MODEL PREDICTIVE CONTROL SOLUTIONS FOR VEHICULAR POWER TRAIN SYSTEMS 1

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1 BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică Gheorghe Asachi din Iaşi Tomul LVI LX Fasc. 4 SecŃia AUTOMATICĂ şi CALCULATOARE MODEL PREDICTIVE CONTROL SOLUTIONS FOR VEHICULAR POWER TRAIN SYSTEMS BY CLAUDIA-ADINA DRAGOŞ* STEFAN PREITL* RADU-EMIL PRECUP* CRISTIAN-SORIN NEŞ** and EMIL M. PETRIU*** Abstract. The paper presents theoretical and implementation aspects concerning Model Predictive Control MPC solutions using the polynomial euivalent structure. Generalized Predictive Control GPC solutions are also developed in this paper. A mechatronic application is included to show how the MPC solutions can be implemented. The paper presents details on the modelling of a class of vehicular power train systems with Continuously Variable Transmission CVT with focus on an analytical analysis of systems components. Digital simulation results for an economic driving scenario are pointed out to validate the MPC solutions. Key words: Continuously Variable Transmission Generalized Predictive Control mechatronic application Model Predictive Control vehicular power train system. Mathematics Subject Classification: 34H5 49J5 93C55 93C83 93C85.. Introduction To increase the performance of mechatronic applications it becomes necessary to adapt the classical control strategies to more complex structures. The Model Predictive Control MPC is the second most popular control method after PID control. Generalized Predictive Control GPC is one of the most popular MPC algorithms and it is successfully implemented in mechatronic applications []. Predictive controllers are very convenient for strongly nonlinear systems as they prove good performance and a certain degree of robustness with respect to the model mismatch and to un-modelled dynamics This is an extended version of the paper: Dragoş C.A. et al. One- and Two-Degree-of Freedom Fuzzy Control of an Electromagnetic Actuated Clutch. Proc. ICSTC 995.

2 8 Claudia-Adina Dragoş et al. [] [3]. In recent years several applications of predictive vehicle speed control were reported in [] [4]. The basic idea of GPC is to calculate the present and the future control actions in the way to minimize the cost function defined over a certain time horizon. The cost function is the difference between the predicted plant output and the reference signal [6]. The paper is organized as follow: Section describes the theoretical concept of MPC algorithms including GPC algorithms. The modelling of a vehicular power train system with Continuously Variable Transmission CVT is presented in Section 3. Some details on the design of MPC algorithms are also emphasized in Section 3. Digital simulations of the predictive control systems and a comparative analysis between MPC and GPC are highlighted in Section 4. The conclusions are presented in Section 5.. Model Predictive Control Algorithms.. Problem Setting In order to show how MPC can be implemented the nonlinear mathematical model of the process is discretized to obtain the ARX model [] [] [7]: A y k = B u k where: ut is the control signal; yt the controlled output; et the zero mean value noise. The polynomials A and B are functions of the backward shift operator - : na = na A a a nb = + + nb B b b. The cost function that is minimized in order to obtain the control signal to be applied to the process in the framework of the MPC algorithm is defined as follows: 3 λ J y ˆ k r k u k [ ] = where: λ is the weighting parameter. The minimization of the cost function defined in e. 3 leads to the following control algorithm referred to as MPC algorithm:

3 Bul. Inst. Polit. Iaşi t. LVI LX f * k e BS AR D CS k r BS AR AT k u = where: r is the reference input. Further the MPC algorithm can be transformed into a two-degree-offreedom -DOF RST polynomial structure referred to as MPC structure illustrated in Fig. []. Fig. -DOF RST polynomial control system structure []. The control algorithm presented in e. 4 is reuired to be transformed into the following form: 5 t y S t r T t u R = where: R S and T are polynomials in - with the expressions []: 6. ˆ / ˆ ˆ ˆ ˆ / ˆ ˆ * b T T b D A T S b D B R = = = The characteristic euation of the MPC structure presented in Fig. is: 7 = + S B R A.

4 3 Claudia-Adina Dragoş et al. The polynomial presented in e. 7 can be used in the stability analysis of the MPC structure. In the case of using the objective function defined in e. 3 the steadystate control error can be cancelled only if λ= or if the process contains an integral component. The expression of the steady-state control error is: 8 ε s = r k+ y s λ = B λ+ b A λ r k+ = λ+ k b r k+... Generalized Predictive Control Algorithm In the case of GPC the process model is supposed to be of CARIMA type []: 9 A y t = z B u t + T e d t where the expressions of the polynomials A and B are given in e. the expression of the polynomial C is: C nc = + c + + c nc = and d is the time delay. For simplicity the C polynomial is chosen to be. The GPC algorithm consists of applying a control seuence that minimizes a cost function expressed as: N J = δ j [ yˆ t+ j t r t+ j ] + λ j [ u t+ j ] j= N where: δj and λj are weighting factors y ˆ t+ j t an optimum j step ahead prediction of the system output on data up to time t rt+j the future reference input trajectory N and N are the limits of the prediction horizon and N u the control horizon. After the minimization the control algorithm is expressed in terms of: N u j=

5 Bul. Inst. Polit. Iaşi t. LVI LX f. 4 3 t = K r t f t = ki[ r t+ i f t+ i ] u. where: K is the first row of the matrix G T G+ λi - G T. The closed-loop system can be transformed into the general classical pole-placement structure illustrated in Fig.. Therefore the expression of the control law is given in e. 5 [] [] [7]. N i= N y d r _ e T R u ZOH P y S T Fig. General -DOF RST control system structure []. Solving specific Diophantine euations given in [] and choosing T = for simplicity the expressions of the polynomials R and S are: 3 N T + kiii i= N = N k i= N i R N kifi i= N =. N k i= N i S where: F i and I i are some elements of the Diophantine euations i.e. I i are the rows of the vector G T. 3. Vehicular Power Train System Modelling 3.. Nonlinear and the Linearized Mathematical Models of the Process In order to show how MPC generally and the particular GPC can be implemented a mechatronic application dealing with a vehicular power train

6 3 Claudia-Adina Dragoş et al. system with CVT is presented in this section. The schematic structure of a vehicular power train system for a front-engine front-wheel-driven automobile with CVT is illustrated in Fig. 3 [9] [] [6]. It consists of: Fig. 3 The schematic structure of a vehicular power train system [6]. A. The Internal Combustion Engine: it generates the driving torue T eng by combustion and produces a maximum power of 3 PS DIN 83 kw at 63 rpm rad/sec. The mechanical characteristic of engine is presented in [6]. The maximum engine torue is 43 Nm at 48 rpm 5.65 rad/sec. The engine dynamics is characterized by [7]: J ω ɺ = T T eng eng eng c T T T =Γ throttle ω = T ω ω throttle max p 4 eng eng max eng M ωpωm % throttle %. where: J eng mass moment of inertia; T eng engine torue [N m]; ω eng engine speed [rad/sec]; T c external load from the clutch [N m]; T max engine maximum torue [N m]; T P engine torue at maximum engine power [N m]; ω P engine speed at maximum engine power [rpm]; ω M engine speed at

7 Bul. Inst. Polit. Iaşi t. LVI LX f maximum torue [rpm]; throttle throttle position [%]; Γ nonlinear function of throttle and ω eng. B. The Torue Converter: is usually modelled by using the capacity factor k and the torue ratio i t versus speed ratio steady-state curves Γ and Γ 3 [6] [3]: 5 T = ω k k =Γ ω ω i =Γ ω ω. c eng / c eng t 3 C. The Continuously Variable Transmission CVT: it allows the change of an infinite number of effective gear ratios i CVT between the minimum and the maximum limits.77 i CVT 3.43 [4] [8]. The CVT euations are: c eng 6 icvt =Γ 4 vv Ttr = icvt it Tc T = i T ω = i ω. w frg tr c CVT w where: Γ 4 function of vehicle speed v v T tr transmission input torue [N m] T w transmission output torue [N m] i frg final drive ratio. D. The Vehicle Dynamics: the euations governing the vehicle dynamics are: ɺ ω = T T T. 7 veh w w Drag Roll J where: J veh wheels moment of inertia; ω w wheels speed [rad/sec]; T w transmission output torue [N m]. The euations of the air drag torue T Drag and the rolling resistance torue T Roll are: 8 ωw Drag = ρ ωw w T c A sgn r 3 T = k m g r v = 3.6 r ω. Roll r v w v w w where: k r is a constant depending on the tire and the tire pressure; ρ the air density; A the maximum vehicle cross-section area; c the drag coefficient; m v the vehicle mass; g the gravitational constant; r w the effective wheel radius. The following nonlinear mathematical model:

8 34 Claudia-Adina Dragoş et al. T T x xɺ [ T x ] u max p = max ωm J eng ωpωm Jeng k 9 x FRG t 4 v w Jveh k 3 xɺ = i i Γ v c ρ A sgn x r x 5 y= 3.6 r x w was determined on the basis of es. 4 8 for each subsystem of the vehicular power train system where the characteristic variables are: the control signal input variable u = throttle; the state variables x = ω eng and x = ω w ; the controlled output output variable y = v v. Due to the nonlinearities of the systems the nonlinear model presented in e. 6 was linearized around several operating points which were chosen depending on the values of the throttle: 5% 5% 5% 75% and %. The linearized state space model and the obtained transfer functions t.f.s of the system for each operating point are presented in [5]. 3.. Development of Model Predictive Control Algorithm In this paper the economic driving scenario defined in Section 4 was tested considering that the operating point is stabilized depending on the fuel combustion. Therefore on the basis of fuel combustion characteristics illustrated in [5] the optimum fuel combustion is realized around a cruise engine speed of 35 rpm. In this case the ARX model of the process leads to the discrete t.f.: P z =.6 z +.5 z /.57 z +.57 z where: h=. s and the zero-order-hold ZOH is included. The parameters λ and c with favourable values λ =.75 and c = and e. 6 were used in order to calculate the values of the parameters R S and T of the standard form of the MPC structure:

9 Bul. Inst. Polit. Iaşi t. LVI LX f R= B+λ / b +λ = +.4 S = [ b A] / b +λ =.76.8 * T = b c / b +λ =.5. In the case of GPC the modifications of the tuning parameters N N N u and λ can affect the system s stability through the poles of the closed-loop system. 4. Digital Simulation Results A modern automatic transmission allows the driver to choose from several operating strategies: normal mode sport mode economic mode and winter mode. The main parameter which decides these strategies is the engine speed. A new parameter appears in the form of the fuel consumption curve which correlates the fuel consumption with the engine speed. The normal mode strategy consists in accelerating the car while maintaining the engine revolution speed at a level that corresponds to the desired vehicle speed. The sport mode strategy allows the engine speed to reach the maximum torue regime. Even at the desired final speed the maximum torue regime is maintained if the transmission ratio hasn t reached the minimum value. The economical mode strategy consists in maintaining the engine speed within the most convenient interval concerning the fuel consumption. The winter mode strategy functions similarly to the normal strategy but limits the wheel acceleration to a specified value. The MPC algorithm has been designed for all driving scenarios but in this paper only the economic mode strategy was tested on the vehicular power train system with continuously variable transmission. The simulation scenarios use a repeating seuence interpolation for the control signal-throttle in order to obtain the desired vehicle velocity. It is a reduced form of the NEDC and it is detailed in []. To confirm the appropriate solution two simulation scenarios have been performed in Matlab&Simulink taking into account the behaviour of four variables: throttle the engine speed the gear ratio and the vehicle s velocity: The simulation results with respect to the behaviour of the vehicular power train system in the economic operating scenarios are illustrated in Fig. 4; the settling time of 4 sec is necessary to achieve the reference vehicle speed of 3 km/h corresponding to the economic fuel combustion.

10 36 Claudia-Adina Dragoş et al. Fig. 4 Behaviour of the vehicular power train system: a throttle b engine speed c gear ratio and d vehicle speed. The simulation results with respect to the behaviour of the MPC using onestep ahead uadratic objective function designed for the vehicular power train system in the economic operating scenarios are illustrated in Fig. 5; in this case the settling time is slightly improved; therefore 9 sec is necessary to achieve the reference vehicle speed of 3 km/h corresponding to the economic fuel combustion. The simulation results with respect to the behaviour of the MPC using one-step ahead uadratic objective function designed for the vehicular power train system in the economic operating scenarios are illustrated in [5]. Fig. 5 Behaviour of the vehicular power train system with MPC: a throttle b engine speed c gear ratio and d vehicle speed.

11 Bul. Inst. Polit. Iaşi t. LVI LX f Conclusions. The paper presents the theoretical aspects and the design of two predictive control structures for a mechatronic application dealing with a vehicular power train system with CVT.. In order to design predictive control strategies the nonlinear mathematical model of the vehicular power train system was linearized around several operating points. 3. The Model Predictive controller has been designed using the one step-ahead uadratic objective function. 4. For the presented classes of systems both MPC and GPC structures were transformed into its euivalent RST polynomial structure which ensures the stable behaviour of the closed-loop system. 5. The future research will be dedicated to the improvement of the control systems behaviours at several operating points by inserting additional controller functionalities [] [4] [8] [] [5] [9] [] [3]. A c k n o w l e d g e m e n t s. This work was supported by the CNMP and CNCSIS of Romania. This work was partially supported by the strategic grant POSDRU 6/.5/S/3 8 of the Ministry of Labour Family and Social Protection Romania co-financed by the European Social Fund Investing in People. Received: November * Politehnica University of Timişoara Department of Automation and Applied Informatics claudia.dragos@aut.upt.ro ** Politehnica University of Timişoara Department of Strength of Materials cristianedonis@yahoo.com ***University of Ottawa School of Information Technology and Engineering Ottawa Ontario Canada petriu@site.uottawa.ca R E F E R E N C E S. Bălău A.E. Căruntu C.F. Pătraşcu D.I. Lazăr C. Matcovschi M.H. Păstrăvanu O. Modelling of a Pressure Reducing Valve Actuator for Automotive Applications. In Proc. 3rd IEEE Multi-conference on Systems and Control MSC 9 Saint Petersburg Russia Camacho E.F. Bordons C. Model Predictive Control. nd Ed. Springer-Verlag Berlin Heidelberg New York Chaikin D. How it Works: Torue Converter. Popular Mechanics Vol Derr K. Manic M. Multi-Robot Multi-Target Particle Swarm Optimization Search in Noisy Wireless Environments. In Proc. nd Conference on Human System Interaction HSI 9 Catania Italy

12 38 Claudia-Adina Dragoş et al. 5. Dragoş C.-A. Mathematical Modelling of Mechatronic Applications. Designing Modern Low-Cost Control Solutions for Mechatronic Applications in Romanian. Ph. D. Report No. Politehnica University of Timişoara 9 unpublished. 6. Dragoş C.-A. Preitl S. Precup R.-E. Pîrlea D. Neş C.-S. Petriu E.M. Pozna Cl. Modeling of a Vehicle with Continuously Variable Transmission. in Proc. 9 th International Workshop on Robotics in Alpe-Adria-Danube Region RAAD Budapest Hungary Isermann R. Mechatronic Systems: Fundamentals. Springer-Verlag Berlin Heidelberg New York Johanyák Z.C. Kovács S. Distance Based Similarity Measures of Fuzzy Sets. in Proc. 3 rd Slovakian-Hungarian Joint Symposium on Applied Machine Intelligence SAMI 5 Herl any Slovakia Kiencke U. Nielsen L. Automotive Control Systems for Engine Driveline and Vehicle. nd Ed. Springer-Verlag Berlin Heidelberg New York 5.. Lazăr C. Model-Based Predictive Control. Edit. Matrix Rom Bucharest in Romanian Lazăr C. Real-Time Informatics Technologies for Embedded-System-Control of Power-Train in Automotive Design and Applications in Romanian. Research Report. of the SICONA CNMP Grant Gheorghe Asachi Technical University of Iaşi 9.. Lazăr M. Heemels W.P.M.H. Weiland S. Bemporad A. Stabilizing Model Predictive Control of Hybrid Systems. IEEE Trans. Autom. Control Vol LepetIč M. Škrjanc I. Chiacchiarini H.G. Matko D. Predictive Functional Control Based on Fuzzy Model: Magnetic Suspension System Case Study. Eng. Appl. Artif. Intell. Vol Modak G.S. Sane S.S. Mechanical Continuously Variable Transmission CVT for Parallel Hybrid Vehicle. in Proc. IEEE Conference on Electric and Hybrid Vehicles ICEHV 6 Pune India Petres Z. Baranyi P. Hashimoto H. Approximation and Complexity Trade-off by TP Model Transformation in Controller Design: A Case Study of the TORA System. Asian J. Control Vol Preitl Z. Bars R. Vajk I. Haber R. Internal Model Representation for Generalized Predictive Control with Constraint Handling. in Proc. IEEE 4 th International Conference on Intelligent Systems Design and Application ISDA 4 Budapest Hungary Preitl Z. Model Based Design Methods for Speed Control Applications. Ph. D. Diss. Politehnica University of Timişoara Setlur P. Wagner J.R. Dawson D.M. Samuels B. Nonlinear Control of a Continuously Variable Transmission CVT. IEEE Trans. Contr. Syst. Tech. Vol Škrjanc I. Blažič S. Predictive Functional Control Based on Fuzzy Model: Design and Stability Study. J. Intell. Robot. Syst. Vol Song J. Wang C. Modeling and Simulation of Hydraulic Control System for Vehicle Continuously Variable Transmission. in Proc. 3 rd IEEE Conference on Industrial Electronics and Applications ICIEA 8 Singapore

13 Bul. Inst. Polit. Iaşi t. LVI LX f Sumar R.R. Coelho A.A.R. Coelho L.D. Computational Intelligence Approach to PID Controller Design Using the Universal Model. Inf. Sci. Vol Vaščák J. Hirota K. Mikloš M. Hybrid Fuzzy Adaptive Control of LEGO Robots. Int. J. Fuzzy Logic Intell. Syst. Vol Zhang Z. Shi P. Xia Y. Robust Adaptive Sliding-Mode Control for Fuzzy Systems with Mismatched Uncertainties. IEEE Trans. Fuzzy Syst. Vol Zhou M. Wang X. Zhou Y. Modeling and Simulation of Continuously Variable Transmission for Passenger Car. in Proc. st International Forum on Strategic Technology Ulsan Korea 3 6. SOLUłII DE REGLARE AUTOMATĂ CU PREDICłIE DEDICATE UNOR SISTEME DE TRANSMISIE A PUTERII LA AUTOVEHICULE Rezumat În cadrul lucrării sunt prezentate aspecte teoretice privind dezvoltarea algoritmilor de reglare automată cu predicńie folosind structura standard de reglare automată cu predicńie cu două grade de libertate RST. Lucrarea se referă la algoritmul de reglare automată cu predicńie bazată pe model MPC şi la cel generalizat GPC. Algoritmul de reglare automată cu predicńie a fost obńinut prin minimizarea unei funcńii obiectiv pătratice pe un pas. Dezvoltarea algoritmilor de reglare automată cu predicńie a fost exemplificată pe o aplicańie mecatronică şi anume sistemul de transmisie a puterii la autovehicule cu transmisie cu variańie continuă CVT. Lucrarea pune în evidenńă aspecte privind modelarea matematică a sistemului inclusiv o analiză a subsistemelor componente: motorul cu ardere internă convertorul de cuplu transmisia cu variańie continuă şi rońile. Datorită neliniarităńii sistemului de transmisie a puterii la autovehicule modelul neliniar a fost liniarizat în vecinătatea unor puncte de funcńionare stańionară constantă. Proiectarea algoritmilor de reglare automată a fost efectuată pentru un model matematic liniarizat în jurul unui punct de funcńionare situate pe caracteristica de consum astfel încât să fie asigurat un consum mediu optim dintr-un anumit punct de vedere o anumită funcńie obiectiv. Pentru testarea validarea şi compararea algoritmilor de reglare automată cu predicńie au fost efectuate simulări numerice privind funcńionarea vehiculului în modul economic. Au fost înregistrate grafice privind evoluńiile în timp ale accelerańiei turańiei motorului raportului de transmisie a vitezelor şi vitezei vehiculului. Au efectuat simulări atât pentru procesul condus cât şi pentru sistemul de reglare automată cu regulator cu predicńie. A fost pusă în evidenńă îmbunătăńirea peformanńelor sistemului de reglare automată în cel de-al doilea caz prin reducerea timpului de reglare.