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Integrated design for a CVT: dynamical optimization of actuation and control
9th IFAC International Symposium on Advances in Automotive 9th IFAC International Symposium on Advances in Automotive 9th IFAC IFAC International Symposium on Advances Advances in in Automotive Automotive Control 9th online at www.sciencedirect.com Control International Symposium onAvailable 9th IFAC France, International Control Orléans, June Symposium 23-27, 2019 on Advances in Automotive Control Orléans, France, June 23-27, 2019 Control Orléans, France, June 23-27, 2019 Orléans, France, June 23-27, 2019 Orléans, France, June 23-27, 2019
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IFAC PapersOnLine 52-5 (2019) 393–398
Integrated design for a CVT: dynamical Integrated design for a CVT: dynamical Integrated design for a CVT: dynamical optimization of actuation and control Integrated design for a CVT: dynamical optimization of actuation and optimization of actuation and control control optimization of actuation and ∗ ∗∗ control C.A. Fahdzyana ∗ T. Hofman ∗∗ C.A. C.A. C.A. C.A.
Fahdzyana ∗ T. Hofman ∗∗ ∗∗ Fahdzyana Fahdzyana ∗∗ T. T. Hofman Hofman ∗∗ Fahdzyana T. Hofman ∗ ∗ Eindhoven University ∗ Eindhoven University ∗ Eindhoven University of T echnology, Eindhoven, theN etherlands(e − University ∗ Eindhoven of T echnology, Eindhoven, theN Eindhoven University of T T echnology, echnology, Eindhoven, theN etherlands(e etherlands(e − − mail : c.a.f [email protected]) of Eindhoven, theN etherlands(e − mail : c.a.f [email protected]) ∗∗ of T echnology, Eindhoven, theN etherlands(e − mail :: of c.a.f [email protected]) Technology, Eindhoven, the Netherlands mail c.a.f [email protected]) ∗∗ Eindhoven University of Technology, Eindhoven, the Netherlands ∗∗ Eindhoven University mail : c.a.f [email protected]) ∗∗ Eindhoven University of Technology, Eindhoven, the Netherlands (e-mail: [email protected]) of Technology, Eindhoven, the Netherlands ∗∗ Eindhoven University (e-mail: [email protected]) Eindhoven University of Technology, Eindhoven, the Netherlands (e-mail: [email protected]) (e-mail: [email protected]) (e-mail: [email protected])
Abstract: Abstract: Abstract: With increasing increasing demands demands on on more energy energy and fuel fuel efficient vehicles, vehicles, one one can achieve achieve the the goal by by Abstract: With Abstract: With increasing demands on more more system. energy and and fuel efficient efficient vehicles, vehicles, one can can achieve achieve the goal goal by improving the vehicle vehicle powertrain A continuously continuously variable transmission transmission (CVT) allows With increasing demands on more energy and fuel efficient one can the goal by improving the powertrain system. A variable (CVT) allows With increasing demands on more energyon and fuel efficient vehicles, one can the goal by improving theelectric vehicle powertrain system. Aitscontinuously continuously variable transmission (CVT) allows the engine engine or machine to operate operate optimal operation points. Theachieve optimal operation improving the vehicle powertrain system. A variable transmission (CVT) allows the or electric machine to on its optimal operation points. The optimal operation improving the vehicle powertrain system. A continuously variable transmission (CVT) allows the engine or electric machine to operate on its optimal operation points. The optimal operation points are high high efficiency pointstothat that lead to to reduced energy consumption ofThe theoptimal vehicle. operation However, the engine or electric machine operate onreduced its optimal operation points.of points are efficiency points lead energy consumption the vehicle. However, the engine or transmission electric machine tothat operate onrelatively its optimal operation points. The optimal operation points are high efficiency points lead to reduced energy consumption of the vehicle. However, this type of may still have high actuation losses (depending on the points are high efficiency points that lead to reduced energy consumption of the vehicle. However, this type of may still have relatively high actuation (depending on points are type), high efficiency points that lead reduced energy consumption of plant the vehicle. However, this type of transmission transmission maythe stillenergy havetosaving relatively high Classically, actuation losses losses (depending on the the actuation which hinders hinders benefits. the (e.g., actuation actuation this type of transmission may still have relatively high actuation losses (depending on the actuation type), which the energy saving benefits. Classically, the plant (e.g., this typevariator) of transmission may still have saving relatively high actuation losses (depending on the actuation type), which hinders the energy saving benefits. Classically, the plant (e.g., actuation system, of the CVT was separately designed from the control design. In this paper, an actuation type), which hinders the energy benefits. Classically, the plant (e.g., actuation system, variator) of the CVT was separately designed from the control design. In this paper, an actuation type), which hinders theand energy saving benefits. Classically, the plant (e.g., actuation system, variator) of the CVT was separately designed from the control design. In this paper, an integrated optimal CVT variator actuation control design is presented. The aim of the new system, variator) of the CVT was separately designed from the control design. In this paper, an integrated optimalofCVT variator and actuation control design is presented. The of the new system, thethe CVT wasmass separately thetracking design. Inaim this an integrated CVT variator and actuation control design is presented. The aim of the new design is isvariator) tooptimal minimize CVT (pulleydesigned sheaves, belt), error and control effort. integrated optimal CVT variator and actuation controlfrom design iscontrol presented. The aim ofpaper, the new design to minimize the CVT mass (pulley sheaves, belt), tracking error and control effort. integrated optimal CVT variator and actuation control design is presented. The aim of the new design is to to this minimize the CVT optimization mass (pulley (pulleyframework sheaves, belt), belt), tracking error error and control control effort. To achieve goal, a nested is implemented to obtain an optimal design is minimize the CVT mass sheaves, tracking and effort. To achieve goal, a nested optimization framework is implemented to obtain an design is to this minimize CVT (pulley sheaves, belt), tracking andoptimized control effort. To achieve this goal,design athe nested optimization framework isresults implemented to the obtain an optimal optimal transmission system over optimization amass selected drive cycle. The The show error that CVT To achieve this goal, a nested framework is implemented to obtain an optimal transmission system design over a selected drive cycle. results show that the optimized CVT To achieve this goal, a nested optimization framework is implemented to obtain an optimal transmission system design over a selected drive cycle. The results show that the optimized CVT design yields non-compromising tracking performance, however, with much smaller variator mass transmission system design over a selected drive cycle. The results show that the optimized CVT design yields non-compromising however, with much variator mass transmission systemeffort design(-62%). over tracking a selectedperformance, drive cycle. The results show thatsmaller the optimized design non-compromising tracking performance, however, with much smaller variator mass (-46%)yields and control control design yields non-compromising tracking performance, however, with much smaller variator CVT mass (-46%) and effort (-62%). design non-compromising tracking performance, however, with much smaller variator mass (-46%)yields and control control effort (-62%). (-62%). (-46%) and effort © 2019, and IFACcontrol (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. (-46%) effort (-62%). Keywords: Co-design, Co-design, optimization, optimization, continuous continuous variable variable transmission, transmission, vehicle vehicle powertrains powertrains Keywords: Keywords: Co-design, Co-design, optimization, optimization, continuous continuous variable variable transmission, transmission, vehicle vehicle powertrains powertrains Keywords: Keywords: Co-design, optimization, continuous variable transmission, vehicle powertrains 1. INTRODUCTION INTRODUCTION A vehicle powertrain is aa complex dynamical system, 1. A vehicle powertrain is complex dynamical system, 1. INTRODUCTION INTRODUCTION A vehicle powertrain is aacomponents complex dynamical system, which consists of multiple at the subsystem 1. A vehicle powertrain is complex dynamical system, which consists of multiple components at the subsystem 1. INTRODUCTION A vehicle powertrain is a complex dynamical system, which consists of multiple components at the subsystem level, e.g., engine, alternator, and a transmission. On the which consists of multiple components at the subsystem With systems becoming more complex, achieving a true level, e.g., engine, alternator, and aa transmission. On the With systems becoming more complex, achieving aa true which consists of multiple components at the subsystem level, e.g., engine, alternator, and transmission. Onhigh the With systems becoming more complex, achieving true present day, it is desirable to design a vehicle with level, e.g., engine, alternator, and a transmission. On the optimal system design can be a challenge due to the With systems becoming morebecomplex, achieving a true present day, it is desirable to design a vehicle with high optimal system design can a challenge due to the level, e.g., engine, alternator, and a transmission. On the With systems becoming more complex, achieving a true present efficiency. day, it it is is A desirable to designthe a vehicle vehicle with high optimal system design designcoupling can be be (e.g. a challenge challenge duevariables to the the present energy way to improve energy efficiency day, desirable to design a with high existing mathematical via shared optimal system can a due to energy efficiency. A way to improve the energy efficiency existing mathematical coupling (e.g. via shared variables present day, it is desirable to design a vehicle with high optimal system design can be a challenge due to the energy efficiency. A way to improve the energy efficiency existing mathematical coupling (e.g. via shared variables is by utilizing aa continuously variable transmission (CVT) efficiency. A way to improve the energy efficiency in constraints constraints and objectives) objectives) between the corresponding existing mathematical couplingbetween (e.g. viathe shared variables energy is by utilizing variable transmission (CVT) in and corresponding energy efficiency. AAway to improve the energy efficiency existing mathematical coupling (e.g. viathe shared variables is by utilizing aa continuously continuously variable transmission (CVT) in constraints andbetween objectives) between the corresponding in the drivetrain. CVT allows smooth shifting peris by utilizing continuously variable transmission (CVT) subsystems (e.g., the controller and the plant). in constraints and objectives) between corresponding in the drivetrain. A CVT allows smooth shifting persubsystems (e.g., between the controller and the plant). is by utilizing a continuously variable transmission (CVT) in constraints andbetween objectives) the corresponding in the the drivetrain. drivetrain. A aCVT CVT allows smooththe shifting persubsystems (e.g., between the between controller and the plant). formance. Moreover, CVT can operate engine (or in A allows smooth shifting perCombined optimal plant and control design has been subsystems (e.g., the controller and the plant). formance. Moreover, a CVT can operate the engine (or Combined optimal plant and control design has been in the drivetrain. A CVT allows smooth shifting persubsystems (e.g., between the controller and the plant). formance. Moreover, a CVT CVT can operation operate the the engine (or Combined optimal plant and control control design has been been electric machine) at the optimal points, which formance. Moreover, a can operate engine (or shown to result in an improved system performance [1], Combined optimal plant and design has electric machine) at the optimal operation points, which shown to result in an improved system performance [1], formance. Moreover, a CVT can operation operate the engine (or Combined optimal plant and control design has might been machine) at the optimal points, which shown to result result in an an improved system performance [1], electric results in a more efficient energy consumption. However, electric machine) at the optimal operation points, which [2]. However, a combined system-level optimization shown to in improved system performance [1], results in aa more efficient energy consumption. However, [2]. However, a combined system-level optimization might electric machine) at the optimal operation points, which shown to result in an improved system performance [1], results in more efficient energy consumption. However, [2]. However, a combined system-level optimization might the efficiency of this type of transmission strongly depends in a more efficient energy consumption. However, result in a nonconvex nonconvex optimization problem, even when when the results [2]. However, a combined system-level optimization might the efficiency of type of transmission strongly depends result in optimization problem, even the in a more efficient consumption. However, [2]. However, a and combined system-level optimization might the its efficiency of this this type which ofenergy transmission strongly depends result in aa aplant nonconvex optimization problem, even when when the on actuation design, hinders the energy saving the efficiency of this type of transmission strongly depends separate control optimization subproblems are results result in nonconvex optimization problem, even the on its actuation design, which hinders the energy saving separate plant and control optimization subproblems are the efficiency of this type of transmission strongly depends result in a nonconvex optimization problem, even when the on its actuation design, which hinders the energy saving separate plant and control control optimization subproblems are capability that it potentially offers. on its actuation design, which hinders the energy saving convex [3]. Therefore, a suitable design framework must be separate plant and optimization subproblems are capability that it potentially offers. convex [3]. Therefore, a suitable design framework must be on its actuation design, which hinders the energy saving separate plant and control optimization subproblems are capability that it potentially offers. convex [3]. Therefore, a suitable design framework must be capability that it potentially offers. developed in order to overcome this challenge. The method convex [3]. Therefore, a suitable design framework must be developed in order to overcome this challenge. The method For automotive applications, aa higher power density capability that it potentially offers. convex [3]. Therefore, a suitable design framework must be developed in order to overcome this challenge. The method For automotive applications, higher power density for combined combined plantto and control this optimization isThe referred as For automotive applications, a higher power density developed in order overcome challenge.is method for plant and control optimization referred as (kW/kg) and reduced size of transmissions are automotive applications, a higher (packaging) power density developed in order overcome challenge.is method for combined plantto and control this optimization isThe referred as For (kW/kg) and reduced size of transmissions (packaging) are co-design. for combined plant and control optimization referred as For automotive applications, a higher are power density (kW/kg) and reduced size of transmissions (packaging) are co-design. desired. Smaller transmission dimensions also benefi(kW/kg) and reduced size of transmissions (packaging) are for combined plant and control optimization is referred as co-design. desired. Smaller transmission dimensions are also benefico-design. (kW/kg) and reduced size of transmissions (packaging) are desired. Smaller transmission dimensions are also beneficial in terms of production cost. Furthermore, as investidesired. Smaller transmission dimensions are also benefiThe combined plant and control design has been a subject co-design. cial in terms of production cost. Furthermore, as investiThe combined plant and control design has been subject desired. Smaller transmission dimensions aredependent also beneficial in ininterms terms of production production cost. Furthermore, as investiinvestiThe combinedfor plant and control design Generally, has been been aa a the subject gated [4], the power capacity of a CVT is on cial of cost. Furthermore, as of discussion the past few years. coThe combined plant and control design has subject gated [4], the power capacity aa CVT is dependent on of discussion for the past few years. Generally, the coinin terms of production cost.of Furthermore, as investiThe combined plant and control design has been a the subject gated in [4], the power capacity of CVT is dependent on of discussion for the past few years. years. Generally, the co- cial the ratio coverage of the transmission, which is determined gated in [4], the power capacity of a CVT is dependent on design formulation can be expressed as the weighted sum of discussion for the past few Generally, cothe ratio coverage of the transmission, which is determined design formulation can be expressed as the weighted sum gated in [4], the power capacity of a CVT is dependent on of discussion for the past few years. Generally, the cothe ratio coverage of the transmission, which is determined design formulation can be expressed as the weighted sum by the physical parameters of the CVT, such as the pulley the ratio coverage of the transmission, which is determined of the plant and control objectives [3]: design formulation can be expressed as the weighted sum by the physical parameters of the CVT, such as the pulley of the plant and control objectives [3]: the ratio coverage of the transmission, which is determined design formulation can be expressed as the weighted sum by the physical parameters of the CVT, such as the pulley of the plant and control objectives [3]: center distance and wedge angle. Other than increasing the by the physical parameters of the CVT, such as the pulley of the plant and objectives wp J wc[3]: J (1) mincontrol center wedge angle. Other than increasing the p (x p) c (xp ,, x c) , by the distance physical and parameters of the CVT, such as the pulley )) + + J x (1) min of the plant and p (x center distance and wedgewedge angle. Other than increasing the xp ⊆R, xccontrol ⊆R w power capacity, reduced angle gives added benefits, distance and wedge angle. Other than increasing the wppp J Jobjectives (xp +w wccc[3]: Jccc (x (xp xccc ))) ,,, (1) center min p (x p) p ,, x xp ⊆R, xc ⊆R w min J + w J (x (1) power capacity, reduced wedge angle gives added benefits, p p p distance and wedge angle.angle Other thanadded increasing the x ⊆R, x power capacity, reduced wedge gives benefits, xp xcc ⊆R ⊆R wp Jp (xp ) + wc Jc (xp , xc ) , min (1) center such as reduced size and increased efficiency. p ⊆R, power capacity, reduced wedge angle gives added benefits, subject to: such as reduced size and increased efficiency. subject to: xp ⊆R, xc ⊆R power capacity, reduced wedge angle gives added benefits, such as reduced size and increased efficiency. subject to: to: such as reduced size and increased efficiency. subject ≤ 0 ,, h = 0 ,, g In this work, an design method of a CVT p (x p) p (x p) such as reduced sizeintegrated and increased efficiency. ≤ 0 h = 0 g subject to: In this work, an integrated design method of a CVT p (x p) p (x p) (x ) ≤ 0 , h (x ) = 0 , g In this work, an integrated design method of a CVT p p p p , (x hpc (x ) = 0, gxpp(x variator (pulleys and v-belt) and its control system is In this work, an integrated design method of a CVT p ) 0≤, 0 p g (x , ) ≤ h , x c c c p variator (pulleys and v-belt) and its control system is g (x , x ) ≤ 0 , h (x , x , (x ) ≤ 0 , h (x ) = 0 g c c p c c p In this work, an integrated design method of a CVT p )≤ p 0 , h (xp , xp ) = 0 , variator (pulleys and v-belt) and its control system is g (x , x proposed. The CVT design will be optimized dynamically c (xdesign c , xp p) ≤ c (xc c , xg p ) and variator (pulleys and v-belt) and its control system is g 0 , h = 0 , c c c p are the parameters, h are the inwhere x proposed. The CVT design will be optimized dynamically i are g i i variator (pulleys and v-belt) and its control system is the design parameters, g and h are the inwhere x (x , x ) ≤ 0 , h (x , x ) = 0 , i i i proposed. The CVT design will be optimized dynamically c c p c c p over a complete drive cycle profile. In this work, we limit proposed. The CVT design will be optimized dynamically are equality the design design parameters, giiJiand and hthe are the inin- over a complete drive cycle profile. In this work, we limit where xiiand ii are equality constraints, and are objective are the parameters, g h the where x proposed. The CVT design will be optimized dynamically equality and equality constraints, and J are the objective i are h over aa complete complete drive cycle cycle profile. In pulley this work, work, we limit limit are the design parameters, the in- over where xiand ourselves to optimizing the variator sheaves and drive profile. In this we i are equality constraints, and objective functions, w are the weighting parameters, for the plant equality and equality constraints, andgiJ Jiiand are the the objective ourselves to optimizing the variator sheaves and i equality over a complete drive cycle profile. In pulley this work, we limit functions, w are the weighting parameters, for the plant i equality ourselves to optimizing the variator pulley sheaves and equality and constraints, and J are the objective ourselves to optimizing the variator pulley sheaves and i functions, w are the weighting parameters, for the plant i are thei ∈ and control system, {p, c} with w ∈ R, respectively. functions, w weighting parameters, for the plant i i ourselves to optimizing the variator pulley sheaves and and control system, i ∈ {p, c} with w ∈ R, respectively. i functions, w are the weighting parameters, for the plant i and control control system, system, ∈ {p, {p, c} c} with with w wii ∈ ∈ R, R, respectively. respectively. and ii ∈ and control system, i ∈ {p, c} with w ∈ R, respectively. i 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 IFAC 393 Copyright © under 2019 IFAC 393 Control. Peer review responsibility of International Federation of Automatic Copyright © 393 Copyright © 2019 2019 IFAC IFAC 393 10.1016/j.ifacol.2019.09.063 Copyright © 2019 IFAC 393
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belt. The optimization of the actuation system (motors) and mechanism is seen as future work. This paper is arranged as follows. Section 2 discusses the plant design modeling. The co-design formulation for the CVT is explained in Section 3. The results of the proposed co-design for CVT on a vehicle powertrain over a drive cycle is elaborated in Section 4. Lastly, the conclusion and future work are presented in Section 5.
with
R p − Rs . (6) ϕ = sin a where a is the center distance between the primary and secondary pulley, ϕp and ϕs are the belt wrap angles, respectively. −1
dp
h β
A
2. PLANT MODEL: ELECTROMECHANICALLY ACTUATED CONTINUOUS VARIABLE TRANSMISSION
ds A δa
rg > 1
R1
' R2
In this work, the CVT that will be studied is the electromechanically actuated continuously variable transmission (EMPACT). This type of CVT, as shown in Fig. 1, has electric servomotors as the actuation system, which prompt the pulley sheaves on the primary and secondary side. In this section, the behaviour and mathematical formulations of EMPACT will be discussed.
dp
Rp
a
!p
!s
Rs
A-A view Rin
Fig. 2. CVT variator diagram Due to friction, there exists a slip v in the variator, which is the difference between the tangential speed on the primary ω R −ω R and secondary side, v = p ωpp Rps s . Here, the slip is considered to be small, such that ωp Rp ≈ ωs Rs , and therefore, rg ≈ rs .
Actuation
2.2 Variator dynamics and clamping forces Several models have been derived to describe the transient behaviour of the CVT variator. Here, the Carbone, Mangialardi, and Mantriota (CMM) model will be utilized to describe the behaviour of the variator ratio dynamics [6]. This mathematical model describes the behaviour of the ratio change as a function of the clamping force and also the pulley deformation. The rate of change of the speed ratio is then given by, 1 + cos2 β Fp,ss Fp r˙g = 2 ωp ∆ c(rg ) ln −ln , (7) sin(2β) Fs Fs,ss
ds
variator
Fig. 1. Schematic diagram of EMPACT CVT [5] 2.1 CVT geometric model The schematic diagram of the CVT variator is depicted in Fig. 2. The CVT geometric ratio is defined as the ratio between the running radii on the primary and the secondary side, Rp rg = . (2) Rs Additionally, the speed ratio is defined as, ωs rs = , (3) ωp where ωp and ωs are the primary and secondary rotational speeds, respectively. The movable pulley positions dp and ds determine the corresponding running radii of the CVT at the primary and secondary side, dp −ds + Ro ; Rs = + Ro , Rp = (4) 2 tan β 2 tan β where β is the pulley wedge angle and Ro is the running radius at rg = 1. The position of the movable pulley sheave results in different wrap angles ϕi , which are the span of the belt on the pulley sheaves, and are formulated as: (5) ϕp = π + 2ϕ ; ϕs = π − 2ϕ , 394
where ωp is the rotational speed of the driving (primary) pulley, ∆ is the pulley deformation, β is the pulley wedge angle. Fp and Fs are the primary and secondary clamping forces, and the subscript ss indicates the steady-state value when r˙g = 0. The term c(rg ) is a function of rg that relates the dimensionless speed ratio with the difference of the logarithmic steady state and applied clamping forces. The pulley deformation ∆ is affected by the clamping forces, and can be expressed as [7]: ∆ = (1 + 0.02(Fs − 20)) · 10−3 ,
(8)
where Fs is the secondary clamping force in kN. The clamping forces Fp and Fs are given by, Si (θ) − Ci (θ) − Fb − Fc Fi = dθ , (9) 2 sin β ϕi
where i ∈ {p, s}, Si and Ci are the tension and compression forces over the pulleys, Fb and Fc are the centrifugal forces of the band and belt blocks, Fb = ρb vb2 and Fc = ρc vb2 , respectively [8]. The belt velocity vb is assumed to be vb = ωp Rp = ωs Rs . The calculation of the clamping forces are also dependent on ρb and ρc , which are the mass per unit length of the bands and the blocks, respectively.
2019 IFAC AAC Orléans, France, June 23-27, 2019
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3. CO-DESIGN FORMULATION A co-design strategy is proposed to obtain an improved EMPACT plant and control design for a specific drive cycle. The key performance indicators (KPIs) are packaging (dimensions), efficiency, performance, and cost. Here, we focus on minimizing the variator mass, which is expressed as a function of the variator dimensions (plant objective) and the tracking performance and actuation effort (control objective) of the CVT system. The KPIs can be interpreted as the system wide objectives. Further, we use optimal control techniques in order to not compromise the tracking performance of the control system. The CVT actuation controller should provide the corresponding signal that minimizes the tracking error and control effort for the actuation system. However, due to the presence of the coupling between the geometrical properties and the reference for control, the response of the CVT is also dependent on the choice of the plant design parameters, namely β and a, which becomes a challenge in finding the combined plant and control optimal design. 3.1 Plant Design Problem To derive the plant design objective Jp , the CVT pulley sheaves are considered to be identical truncated cones and the metal belt is assumed as a v-belt. The mathematical formulation of the variator mass is defined as, Mv = ρpu Vpu + ρb Vb . (10) Vpu and Vb are the pulleys and belt volume of the variator, given by 1 2 , Vpu = 4π(R2 − R1 ) tan β (R12 + R1 R2 + R22 ) − Rin 3 (11) (b1 + b2 )b3 Vbe = Lb , (12) 2 where R1 and R2 are the pulley top and bottom radii, Rin is the radius of the shaft, Lb is the length of the belt, and b1 , b2 , and b3 are the belt parameters, respectively, as depicted in Figs. 2 and 3. b2 b2 Lb
b3
b3
β
b1 b1
Fig. 3. CVT belt geometries As seen in Fig. 2, the sheave radius R2 determines the value of the center distance a. Hence, the plant design objective is mathematically expressed as, min Jp = Mv (xp ) , (13) xp ⊆Xp ⊆R
subject to:
hmin
β − βmax ≤ 0 −β + βmin ≤ 0 ≤ tan β(R2 − R1 ) ≤ hmax Ri,min ≤ Ri ≤ Ri,max 2R2 − a + δa ≤ 0
395
and Ri,max are the minimum and maximum pulley radii for i = {1, 2}; a is the pulley center distance; δa is the minimum pulley distance; and xp are the plant design parameters, xp = {R1 , R2 , β} , where the set xp is a subset of set Xp , given by, Xp = {xp ∈ R | gp (xp ) ≤ 0 , hp (xp ) = 0} . 3.2 Control Design Problem This work focuses only on the speed ratio control of CVT. The reference for the primary servomotor to realize the desired ratio values is influenced by the plant design parameters (i.e., β, R1 , R2 ), which indicates the coupling that exists between the plant parameters and the control. Here, firstly, the primary servomotor reference design model for the primary rotational position is discussed. This serves as input to the servomotor control problem accordingly. The ratio rg is not directly measurable, and must, therefore, be estimated from the measureable variables. For the EMPACT, the measured variables are the servomotor rotations θmp and θms and the primary ωp and secondary ωs shaft speeds. The electric servomotors actuate the movable pulley sheaves to realize the required clamping forces necessary to perform ratio shifting. Hence, the ratio control of the EMPACT CVT can be approached as a servomotor control problem [5]. 3.2.1 Servomotor reference design Mathematically, the pulley sheave positions can be expressed as a function of the servomotor positions, θmp sz (19a) rw θms sz (19b) d s = dp + r r rc where θmp and θms are the motor angular positions of the primary and secondary actuations; rr , rc , and rw are the reduction gear constants; s is the screw pitch, and z is the ratio between the sun and annulus of the planetary gear set connecting the servomotor and the screw. dp =
The desired reference ratio trajectory rg,r can be translated as the reference for the primary servomotor θmp,r . As rg,r is known a priori, r˙g,r can be obtained using the trajectory. Then, using the CMM model that is described F in (7), the required clamping force ratio Fps r that yields the desired ratios can be calculated by the relation: Fp Fp,ss sin(2β) · = exp r ˙ . (20) g Fs r 2 ωp ∆ (1 + cos2 β) c(rg ) Fs,ss F
(14) (15) (16) (17) (18)
where hmin and hmax and βmin and βmax are the minimum and maximum pulley heights and wedge angles; Ri,min 395
as a function The steady state clamping force ratios Fp,ss s,ss of gear ratio values can be computed a priori using the geometrical and physical variator model as described by [6]. To reduce the calculation time for the design optimization, some models are reduced as parametric representations of the relevant design parameters. The steady-state clamping force ratio can be approximated as, Fp,ss a1 βrg + a2 β + a3 , for rg ≤ 1 (21) = for rg > 1 b1 + b2 β 2 ln(rg ) , Fs,ss
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with fitted constants a1 = 1.2204, a2 = −1.65, and a3 = 1.0738, and b1 = 0.9968 and b2 = 6.2373. F
Similarly, the resulting clamping force ratio Fps can be approximated as a function of the primary wrap angle ϕp and β, Fp for rg ≤ 1 , 0.6793ϕp − 2.1303β − 0.47 , (22) = 0.8409ϕp − 3.4886β − 0.456 , for rg > 1 Fs Using (4), (5), (6), and (22), the corresponding dp,r that F yields the desired clamping force ratio Fps r can be obtained. Furthermore, via the relation in (19a), the desired reference trajectory for the primary servomotor actuation θmp,r can be determined. 3.2.2 Servomotor control design The primary servomotor type is selected to be a DC motor. The dynamics is given by: θmp 0 1 0 0 d θmp ωmp = 0 −b/I Kt /I ωmp + 0 u , (23) dt 0 −Ke /L −R/L i i 1/L A
ξ
θmp y = [1 0 0] ωmp i C
B
(24)
where θmp and ωmp are the primary servomotor angular position and rotational speed, i is the motor current, Kt and Ke are the torque and voltage constants, b is the friction constant, L is the inductance, and R is the armature resistance. I is the rotor inertia, u is the control input which is in voltage, and finally the state variables are given by ξ = [θmp , ωmp , i] . In order to design a controller such that the error between the desired and actual trajectory is minimized, the state space model of the system is augmented such that the T error ε = 0 (θmp,r (t) − θmp (t)) dt is included as an additional state. The feedback gains are found by utilizing an LQR formulation. A feedforward term kf f is included to minimize the steady state error. The augmented state is expressed as: d ξ B A0 ξ 0 + = u+ θ , (25) 0 C 0 ε −1 mp,r dt ε ξa
Aa
Ba
where the control input is given by (26) u = −kfb · ξ + kf f · ε , where kfb and kf f are the control gains. The proposed control structure is depicted in Fig. 4.
The primary actuation controller calculates the required voltage u to the servomotor such that the desired ratio is tracked, while still minimizing the energy consumption. The cost function for the controller is given by tf ξa Qξa + u Ru dt , (27) Jc = 0
where the minimum of the cost function can be found by solving the algebraic Riccati equation. The weight matrices Q and R are commonly chosen by the user to have
396
acceptable balance between minimizing error and control effort that yields satisfactory performance. However, as in the co-design framework the plant design parameters (i.e., β, R1 , R2 ) will be varied, the corresponding reference trajectory for the servomotor reference θmp,r are also changed. Hence, the weight matrices Q and R are selected such that the error between the desired and actual ratio trajectory is minimized for a plant design. The optimal solution of (27) is found by solving the Riccati equation, −1 Ba P + Q = 0 , (28) A a P + PAa − PBa R as long as the conditions P > 0 , Q ≥ 0 , R > 0 are satisfied. P is a positive definite matrix that satisfies the algebraic Riccati equation. The weight matrices Q is a m × m matrix and R is a n × n matrix, where m is the rank of Aa and n is the rank of the input u. The gains kfb and kf f are determined by solving the Riccati equation [kfb kf f ] = R−1 B a P.
In the EMPACT CVT, the primary side is often used to perform ratio tracking, while the secondary side is used to regulate the slip in the variator. The minimum required secondary clamping force to ensure torque transfer is given by α Tsi cos β Fs,r = , (29) 2 µ Rs where α is a safety factor, commonly chosen to be 1.3, Tsi is the belt internal torque exerted on the secondary pulley, and µ is the effective traction coefficient. The design of secondary actuation controller is outside of the research scope. 3.3 Co-design problem formulation The combined plant and control design problem formulation of the CVT is written as: tf wp Mv (xp ) + wc ξa Qξa + u Ru dt , min xp ⊆Xp ,xc ⊆Xc
0
(30)
subject to: (15)-(18) and P > 0 , Q ≥ 0 , R > 0.
The optimized design parameters are defined as xp = {R1 , R2 , β} and xc = {Q, R}, and xp ⊆Xp , xc ⊆Xc . The selection of the weights wp and wc are important in finding the Pareto front of the optimization problem. In this work, the weights wi for i = p, c are selected to be 1. Furthermore, in this specific problem, the parameters β and a serve as couplings between the plant design (minimizing variator mass, a more compact CVT) and the control design (minimizing speed ratio tracking error) problem. 4. NESTED OPTIMIZATION APPROACH A nested optimization approach will be implemented to dynamically obtain an optimal CVT design for the selected drive cycle. This is because the plant design parameters (i.e., β, R1 , R2 ) have influence on the control design (minimizing ratio tracking error). The nested approach has an outer loop that optimizes the combined design objective (30) by varying the plant design parameters, and an inner loop that optimizes the control performance. For this problem, the ratio control problem is solved using an
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397
β
rg;r
ss clamp. (21)
Fp Fs ss
β !p
primary actuation
β r_g;r
CMM−1 (7)
Fp Fs r
β
desired 'p 'p;r
desired dp dp;r (4), (6)
(22)
β
a rw sz
R
θmp;r
"
u(t)
kf f
ξ_ = Aξ + Bu y = Cξ
(19a)
(23)-(24)
-
-
kfb
θmp
sz rw
dp
(19a)
β
a
dp to 'p (4), (6)
'p
β Fp
clamp. ratio (22)
x
CMM (7)
r_g
R
rg
ξ Fs
µ Tsi
minimal clamp. (29)
β
Fs;r
Rs
Fig. 4. Control scheme for EMPACT ratio control LQR formulation with augmented state. Additionally, the weight matrices {Q, R} are obtained by optimization. The proposed framework is depicted in Fig. 5. outer loop: combined problem min wp Jp (xp ) + wc Jc (xp ; x∗c ) xp
s.t. : gp;1;:::;6 x∗c (xp )
xp inner loop: control problem
xc ∗ = argmin Jc = argmin xc xc s.t. :
R tf to
ξa> Qξa + u> Ru dt
Q ≥ 0;R > 0
x∗p ; x∗c
Fig. 5. Nested optimization framework for the EMPACT CVT 4.1 Reference speed trajectory
Pwh Twh ωwh = . (33) ηt ηt The engine optimal line is found to be the combination of engine speed and torque ωe∗ and Te∗ that yield the minimum fuel consumption. Using the optimal operation points, the desired ratio reference that yields optimal fuel consumption can be calculated. The desired speed ratio rg,r is defined as, ωwh iF DR rg,r = min max rg,min , , rg,max , (34) ωe∗ where ωe∗ is the engine speed at the optimal operation line, and iF DR is the final drive ratio. Pe,r =
The obtained ratio reference is used to perform co-design on the CVT variator and ratio controller dynamically for the complete NEDC profile. The vehicle parameters used throughout the simulation study are summarized in Table 1. Table 1. Simulation Parameters
The performance of the optimized CVT throughout a given driving profile will be analyzed. The reference CVT ratio rg,r is determined such that the engine is operated near, if not at, the optimal operation line of the engine. The NEDC drive cycle is selected to investigate the benefits of utilizing CVT in a traditional vehicle powertrain with a 2.0 L gasoline engine. The NEDC drive cycle is used to determine the required CVT ratio values such that the vehicle is operated at the engine optimal operation line.The wheel speed ωwh is defined as vwh ωwh = , (31) Rwh where vwh is the vehicle speed from the drive cycle and Rwh is the wheel radius. The torque demand at the wheel is calculated using (32) Twh = (Fr + Fa + Fi ) Rwh , where Fr is the rolling resistance, Fa is the aerodynamic drag, and Fi is the force due to inertia of the vehicle. Furthermore, the ratio is selected such that the required engine power to drive the vehicle is generated with minimum fuel consumption. To determine the ratio, several assumptions are considered. At ωwh = 0, the engine is at idle, ωe = ωid . Throughout the CVT operation over the NEDC drive cycle, the efficiency of the transmission is treated to be constant ηt (t) = ηt . The power demand at the wheels is used to determine the power required from the engine, 397
Vehicle mass Roll. resistance Drag. resistance Vehicle frontal area Air density Rotational mass Wheel radius Final drive ratio Engine idle speed CVT efficiency
Symbol ms cr cd Af ρa mro Rwh iF DR ωid ηt
Value 1180 0.0174 0.29 2.38 1.225 2 0.35 3 78.53 0.9
Units kg − − m2 kg/m3 % m − rad/s −
4.2 Design results The CVT variator design optimization and the speed ratio controller over a complete drive cycle are demonstrated. The simulation parameters used in this study is summarized in Table 2. This subsection discusses the results obtained from the study of optimizing the CVT design compared to a baseline model 1 . It can be seen from the results that the optimized CVT is able to follow the desired ratio trajectory despite having a lower variator dimensions throughout the drive cycle, as summarized in Table 3. Using the proposed nested optimization framework, the mass of the pulley sheaves 1 Variator geometries from Jatco CK2 with EMPACT actuation system for reasons of comparison
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Table 2. Simulation Parameters Motor inertia Motor friction constant Motor inductance Motor torque constant Motor voltage constant Motor armature resistance Range of β Range of R1 Range of R2 Inner radius Range of ratio Weight Weights for i ∈ {c, 1, 2} Belt width Belt thickness Screw pitch Reduction gear Initial ratio Planetary gear set ratio Mass density
Symbol I b L Kt Ke Ra [βmin , βmax ] [R1,min , R1,max ] [R2,min , R2,max ] Rin [rg,min , rg,max ] wp wi b1 b3 s rw rg,o z ρb , ρpu
Value 31.4 · 10−6 2.1 · 10−3 2.75 52.96 · 10−3 5.5 0.47 [6,15] [15,30] [60,90] 7.5 [0.5,2.5] 10 1 28 24 4/(2π · 1000) 16 0.5 2 7850
implemented. The proposed approach calculates the minimized CVT variator’s mass, as well as the corresponding actuation control parameters that yields the best possible ratio tracking performance.
Units kg · m2 N · m/krpm mH Nm/A V/krpm Ω ◦
mm mm mm − − − mm mm rad/m − − − kg/m3
can potentially be reduced from 6.86 to 3.69 kg (-46%). The optimized parameter β for the new variator is found to be 6◦ , R1 to be 22.8 mm, R2 to be 66.7 mm, and the pulley center distance a to be 136.4 mm. Despite the reduced dimensions, the optimized CVT variator yields comparable performance to the baseline. The resulting ratio tracking performance is depicted in Fig. 6. Optimized variator performance for NEDC cycle
120
NEDC speed profile Reference Actual
3
80
2.5
60
2
40
1.5
20
1
0 200
400
600
800
1000
In this work, the CVT efficiency is not modeled. In the future, the actual CVT efficiency as a function of the relevant physical design parameters should be included in the co-design framework. Currently, only ratio controller is considered in this work as part of the co-design framework for CVT. The framework could also be extended to include slip controller on the secondary side of the variator. Lastly, in the future, the potential of using CVT in a full electric vehicle powertrain will be investigated. REFERENCES
Ratio rg [-]
Vehicle speed v wh [km/h]
100
Based on the results, it is seen that the variator design could be reduced in terms of size and mass without compromising the ratio tracking performance. At the same time, the corresponding optimal controller for the new optimized design is found using the proposed framework. Using the co-design framework explained in this paper, the mass of the pulley sheaves can potentially be reduced up to 46.2%, from 6.86 to 3.69 kg. It was also obtained that the wedge angle β for the CVT variator can be reduced from 11◦ to 6◦ , and center distance a from 169 to 136.4 mm. Furthermore, with the new optimized parameters, the ratio coverage of the CVT is reduced from 5.33 to 4.23. It is seen that the optimized CVT design yields comparable performance to the baseline CVT, despite the smaller size.
[1]
[2]
0.5 1200
Time [s]
Fig. 6. Results for the optimized model for NEDC drive cycle Table 3. Nested co-design results for NEDC drive cycle J
xp
xc
K, rcov , a
Plant objective Control objective Variator volume Wedge angle Top outer radius Bottom outer radius LQR weight LQR weight LQR weight LQR weight LQR weight Center distance Ratio coverage Feedback gain Feedback gain Feedback gain Feedforward gain
Symbol Jp Jc Vv β R1 R2 Q(1) Q(2) Q(3) Q(4) R a rcov kf b (1) kf b (2) kf b (3) kf f
Baseline 6.86 7.6 8.74 11.0 26.3 82.5 2.065 0.001 0.001 104 0.01 168 5.33 2.4439 0.035 3.54 62.38
Optimized 3.69 2.89 4.70 6 22.8 66.7 0.768 0.001 0.001 104 0.01 136.4 4.23 2.355 0.035 3.497 63.88
[3]
[4]
Unit kg 1010 −4 10 m3 ◦
mm mm − − − − − mm − − − 10−4 −
[5]
[6]
[7]
5. CONCLUSIONS AND FUTURE WORK In this study, a nested optimization framework for the EMPACT CVT to obtain a combined plant and control design for the NEDC drive cycle has been succesfully 398
[8]
H. K. Fathy, P. Y. Papalambros, A. G. Ulsoy, and D. Hrovat, Nested plant/controller optimization with application to combined passive/active automotive suspensions, in Proc. American Control Conference, Denver, USA, 2003. J. A. Reyer and P. Y. Papalambros, Combined optimal design and control with application to an electric DC motor, in Journal of Mechanical Design, vol. 124, pp. 183–191, 2002. H. K. Fathy, J. Reyer, P. Papalambros, G. Ulsoy, On the coupling between the plant and controller optimization problems, in Proc. American Control Conference, Virginia, USA, 2001. A. Brandsma, J. van Lith, E. Hendriks, Push belt CVT developments for high power applications, in International Congress on Continuously Variable Power Transmission CVT, Eindhoven, The Netherlands, 1999. T. W. G. L. Klaassen, The EMPACT CVT : dynamics and control of an electromechanically actuated CVT, PhD Thesis, Eindhoven University of Technology, 2007. G. Carbone, L. Mangialardi, G. Mantriota, The Influence of Pulley Deformations on the Shifting Mechanism of Metal Belt CVT, Journal of Mechanical Design, vol. 127, pp. 103–113, 2005. G. Carbone, L. Mangialardi, B. Bonsen, C. Tursi, P. A. Veenhuizen, CVT dynamics: Theory and experiments, Mechanism and Machine Theory, vol. 42, pp. 409–428, 2007. B. G. Vroemen, Component control for the Zero Inertia powertrain, PhD thesis, Eindhoven University of Technology, 2001.
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Integrated design for a CVT: dynamical Integrated design for a CVT: dynamical Integrated design for a CVT: dynamical optimization of actuation and control Integrated design for a CVT: dynamical optimization of actuation and optimization of actuation and control control optimization of actuation and ∗ ∗∗ control C.A. Fahdzyana ∗ T. Hofman ∗∗ C.A. C.A. C.A. C.A.
Fahdzyana ∗ T. Hofman ∗∗ ∗∗ Fahdzyana Fahdzyana ∗∗ T. T. Hofman Hofman ∗∗ Fahdzyana T. Hofman ∗ ∗ Eindhoven University ∗ Eindhoven University ∗ Eindhoven University of T echnology, Eindhoven, theN etherlands(e − University ∗ Eindhoven of T echnology, Eindhoven, theN Eindhoven University of T T echnology, echnology, Eindhoven, theN etherlands(e etherlands(e − − mail : c.a.f [email protected]) of Eindhoven, theN etherlands(e − mail : c.a.f [email protected]) ∗∗ of T echnology, Eindhoven, theN etherlands(e − mail :: of c.a.f [email protected]) Technology, Eindhoven, the Netherlands mail c.a.f [email protected]) ∗∗ Eindhoven University of Technology, Eindhoven, the Netherlands ∗∗ Eindhoven University mail : c.a.f [email protected]) ∗∗ Eindhoven University of Technology, Eindhoven, the Netherlands (e-mail: [email protected]) of Technology, Eindhoven, the Netherlands ∗∗ Eindhoven University (e-mail: [email protected]) Eindhoven University of Technology, Eindhoven, the Netherlands (e-mail: [email protected]) (e-mail: [email protected]) (e-mail: [email protected])
Abstract: Abstract: Abstract: With increasing increasing demands demands on on more energy energy and fuel fuel efficient vehicles, vehicles, one one can achieve achieve the the goal by by Abstract: With Abstract: With increasing demands on more more system. energy and and fuel efficient efficient vehicles, vehicles, one can can achieve achieve the goal goal by improving the vehicle vehicle powertrain A continuously continuously variable transmission transmission (CVT) allows With increasing demands on more energy and fuel efficient one can the goal by improving the powertrain system. A variable (CVT) allows With increasing demands on more energyon and fuel efficient vehicles, one can the goal by improving theelectric vehicle powertrain system. Aitscontinuously continuously variable transmission (CVT) allows the engine engine or machine to operate operate optimal operation points. Theachieve optimal operation improving the vehicle powertrain system. A variable transmission (CVT) allows the or electric machine to on its optimal operation points. The optimal operation improving the vehicle powertrain system. A continuously variable transmission (CVT) allows the engine or electric machine to operate on its optimal operation points. The optimal operation points are high high efficiency pointstothat that lead to to reduced energy consumption ofThe theoptimal vehicle. operation However, the engine or electric machine operate onreduced its optimal operation points.of points are efficiency points lead energy consumption the vehicle. However, the engine or transmission electric machine tothat operate onrelatively its optimal operation points. The optimal operation points are high efficiency points lead to reduced energy consumption of the vehicle. However, this type of may still have high actuation losses (depending on the points are high efficiency points that lead to reduced energy consumption of the vehicle. However, this type of may still have relatively high actuation (depending on points are type), high efficiency points that lead reduced energy consumption of plant the vehicle. However, this type of transmission transmission maythe stillenergy havetosaving relatively high Classically, actuation losses losses (depending on the the actuation which hinders hinders benefits. the (e.g., actuation actuation this type of transmission may still have relatively high actuation losses (depending on the actuation type), which the energy saving benefits. Classically, the plant (e.g., this typevariator) of transmission may still have saving relatively high actuation losses (depending on the actuation type), which hinders the energy saving benefits. Classically, the plant (e.g., actuation system, of the CVT was separately designed from the control design. In this paper, an actuation type), which hinders the energy benefits. Classically, the plant (e.g., actuation system, variator) of the CVT was separately designed from the control design. In this paper, an actuation type), which hinders theand energy saving benefits. Classically, the plant (e.g., actuation system, variator) of the CVT was separately designed from the control design. In this paper, an integrated optimal CVT variator actuation control design is presented. The aim of the new system, variator) of the CVT was separately designed from the control design. In this paper, an integrated optimalofCVT variator and actuation control design is presented. The of the new system, thethe CVT wasmass separately thetracking design. Inaim this an integrated CVT variator and actuation control design is presented. The aim of the new design is isvariator) tooptimal minimize CVT (pulleydesigned sheaves, belt), error and control effort. integrated optimal CVT variator and actuation controlfrom design iscontrol presented. The aim ofpaper, the new design to minimize the CVT mass (pulley sheaves, belt), tracking error and control effort. integrated optimal CVT variator and actuation control design is presented. The aim of the new design is to to this minimize the CVT optimization mass (pulley (pulleyframework sheaves, belt), belt), tracking error error and control control effort. To achieve goal, a nested is implemented to obtain an optimal design is minimize the CVT mass sheaves, tracking and effort. To achieve goal, a nested optimization framework is implemented to obtain an design is to this minimize CVT (pulley sheaves, belt), tracking andoptimized control effort. To achieve this goal,design athe nested optimization framework isresults implemented to the obtain an optimal optimal transmission system over optimization amass selected drive cycle. The The show error that CVT To achieve this goal, a nested framework is implemented to obtain an optimal transmission system design over a selected drive cycle. results show that the optimized CVT To achieve this goal, a nested optimization framework is implemented to obtain an optimal transmission system design over a selected drive cycle. The results show that the optimized CVT design yields non-compromising tracking performance, however, with much smaller variator mass transmission system design over a selected drive cycle. The results show that the optimized CVT design yields non-compromising however, with much variator mass transmission systemeffort design(-62%). over tracking a selectedperformance, drive cycle. The results show thatsmaller the optimized design non-compromising tracking performance, however, with much smaller variator mass (-46%)yields and control control design yields non-compromising tracking performance, however, with much smaller variator CVT mass (-46%) and effort (-62%). design non-compromising tracking performance, however, with much smaller variator mass (-46%)yields and control control effort (-62%). (-62%). (-46%) and effort © 2019, and IFACcontrol (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. (-46%) effort (-62%). Keywords: Co-design, Co-design, optimization, optimization, continuous continuous variable variable transmission, transmission, vehicle vehicle powertrains powertrains Keywords: Keywords: Co-design, Co-design, optimization, optimization, continuous continuous variable variable transmission, transmission, vehicle vehicle powertrains powertrains Keywords: Keywords: Co-design, optimization, continuous variable transmission, vehicle powertrains 1. INTRODUCTION INTRODUCTION A vehicle powertrain is aa complex dynamical system, 1. A vehicle powertrain is complex dynamical system, 1. INTRODUCTION INTRODUCTION A vehicle powertrain is aacomponents complex dynamical system, which consists of multiple at the subsystem 1. A vehicle powertrain is complex dynamical system, which consists of multiple components at the subsystem 1. INTRODUCTION A vehicle powertrain is a complex dynamical system, which consists of multiple components at the subsystem level, e.g., engine, alternator, and a transmission. On the which consists of multiple components at the subsystem With systems becoming more complex, achieving a true level, e.g., engine, alternator, and aa transmission. On the With systems becoming more complex, achieving aa true which consists of multiple components at the subsystem level, e.g., engine, alternator, and transmission. Onhigh the With systems becoming more complex, achieving true present day, it is desirable to design a vehicle with level, e.g., engine, alternator, and a transmission. On the optimal system design can be a challenge due to the With systems becoming morebecomplex, achieving a true present day, it is desirable to design a vehicle with high optimal system design can a challenge due to the level, e.g., engine, alternator, and a transmission. On the With systems becoming more complex, achieving a true present efficiency. day, it it is is A desirable to designthe a vehicle vehicle with high optimal system design designcoupling can be be (e.g. a challenge challenge duevariables to the the present energy way to improve energy efficiency day, desirable to design a with high existing mathematical via shared optimal system can a due to energy efficiency. A way to improve the energy efficiency existing mathematical coupling (e.g. via shared variables present day, it is desirable to design a vehicle with high optimal system design can be a challenge due to the energy efficiency. A way to improve the energy efficiency existing mathematical coupling (e.g. via shared variables is by utilizing aa continuously variable transmission (CVT) efficiency. A way to improve the energy efficiency in constraints constraints and objectives) objectives) between the corresponding existing mathematical couplingbetween (e.g. viathe shared variables energy is by utilizing variable transmission (CVT) in and corresponding energy efficiency. AAway to improve the energy efficiency existing mathematical coupling (e.g. viathe shared variables is by utilizing aa continuously continuously variable transmission (CVT) in constraints andbetween objectives) between the corresponding in the drivetrain. CVT allows smooth shifting peris by utilizing continuously variable transmission (CVT) subsystems (e.g., the controller and the plant). in constraints and objectives) between corresponding in the drivetrain. A CVT allows smooth shifting persubsystems (e.g., between the controller and the plant). is by utilizing a continuously variable transmission (CVT) in constraints andbetween objectives) the corresponding in the the drivetrain. drivetrain. A aCVT CVT allows smooththe shifting persubsystems (e.g., between the between controller and the plant). formance. Moreover, CVT can operate engine (or in A allows smooth shifting perCombined optimal plant and control design has been subsystems (e.g., the controller and the plant). formance. Moreover, a CVT can operate the engine (or Combined optimal plant and control design has been in the drivetrain. A CVT allows smooth shifting persubsystems (e.g., between the controller and the plant). formance. Moreover, a CVT CVT can operation operate the the engine (or Combined optimal plant and control control design has been been electric machine) at the optimal points, which formance. Moreover, a can operate engine (or shown to result in an improved system performance [1], Combined optimal plant and design has electric machine) at the optimal operation points, which shown to result in an improved system performance [1], formance. Moreover, a CVT can operation operate the engine (or Combined optimal plant and control design has might been machine) at the optimal points, which shown to result result in an an improved system performance [1], electric results in a more efficient energy consumption. However, electric machine) at the optimal operation points, which [2]. However, a combined system-level optimization shown to in improved system performance [1], results in aa more efficient energy consumption. However, [2]. However, a combined system-level optimization might electric machine) at the optimal operation points, which shown to result in an improved system performance [1], results in more efficient energy consumption. However, [2]. However, a combined system-level optimization might the efficiency of this type of transmission strongly depends in a more efficient energy consumption. However, result in a nonconvex nonconvex optimization problem, even when when the results [2]. However, a combined system-level optimization might the efficiency of type of transmission strongly depends result in optimization problem, even the in a more efficient consumption. However, [2]. However, a and combined system-level optimization might the its efficiency of this this type which ofenergy transmission strongly depends result in aa aplant nonconvex optimization problem, even when when the on actuation design, hinders the energy saving the efficiency of this type of transmission strongly depends separate control optimization subproblems are results result in nonconvex optimization problem, even the on its actuation design, which hinders the energy saving separate plant and control optimization subproblems are the efficiency of this type of transmission strongly depends result in a nonconvex optimization problem, even when the on its actuation design, which hinders the energy saving separate plant and control control optimization subproblems are capability that it potentially offers. on its actuation design, which hinders the energy saving convex [3]. Therefore, a suitable design framework must be separate plant and optimization subproblems are capability that it potentially offers. convex [3]. Therefore, a suitable design framework must be on its actuation design, which hinders the energy saving separate plant and control optimization subproblems are capability that it potentially offers. convex [3]. Therefore, a suitable design framework must be capability that it potentially offers. developed in order to overcome this challenge. The method convex [3]. Therefore, a suitable design framework must be developed in order to overcome this challenge. The method For automotive applications, aa higher power density capability that it potentially offers. convex [3]. Therefore, a suitable design framework must be developed in order to overcome this challenge. The method For automotive applications, higher power density for combined combined plantto and control this optimization isThe referred as For automotive applications, a higher power density developed in order overcome challenge.is method for plant and control optimization referred as (kW/kg) and reduced size of transmissions are automotive applications, a higher (packaging) power density developed in order overcome challenge.is method for combined plantto and control this optimization isThe referred as For (kW/kg) and reduced size of transmissions (packaging) are co-design. for combined plant and control optimization referred as For automotive applications, a higher are power density (kW/kg) and reduced size of transmissions (packaging) are co-design. desired. Smaller transmission dimensions also benefi(kW/kg) and reduced size of transmissions (packaging) are for combined plant and control optimization is referred as co-design. desired. Smaller transmission dimensions are also benefico-design. (kW/kg) and reduced size of transmissions (packaging) are desired. Smaller transmission dimensions are also beneficial in terms of production cost. Furthermore, as investidesired. Smaller transmission dimensions are also benefiThe combined plant and control design has been a subject co-design. cial in terms of production cost. Furthermore, as investiThe combined plant and control design has been subject desired. Smaller transmission dimensions aredependent also beneficial in ininterms terms of production production cost. Furthermore, as investiinvestiThe combinedfor plant and control design Generally, has been been aa a the subject gated [4], the power capacity of a CVT is on cial of cost. Furthermore, as of discussion the past few years. coThe combined plant and control design has subject gated [4], the power capacity aa CVT is dependent on of discussion for the past few years. Generally, the coinin terms of production cost.of Furthermore, as investiThe combined plant and control design has been a the subject gated in [4], the power capacity of CVT is dependent on of discussion for the past few years. years. Generally, the co- cial the ratio coverage of the transmission, which is determined gated in [4], the power capacity of a CVT is dependent on design formulation can be expressed as the weighted sum of discussion for the past few Generally, cothe ratio coverage of the transmission, which is determined design formulation can be expressed as the weighted sum gated in [4], the power capacity of a CVT is dependent on of discussion for the past few years. Generally, the cothe ratio coverage of the transmission, which is determined design formulation can be expressed as the weighted sum by the physical parameters of the CVT, such as the pulley the ratio coverage of the transmission, which is determined of the plant and control objectives [3]: design formulation can be expressed as the weighted sum by the physical parameters of the CVT, such as the pulley of the plant and control objectives [3]: the ratio coverage of the transmission, which is determined design formulation can be expressed as the weighted sum by the physical parameters of the CVT, such as the pulley of the plant and control objectives [3]: center distance and wedge angle. Other than increasing the by the physical parameters of the CVT, such as the pulley of the plant and objectives wp J wc[3]: J (1) mincontrol center wedge angle. Other than increasing the p (x p) c (xp ,, x c) , by the distance physical and parameters of the CVT, such as the pulley )) + + J x (1) min of the plant and p (x center distance and wedgewedge angle. Other than increasing the xp ⊆R, xccontrol ⊆R w power capacity, reduced angle gives added benefits, distance and wedge angle. Other than increasing the wppp J Jobjectives (xp +w wccc[3]: Jccc (x (xp xccc ))) ,,, (1) center min p (x p) p ,, x xp ⊆R, xc ⊆R w min J + w J (x (1) power capacity, reduced wedge angle gives added benefits, p p p distance and wedge angle.angle Other thanadded increasing the x ⊆R, x power capacity, reduced wedge gives benefits, xp xcc ⊆R ⊆R wp Jp (xp ) + wc Jc (xp , xc ) , min (1) center such as reduced size and increased efficiency. p ⊆R, power capacity, reduced wedge angle gives added benefits, subject to: such as reduced size and increased efficiency. subject to: xp ⊆R, xc ⊆R power capacity, reduced wedge angle gives added benefits, such as reduced size and increased efficiency. subject to: to: such as reduced size and increased efficiency. subject ≤ 0 ,, h = 0 ,, g In this work, an design method of a CVT p (x p) p (x p) such as reduced sizeintegrated and increased efficiency. ≤ 0 h = 0 g subject to: In this work, an integrated design method of a CVT p (x p) p (x p) (x ) ≤ 0 , h (x ) = 0 , g In this work, an integrated design method of a CVT p p p p , (x hpc (x ) = 0, gxpp(x variator (pulleys and v-belt) and its control system is In this work, an integrated design method of a CVT p ) 0≤, 0 p g (x , ) ≤ h , x c c c p variator (pulleys and v-belt) and its control system is g (x , x ) ≤ 0 , h (x , x , (x ) ≤ 0 , h (x ) = 0 g c c p c c p In this work, an integrated design method of a CVT p )≤ p 0 , h (xp , xp ) = 0 , variator (pulleys and v-belt) and its control system is g (x , x proposed. The CVT design will be optimized dynamically c (xdesign c , xp p) ≤ c (xc c , xg p ) and variator (pulleys and v-belt) and its control system is g 0 , h = 0 , c c c p are the parameters, h are the inwhere x proposed. The CVT design will be optimized dynamically i are g i i variator (pulleys and v-belt) and its control system is the design parameters, g and h are the inwhere x (x , x ) ≤ 0 , h (x , x ) = 0 , i i i proposed. The CVT design will be optimized dynamically c c p c c p over a complete drive cycle profile. In this work, we limit proposed. The CVT design will be optimized dynamically are equality the design design parameters, giiJiand and hthe are the inin- over a complete drive cycle profile. In this work, we limit where xiiand ii are equality constraints, and are objective are the parameters, g h the where x proposed. The CVT design will be optimized dynamically equality and equality constraints, and J are the objective i are h over aa complete complete drive cycle cycle profile. In pulley this work, work, we limit limit are the design parameters, the in- over where xiand ourselves to optimizing the variator sheaves and drive profile. In this we i are equality constraints, and objective functions, w are the weighting parameters, for the plant equality and equality constraints, andgiJ Jiiand are the the objective ourselves to optimizing the variator sheaves and i equality over a complete drive cycle profile. In pulley this work, we limit functions, w are the weighting parameters, for the plant i equality ourselves to optimizing the variator pulley sheaves and equality and constraints, and J are the objective ourselves to optimizing the variator pulley sheaves and i functions, w are the weighting parameters, for the plant i are thei ∈ and control system, {p, c} with w ∈ R, respectively. functions, w weighting parameters, for the plant i i ourselves to optimizing the variator pulley sheaves and and control system, i ∈ {p, c} with w ∈ R, respectively. i functions, w are the weighting parameters, for the plant i and control control system, system, ∈ {p, {p, c} c} with with w wii ∈ ∈ R, R, respectively. respectively. and ii ∈ and control system, i ∈ {p, c} with w ∈ R, respectively. i 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2019 IFAC 393 Copyright © under 2019 IFAC 393 Control. Peer review responsibility of International Federation of Automatic Copyright © 393 Copyright © 2019 2019 IFAC IFAC 393 10.1016/j.ifacol.2019.09.063 Copyright © 2019 IFAC 393
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belt. The optimization of the actuation system (motors) and mechanism is seen as future work. This paper is arranged as follows. Section 2 discusses the plant design modeling. The co-design formulation for the CVT is explained in Section 3. The results of the proposed co-design for CVT on a vehicle powertrain over a drive cycle is elaborated in Section 4. Lastly, the conclusion and future work are presented in Section 5.
with
R p − Rs . (6) ϕ = sin a where a is the center distance between the primary and secondary pulley, ϕp and ϕs are the belt wrap angles, respectively. −1
dp
h β
A
2. PLANT MODEL: ELECTROMECHANICALLY ACTUATED CONTINUOUS VARIABLE TRANSMISSION
ds A δa
rg > 1
R1
' R2
In this work, the CVT that will be studied is the electromechanically actuated continuously variable transmission (EMPACT). This type of CVT, as shown in Fig. 1, has electric servomotors as the actuation system, which prompt the pulley sheaves on the primary and secondary side. In this section, the behaviour and mathematical formulations of EMPACT will be discussed.
dp
Rp
a
!p
!s
Rs
A-A view Rin
Fig. 2. CVT variator diagram Due to friction, there exists a slip v in the variator, which is the difference between the tangential speed on the primary ω R −ω R and secondary side, v = p ωpp Rps s . Here, the slip is considered to be small, such that ωp Rp ≈ ωs Rs , and therefore, rg ≈ rs .
Actuation
2.2 Variator dynamics and clamping forces Several models have been derived to describe the transient behaviour of the CVT variator. Here, the Carbone, Mangialardi, and Mantriota (CMM) model will be utilized to describe the behaviour of the variator ratio dynamics [6]. This mathematical model describes the behaviour of the ratio change as a function of the clamping force and also the pulley deformation. The rate of change of the speed ratio is then given by, 1 + cos2 β Fp,ss Fp r˙g = 2 ωp ∆ c(rg ) ln −ln , (7) sin(2β) Fs Fs,ss
ds
variator
Fig. 1. Schematic diagram of EMPACT CVT [5] 2.1 CVT geometric model The schematic diagram of the CVT variator is depicted in Fig. 2. The CVT geometric ratio is defined as the ratio between the running radii on the primary and the secondary side, Rp rg = . (2) Rs Additionally, the speed ratio is defined as, ωs rs = , (3) ωp where ωp and ωs are the primary and secondary rotational speeds, respectively. The movable pulley positions dp and ds determine the corresponding running radii of the CVT at the primary and secondary side, dp −ds + Ro ; Rs = + Ro , Rp = (4) 2 tan β 2 tan β where β is the pulley wedge angle and Ro is the running radius at rg = 1. The position of the movable pulley sheave results in different wrap angles ϕi , which are the span of the belt on the pulley sheaves, and are formulated as: (5) ϕp = π + 2ϕ ; ϕs = π − 2ϕ , 394
where ωp is the rotational speed of the driving (primary) pulley, ∆ is the pulley deformation, β is the pulley wedge angle. Fp and Fs are the primary and secondary clamping forces, and the subscript ss indicates the steady-state value when r˙g = 0. The term c(rg ) is a function of rg that relates the dimensionless speed ratio with the difference of the logarithmic steady state and applied clamping forces. The pulley deformation ∆ is affected by the clamping forces, and can be expressed as [7]: ∆ = (1 + 0.02(Fs − 20)) · 10−3 ,
(8)
where Fs is the secondary clamping force in kN. The clamping forces Fp and Fs are given by, Si (θ) − Ci (θ) − Fb − Fc Fi = dθ , (9) 2 sin β ϕi
where i ∈ {p, s}, Si and Ci are the tension and compression forces over the pulleys, Fb and Fc are the centrifugal forces of the band and belt blocks, Fb = ρb vb2 and Fc = ρc vb2 , respectively [8]. The belt velocity vb is assumed to be vb = ωp Rp = ωs Rs . The calculation of the clamping forces are also dependent on ρb and ρc , which are the mass per unit length of the bands and the blocks, respectively.
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3. CO-DESIGN FORMULATION A co-design strategy is proposed to obtain an improved EMPACT plant and control design for a specific drive cycle. The key performance indicators (KPIs) are packaging (dimensions), efficiency, performance, and cost. Here, we focus on minimizing the variator mass, which is expressed as a function of the variator dimensions (plant objective) and the tracking performance and actuation effort (control objective) of the CVT system. The KPIs can be interpreted as the system wide objectives. Further, we use optimal control techniques in order to not compromise the tracking performance of the control system. The CVT actuation controller should provide the corresponding signal that minimizes the tracking error and control effort for the actuation system. However, due to the presence of the coupling between the geometrical properties and the reference for control, the response of the CVT is also dependent on the choice of the plant design parameters, namely β and a, which becomes a challenge in finding the combined plant and control optimal design. 3.1 Plant Design Problem To derive the plant design objective Jp , the CVT pulley sheaves are considered to be identical truncated cones and the metal belt is assumed as a v-belt. The mathematical formulation of the variator mass is defined as, Mv = ρpu Vpu + ρb Vb . (10) Vpu and Vb are the pulleys and belt volume of the variator, given by 1 2 , Vpu = 4π(R2 − R1 ) tan β (R12 + R1 R2 + R22 ) − Rin 3 (11) (b1 + b2 )b3 Vbe = Lb , (12) 2 where R1 and R2 are the pulley top and bottom radii, Rin is the radius of the shaft, Lb is the length of the belt, and b1 , b2 , and b3 are the belt parameters, respectively, as depicted in Figs. 2 and 3. b2 b2 Lb
b3
b3
β
b1 b1
Fig. 3. CVT belt geometries As seen in Fig. 2, the sheave radius R2 determines the value of the center distance a. Hence, the plant design objective is mathematically expressed as, min Jp = Mv (xp ) , (13) xp ⊆Xp ⊆R
subject to:
hmin
β − βmax ≤ 0 −β + βmin ≤ 0 ≤ tan β(R2 − R1 ) ≤ hmax Ri,min ≤ Ri ≤ Ri,max 2R2 − a + δa ≤ 0
395
and Ri,max are the minimum and maximum pulley radii for i = {1, 2}; a is the pulley center distance; δa is the minimum pulley distance; and xp are the plant design parameters, xp = {R1 , R2 , β} , where the set xp is a subset of set Xp , given by, Xp = {xp ∈ R | gp (xp ) ≤ 0 , hp (xp ) = 0} . 3.2 Control Design Problem This work focuses only on the speed ratio control of CVT. The reference for the primary servomotor to realize the desired ratio values is influenced by the plant design parameters (i.e., β, R1 , R2 ), which indicates the coupling that exists between the plant parameters and the control. Here, firstly, the primary servomotor reference design model for the primary rotational position is discussed. This serves as input to the servomotor control problem accordingly. The ratio rg is not directly measurable, and must, therefore, be estimated from the measureable variables. For the EMPACT, the measured variables are the servomotor rotations θmp and θms and the primary ωp and secondary ωs shaft speeds. The electric servomotors actuate the movable pulley sheaves to realize the required clamping forces necessary to perform ratio shifting. Hence, the ratio control of the EMPACT CVT can be approached as a servomotor control problem [5]. 3.2.1 Servomotor reference design Mathematically, the pulley sheave positions can be expressed as a function of the servomotor positions, θmp sz (19a) rw θms sz (19b) d s = dp + r r rc where θmp and θms are the motor angular positions of the primary and secondary actuations; rr , rc , and rw are the reduction gear constants; s is the screw pitch, and z is the ratio between the sun and annulus of the planetary gear set connecting the servomotor and the screw. dp =
The desired reference ratio trajectory rg,r can be translated as the reference for the primary servomotor θmp,r . As rg,r is known a priori, r˙g,r can be obtained using the trajectory. Then, using the CMM model that is described F in (7), the required clamping force ratio Fps r that yields the desired ratios can be calculated by the relation: Fp Fp,ss sin(2β) · = exp r ˙ . (20) g Fs r 2 ωp ∆ (1 + cos2 β) c(rg ) Fs,ss F
(14) (15) (16) (17) (18)
where hmin and hmax and βmin and βmax are the minimum and maximum pulley heights and wedge angles; Ri,min 395
as a function The steady state clamping force ratios Fp,ss s,ss of gear ratio values can be computed a priori using the geometrical and physical variator model as described by [6]. To reduce the calculation time for the design optimization, some models are reduced as parametric representations of the relevant design parameters. The steady-state clamping force ratio can be approximated as, Fp,ss a1 βrg + a2 β + a3 , for rg ≤ 1 (21) = for rg > 1 b1 + b2 β 2 ln(rg ) , Fs,ss
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with fitted constants a1 = 1.2204, a2 = −1.65, and a3 = 1.0738, and b1 = 0.9968 and b2 = 6.2373. F
Similarly, the resulting clamping force ratio Fps can be approximated as a function of the primary wrap angle ϕp and β, Fp for rg ≤ 1 , 0.6793ϕp − 2.1303β − 0.47 , (22) = 0.8409ϕp − 3.4886β − 0.456 , for rg > 1 Fs Using (4), (5), (6), and (22), the corresponding dp,r that F yields the desired clamping force ratio Fps r can be obtained. Furthermore, via the relation in (19a), the desired reference trajectory for the primary servomotor actuation θmp,r can be determined. 3.2.2 Servomotor control design The primary servomotor type is selected to be a DC motor. The dynamics is given by: θmp 0 1 0 0 d θmp ωmp = 0 −b/I Kt /I ωmp + 0 u , (23) dt 0 −Ke /L −R/L i i 1/L A
ξ
θmp y = [1 0 0] ωmp i C
B
(24)
where θmp and ωmp are the primary servomotor angular position and rotational speed, i is the motor current, Kt and Ke are the torque and voltage constants, b is the friction constant, L is the inductance, and R is the armature resistance. I is the rotor inertia, u is the control input which is in voltage, and finally the state variables are given by ξ = [θmp , ωmp , i] . In order to design a controller such that the error between the desired and actual trajectory is minimized, the state space model of the system is augmented such that the T error ε = 0 (θmp,r (t) − θmp (t)) dt is included as an additional state. The feedback gains are found by utilizing an LQR formulation. A feedforward term kf f is included to minimize the steady state error. The augmented state is expressed as: d ξ B A0 ξ 0 + = u+ θ , (25) 0 C 0 ε −1 mp,r dt ε ξa
Aa
Ba
where the control input is given by (26) u = −kfb · ξ + kf f · ε , where kfb and kf f are the control gains. The proposed control structure is depicted in Fig. 4.
The primary actuation controller calculates the required voltage u to the servomotor such that the desired ratio is tracked, while still minimizing the energy consumption. The cost function for the controller is given by tf ξa Qξa + u Ru dt , (27) Jc = 0
where the minimum of the cost function can be found by solving the algebraic Riccati equation. The weight matrices Q and R are commonly chosen by the user to have
396
acceptable balance between minimizing error and control effort that yields satisfactory performance. However, as in the co-design framework the plant design parameters (i.e., β, R1 , R2 ) will be varied, the corresponding reference trajectory for the servomotor reference θmp,r are also changed. Hence, the weight matrices Q and R are selected such that the error between the desired and actual ratio trajectory is minimized for a plant design. The optimal solution of (27) is found by solving the Riccati equation, −1 Ba P + Q = 0 , (28) A a P + PAa − PBa R as long as the conditions P > 0 , Q ≥ 0 , R > 0 are satisfied. P is a positive definite matrix that satisfies the algebraic Riccati equation. The weight matrices Q is a m × m matrix and R is a n × n matrix, where m is the rank of Aa and n is the rank of the input u. The gains kfb and kf f are determined by solving the Riccati equation [kfb kf f ] = R−1 B a P.
In the EMPACT CVT, the primary side is often used to perform ratio tracking, while the secondary side is used to regulate the slip in the variator. The minimum required secondary clamping force to ensure torque transfer is given by α Tsi cos β Fs,r = , (29) 2 µ Rs where α is a safety factor, commonly chosen to be 1.3, Tsi is the belt internal torque exerted on the secondary pulley, and µ is the effective traction coefficient. The design of secondary actuation controller is outside of the research scope. 3.3 Co-design problem formulation The combined plant and control design problem formulation of the CVT is written as: tf wp Mv (xp ) + wc ξa Qξa + u Ru dt , min xp ⊆Xp ,xc ⊆Xc
0
(30)
subject to: (15)-(18) and P > 0 , Q ≥ 0 , R > 0.
The optimized design parameters are defined as xp = {R1 , R2 , β} and xc = {Q, R}, and xp ⊆Xp , xc ⊆Xc . The selection of the weights wp and wc are important in finding the Pareto front of the optimization problem. In this work, the weights wi for i = p, c are selected to be 1. Furthermore, in this specific problem, the parameters β and a serve as couplings between the plant design (minimizing variator mass, a more compact CVT) and the control design (minimizing speed ratio tracking error) problem. 4. NESTED OPTIMIZATION APPROACH A nested optimization approach will be implemented to dynamically obtain an optimal CVT design for the selected drive cycle. This is because the plant design parameters (i.e., β, R1 , R2 ) have influence on the control design (minimizing ratio tracking error). The nested approach has an outer loop that optimizes the combined design objective (30) by varying the plant design parameters, and an inner loop that optimizes the control performance. For this problem, the ratio control problem is solved using an
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397
β
rg;r
ss clamp. (21)
Fp Fs ss
β !p
primary actuation
β r_g;r
CMM−1 (7)
Fp Fs r
β
desired 'p 'p;r
desired dp dp;r (4), (6)
(22)
β
a rw sz
R
θmp;r
"
u(t)
kf f
ξ_ = Aξ + Bu y = Cξ
(19a)
(23)-(24)
-
-
kfb
θmp
sz rw
dp
(19a)
β
a
dp to 'p (4), (6)
'p
β Fp
clamp. ratio (22)
x
CMM (7)
r_g
R
rg
ξ Fs
µ Tsi
minimal clamp. (29)
β
Fs;r
Rs
Fig. 4. Control scheme for EMPACT ratio control LQR formulation with augmented state. Additionally, the weight matrices {Q, R} are obtained by optimization. The proposed framework is depicted in Fig. 5. outer loop: combined problem min wp Jp (xp ) + wc Jc (xp ; x∗c ) xp
s.t. : gp;1;:::;6 x∗c (xp )
xp inner loop: control problem
xc ∗ = argmin Jc = argmin xc xc s.t. :
R tf to
ξa> Qξa + u> Ru dt
Q ≥ 0;R > 0
x∗p ; x∗c
Fig. 5. Nested optimization framework for the EMPACT CVT 4.1 Reference speed trajectory
Pwh Twh ωwh = . (33) ηt ηt The engine optimal line is found to be the combination of engine speed and torque ωe∗ and Te∗ that yield the minimum fuel consumption. Using the optimal operation points, the desired ratio reference that yields optimal fuel consumption can be calculated. The desired speed ratio rg,r is defined as, ωwh iF DR rg,r = min max rg,min , , rg,max , (34) ωe∗ where ωe∗ is the engine speed at the optimal operation line, and iF DR is the final drive ratio. Pe,r =
The obtained ratio reference is used to perform co-design on the CVT variator and ratio controller dynamically for the complete NEDC profile. The vehicle parameters used throughout the simulation study are summarized in Table 1. Table 1. Simulation Parameters
The performance of the optimized CVT throughout a given driving profile will be analyzed. The reference CVT ratio rg,r is determined such that the engine is operated near, if not at, the optimal operation line of the engine. The NEDC drive cycle is selected to investigate the benefits of utilizing CVT in a traditional vehicle powertrain with a 2.0 L gasoline engine. The NEDC drive cycle is used to determine the required CVT ratio values such that the vehicle is operated at the engine optimal operation line.The wheel speed ωwh is defined as vwh ωwh = , (31) Rwh where vwh is the vehicle speed from the drive cycle and Rwh is the wheel radius. The torque demand at the wheel is calculated using (32) Twh = (Fr + Fa + Fi ) Rwh , where Fr is the rolling resistance, Fa is the aerodynamic drag, and Fi is the force due to inertia of the vehicle. Furthermore, the ratio is selected such that the required engine power to drive the vehicle is generated with minimum fuel consumption. To determine the ratio, several assumptions are considered. At ωwh = 0, the engine is at idle, ωe = ωid . Throughout the CVT operation over the NEDC drive cycle, the efficiency of the transmission is treated to be constant ηt (t) = ηt . The power demand at the wheels is used to determine the power required from the engine, 397
Vehicle mass Roll. resistance Drag. resistance Vehicle frontal area Air density Rotational mass Wheel radius Final drive ratio Engine idle speed CVT efficiency
Symbol ms cr cd Af ρa mro Rwh iF DR ωid ηt
Value 1180 0.0174 0.29 2.38 1.225 2 0.35 3 78.53 0.9
Units kg − − m2 kg/m3 % m − rad/s −
4.2 Design results The CVT variator design optimization and the speed ratio controller over a complete drive cycle are demonstrated. The simulation parameters used in this study is summarized in Table 2. This subsection discusses the results obtained from the study of optimizing the CVT design compared to a baseline model 1 . It can be seen from the results that the optimized CVT is able to follow the desired ratio trajectory despite having a lower variator dimensions throughout the drive cycle, as summarized in Table 3. Using the proposed nested optimization framework, the mass of the pulley sheaves 1 Variator geometries from Jatco CK2 with EMPACT actuation system for reasons of comparison
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Table 2. Simulation Parameters Motor inertia Motor friction constant Motor inductance Motor torque constant Motor voltage constant Motor armature resistance Range of β Range of R1 Range of R2 Inner radius Range of ratio Weight Weights for i ∈ {c, 1, 2} Belt width Belt thickness Screw pitch Reduction gear Initial ratio Planetary gear set ratio Mass density
Symbol I b L Kt Ke Ra [βmin , βmax ] [R1,min , R1,max ] [R2,min , R2,max ] Rin [rg,min , rg,max ] wp wi b1 b3 s rw rg,o z ρb , ρpu
Value 31.4 · 10−6 2.1 · 10−3 2.75 52.96 · 10−3 5.5 0.47 [6,15] [15,30] [60,90] 7.5 [0.5,2.5] 10 1 28 24 4/(2π · 1000) 16 0.5 2 7850
implemented. The proposed approach calculates the minimized CVT variator’s mass, as well as the corresponding actuation control parameters that yields the best possible ratio tracking performance.
Units kg · m2 N · m/krpm mH Nm/A V/krpm Ω ◦
mm mm mm − − − mm mm rad/m − − − kg/m3
can potentially be reduced from 6.86 to 3.69 kg (-46%). The optimized parameter β for the new variator is found to be 6◦ , R1 to be 22.8 mm, R2 to be 66.7 mm, and the pulley center distance a to be 136.4 mm. Despite the reduced dimensions, the optimized CVT variator yields comparable performance to the baseline. The resulting ratio tracking performance is depicted in Fig. 6. Optimized variator performance for NEDC cycle
120
NEDC speed profile Reference Actual
3
80
2.5
60
2
40
1.5
20
1
0 200
400
600
800
1000
In this work, the CVT efficiency is not modeled. In the future, the actual CVT efficiency as a function of the relevant physical design parameters should be included in the co-design framework. Currently, only ratio controller is considered in this work as part of the co-design framework for CVT. The framework could also be extended to include slip controller on the secondary side of the variator. Lastly, in the future, the potential of using CVT in a full electric vehicle powertrain will be investigated. REFERENCES
Ratio rg [-]
Vehicle speed v wh [km/h]
100
Based on the results, it is seen that the variator design could be reduced in terms of size and mass without compromising the ratio tracking performance. At the same time, the corresponding optimal controller for the new optimized design is found using the proposed framework. Using the co-design framework explained in this paper, the mass of the pulley sheaves can potentially be reduced up to 46.2%, from 6.86 to 3.69 kg. It was also obtained that the wedge angle β for the CVT variator can be reduced from 11◦ to 6◦ , and center distance a from 169 to 136.4 mm. Furthermore, with the new optimized parameters, the ratio coverage of the CVT is reduced from 5.33 to 4.23. It is seen that the optimized CVT design yields comparable performance to the baseline CVT, despite the smaller size.
[1]
[2]
0.5 1200
Time [s]
Fig. 6. Results for the optimized model for NEDC drive cycle Table 3. Nested co-design results for NEDC drive cycle J
xp
xc
K, rcov , a
Plant objective Control objective Variator volume Wedge angle Top outer radius Bottom outer radius LQR weight LQR weight LQR weight LQR weight LQR weight Center distance Ratio coverage Feedback gain Feedback gain Feedback gain Feedforward gain
Symbol Jp Jc Vv β R1 R2 Q(1) Q(2) Q(3) Q(4) R a rcov kf b (1) kf b (2) kf b (3) kf f
Baseline 6.86 7.6 8.74 11.0 26.3 82.5 2.065 0.001 0.001 104 0.01 168 5.33 2.4439 0.035 3.54 62.38
Optimized 3.69 2.89 4.70 6 22.8 66.7 0.768 0.001 0.001 104 0.01 136.4 4.23 2.355 0.035 3.497 63.88
[3]
[4]
Unit kg 1010 −4 10 m3 ◦
mm mm − − − − − mm − − − 10−4 −
[5]
[6]
[7]
5. CONCLUSIONS AND FUTURE WORK In this study, a nested optimization framework for the EMPACT CVT to obtain a combined plant and control design for the NEDC drive cycle has been succesfully 398
[8]
H. K. Fathy, P. Y. Papalambros, A. G. Ulsoy, and D. Hrovat, Nested plant/controller optimization with application to combined passive/active automotive suspensions, in Proc. American Control Conference, Denver, USA, 2003. J. A. Reyer and P. Y. Papalambros, Combined optimal design and control with application to an electric DC motor, in Journal of Mechanical Design, vol. 124, pp. 183–191, 2002. H. K. Fathy, J. Reyer, P. Papalambros, G. Ulsoy, On the coupling between the plant and controller optimization problems, in Proc. American Control Conference, Virginia, USA, 2001. A. Brandsma, J. van Lith, E. Hendriks, Push belt CVT developments for high power applications, in International Congress on Continuously Variable Power Transmission CVT, Eindhoven, The Netherlands, 1999. T. W. G. L. Klaassen, The EMPACT CVT : dynamics and control of an electromechanically actuated CVT, PhD Thesis, Eindhoven University of Technology, 2007. G. Carbone, L. Mangialardi, G. Mantriota, The Influence of Pulley Deformations on the Shifting Mechanism of Metal Belt CVT, Journal of Mechanical Design, vol. 127, pp. 103–113, 2005. G. Carbone, L. Mangialardi, B. Bonsen, C. Tursi, P. A. Veenhuizen, CVT dynamics: Theory and experiments, Mechanism and Machine Theory, vol. 42, pp. 409–428, 2007. B. G. Vroemen, Component control for the Zero Inertia powertrain, PhD thesis, Eindhoven University of Technology, 2001.