implementation than that one baed on hift (q) operator implementation. he -operator i an Euler approximation to the derivative and thi approach ha been known a the difference operator in the field of numerical analyi. Since the publihing of (Middleton & Goodwin, 1986), many reearcher paid attention toward identification and adaptive control uing -operator (Wu et al., 2000; Chadwick et al., 2010). Nowaday, the -operator i widely applied in wideband communication and real time digital control (atu, 2007). he q-dicrete domain i unconnected with the continuou domain, thi i becaue the continuou domain decription cannot be obtained by etting the ampling time = 0. here i a cloe connection between continuou time reult and repreentation. Moreover, the -operator ha certain numerical advantage compared with q hift operator. he paper compare the -operator with the q-operator tate pace realization in term of the effect of the finite word length error on a tranfer function. he reult how that the parameterization in the -domain provide a uperior enitivity performance over thoe in the q-operator. he author Cheng and Chiu (2007) in the paper analyze and propoe upper bound on the ample rate, lower frequency of interet and the available word length for controller repreentation. In thi paper, a predictive control trategy for drivetrain ocillation damping i introduced in the context of fat ampling. he drivetrain ytem under tudy wa invetigated by (Baumann et al., 2006) and i characterized by data taken from a real drive tet vehicle. Firt, a tate pace model of the drivetrain i analyzed and converted in dicrete domain. hereafter, thi model i ued a the bai for the predictive control deign uing the benefit of the operator. he performance of the controller i evaluated by imulation experiment in comparion with the claical dicrete time GPC controller and the conventional dicrete PID controller in the context of floating point and 16 bit fixed point repreentation, repectively. he paper i organized a follow: a brief background to the -domain tate pace model i given in Section 2. he two ubection deal with the tate pace drivetrain ytem and it converion to the -domain. In Section 3, the
Bul. Int. Polit. Iaşi, t. LVII (LXI), f. 4, 2011 35 framework for the GPC deign in the tate pace approach baed on operator i preented. In Section 4, the performance of the equivalent hift and -domain predictive control algorithm are compared in the cenario of fat-ampling and Section 5 conclude the paper. 2. he -Domain State Space Model he -operator i defined a follow in term of q-operator from the definition: q 1 e 1 = =. (1) where: i the ampling time. One iue of thi -operator model i that they can be applied to a wide range of dicrete-time ytem, from ampled data ytem with coare ampling interval to rapidly ampled, near continuou-time ytem. Although there i a linear tranformation between the two dicrete domain, the two operator have ditinct conceptual role: j j j n j = (1 + ) = j ( ) n= 0 q C C n j j! = n!( j n)!. (2) In the cae of continuou time ytem, the table pole hould be in the left half plane of the complex plane, wherea in the dicrete time thi become the interior of the unit circle. In the cae of -domain, the tability region increae a the ample rate i higher, o that the tability region i the interior of a circle centered at ( 1,0) with the radiu 1 (Middleton & Goodwin, 1986). hi property give the delta operator it uperior performance at high ample rate a compared to the claical q-operator. he dicrete tate pace model i: x( k ) = A x( k ) + B u( k ). (3) y( k ) = C x( k ) + D u( k ) he determinitic cae of ingle input ingle output tate pace form in the claical repreentation in the q dicrete domain i:
36 Critina Budaciu and Corneliu Lazăr qx( k ) = A x( k) + B u( k ) q y( k) = C x( k) + D u( k ) q q q (4) where the relationhip between the two dicrete model i given by (Kadirkamanathan, et al. 2009): A q I B q A =, B, q, q. = C = C D = D (5) Middleton and Goodwin have propoed a procedure for obtaining tate pace model directly from the continuou model. hu, they ugget the following relation for converion: A c e I A = = Ω A c, B = Ω B c, C = C c, D = D c. (6) where: A, B, C, D are continuou-time tate pace model matrice and: c c c c 0 2 2 cτ c 1 c c τ c 2! 3! 1 A 1 ( A A A Ω = e d = e I ) A = I + + +... (7) he correpondence between the two domain i emphaized in the limit cae: lim A 0. (8) lim B = B 0 = A c c For the limit cae 0, there i no equivalence between the matrice obtained in the dicrete time q domain and domain a one would expect: lim Aq = I 0. (9) lim B = 0 0 2.1. he Drivetrain Continuou ime State Space Model he powertrain ytem illutrated in Fig.1 conit of the following part: engine, clutch, tranmiion (gear box), final drive, drivehaft and wheel (Căruntu, 2011).
Bul. Int. Polit. Iaşi, t. LVII (LXI), f. 4, 2011 37 Drivehaft Engine Clutch ranmiion Final drive Drivehaft Wheel Fig. 1 he drivetrain tructure. In order to apply the GPC algorithm, an accurate drivetrain model i required to predict the vehicle' repone to a torque input. he linear tate pace model i converted into the -repreentation. hi model will be ued to deign and imulate the predictive control trategy in the -domain. he tate variable of the model conidered in thi paper are a follow: the torion angle between engine and tranmiion define the firt tate, angular peed of the engine i the econd tate and the angular peed of the wheel the third tate. he tate pace linear model i: 1 0 0 i d d 0 1 + 2 c i d 1 1 Ac =, Bc =, C c = 0 1 ij1 J1 ij1 J 1 i. (10) c d d2 + d 0 J2 ij2 J2 he output of the model i the difference between the engine and wheel peed and the input i the torque. J 1 repreent the um of the ma moment of inertia of engine, tranmiion and the final drive, J 2 the ma moment of inertia of the wheel and equivalent ma moment of inertia of the vehicle, d damping of the drive haft, d 1 the damping of tranmiion and final drive, d 2 the damping of the wheel, c tiffne of the drivehaft, i the tranmiion ratio between engine and wheel peed. All thee parameter are taken from (Baumann et al., 2006). In the following a - domain tate pace model i derived baed on the relation (6) and (7). he control action i repreented by the torque that can be generated by the internal combution engine.
38 Critina Budaciu and Corneliu Lazăr 2.2. he - Domain State Space Model he ampling period of 2 m correpond to an acceptable compromie between the indutrial tandard ampling period for uch an application (around 5 m) and the correponding error in term of velocity. he higher the ampling rate ued, the better i the reolution of the reult obtained. It i to be noticed that the eigenvalue of the matrix A are cloed to the value of the Ac continuou time matrix for f = 500 Hz and the value lie in the tability region etablihed by the interior of a circle centered at ( 500, j0) with the radiu of 500. In fact, the tability region approache the left half complex plane a the ample rate i higher. he - domain matrice obtained are calculated: A 0.1850 0.0768 0.9956 0.00385 0 = 1914.627 3.45 43.47, B 49.98, C 0.07722 = = 36.9211 0.0647 1.0132 0.00319 1 In the following, the tate pace formulation analyzed in thi paragraph will be ued to calculate the prediction for the predictive control algorithm formulated in the -domain. 3. he -Domain Predictive Controller Proceed from the model, the j- th order derivative tate are obtained: j 1 j j j i 1 i xk = xk + u k i = 0 A A B, (11) with j = 0, N y, N y being the prediction horizon. Uing the model (19) the following derivative predictor are: min{ j, Ny} 1 j j j i 1 i yk = xk + A uk i= 0 C A C B. (12) In a matrix notation the expreion of derivative predictor are: where u [ [ y ˆ = f + G u, (13) 2 Nu 1 k k k k 2 N y k k k k yˆ = y y y... y ]. = u u u... u ], (14)
Bul. Int. Polit. Iaşi, t. LVII (LXI), f. 4, 2011 39 G i the expanded oeplitz matrix and f the free repone (Budaciu & Lazăr, 2011). he optimal control equence i obtained by minimizing an objective function, knowing the reference trajectory r k+i : J N y Nu 2 2 [ k + i k + i ] λ [ k + i 1 ] i= N i= 1 = yˆ r + u, (15) 1 where: N u i the control horizon, N 1 minimum coting horizon, λ control weighting factor. In order to obtain the optimal control equence in the - domain, the et of vector that arie in criterion function are obtained from mapping the q-domain term into the -domain through binomial expanion. Conidering (11) and the predictor (12) into the criterion function (15) and 1 differentiating with repect to u produce the optimal trategy. he predictive control trategy for determining the optimal control i fully developed in the - domain, o that all the reult are baed on the benefit of the -operator. 4. Simulation Experiment hi ection preent the performance of the GPC trategy deigned in the -domain, invetigated on the automotive drivetrain model. he predictive control trategy elaborated in dicrete time domain ha been teted uing Matlab environment. Fig. 2 illutrate the comparion of engine torque and wheel peed uing GPC and claical dicrete time hift operator GPC trategy. Moreover, the bet reult obtained with a conventional PID controller are illutrated comparatively with the two above dicrete time algorithm. he control parameter for GPC were et a follow: ample frequency f = 500 Hz, the control horizon i fixed to N u = 1, and the prediction horizon i fixed to N y = 3 and the control weighting parameter i λ = 0.152. For the claical predictive control trategy the tuning parameter λ wa changed to the value λ = 0.01 in order to illutrate comparatively the bet reult obtained with each algorithm. he engine torque repreented in the firt figure i referred a input ignal and the output, the peed difference which mut follow the reference trajectory drawn with dotted line, in the econd plot. A it i hown, the GPC controller perform much moother than the claical GPC algorithm. he peed difference between the wheel and the engine reache the deired value with very mall ocillation a i hown with olid line in the econd plot of Fig. 2. From application reaon point of view it i deirable to get an analogy of the deign controller to a claical controller. herefore, a dicrete PID controller wa deigned with the parameter proportional gain k R = 0.9 and integral gain k R = 0.7.
40 Critina Budaciu and Corneliu Lazăr Fig. 2 Comparion of control ignal and peed difference uing PID, and q GPC trategy repectively. he GPC algorithm i able to offer better performance via manipulation of prediction horizon N y and weighting factor λ. he cope of thi tudy i to highlight the performance of the propoed algorithm for fat dynamic reult under a fat ampling frequency. In order to illutrate the performance of the predictive control in - domain, ome imulation have been performed, conidering tip in, tip out maneuver. In Fig. 3, the amplitude of the peed difference i increaed during tip out, o that the torque i fully applied at the beginning of the tep. he peed difference of the controlled ytem uing the claical GPC trategy illutrated in Fig. 3 with dotted line how more ripple than in the cae of delta GPC approach for the ame predictive control parameter. Fig. 3 Comparion of control ignal and peed difference uing and q GPC trategy for tip out maneuver.
Bul. Int. Polit. Iaşi, t. LVII (LXI), f. 4, 2011 41 It i to be mentioned that the claical GPC algorithm might improve the overall control performance with the price of a higher value of prediction horizon. However, lower value of prediction horizon implie a lower computational complexity and lower computing time, epecially when the algorithm i implemented uing finite word length repreentation. 5. Concluion he paper i addreed to thoe application for which the implementation of predictive control trategy on finite word length digital hardware would bring both numerical and conceptual advantage in term of control performance. he predictive control trategy in the dicrete -domain can bring ignificant improvement, epecially in the context of fat ampling. In thi paper, the predictive controller deign in -domain wa uccefully applied on a drivetrain ytem model. he paper illutrate the performance of the propoed olution in comparion with the equivalent claical GPC algorithm and PID trategy, repectively in order to reduce the ocillation and to improve the vehicle paenger comfort. he paper demontrate that the propoed olution propoed for peed difference control perform better than the claical GPC controller at high ampling rate and i alo able to offer lower computational complexity due to mall prediction horizon. Acknowledgement. hi work wa partially upported by CNCS - UEFISCDI, project number PN II-RU PD 331/2010. REFERENCES Aoki., Implementation of Fixed-Point Control Algorithm Baed on the Modified Delta Operator and Form for Intelligent Sytem. Journal of Advanced Computational Intelligence and Intelligent Informatic, 11, 6, 709 714, 2007. Bălău A.E., Căruntu C.F., Lazăr C., Simulation and Control of an Electro-Hydraulic Actuated Clutch. Mechanical Sytem and Signal Proceing, 19, 845 857, 2011 (in pre). Baumann J., orkzadeh D.., Ramtein A., Kiencke U., Schlegl., Model Baed Predictive Anti-Jerk Control. Control Engineering Practice, 14, 259 266, 2006. Budaciu C., Lazăr C., State Space Delta GPC for Automotive Powertrain Sytem. 16 IEEE International Conference on Emerging echnologie and Factory Automation, September 5-9, 2011, ouloue, France. Căruntu C.F., Bălău A.E., Lazăr M., van den Boch P.P.J., Di Cairano S., A Predictive Control Solution for Driveline Ocillation Damping. In: Hybrid Sytem: Computation and Control, Chicago, USA, 181 190, 2011. Chadwick M.A., Anderon S.R., Kadirkamanathan V., An Iterative Kalman Smoother/Leat-Square Algorithm for the Identification of Delta-ARX Model. International Journal of Sytem Science, 41, 7, 839 851, 2010.
42 Critina Budaciu and Corneliu Lazăr Cheng H-M., Chiu G.., Coupling Between Sample Rate and Required Wordlength for Finite Preciion Controller Implementation with Delta ranform. Proceeding of the American Control Conference, New York, July 11-13, 2007. Hrovat D., Agari J., Fodor M., Automotive Mechatronic Sytem. Mechatronic Sytem echnique and Application, Vol. 2, Gordon and Breach Science Publiher, Inc., 2000. Kadirkamanathan V., (Hălăucă) Budaciu C., Anderon S., Predictive Control of Fat- Sampled Sytem Uing the Delta-Operator. International Journal of Sytem Science, 40, 7, 745 756, 2009. Middleton R., Goodwin G., Improved Finite Word Length Characteritic in Digital Control Uing Delta Operator. IEEE ranaction on Automatic Control, 31, 1015 1021, 1986. Wu J., Chen S., Li G., Itepanian R.H., Chu J., Shift and Delta Operator Realization for Digital Controller with Finite-Word-Length Conideration. IEEE Proc., Control heory Application, 147, 6, 664 672, 2000. CONROL PREDICIV ÎN DOMENIUL DISCRE DELA PENRU AMORIZAREA OSCILAłIILOR ÎNR-UN SISEM DE RANSMISIE (Rezumat) Lucrarea prezintă o trategie de reglare predictivă proiectată în domeniul dicret de timp delta cu aplicabilitate în itemele rapide. Regulatorul predictiv implementat în domeniul delta aduce avantaje numerice şi conceptuale în ceea ce priveşte performanńele de reglare în lanńul de tranmiie exitent în itemele auto. Obiectivul principal contă în reducerea ocilańiilor ce apar la autovehiculele cu motor dieel unde e generează un cuplu motor de valoare mare la o viteză mică a motorului. O măură a acetor ocilańii ete dată de diferenńa între viteza motorului şi viteza rońilor, coniderată mărime de ieşire ce e doreşte a fi reglată în funcńie de cuplul motor. Modelul matematic într-o reprezentare intrare-tare-ieşire a proceului a fot dicretizat în domeniul delta, valorile proprii ale matricei itemului având valori apropiate de valorile numerice corepunzătoare domeniului continuu de timp. Regulatorul propu a fot tetat prin imulare numerică coniderând diferite valori pentru orizontul de predicńie şi factorul de ponderare, aceşti indici de proiectare pătrându-şi rolul şi în domeniul dicret delta. Analiza e extinde prin comparańii realizate între algoritmul predictiv în reprezentarea delta şi trategia proiectată în domeniul claic de dicretizare în ituańia utilizării unei perioade mici de eşantionare. În urma imulărilor efectuate cu o perioadă de eşantionare de 2 m, regulatorul predictiv propu aduce îmbunătăńiri în enul amortizării ocilańiilor ce apar la manevre de tip tip in, tip out. Mai mult, algoritmul demontrează performanńe mai bune pentru perioade mici de eşantionare, fiind mai puńin enibil la variańiile parametrilor de acord pecifici algoritmului predictiv. Pentru obńinerea unor performanńe aemănatoare cu cele obńinute de algoritmul predictiv în domeniul delta, în cazul algoritmului predictiv claic e poate creşte valoarea orizontului de predicńie, preńul de plătit fiind mărirea dimeniunilor matricilor algoritmului şi prin urmare creşterea complexităńii de calcul.