Data-Driven Predictive Control Applied to Gear Shifting for ... - MDPI

energies Article

Data-Driven Predictive Control Applied to Gear Shifting for Heavy-Duty Vehicles Xinxin Zhao * and Zhijun Li School of Mechanical Engineering, University of Science & Technology Beijing, Beijing 100083, China; [email protected] * Correspondence: [email protected]; Tel.: +86-010-62-333-221 Received: 25 July 2018; Accepted: 14 August 2018; Published: 16 August 2018







Abstract: In this paper, the data-driven predictive control method is applied to the clutch speed tracking control for the inertial phase of the shift process. While the clutch speed difference changes according to the predetermined trajectory, the purpose of improving the shift quality is achieved. The data-driven predictive control is implemented by combining the subspace identification with the model predictive control. Firstly, the predictive factors are constructed from the input and output data of the shift process via subspace identification, and then the factors are applied to a prediction equation. Secondly, an optimization function is deduced by taking the tracking error and the increments of inputs into accounts. Finally, the optimal solutions are solved through quadratic programming algorithm in Matlab software, and the future inputs of the system are obtained. The control algorithm is applied to the upshift process of an automatic transmission, the simulation results show that the algorithm is in good performance and satisfies the practical requirements. Keywords: subspace identification; model predictive control; data-driven control; shift control

1. Introduction Compared with the manual transmission, the automatic hydraulic transmission (AT) not only reduces the intensity of driver’s work, but improves the labor productivity for heavy duty vehicles, especially in characteristics with power shifting ability [1]. The traditional AT is a kind of clutch-to-clutch transmission which the original clutch is separated and the on-coming clutch combined during shifting process. The clutch alternation process should avoid the power interruption, shift overlapping and shocks [2]. The shift process is usually defined as including torque phase and inertia phase. The transmitted torque would change with the oil pressure between the off-going clutch and the on-coming clutch in the torque phase. When the torque transmitted by off-going clutch transfers to the on-coming clutch completely, finally the inertia phase occurs. In this phase, the accurate oil pressure would make the output speed and input speed of transmission synchronous. Based on the experimental data the input speed and output speed changes a little during the torque phase, whereas, it varies intensely in the inertia phase. Therefore, it is crucial to control the clutch speed during the inertial phase for improving shifting quality. For improving shifting quality several researchers have carried on a large amount of research about the clutch speed tracking control during the inertia phase, and they have already reported successes. As we all known that the mature PID method has been widely used in industry control. Meng et al. [3] used the PID method for improving system characteristics by using a robust 2-D controller defining optimal control parameters, which tracked the desired speed difference trajectory. Mishra and Srinivasan [4] combined adaptive feedback linearization control and sliding mode control to track reference trajectory accurately in inertia phase. The adaptive feedback linear controller calculated the reference oil pressure of the clutch based on the desired clutch speed and real speed, Energies 2018, 11, 2139; doi:10.3390/en11082139

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and the collected real pressure and the reference one would be used for input. Gao et al. [5] designed a nonlinear controller with stable input performance using a backstepping method considering system nonlinearity and uncertainty for the inertia phase. Depraetere et al. [6] have designed a two level controller for shifting, the low level of which solved the control variables through building an optimal problem of objective function with constraints, whereas, the high level controller utilized the ILC (Iterative Learning Control) method to update the model and constraints based on the measured data, and the control variables were calculated by the low level controller. Dutta et al. [7–9] solved the clutch engagement control problem by using Model Predictive Control, and they emphatically discussed the learning strategy for clutch engagement. Shi et al have developed the way of maintaining the shift quality of automatic transmissions consistent with adaptive control in mass production and with mileage accumulation [10]. In this paper the authors developed model-based learning control (two level nonlinear model predictive control, two level iterative learning control and iterative optimization) and model-free learning control (genetic algorithm and reinforcement learning), and those controllers have been verified by experiments with good control effectiveness. Most of the mentioned methods for shifting control depend on a detailed model. Since shifting is a complicated and nonlinear process, it is hard to build a theoretical model accurately. The higher order system is also difficult to solve, even when a precise model is obtained [11]. With the rapid development of computer science and technology, a large amount data of during system working had been collected, which includes the system information. When a precise theoretical model is hard to build, data-driven methods could be used to obtain the characteristic information from off-line data and real time date for realizing the optimization control, forecast and evaluation of the system process, which had received a great deal of attention among the control community [12–15]. In addition, due to some natural advantages, more and more control systems are applied to model predictive control. In this paper the subspace identification and model predictive control are combined, and a data-driven predictive controller has been designed for gear shifting of ATs. Firstly, the dynamic equation of a planetary gear box was induced by the Lagrange method. Actually, the controller is a kind of model-free control which obtains the system characteristic information to establish the predictive controller from the input and output data by using subspace identification. In the process of clutch speed tracking control, it is necessary to balance the conflicting control requirements. Therefore, the tracking control was changed to a multi-objective optimization problem which is solved by predictive control with considering the constraints in real mechanical system and the improved particle swarm optimization algorithm was utilized for searching optimal solution. 2. Topology Structure of Powertrain System The powertrain system divided into several subsystems in the simulation model, which includes diesel engine model, torque converter, transmission, drive shaft, longitudinal vehicle model and hydraulic model. 2.1. Diesel Engine Model By using linear function, sine function and hyperbolic function, the full range of continuous speed regulation characteristic function model was constructed, and it was applied to the engine model shown in Figure 1 [16].

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Figure 1. Powertrain Powertrain system. system. Figure Figure1. 1. Powertrain system.

2.2. Torque Converter Model Converter Model 2.2. Torque Converter Model λpp and K K, characteristic and torque ratio the characteristic curve of Based on pump the pump capacity factor Based on capacity factor λ p λand torque ratio K, the curve of thethe torque torque ratio , the characteristic curve of the on the the pump capacity factor torque converter is determined by the test data which is utilized in the torque converter model [17]. converter is determined by the test data which is utilized in the torque converter model [17]. torque converter is determined by the test data which is utilized in the torque converter model [17]. 2.3. Transmission Model 2.3. Transmission Model 2.3. Transmission Model In this study the automatic transmission consists consists of four planetary systems, twotwo clutches and and In this study the automatic transmission of planetary systems, clutches In this study themain automatic transmission consists of four four planetary systems, two model clutches and four brakes. The work of modelling the AT is focused on definition of the friction of four brakes. The main main work work of of modelling modelling the the AT AT is is focused focused on on definition of the friction model of four clutch brakes. The definition of the friction model of and the dynamic performances of planetary. For the clutch friction model, the Woods static clutch and the dynamic performances of planetary. For the clutch friction model, the Woods static clutch and the dynamic of used, planetary. the clutch friction model,logic the Woods and dynamic friction performances model had been whichFor is deemed to be a three-state functionstatic and friction model had been used, which is deemed to be to a three-state logic function related and dynamic dynamic friction model had been used, which is deemed be athethree-state logic function related with the friction torque and reference speed of clutches. Regarding planetary system, the with the friction torque and reference speed of clutches. Regarding the planetary system, the planetary planetary dynamic acquired by thespeed Lagrange method Regarding and virtual work principle, system, which the related with the frictionequations torque and reference of clutches. the planetary dynamic equations acquired by the Lagrange method and virtual work principle, which be introduced be introduced for modelling particularly in [18,19]. planetary dynamic equations acquired by the Lagrange method and virtual work principle, which for modelling particularly [18,19]. transmission The structure of theinautomatic was shown in Figure 2, where the torque converter, be introduced for modelling particularly in [18,19]. hydraulic retarder, planetary system, clutches/brakes, hydraulic system control unit can be The structure of the automatic transmission in 2, where the converter, The structure of the automatic transmission was was shown shown in Figure Figure 2, and where the torque torque converter, seen. The planetary systems represented by P1, P2, P3 and P4. CL, CS and CH indicate the clutches, hydraulic hydraulic system system and and control control unit unit can can be hydraulic retarder, retarder, planetary planetary system, system, clutches/brakes, clutches/brakes, hydraulic be respectively. In addition, BS, BM, BL and BR represent brakes. When theand control unit sendsthe signals seen. The planetary systems represented by P1, P2, P3 and P4. CL, CS CH indicate clutches, seen.toThe planetary systems represented by P1, P2, P3 and P4. CL, CS and CH indicate the clutches, the hydraulic system, the clutches and brakes startbrakes. working in different waysunit to acquire the respectively. In BS, BL BR When control sends signals to respectively. Inaddition, addition, BS,BM, BM, BLand and BRrepresent represent Whenthe the control unit sends signals desired ratio. The schedule is shown in Table 1, where ‘√’brakes. means engaging. the hydraulic system, the clutches and brakes start working in different ways to acquire the desired to the hydraulic system, the clutches and brakes √ start working in different ways to acquire the ratio. The schedule is shown in Table 1, where ‘ ’ means engaging. desired ratio. The schedule is shown in Table 1, where ‘√’ means engaging.

Figure 2. Diagram of the automatic transmission.

According to the structure of AT, the Lagrange method was used to establish the dynamic function of the planetary system. Because the fourth planetary didn’t work during shifting from 1st Figure 2. Diagram of the automatic transmission. Figure 2. Diagram of the during automatic transmission. gear to 2nd gear, the reverse gear was neglected deduction of the function to simplify the modelling. Therefore, the number of planetary systems was reduced to three. The connection type According to the structure of AT, the Lagrange method was used to establish the dynamic of the planetary was shown inthe Figure 3. It can be seen that thetoring gear ofthe planetary is According to theset structure of AT, Lagrange method was used establish dynamicP1function

function of the planetary system. Because the fourth planetary didn’t work during shifting from 1st of thetoplanetary Because fourth planetary didn’t work during 1st gearthe to gear 2nd gear,system. the reverse gearthe was neglected during deduction of the shifting functionfrom to simplify 2nd gear, the reverse gear was neglected during deduction of the function to simplify the modelling. modelling. Therefore, the number of planetary systems was reduced to three. The connection type Therefore, the number of planetary systems three. Thethe connection of the planetary of the planetary set was shown in Figurewas 3. Itreduced can betoseen that ring geartype of planetary P1 is set was shown in Figure 3. It can be seen that the ring gear of planetary P1 is connected to the sun

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connected to the sun gear S2 and sun gear S3, which is named r1s2s3, while the carrier C2 gear S2 andring sungear gearR3 S3,which whichisisnamed namedc2r3. r1s2s3, while the carrier C2 connected ring gear R3 which is connected named c2r3. Table 1. Combined clutch/brake schedule. Table 1. Combined clutch/brake schedule.

Gear CS Gear 1 CS √ 2 √ 1 3 √ 2 √ 3 4 4 5 √ √ 5 6 6 √ R1 √ R1 R2 R2

BS BS

√

CH CH

√

√√ √√ √

√ √ √ √

√

BM

BL BR Ratio √BL BM BR 4.00Ratio √√ 2.67 4.00 √ √ 2 2.67 √ √√ 1.33 2 1.00 1.33 1.00 0.67 0.67 √ √ −5.00−5.00 √ √ −3.33−3.33

Figure Figure 3. 3. Gear Gear set. set.

Basedonon meshing relationship, the kinematical of planetary system was Based thethe meshing relationship, the kinematical equation of equation planetary system was shown as follows: shown as follows: ( . . . β1 Rs1 α1 Rs1  R− γ−1γRRp1 =  . .  β = α (1) 1 s 1 1 p 1 1. Rs1  β1 (Rs1 + 2R p1 ) + γ1 R p1 = θ 1 Rr1 (1)  β ( R + 2 R p1 ) + γ1 R p1 = θ1 Rr1 ( .  1 s1. . β2 Rs2 − γ2 R p2 = θ1 Rs2 .  β2 Rs 2 − γ2 R p 2 = θ.1 Rs 2 . (2)  β2( Rs2 + 2R p2 ) + γ2 R p2 = θ 2 Rr2 (2)   ( .  β2 ( Rs 2. + 2 R p 2 ) +. γ2 R p 2 = θ2 Rr 2 β3 Rs3 − γ3 R p3 = θ 1 Rs3 . . (3) . R β3 Rs+ − γ3 R p)3 + = θγ 1 R β3( R s3 3 2R p3 3 s 3 p3 = β2 Rr3 (3)  Rs 3 +angular γ3 R p 3 = β2 Rr 3 of the sun gear, carrier and pinion 2 R p 3 ) + displacement  β3 (the where, the terms α , β and γ represent 1

1

1

in the P1 planetaryαsystem, respectively, θ1 represents the angular displacement of the ring gear in where, the terms 1 , β1 and γ 1 represent the angular displacement of the sun gear, carrier and P1, sun gear in P2 and sun gear in P3, β 2 represents the angular displacement of the carrier in P2 pinion in the P1 planetary system, respectively, θ1 represents the angular displacement of the ring and ring gear in P3, θ2 represents the angular displacement of carrier in P2, γ2 represents the angular gear in P1, sun in P2 andβsun gear in P3, β 2 represents the angular displacement of the carrier displacement ofgear pinion in P2, 3 presents the angular displacement of carrier in P3 and γ3 represents in P2 and ring gear in P3,ofθthe the angular displacement of carrier in P2, γ 2 represents 2 represents the angular displacement pinion in P3. . . . β presents the angular displacementofofoperation, pinion in αP2, the angular displacement of carrier in P3 For the convenience 1 , β 3and β were selected as the independent coordinate 1

3

and γ 3 After represents the angular displacement of are thesolved, pinion in system. the above six constraint equations theP3. equation was written in matrix form:    α β β , and were selected as the independent For the convenience of operation,  . 1 1 3 θ 1 coordinate system. After the above six  constraint are solved, the equation was written in .  equations  .    β matrix form: α1  .2   θ   .   .2  (4)  = A β. 1    θ  γ1 1  β3  . β   γ2 2   α1  .   γ3 θ2     = A  β1  (4)  γ  1   β3   γ   2  γ3 

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where A is the coefficient matrix: where A is the coefficient matrix: − A1 1 + A1 0      −A 1 + A 0 −(1 + A1 ) A21 A11A2 1 + A2    A1 A2 −(1 + A1 ) A2 1+A  −(1 + A1 )( A2 + A2 A (1 + A2 )(1 + A3 )  2 1 ( A2 + A2 A3 + A3 ) 3 + A3 )  A (A A = A + A2 A3 + A3 ) −(1 + A1 )( A2 + A2 A3 + A3 ) (1 + A2 )(1 + A3 ) A =  1 2 −A A4 A 0   − A4 4 0 4    + − + + + A A A A A A A A (1 ) (1 )(1 ) (1 )  A (11 + A2 )2 A55 −(1 +1A1 )(1 + 2 A52 ) A5 2(1 + 5 A2 ) A5  1  A1AA1 6A6 +1 )A A6 −−( (1 +1 A A16 ) A6 A6 

        

(5)

(5)

During thethe gear shifting, thethe external force from thethe hydraulic system is added to to thethe planetary During gear shifting, external force from hydraulic system is added planetary system. The virtual work generated by external torques in virtual displacements is given as follows: system. The virtual work generated by external torques in virtual displacements is given as follows:

δ W = T δα + T δβ + T

δθ + T

δβ + T δθ + T δβ

1s 2 s 3 1 c 2r 3 2 r2 2 c3 3 δW = Ts1s1δα11+ Tcc11 δβ11 + rT r1s2s3 δθ1 + Tc2r3 δβ 2 + Tr2 δθ2 + Tc3 δβ 3

(6) (6)

It can be seen that the torque would be change when the gear shifts to different gear. Towards It can be seen that the torque would be change when the gear shifts to different gear. Towards the the torque of the sun gear in P1, Ts1 , it codetermined by the friction torque T f ,CS and T f , BS torque of the sun gear in P1, Ts1 , it codetermined by the friction torque T f ,CS and T f ,BS induced by the induced by the clutch CS and brake BS. The torque of carrier in P1, Tc1 , it related with the input clutch CS and brake BS. The torque of carrier in P1, Tc1 , it related with the input torque of the turbine Tin andfrom torque of the the friction torque fromto the clutch CS.carrier Due toin the the carrierwith in P1the T and theturbine friction torque the clutch CS. Due the fact the P1fact is connected in

is connected the P3, sunthe gears in P2torque and P3, the external torque of this component is determined sun gears inwith P2 and external of this component is determined by the friction torque from byclutch the friction torque from clutch CH. The carrier in P2 and ring gear in P3 are connected together,by CH. The carrier in P2 and ring gear in P3 are connected together, and the torque determined and the torque determined by the friction torque of brake BL. The external torque of the ring gear the friction torque of brake BL. The external torque of the ring gear in P2 was codetermined byinthe P2friction was codetermined by the torque clutchthe CStorque and brake BM. Finally, the torque carrier torque of clutch CS friction and brake BM. of Finally, of carrier in P3 is related withofthe output T in torque P3 is related with the output torque of transmission, . out of transmission, Tout . Applying thethe Lagrange equation to to thethe system, thethe total energy was taken thethe derivative with Applying Lagrange equation system, total energy was taken derivative with respect to time and calculated by the partial differentiation. The dynamic equation of the AT500 respect to time and calculated by the partial differentiation. The dynamic equation of the AT500 .. .. ..  planetary system was acquired about thethe generalized coordinates, and β3β:3 : planetary system was acquired about generalized coordinates,αα β11 and 1 ,1 , β   Tin T  T in f , CST f ,CS  ..     T f , BSTf ,BS αα11    ..    D D ββ11 == B BT f ,CHT f,CH ..  T  T f,BM ββ33   f , BMT  Tf , BL f ,BL  T Tout  out 

           

(7) (7)

According Equation angular speed coordinate components was solved when According to to Equation (7),(7), thethe angular speed of of thethe coordinate components was solved when the friction torque of clutches and brakes were known. In addition, the kinematic relationship the friction torque of clutches and brakes were known. In addition, the kinematic relationship of of allall components AT500 was shown following Figure components of of AT500 was shown in in thethe following Figure 4. 4.

Figure 4. Dynamic of gear gear set. set. Figure 4. Dynamic characteristic characteristic solver solver of

2.4. Output Shaft and Longitudinal Vehicle Model 2.4. Output Shaft and Longitudinal Vehicle Model For modelling convenience, the longitudinal vehicle model was simplified [20]. The main For modelling convenience, the longitudinal vehicle model was simplified [20]. The main reducer reducer was considered a simple transmission, and the connection between the reducer and vehicle was considered a simple transmission, and the connection between the reducer and vehicle body was body was considered as a torsion spring with a certain stiffness which is reduced to a mass block with a certain moment of inertia.

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considered a torsion spring with a certain stiffness which is reduced to a mass block with a certain 2.5. Control as Scheme moment of inertia. For achieving speed synchronization of two relative rotating objects, Meng et al. [1] utilized the piecewise 2.5. Controlfunction Scheme method to fit the speed difference curve, which the quadratic function was applied in two terminals and the linear function in the middle part. This way insured the difference For achieving speed synchronization of two relative rotating objects, Meng et al. [1] utilized the curve derivative, furthermore, the relative speed variation maintained as zero at the beginning and piecewise function method to fit the speed difference curve, which the quadratic function was applied last moment. Additionally, some research papers have introduced that the cubic function can be in two terminals and the linear function in the middle part. This way insured the difference curve used as the curve [21–25]. Over those designations of various difference curves, they aimed to derivative, furthermore, the relative speed variation maintained as zero at the beginning and last guarantee the relative speed variation as zero and the shifting time fulfilled the requirement since moment. Additionally, some research papers have introduced that the cubic function can be used in tracking problems the error between the reference trajectory and real speed is the focus rather as the curve [21–25]. Over those designations of various difference curves, they aimed to guarantee than the selection of a reference trajectory. In this paper, the cubic function was used as the the relative speed variation as zero and the shifting time fulfilled the requirement since in tracking reference curve and it expressed as below: problems the error between the reference trajectory and real speed is the focus rather than the selection 2Δωthe 3Δω0 was used of a reference trajectory. In this paper, cubic function as the reference curve and it 0 Δω ∗ (t ) = (t − t0 )3 − (t − t0 ) 2 + Δω0 (8) expressed as below: (t f − t0 )3 (t f − t0 ) 2 2∆ω0 trajectory3 of speed 3∆ω0 difference, Δω reference where, Δω∗ represents ∆ω ∗the (t) = ( t − t0 ) − (t − t0 )2 + ∆ω0 0 represents the speed (8) 3 2 (t f − t0phase ) t0 ) difference at start moment of inertia and t0 ,(ttf f −represented the start time and end time

respectively. We focused on the shifting process from first gear to second gear in this paper which where, ∆ω ∗ represents the reference trajectory of speed difference, ∆ω0 represents the speed difference is the alternating process between off-going clutch CS and on-coming clutch BS. When the gear shift at start moment of inertia phase and t0 , t f represented the start time and end time respectively. starts, the pressure of brake BS is increased gradually. The external torque for AT500 is gradually We focused on the shifting process from first gear to second gear in this paper which is the alternating applied step by step from the clutch CS to the brake BS and it would be split automatically. When process between off-going clutch CS and on-coming clutch BS. When the gear shift starts, the pressure the clutch CS is separated totally, it means that the inertia phase starts. Therefore, the target for the of brake BS is increased gradually. The external torque for AT500 is gradually applied step by step pressure control of brake BS is to track the reference trajectory deducing by Equation (8) which is from the clutch CS to the brake BS and it would be split automatically. When the clutch CS is separated shown in Figure 5. totally, it means that the inertia phase starts. Therefore, the target for the pressure control of brake BS To optimize the shift quality, the current of electric valve controlled the brake BS i and is to track the reference trajectory deducing by Equation (8) which is shown in Figure 5. throttle angle θ are the control variables, which fulfil 0 ≤ i ≤ 0.6 A and 0 ≤ θ ≤ 1 . Considering To optimize the shift quality, the current of electric valve controlled the brake BS i and throttle angle the method utilized in the weConsidering set the range control variables θ areincremental the control variables, which fulfil 0 ≤controller i ≤ 0.6 A designation, and 0 ≤ θ ≤ 1. the of incremental method di d θ di utilized in the designation, of control variables satisfying 0 ≤for 3 A/s and 0 ≤controller ≤ 3 A/s 4%range satisfying and 0 ≤we set≤ the /s, and the increment constraints the control dt ≤ dt dt 0 ≤ dθ 4%/s, and the increment constraints for the control variables would be calculated according to ≤ dt variables would calculated according the step length of simulation and its change rate. the step length ofbe simulation and its changetorate.

Speed difference 相相相相Δω rad/s

Δω0

0

t0

时时t s time

tf

Figure 5. 5. The The reference reference trajectory trajectory of of the the clutch clutch speed speed difference. difference. Figure

3. Shift Shift Controller Controller Design Design 3. mechanism of the system model structure structure is unnecessary, while the subspace Looking into the mechanism identification just needs a linear algebra tool such as QR without iterative optimization. Therefore, Therefore, identification it was easy to achieve and thus has developed rapidly [26]. The main idea is to design an appropriate was easy to achieve and thus has developed rapidly [26]. The main idea is to design an driving sign driving for the system, it to acquire output data. data normalization, appropriate sign forand the use system, and useinput it to and acquire input andAfter output data. After data the Hankel matrix built from the input and output data was used to divide it by the QR method normalization, the Hankel matrix built from the input and output data was used to divide it byand the singular value estimating the vectorsthe of state the system. the state space QR method anddecomposition singular valuefor decomposition forstate estimating vectorsFinally, of the system. Finally, model matrixes A, B, Cmatrixes and D were least squares method, themethod, model predictive the state space model A, B,solved C andby Dthe were solved by the least while squares while the

model predictive method was usually used to calculate the future control sequence minimizing the cost function based on the latest measurement at each sampling time in a determined model. Using

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the first column of those acquired control sequence into the system, the above procedure would be repeated in the next sampling time until the end of the shift. To utilize the data drive predictive Energies 2018, 11, 2139 7 of 12 method for shifting control, the subspace identification and model predictive control are combined which is presented in the Figure 6. According the past future input and output date, the subspace predictors infunction the prediction method was usually usedand to calculate the future control sequence minimizing the cost based equations were acquired by the least squares [27–31]. model. The prediction was on the latest measurement at each sampling time method in a determined Using theequation first column optimized to obtain the future control sequence. Compared with the subspace identification of those acquired control sequence into the system, the above procedure would be repeatedand in the model predictive it end can of bethe seen thatTothe datathe predictive control is seen as a for model-free next sampling timecontrol, until the shift. utilize data drive predictive method shifting predictive method without and solving thepredictive coefficientcontrol matrixes the state which space model, but the control, thecontrol subspace identification model areofcombined is presented in predictive equation was acquired by input and output date directly. the Figure 6.

Figure 6. 6. Principle Principle of of data-driven data-driven predictive predictive control control algorithm. Figure algorithm.

The description of data-drive as below:predictors in the prediction According the past and futurepredictive input andcontrol outputmethod date, theissubspace equations were acquired by the least squares method [27–31]. The prediction equation was optimized (1) Design an appropriate input to acquire the input data u and output data y; to obtain the future control sequence. Compared with the subspace identification and the model (2) Formulate the Hankel matrix U p , U f , Yp and Y f based on those input and output data; predictive control, it can be seen that the data predictive control is seen as a model-free predictive L and Lu by least (3) Solving thewithout subspace prediction equation matrixes for acquiring control method solving the coefficient of thepredictive state spacefactor model,wbut the predictive squares method; equation was acquired by input and output date directly. (4) Rewrite the predict equation as the control incremental and The description of data-drive predictive method is asbuild below:the initial relatively speed y Δ  p Re data Δwp = an appropriate sequence .  and reference (1) Design input output to acquire the input u and output data y;  Δu p  (2) Formulate the Hankel matrix U p , U f , Yp and Y f based on those input and output data; (5) Building the objective function J and defining the initial constraints; (3) Solving the subspace prediction equation for acquiring predictive factor Lw and Lu by least (6) Solving the objective function by QP method for acquiring the sequence of the future control squares method; variables Δu . The first one would be used into the system input. (4) Rewrite the f predict equation as the incremental and build the initial relatively speed " #  Δy p  ∆y p wp = sequence (7) Updating speed Δ ∆w p = the relatively and reference output  andRe .Re cbased on the system output. ∆u p  Δu p  (8) Back to the sixth step until the end of defining shifting. the initial constraints; (5) Building the objective function J and (6) Solving the objective function by QP method for acquiring the sequence of the future control 3.1. Data-Driven Shift Predictor variables ∆u f . The first one would be used into the system input. " # Considering the optimal control for the ∆y inertia phase of shifting, the current of electric valve p (7) Updating the relatively speed ∆w p = and Re cbased on the system output. u and the relatively speed of clutch Δω was and throttle angle were regard as the system ∆u inputs p seen as the system output . To stimulate the dynamic characteristics of system, the simulation was y (8) Back to the sixth step until the end of shifting. carried out in the inertia phase of shifting by using the random input signals. Usually, theShift linear time-invariant system is usually described as a state space: 3.1. Data-Driven Predictor Considering the optimal control for the inertia phase of shifting, the current of electric valve and throttle angle were regard as the system inputs u and the relatively speed of clutch ∆ω was seen as the system output y. To stimulate the dynamic characteristics of system, the simulation was carried out in the inertia phase of shifting by using the random input signals.

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Usually, the linear time-invariant system is usually described as a state space: xk+1 = Axk + Buk yk = Cxk + Duk

(9)

where, uk , yk and xk represent the inputs, outputs and state variables, respectively. Assuming known the inputs uk and outputs yk at any moment, the Hankel matrix could be defined which was introduced in the thesis [22]. According to the system identification, the output estimate matrix was deduced as follows: Yˆ f = Lw Wp + Lu U f (10) The predictive factor Lw and Lu could be acquired by solving the constructed least squares problem. To verify the effect of predictive factors, the first column of the future output matrix calculating by Equation (10) is compared with the test output data based on the simulation model. The simulation results show that the error between the estimated value and test value was very small, and it illustrates that the predictor was useful for estimating the future output date. 3.2. Data-Driven Shift Controller We substituted the future output sequence into the objective function with considering the constraints. The optimal problem could be deal with as a quadratic programming problem: min

∆u f (k )

T 1 2 ∆u f ( k ) H∆u f ( k ) +

G T ∆u f (k)

(11)

C∆u f (k) ≤ b

s.t.

The increment sequence of future control variables is the solution of the above quadratic programming problem. The first incremental and the control variable of last step are used together to the system, and it will be repeated until the end of shifting. 4. Simulation Results and Discussion Simulation of controlled object and controller were built by the Matlab/Simulink in this article. Due to the similar process for different gear, we just focus on the shifting from the first gear to the second gear in various working conditions which aimed to verify the effect of controller. In the simulation, the parameters of vehicle and road were changes along with the different conditions, while the parameters of controller maintained as same value. The main parameters of shifting controller included the number of rows for Hankel matrix i = 50, the number of columns j = 1100, the control horizon of the predictor Nu = 4 and the prediction horizon Np = 50. Additionally, the control sequence and incremental constraints were the same as mentioned before. The weight vector, Γu and Γw , represented the row vectors after normalization, of which dimensions were 8 and 50, respectively. Because of the high dimensions of weight vector Γw , only the weight vector is shown: Γu =

h

0.2789

0.2789

0.1317

0.1317

0.0609

0.0609

0.0285

0.0285

i

(12)

where, Γu reflects the impact on the objective function of different control sequences in future time. It means that the future control sequence was closer to the current time and the cost functions were influenced easier. We assume that the influence of the throttle angle and the current of electric valve were the same in the same moment. In this simulation, we set the start moment of the inertia phase as t0 = 4 s and it would last about 0.5 s. Figure 7 reveals that the simulation results in empty condition without slope, where the mass of ◦ vehicle mv = 55, 000, the slope of road α = 0 , the initial throttle angle θ = 0.9, the initial current of electric valve i = 370.2 and the initial relative speed difference of the brake BS ∆ω0 = 118.8.

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current 电电i A current 电电i AThrottle angle 油油油油 θ Throttle angle 油油油油 θ

0.4 0.45 0.35 0.4

* desired 理理相相转 Δω simulation Δω 仿仿仿仿 * desired 理理相相转 Δω simulation Δω 仿仿仿仿

100 50

0.9 0.8 0.5 0.8 0.45 0.5

0.35

相相转 rad/s rad/s difference 相相转 error rad/s error rad/sdifference 误转 误转

100

1 1 0.9

4

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50 0 10 0 105 50 -5 0 -5

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4.25

4.3

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4.5

4.3

4.35

4.4

4.45

4.5

time s 时时t

Figure7.7.Simulation Simulationresults results(vehicle (vehicle massmm==5555t; roadslope slopeαα = = 00◦ ).). t ; road Figure mass Figure 7. Simulation results (vehicle mass m = 55t ; road slope α = 0  ).

current angle 电电icurrent A 油油油油 Throttle θangle θ 电电i A Throttle 油油油油

1

1 0.9 0.9 0.8 0.7 0.8 0.6 0.7 0.5 0.6 0.4 0.5 0.4

4

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相相转 rad/s rad/s difference 相相转 误转 error rad/s error 误转 rad/s difference

According to the simulation results, the effect the designed controller controller fulfilled thefulfilled requirements, According to the simulation results, the of effect of the designed the requirements, and the maximum tracking error for angular speed of clutch is about 5.3 rad/s. The and theAccording maximumtotracking error for results, angularthe speed of clutch about 5.3 controller rad/s. Thefulfilled throttle the angle the simulation effect of theis designed throttle angle and the maximum current electric valve fluctuations beginning of5.3 inertia phase requirements, tracking errorhave foratangular speedatofthe clutch is about rad/s. The and the current of electric valveofhave fluctuations the beginning of inertia phase that lead to the that leadangle to theand unsmooth curve tracking error. Reference [22] has that the speed throttle the current ofofelectric valve have at revealed the of turbine inertia unsmooth curve of tracking error. Reference [22] hasfluctuations revealed that thebeginning turbine speed was phase affected was affected easily by the control sequence at the beginning of inertia phase. Thereafter, the control that lead to the unsmooth curve of tracking error. Reference [22] has revealed that the turbine speed easily by the control sequence at the beginning of inertia phase. Thereafter, the control sequence and sequence andeasily speed start toat become smooth. there isthe no simulation error Δsequence ωsmooth. was affected bydifference the the beginning inertia phase. Thereafter, thebetween control speed difference ∆ω start tocontrol become Especially, thereof isEspecially, no error between speed the simulation speed difference and the reference curve at the end of inertia phase. Comparing with sequenceand andthe speed difference to become smooth. there is the no error between Δ ωat start difference reference curve the end of inertia phase.Especially, Comparing with limiting change limiting change rate of speed at beginning and end of inertia phase, the maximum tracking theofsimulation speed difference and theofreference curve at the end of inertia phase. Comparing with rate speed at the beginning and end inertia phase, the maximum tracking error occurred at the error occurred at therate middle part.atDue to the requirement only for those two parts, the controller the limiting change of speed the beginning and end of inertia phase, the maximum tracking middle part. Due to the requirement only for those two parts, the controller has achieved the desire has achieved effect which could for shifting. error occurredthe atdesire the middle part. Due to be theused requirement only for those two parts, the controller effect which could be used for shifting. Since the vehicle works under time-varied conditions, has achieved the desire effect which could be used for shifting. it should be tested under other Since the vehicle works under time-varied conditions, it should be tested under other conditions. conditions. The vehicle simulation results with time-varied empty loss inconditions, slope road it were shown the Figure 8, where Since the works under should beintested under other The simulation results with empty loss in slope road were shown Figure 8, where the mass of =in5°the 55,000 with kg and the loss slope road is αwere . In thisincondition, initial the mass ofThe vehicle is mv =results conditions. simulation empty inof slope shown the Figurethe 8, where ◦ road vehicle is mv = 55, 000 kg and the slope of road is α = 5 . In this the initial electric valve α =condition, 5difference ° . In this iscondition, mv = 55to ,000 Δω0 = 48.35 i = kg 423.2 andand the the slope of road isspeed the initial the mass of vehicle electric valve currentischanged initial clutch rad/s. current changed to i = 423.2 and the initial clutch speed difference is ∆ω0 = 48.35 rad/s. It shows ω = about 48.35 0.06 i = 423.2 and It showsvalve clearly that changed the control have large fluctuations s electric current to variables and output the initial clutch speed differenceand is Δtakes rad/s. clearly that the control variables and output have large fluctuations and takes about0 0.06 s to become toshows become smooth. addition, maximum tracking error is 4.2 rad/s. Theand results of about the full losss It clearly thatInthe control the variables and output have large fluctuations takes 0.06 smooth. In addition, the maximum tracking error is 4.2 rad/s. The results of the full loss in slope mv =results 72, 000ofkg, in become slope aresmooth. presented in Figure the 9, where the mass of vehicle to The to In addition, maximum tracking error increases is 4.2 rad/s. thethe fullslope loss are presented in Figure 9, where the mass of vehicle increases to mv = 72, 000 kg, the slope of road is = 2° , the in θ = 0.9 in Figure 9, where of vehicle increases toofmelectric kg, the slope initial throttle anglethe is mass , the initial current valve changed to of slope road isareαpresented ◦ v = 72, 000 α = 2 , the initial throttle angle is θ = 0.9, the initial current of electric valve changed to i = 413.5 and α =the 2° ,initial Δω, 0the = 73.84 i =road 413.5is and the initial throttle is θ = initial rad/s. currentThe of electric valve changed to of clutch speedangle difference is 0.9 simulation results in this the initial clutch speed difference is ∆ω0 = 73.84 rad/s. The simulation results in this condition falls in 73.84 the i = 413.5 and condition fallsthe in between the two mentioned conditions maximum error is 3.3 rad/s. initial clutch speed difference is Δω0 =where rad/s. The simulation results in this between the two mentioned conditions where the maximum error is 3.3 rad/s. condition falls in between the two mentioned conditions where the maximum error is 3.3 rad/s. 60 * desired 理理相相转 Δω

40 60

simulation Δω 仿仿仿仿 * desired 理理相相转 Δω simulation 仿仿仿仿 Δω

20 40 200 5 0 5 0 0 -5 -5

4

4.05

4.1

4.15

4.2

4

4.05

4.1

4.15

4.2

4.25 time s 时时t

4.25

4.3

4.35

4.4

4.45

4.5

4.3

4.35

4.4

4.45

4.5

time s 时时t

Figure 8. Simulation results (vehicle mass m = 55t ; road slope α = 5°◦ ). Figure8.8.Simulation Simulationresults results (vehicle massmm==5555t; roadslope slopeαα ==55° ).). t ; road Figure (vehicle mass

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rad/s

* desired 理理相相转 Δω

相相转 difference

油油油油 Throttle angle θ

0.9

60 40

simulation 仿仿仿仿

20

误转 error rad/s

80 1

2

0.8

电电i A current

0.55 0.5 0.45 0.4

4

4.05

4.1

4.15

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4.25 time 时时t s

4.3

4.35

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Δω

0 4

0 -2

4

4.05

4.1

4.15

4.2

4.25

4.3

4.35

4.4

4.45

4.5

time s 时时t

◦

Figure 9. 9.Simulation Simulationresults results(vehicle (vehicle massmm== roadslope slope 7272t; t ; road α α= = 2° 2 ).). Figure mass

Conclusions 5. 5. Conclusions reduce dependency on detailed the detailed model, the subspace identification and predictive model ToTo reduce thethe dependency on the model, the subspace identification and model predictive control are combined in this paper. The data-driven predictive control method has been to control are combined in this paper. The data-driven predictive control method has been applied applied to optimize the shifting quality for the inertial phase and it changed the optimization to optimize the shifting quality for the inertial phase and it changed the optimization to track reference track reference desired curve. The model-free algorithm utilized the input and output signal data desired curve. The model-free algorithm utilized the input and output signal data to implementtothe implement the predictive control by the predictive equation. predictive control by the predictive equation. When the control variables θ and i were solved for ensuring the two contradictory When the control variables θ and i were solved for ensuring the two contradictory indicators indicators that are the shift time and the shift impact, the objective function has been designed that are the shift time and the shift impact, the objective function has been designed considering the considering the constraints for control variables and change rate. The controller was designed in the constraints for control variables and change rate. The controller was designed in the Matlab/Simulink Matlab/Simulink software, which works out the objective function by the QP with constraints. We software, the objective function by the QP the with constraints. We the used the solved used thewhich solvedworks controlout variables as the input for shifts during inertial phase, and simulation control variables the input for shifts the inertial andthe the reference simulationcurve resultsaccurately. showed that results showedasthat the speed of during on-coming clutch phase, tracking theMoreover, speed of on-coming clutch tracking the reference curve accurately. Moreover, the simulation result the simulation result verified the effect of the controller and has robust performance verified the effect of the controller and has robust performance when the mass and road slope change. when the mass and road slope change. InIn conclusion, thethe data-driven predictive control not only improves the shift but provides conclusion, data-driven predictive control not only improves the quality shift quality but a theoretical for real-time by hardware, could be utilized in shifting problems provides a basis theoretical basis fortesting real-time testing by which hardware, which could be utilized in shifting for traditional hybrid vehicles, vehicles and electric vehicles the transmission problems vehicles, for traditional hybrid vehicles and without electric building vehicles without building model. the transmission The results also lay afor technical foundation for hardware-in-the-loop The results alsomodel. lay a technical foundation hardware-in-the-loop tests and real vehicletests tests.and real vehicle tests. Author Contributions: Conceptualization, X.Z. and Z.L.; Methodology, X.Z.; Software, Z.L.; Validation, X.Z. and Author Contributions: Conceptualization, X.Z. and Z.L.; Methodology, X.Z.; Software, Validation, X.Z. Z.L.; Formal Analysis, X.Z.; Investigation, Z.L.; Resources, X.Z.; Data Curation, X.Z.; Z.L.; Writing-Original Draft and Z.L.; Formal Analysis, X.Z.; Investigation, Z.L.;Visualization, Resources, X.Z.; Data Curation, X.Z.; Writing-Original Draft Preparation, X.Z.; Writing-Review & Editing, X.Z.; Z.L.; Supervision, X.Z.; Project Administration, X.Z.; Funding Acquisition, X.Z. Preparation, X.Z.; Writing-Review & Editing, X.Z.; Visualization, Z.L.; Supervision, X.Z.; Project Administration, X.Z.; Funding Acquisition, Fundamental Research Funds for the Central University of China. Funding: This research was funded by the X.Z. (Grant number: Funding: This FRF-TP-18-036A1). research was funded by the Fundamental Research Funds for the Central University of China. (Grant number: FRF-TP-18-036A1). Acknowledgments: The authors gratefully acknowledge the support for this work provided by University of Science and Technology Beijing. Acknowledgments: The authors gratefully acknowledge the support for this work provided by University of Conflicts Interest: The Beijing. authors declare no conflict of interest. Science of and Technology Conflicts of Interest: The authors declare no conflict of interest.

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Li, T. Data-Driven Dual-Clutch Transmission Shift Control. Master Thesis, Jilin University, Changchun, China, 2014. Zhang, Z. Data-Driven Predictive Control of Gearshifts and FPGA Implementation for Dual-Clutch Transmissions. Master Thesis, Jilin University, Changchun, China, 2013. © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).