- Email: [email protected]

Contents lists available at ScienceDirect

Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt

Optimal gearshift control for a novel hybrid electric drivetrain Arash M. Gavgani a, Aldo Sorniotti a,⁎, John Doherty a, Carlo Cavallino b a b

University of Surrey, United Kingdom Oerlikon Graziano SpA, Italy

a r t i c l e

i n f o

Article history: Received 24 October 2015 Received in revised form 12 June 2016 Accepted 19 June 2016 Available online xxxx Keywords: Hybrid electric drivetrain Optimal control Clutch energy dissipation Torque gap ﬁlling

a b s t r a c t Torque-ﬁll capability during gearshifts is an important customer requirement in automated transmission systems. This functionality can be achieved through transmission system layouts (e.g., based on dual-clutch technology) characterized by signiﬁcant mechanical complexity, and hence with relatively high cost and mass. This paper describes a parallel hybrid electric drivetrain concept, based on the integration of an electric motor drive into a relatively simple six-speed automated manual transmission. The resulting hybrid electric drivetrain actuates the torque-ﬁll function through control of the electric motor torque during the gearshifts on the engine side of the drivetrain. An optimal controller, based on the off-line computation of the control gain proﬁles, is presented for the clutch re-engagement phase. The novel controller allows computationally efﬁcient consideration of clutch energy dissipation during the clutch reengagement phase of the gearshift. The performance with the optimal controller is contrasted with that of two conventional clutch engagement controllers, along a set of gearshifts simulated with an experimentally validated vehicle model. © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BYNC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction In recent years the automotive industry has experienced an increasing demand for comfort, energy efﬁciency and safety. In the area of mechanical transmission systems, these requirements have been addressed through robotized manual transmissions (MTs), i.e., the so-called automated manual transmissions (AMTs) and dual-clutch transmissions (DCTs). DCTs [1] are gaining significant use especially in the European market, as they combine energy efﬁciency and substantial reduction of the torque gap during gearshifts, at the price of increased mechanical complexity compared to AMTs. With the progressive electriﬁcation of automotive drivetrains, novel layouts are being proposed, including the parallel hybrid electric architecture of this paper. This set-up represents a hybridized AMT conﬁguration, with the electric motor (EM) drive used to ﬁll in the torque gap during the gearshifts on the engine side of the drivetrain. In addition, it provides the energy efﬁciency beneﬁts (e.g., regeneration during braking) typical of hybrid electric powertrains. Similar functionality is also achievable with ﬂywheel-based hybrid drivetrains [2], with the main drawbacks being associated with hardware implementation complexity, safety and cost. One of the tasks of an AMT gearshift control unit is to accomplish clutch engagement by synchronizing clutch and engine speeds. Due to the limited amount of engine braking torque, engine deceleration during upshifts becomes a crucial objective for the transmission control unit (TCU), as the speed reduction needs to be achieved through clutch control. The clutch engagement process must satisfy several conﬂicting criteria, such as smoothness, rapid engagement and minimization of energy

⁎ Corresponding author. E-mail address: [email protected] (A. Sorniotti). http://dx.doi.org/10.1016/j.mechmachtheory.2016.06.016 0094-114X/© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-ncnd/4.0/).

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

Nomenclature a, b, c, d A Â A Ah AMT B B̂ Bh C C1, C2 Chs DCT DMF DP DTD e ê ecl,w eslip EM F G H Ĥ ICE J Jd JDMF,1 JDMF,2 JE JE,s Jeq,s1 Jeq,s2 Jeq,s3 Jeq,sim JEM Jp Js Jv Jv,l Jw Khs LQ m M MPC MT N PE,max PEM,max PID Q Qe qr qx qy

coefﬁcients deﬁning the reference slip speed dynamic matrix of the simpliﬁed model dynamic matrix of the simpliﬁed model augmented with the integral of the tracking error dynamic matrix for the reformulated cost function upper left sub-matrix of the Hamiltonian automated manual transmission input matrix of the simpliﬁed model input matrix of the simpliﬁed model augmented with the integral of the tracking error upper right sub-matrix of the Hamiltonian dynamic matrix of the reference trajectories sub-matrices of C half-shaft damping coefﬁcient dual-clutch transmission dual-mass ﬂywheel dynamic programming driver torque demand vector of tracking errors augmented vector of tracking errors clutch torque tracking error slip speed tracking error electric motor constant matrix of the simpliﬁed model formulation coefﬁcient matrix of the reference vector in the augmented system formulation output matrix output matrix in the augmented system formulation internal combustion engine cost function mass moment of inertia of the differential mass moment of inertia of the primary inertia of the dual-mass ﬂywheel mass moment of inertia of the secondary inertia of the dual-mass ﬂywheel mass moment of inertia of the moving parts of the internal combustion engine mass moment of inertia of the internal combustion engine and dual-mass ﬂywheel mass moment of inertia of the drivetrain in the ﬁrst state mass moment of inertia of the drivetrain in the second state mass moment of inertia of the drivetrain in the third state mass moment of inertia of the drivetrain in the simpliﬁed model mass moment of inertia of the rotating parts of the electric motor mass moment of inertia of the transmission primary shaft mass moment of inertia of the transmission secondary shaft apparent mass moment of inertia of the vehicle apparent mass moment of inertia of the vehicle including wheels in the simpliﬁed model mass moment of inertia of the driven wheels half-shaft stiffness linear quadratic vehicle mass coefﬁcient matrix of the reference vector in the augmented tracking error formulation model predictive control manual transmission penalty matrix of the clutch dissipation energy maximum internal combustion engine power maximum electric motor power proportional integral derivative penalty matrix of the tracking error penalty matrix of the augmented state vector transformation matrix from Y to r transformation matrix from Γ to X transformation matrix from Γ to Y

353

354

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

QP quadratic programming r vector of reference trajectories R control penalty matrix clutch torque reference trajectory rcl,w slip speed reference trajectory rslip wheel radius Rw s Laplace operator S ﬁnal time tracking error penalty multiplication matrix for feedback gain calculation Stf t time aerodynamic drag torque Taero clutch torque Tcl Tcl,transm transmissible clutch torque Tcl,transm,max maximum transmissible clutch torque dual-mass ﬂywheel torque TDMF torque of the friction elements of the dual-mass ﬂywheel TDMF,fr TDMF,hys amplitude of the clutch damper hysteresis TDMF,sp torque through the elastic elements of the dual-mass ﬂywheel internal combustion engine torque TE electric motor torque TEM maximum torque of the internal combustion engine TE,max TEM,en.man electric motor torque demand from the energy management system TEM,max maximum electric motor torque reference electric motor torque TEM,ref reference internal combustion engine torque TE,ref ﬁnal time and time at step n tf, tn half-shaft torque Ths load torque Tl Tl,rolling_resistance rolling resistance torque rolling resistance torque of the driven wheels TRR,dr rolling resistance torque of the undriven wheels TRR,udr synchronizer torque Tsyn traction torque Tt TCU transmission control unit TPBVP two-point-boundary-value-problem u input matrix of the optimal controller e u input matrix of the reformulated cost function initial vehicle speed v0 x state vector of the simpliﬁed model X, X_ augmented state vector and its time derivative slip speed (i.e., the ﬁrst element of x) x1 Y, Y_ state vector of reference trajectories and its time derivative Z multiplicative matrix of F α corner frequency of the transfer function of the electric motor torque dynamics γ clutch torque factor Γ, Γ_ augmented system states and their time derivatives Δt time step Δθhs, Δθ_ hs half-shaft angular deﬂection and deﬂection rate θbacklash backlash angle θE, θ_ E , €θE angular displacement, speed and acceleration of the engine crankshaft θd, θ_ d , €θd angular displacement, speed and acceleration of the differential case θDMF,2, θ_ DMF ,2, angular displacement and speed of the secondary mass of the dual-mass ﬂywheel θp, θ_ p , €θp angular displacement, speed and acceleration of the transmission primary shaft θw, θ_ w ,€θw angular displacement, speed and acceleration of the driven wheels θv apparent angular displacement, speed and acceleration of the vehicle θv , θ_ v , € λ, λ_ co-state in the optimal control formulations and its derivative Ω state transition matrix ﬁnal drive ratio τd gear ratio from the electric motor to the secondary shaft τEM

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

355

τg gearbox ratio τg,ingoing gear ratio of the ingoing gear ωslip,initial initial value of the clutch slip speed

dissipation. In particular, as the transmitted clutch torque can vary before and after the engagement [3], special care must be taken by the controller to avoid undesirable drivetrain oscillations. Many previous studies have investigated this problem and have proposed various solutions. For example, based on a twodegree-of-freedom model of the drivetrain, [3] analytically proves that the vehicle acceleration discontinuity, experienced at the clutch engagement, depends on the slip speed rate at the engagement instant. Hence, [3] presents a controller decoupling the dynamics of engine speed and clutch slip speed. The engagement problem is subdivided into two independent speed tracking tasks, which are accomplished by two proportional integral derivative (PID) controllers. An improved version of the decoupled controller, which results in a smoother driveshaft torque, is proposed in [4]. [5] discusses the limitations of the no-lurch condition in [3] and obtains a more accurate lurch-free engagement condition based on a three-degree-of-freedom model of the drivetrain. In particular, in addition to the condition in [3], the deﬂection rate of the driveshaft at the engagement instant should be as small as possible. Moreover, through quadratic programming (QP) [5] pre-optimizes the system trajectories (i.e., slip speed and clutch torque) with the assumption of constant engine torque. A low-level controller enforces the system to follow the optimized trajectories. [6] exploits Linear Quadratic (LQ) control for clutch engagement. The proposed methodology beneﬁts from penalizing the clutch torque rate within the LQ cost function. This can lead to a smoother vehicle acceleration proﬁle throughout the engagement, however the engine torque is assumed to be constant and non-controllable. [7] suggests a combination of optimal and decoupled control. A ﬁnite time optimal controller generates a reference clutch speed proﬁle for the decoupled controller. A constant value is used for the engine speed reference in order to avoid engine shut-off. [8] formulates the engagement as an optimal control problem, with physical constraints considered directly within the cost function, while the control inputs are determined by using a gradient iteration algorithm. Studies dealing with numerical dynamic programming (DP) [9], analytical DP [10], model predictive control (MPC) [11], hybrid control based on explicit MPC [12–13] and combinations of fuzzy logic and optimal control [14] are also available for the problem of clutch engagement control. In a typical TCU, the torque demanded from the internal combustion engine (ICE) is delivered to the driver after clutch engagement. Hence, if there is a substantial difference between the clutch torque at the engagement instant and the torque demanded by the driver, this would affect comfort. This implies that in addition to the smooth engagement associated with the no-lurch condition [3], the clutch engagement controller should ensure that the clutch torque at the engagement instant is close to the driver demand. With the exception of [10], the impact of this phenomenon has been neglected within the many studies published, and the controllers proposed have mainly focused on satisfying the no-lurch condition in [3]. From another viewpoint, clutch slip during the engagement phase and the resulting heat reduce component life and increase fuel consumption. Thus, heat dissipation within the clutch assembly should be limited. To the best of their knowledge, the authors believe that [8,9] are the only efforts to directly minimize clutch energy dissipation. However, these previous solutions were based on gradient optimization and DP, which prevents the online implementation of these controllers within the hardware of existing TCUs. The main contributions of this paper are: • A computationally efﬁcient and easily tunable optimal controller for low energy dissipation clutch engagement. The control output is computed through a multiple-step algorithm similar to the one presented in [15] for an aerospace system. • The application of the controller to a novel hybrid electric drivetrain, including torque gap ﬁlling during gearshifts, achieved through an electric motor drive adopted as a secondary power source. • The performance evaluation of the proposed technique using an experimentally validated non-linear vehicle model for gearshift simulation. The novel controller is compared with the controller currently running in the vehicle demonstrator being used to showcase the new drivetrain, and with the PID-based control scheme in [16]. The paper is organized as follows. In Section 2 the vehicle model and the novel hybrid electric drivetrain are explained, including the description of the current controller and its experimental validation. Section 3 discusses the proposed optimal clutch controller, which is evaluated in Section 4. Finally, the main conclusions are reported in Section 5. 2. Non-linear model with baseline controller 2.1. Drivetrain conﬁguration and non-linear model layout The novel hybrid electric drivetrain consists of a six-speed AMT with a dry clutch, a primary shaft and a secondary shaft (see Fig. 1, which includes the drivetrain schematic and a picture of a physical prototype). An EM is integrated into the transmission, and can be linked to the secondary shaft through an epicyclical gearset, a sequence of two drops and a hybrid coupling. Two gear ratios are available between the electric motor and the secondary shaft, to provide ﬂexibility in terms of the maximum achievable wheel torque and speed. In addition to the energy efﬁciency-related functions of a typical hybrid electric drivetrain, this conﬁguration allows full or partial compensation of the torque gap during gearshifts, by controlling the EM torque [17].

356 A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

Fig. 1. a) The layout of the novel hybrid electric drivetrain; b) Its physical implementation on a high-performance vehicle demonstrator.

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

357

Fig. 2. Simpliﬁed schematic of the vehicle and drivetrain model for control system assessment.

The simpliﬁed schematic of the corresponding drivetrain model is presented in Fig. 2. A torsional damping device, e.g., a DualMass Flywheel (DMF) between the ICE crankshaft and the clutch input, is used to reduce the torque oscillations. Since this study is focused on the ICE gearshift analysis, it assumes that the EM operates in constant gear (i.e., its ﬁrst gear is used throughout the paper). The differential is modeled only in terms of ﬁnal reduction ratio, without considering its internal components. A lumped backlash model is included at the differential output. An equivalent model is adopted for the left and right half-shafts, as well as the rear driven wheels, which are assumed to be lumped. The vehicle body and undriven wheels are modeled as an apparent rotating inertia, interacting with the driven wheels through the tire torque. The tire model is based on Pacejka magic formula [18], including relaxation dynamics, with the relaxation length being parameterized as a function of slip ratio and vertical load. The next section reports the mathematical formulation of the simulation model. 2.2. Non-linear model formulation Depending on the status of the clutch and synchronizer, three different states occur during a gearshift sequence: 1) engaged clutch and engaged synchronizer; 2) slipping clutch and engaged synchronizer; and 3) disengaged clutch and disengaged synchronizer. In the ﬁrst state the system model is characterized by four degrees of freedom, while in the second and third states the degrees of freedom are ﬁve and six, respectively. 2.2.1. Engaged clutch and engaged synchronizer A ﬁrst order transfer function is used to model the engine torque dynamics (e.g., caused by the air intake system). The engine shaft dynamics are governed by: T E −T DMF ¼ J E þ J DMF;1 €θE

ð1Þ

JE and JDMF,1 are respectively the mass moments of inertia of the engine and primary side of the DMF. €θE represents the angular acceleration of the crankshaft. TE is the actual engine torque, output by the engine transfer function. TDMF is the DMF torque, given by the sum of the contributions caused by the internal springs, TDMF ,sp, and Coulomb friction, TDMF,fr, among the two inertias of the DMF. The stiffness and frictional terms have non-linear behavior as a function of DMF deﬂection, θE − θDMF,2, and deﬂection rate, θ_ E −θ_ DM F;2 , i.e., TDMF,sp = TDMF,sp(θE − θDMF,2) and T DMF;fr ¼ T DM F;hys ðθE −θDM F;2 Þsignðθ_ E −θ_ DM F;2 Þ. Hence it is: T DMF ¼ T DMF;sp þ T DM F; f r

ð2Þ

with θDMF ,2 being the angular displacement of the secondary side of the DMF. The drivetrain dynamics are expressed by: T DMF τg τd þ T EM τ EM τd −T hs ¼ J eq;s1 €θd

ð3Þ

€ θd is the angular acceleration of the differential case. τg, τd and τEM are respectively the active gearbox ratio (i.e., the gear ratio corresponding to the engaged gear on the ICE side of the drivetrain), the ﬁnal reduction ratio, and the overall gear ratio from the electric motor output to the secondary shaft. TEM and Ths represent the EM torque and half-shaft torque, respectively. TEM is obtained from the motor torque demand, through a transfer function which models the motor drive dynamics. Jeq , s1 is the

358

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

equivalent mass moment of inertia for this drivetrain state: 2 2 2 2 2 J eq;s1 ¼ J d þ J s τ d þ J p þ J DMF;2 τ g τd þ J EM τ EM τd

ð4Þ

Js, Jd, Jp, JEM and JDMF,2 are the mass moments of inertia of the secondary shaft, differential assembly, primary shaft, electric motor and secondary part of the DMF, respectively. The half-shaft torque can be modeled as the sum of stiffness and damping components: ð5Þ T hs ¼ K hs Δθhs þ C hs Δθ_ hs Δθhs is the half-shaft torsional deﬂection, while Khs and Chs are the equivalent stiffness and damping (the internal damping of the material is very low) coefﬁcients. Due to the lumped backlash model, the half-shaft deﬂection is calculated through the following equation, where θw is the angular displacement of the driven wheel and θbacklash represents half of the equivalent magnitude of the transmission backlash (reported at the transmission output): Δθhs ¼

0 if jθd −θw j b θbacklash ðjθd −θw j−θbacklash Þsignðθd −θw Þ if jθd −θw j ≥ θbacklash

ð6Þ

The driven wheel and vehicle dynamics are expressed as: T hs −T RR;dr −T t ¼ J w €θw

ð7Þ

T t −T RR;udr −T aero ¼ ð J w þ J v Þ€θv

ð8Þ

Tt is the tire torque, i.e., the output of the relaxation model. TRR,dr and TRR,udr are the rolling resistance torques of the driven and undriven wheels, formulated as quadratic functions of speed [19]. Taero is the wheel torque corresponding to the aerodynamic drag, and Jw and Jv = mR2w are the mass moments of inertia of the wheels (lumped within each axle) and vehicle body (apparent inertia), with Rw being the tire radius, m the mass of the vehicle, and €θv its equivalent angular acceleration. 2.2.2. Slipping clutch and engaged synchronizer The transition from the conditions of engaged clutch to the condition of slipping clutch happens when the absolute value of the torque transmitted by the clutch, Tcl, is larger than the transmissible clutch torque, Tcl ,transm (i.e., when |Tcl | N |Tcl, transm |). In this state Tcl can be controlled independently from the engine torque, by varying the axial force on the friction surfaces, and is approximated with Eq. (9): T cl ¼ γT cl;transm; max sign θ_ DMF;2 −θ_ p

ð9Þ

γ is a non-dimensional factor varying between 0 and 1, which is dependent on the actuation force, while Tcl,transm,max is the maximum value of Tcl,transm, and θ_ p is the angular speed of the primary shaft. The transmission dynamics are given by: T cl τg τd þ T EM τ EM τd −T hs ¼ J eq;s2 €θd

ð10Þ

Jeq,s2 is the equivalent inertia of the transmission in this state according to Eq. (11): 2

2 2

2

2

J eq;s2 ¼ J d þ J s τ d þ J p τg τ d þ J EM τEM τd

ð11Þ

2.2.3. Disengaged clutch and disengaged synchronizer In this state the dynamics of the primary and secondary shafts are decoupled. The transmission dynamics are described by: T cl −

T syn ¼ Jp € θp τg

ð12Þ

T syn τ d þ T EM τEM τ d −T hs ¼ J eq;s3 €θd

ð13Þ

Tsyn represents the synchronizer torque. Jeq,s3 is the equivalent mass moment of inertia of the transmission in this state: 2

2

2

J eq;s3 ¼ J d þ J s τ d þ J EM τEM τ d

ð14Þ

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

359

Fig. 3. Simulation of an upshift from gear 2 to gear 3 at 70% of engine torque demand and 0% of electric motor torque demand. In the ﬁgure ‘Engine 2nd’ and ‘Engine 3rd’ refer to the expected speed of the primary shaft (calculated starting from the differential case speed) when gear 2 and gear 3 are engaged, respectively.

2.3. Baseline controller and validation The operating principle of the baseline gearshift controller (i.e., the controller currently running on the prototype TCU) is described with reference to Fig. 3, which shows the time history of an upshift from gear 2 to gear 3 with nil EM torque demand from the energy management system. The gearshift is initiated by ramping down Tcl,transm during phase A. Until |Tcl | b |Tcl ,transm |, TE is kept at the level requested by the energy management system of the hybrid electric drivetrain. Clutch slip starts at the beginning of Phase B. Hence, the engine torque is ramped down with an appropriate rate to avoid an increase of engine speed. The synchronization phase (Phase C) starts when the clutch is fully disengaged. The outgoing synchronizer disengages and then the ingoing synchronizer starts transmitting torque. As soon as the speed of the primary shaft matches that of the target gear, the ingoing synchronizer moves to the fully engaged position. This is highlighted by a dimensionless value of 1 for the synchronizer position in the ﬁgure. Phase D actuates the clutch re-engagement, with open-loop clutch torque control and closed-loop engine torque control as a function of the measured slip speed. At the end of phase D, i.e., when the condition of zero slip speed is veriﬁed, the clutch is engaged. During phase E Tcl,transm is ramped up to Tcl,transm,max and is kept at that value until another gearshift is requested. Throughout phases B to E, there is zero EM torque demand from the energy management system of the hybrid electric vehicle in the speciﬁc maneuver, but a reference electric motor torque, TEM,ref, is generated to compensate for the torque gap caused by the clutch torque reduction. In particular, TEM,ref is determined by the difference between the engine torque demand at the wheels corresponding to the target gear ratio (τg,ingoing) and the actual torque (estimated by the controller) transmitted by the clutch to the wheels:

T EM;ref ¼

T E;ref τ g;ingoing τ d −T cl τg τ d þ T EM;en:man τEM τ d

ð15Þ

where TEM,en.man is the EM torque demand from the energy management system (0 in this speciﬁc case). The novel drivetrain concept is implemented on a high-performance rear-wheel-drive vehicle demonstrator (see Fig. 1 and Table 1). The parameters of the control system implemented in simulation are tuned to provide similar performance to the transmission in the prototype vehicle. Different driving modes can be selected, each of them corresponding to different parameterizations of the TCU, for example in terms of clutch torque rates. Fig. 4 shows an example of experimental validation of the gearshift simulation model, in terms of acceleration proﬁles for the baseline controller with the torque-ﬁlling contribution of the EM.

360

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

Table 1 Main vehicle parameters. Symbol

Variable name

Value

Unit

m Rw Jp Js JEM JE Jd JW/2 τg τd τEM TE,max PE,max TEM, max PEM ,max

Vehicle mass Wheel radius Mass moment of inertia of primary shaft Mass moment of inertia of secondary shaft Mass moment of inertia of electric motor rotor Mass moment of inertia of engine Mass moment of inertia of differential Mass moment of inertia of the individual wheels Gearbox ratios Final drive ratio Gear ratio from electric motor to secondary shaft Maximum engine torque Maximum engine power Maximum electric motor torque Maximum electric motor power

2020 0.33 0.01 0.01 0.1 0.3 0.05 1.01 3.23, 2.17, 1.59, 1.18, 0.90, 0.69 3.90 3.55 643 422 198 120

kg m kg m2 kg m2 kg m2 kg m2 kg m2 kg m2 – – – Nm kW Nm kW

3. Optimal clutch engagement controller 3.1. Simpliﬁed model for control system design Fig. 5 shows the schematic of a simpliﬁed drivetrain model used for control system design, having a reduced number of degrees of freedom with respect to the model in Section 2, and obtained for conditions of permanent clutch slip (as this is the phase relevant to clutch engagement control). The connection between the crankshaft and the clutch input plate is assumed to be rigid, and the DMF and engine inertias are summed into JE,s. J E;s ¼ J E þ J DMF;1 þ J DMF;2

ð16Þ

The backlash non-linearity on the ﬁnal drive is neglected. However, this is not a signiﬁcant assumption as the torque-ﬁll control of the electric motor keeps the transmission in power-on during gearshifts for positive torque demands. The main approximation is related to the fact that the vehicle inertia and the inertias of all wheels are assumed to be lumped: 2

J v;l ¼ 2J w þ mRw

ð17Þ

Hence, the damping effect of the tire on the system dynamics is neglected [20]. This design choice allows a very signiﬁcant simpliﬁcation of the control system implementation, and it was already adopted in many previous studies aimed at transmission control synthesis (e.g., in [5,8,12]). Moreover, the value of Chs in the simpliﬁed model can be tuned to reach the damping level in the overall system response corresponding to the more complex model of Section 2. As the vehicle speed does not change signiﬁcantly throughout the gearshift, it is assumed that the load torque, Tl, is constant during the clutch re-engagement phase. Its value can be determined by considering the wheel torques associated with rolling

Fig. 4. Simulated and experimental longitudinal vehicle acceleration proﬁles during upshifts from gear 3 to gear 4 (top) and from gear 4 to gear 5 (bottom) at 100% of engine torque demand and 0% of electric motor torque demand, from initial speeds of 120 km/h (top) and 160 km/h (bottom).

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

361

Fig. 5. Simpliﬁed schematic of the three-degree-of-freedom drivetrain model for control system design.

resistance, Tl ,rolling_resistance, and aerodynamic drag, Tl,aero, evaluated at the initial vehicle speed (v0). T l ¼ T aero ðv0 Þ þ T l;rolling

resistance ðv0 Þ

ð18Þ

Eqs. (19) to (22) illustrate the governing dynamics of the drivetrain: θE T E −T cl ¼ J E;s €

ð19Þ

T cl τg τd þ T EM τEM τd −T hs ¼ J eq;sim

€θ p τg τd

ð20Þ

T hs −T l ¼ J v;l €θv

T hs ¼ K hs

θp −θv τg τd

ð21Þ ! þ C hs

θ_ p −θ_ v τg τd

! ð22Þ

Jeq,sim is the equivalent inertia of the transmission in the simpliﬁed model. Jeq,sim is given by Eq. (23), and is the same as the mass moment of inertia deﬁned by the previous Eq. (11) for the more complex model. A ﬁrst order transfer function is adopted for considering the EM actuation delay, starting from the reference torque demand, TEM,ref: 2

2 2

2 2

J eq;sim ¼ J eq;s2 ¼ J d þ J s τd þ J p τ g τd þ J EM τEM τd

T EM ¼ T EM;ref

α sþα

ð23Þ

ð24Þ

The previous drivetrain equations can be written in state-space form (Eq. (25)) by considering the following state vector: h iT θ_ x ¼ θ_ E −θ_ p τg pτd −θ_ v T hs T EM . The engine and clutch torques are the ﬁrst and second inputs, i.e., u ¼ ½ T E T cl T : x_ ¼ Ax þ Bu þ F

ð25Þ

362

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

where: 2 60 0 6 6 6 6 60 0 A¼6 6 6 6 6 6 0 K hs 4 0

0

2 3 1 −τEM τ2d τ g 6 7 6 J E;s J eq;sim 7 6 7 ! 6 6 1 1 τEM τd 7 7 6 0 − þ 7 J eq;sim 7; B ¼ 6 J v;l J eq;sim 6 7 6 7 ! 6 7 6 0 1 1 τEM τd 7 6 7 C hs −C hs þ 6 J eq;sim 5 J v;l J eq;sim 4 0 0 −α τg τd J eq;sim

2 2

τg τd − þ J E;s J eq;sim τg τd J eq;sim τg τd C hs J eq;sim τg − α τEM 1

!3

3 2 7 0 7 7 7 6 Tl 7 7 6 7 7 6 J 7 7 6 v;l 7 7 6 7; F ¼ 6 Tl 7 7 7 6 C hs 7 6 J v;l 7 7 7 6 7 4 τg 5 7 T E;ref α 5 τ

ð26Þ

EM

3.2. Optimal controller formulation In order to maintain comfort across clutch engagement, the slip speed x1 (i.e., the ﬁrst element of x) should follow a smooth trajectory. To avoid shufﬂe after the engagement (i.e., when the engine torque demand is handed over to the energy management system), the clutch torque should reach a value corresponding to the driver demand. Moreover, since the clutch torque ﬂuctuations directly contribute to vehicle jerk, clutch torque should have a smooth proﬁle as well. These targets can be achieved by minimizing the following tracking errors on clutch slip speed and clutch torque: eslip ¼ r slip −x1

ð27Þ

ecl;w ¼ r cl;w −T cl τg τd

ð28Þ

A third order polynomial is chosen for expressing the slip speed reference, rslip, as a function of time t. The reference clutch torque at the wheels, rcl,w, is linearly varying from 0 to the value corresponding to the engine torque demand at the expected ﬁnal time, tf (Eq. (29)). The parameters a to d are obtained by imposing the conditions in (Eq. (30)).

r slip r ðt Þ ¼ r cl;w

2 6 ¼4

3 3 2 at þ bt þ ct þ d 7 t 5 T E;ref τg τ d tf

ð29Þ

8 rslip ðt Þ ¼ ωslip;initial ; t ¼ 0 > > > > > d > > < r ðt Þ ¼ 0; t ¼ 0 dt slip r slip ðt Þ ¼ 0; t ¼ t f > > > > > > d > : r ðt Þ ¼ 0; t ¼ t f dt slip

ð30Þ

By neglecting the inertial terms in Eq. (20), a simpliﬁed expression for ecl,w is obtained: ecl;w ¼ r cl;w ðt Þ−ðT hs −T EM τEM τ d Þ

ð31Þ

Consequently, the vector of the tracking errors, e, is formulated as: e¼

eslip ecl;w

¼ r−Hx; with H ¼

1 0

0 0

0 0 1 −τ EM τd

ð32Þ

In order to ensure a better tracking performance, e is augmented with the integral of the tracking error: ê ¼

e −H ¼ ĤX þ Mr with Ĥ ¼ ∫ e dt 024

022 I2 ;M ¼ I2 022

ð33Þ

x where X ¼ consists of the original system states augmented with the integral of the tracking error. Its dynamics are given ∫ e dt by: X_ ¼ A ̂X þ B̂u þ ZF þ Gr; with A ̂ ¼

A −H

B I2 042 042 ; B̂ ¼ ;Z ¼ G¼ 022 022 024 I2

ð34Þ

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

363

The cost function of the optimal controller penalizes the tracking errors, the clutch dissipation energy, the control effort and the ﬁnal values of the errors, êðt f Þ: J¼

i 1 T 1 t f h T T T ê t f S ê t f þ ∫ 0 êðt Þ Q êðt Þ þ uðt Þ Ruðt Þ þ 2X ðt Þ Nuðt Þ dt 2 2

ð35Þ

The clutch dissipation energy is included in the term 2X(t)TNu(t), which, being a product of an input (i.e., the clutch torque) and a system state (i.e., the clutch slip speed), signiﬁcantly increases the complexity of the problem. The matrices S and Q are symmetric and positive semi-deﬁnite (S ≥ 0, Q ≥ 0). The matrix R is positive deﬁnite (R N 0). The cost function (35) can be converte, state penalty matrix Qe and system dynamics are represented into the form reported in Eq. (36), where the new control vector u ed by Eqs. (37)–(39). J¼

i 1 T 1 t f h T e T T T T eT Ru e dt ê t f Sê t f þ ∫ 0 X Q X þ 2r M QH ̂X þ r M QMr þ u 2 2

ð36Þ

e ¼ u þ R−1 NT X u

ð37Þ

T −1 T Qe ¼ H ̂ Q Ĥ−NR N ≥ 0

ð38Þ

̂ ~ þ ZF þ Gr; A ¼ A ̂−B̂R−1 NT X_ ¼ AX þ Bu

ð39Þ

The optimal control problem with the cost function of Eq. (36) and system dynamics of Eq. (39) can be solved by formulating the Hamiltonian of the system and applying the optimality conditions [21]. Hence, the control law and co-state dynamics together with the boundary conditions are determined through Eqs. (40)–(42). T

e ¼ −R−1 B̂ λ u

ð40Þ

−λ_ ¼ Qe X þ H ̂ QMr þ A λ

ð41Þ

T λ t f ¼ H ̂ S ĤX t f þ Mr t f

ð42Þ

T

T

Eqs. (39)–(42) belong to a special class of formulations, called two-point-boundary-value-problems (TPBVPs, [21]). A method to solve these problems is based on numerical backward integration of the resultant Riccati and auxiliary equations, which requires a solution with a high time resolution over the controller activation interval. In this application, the initial value of reference slip speed, rslip, is known only when the clutch engagement phase is activated, hence the on-line backward integration is not feasible. Also, the off-line gain computation for various combinations of initial slip speeds and driver torque demands would result in a very large look-up table, which could easily exceed the affordable TCU memory. As a consequence, the methodology adopted in this study is based on converting the original TPBVP into a homogenous form. Then, with a similar approach to the one in [15], a set of piecewise constant feedback gains can be computed. The dynamics of the reference trajectories introduced by Eq. (29) are as follows: Y_ ¼ CY

ð43Þ

where Y is: " Y¼

r slip

dr slip dt

2

3

d r slip

d r slip

dt 2

dt 3

r cl;w

dr cl;w dt

#T ð44Þ

The formulation of C is provided in the Appendix. Moreover, r can be expressed as r = qrY, with qr ϵ R26 . Assuming a new vector Γ ¼ ½ X ZF Y T where X = qxΓ and Y = qyΓ, with qx ϵR618 and qy ϵR618 , a homogenous TPBVP, represented by Eqs. (45) and (46), is obtained. " Ah Γ_ ¼ ~ −HT̂ QMq q −Qq λ_ x r y

2 # A Γ ; with Ah ¼ 4 066 T λ −A 066 Bh

I6 066 066

2 3 3 −1 T Gqr −B̂R B̂ 066 5; Bh ¼ 4 066 5 C 066

ð45Þ

364

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

T T λ t f ¼ St f Γ t f ; with St f ¼ H ̂ SĤqx þ H ̂ SMqr qy

ð46Þ

In order to derive a set of feedback gains, the control activation interval [0 tf] is discretized into a sequence of time steps with length Δt, thus obtaining the discretized time vector [0 t1 = Δt t2 = 2Δt … tn−1 = (n − 1)Δt tn =nΔt … tf]. The state transition matrix (Eq. (47)) can be computed off-line with signiﬁcant accuracy by employing software packages such as Matlab. The resultant state transition matrix Ω, with submatrices Ωij (i, j ϵ [1,2]), provides a relationship between two adjacent vectors, ½ Γ t n−1 λt n−1 T and ½ Γ t n λtn T , in the discretized time interval. Starting from the ﬁnal point with the a-priori known value Stf from Eq. (46) and utilizing the backward recursive procedure expressed by Eq. (49), a set of Stn can be computed, which, in combination with Eq. (40), provides the feedback gains and the control action throughout the time interval. In formulas: " Ω¼

Γ tn λt n

Ω11 Ω21

¼

Ω12 Ω22 Ω11 Ω21

¼e

Ω12 Ω22

Ah T e −Q qx −H ̂ QMqr qy

Γ t n−1 λt n−1

#

Bh T

−A

dt

ð47Þ

−1 λtn−1 ¼ St n−1 Γ t n−1 ; with St n−1 ¼ Ω22 −St n Ω12 St n Ω11 −Ω21

ð48Þ

ð49Þ

An appropriate selection of the matrices Q , R and N delivers the desired controller performance in terms of dissipated energy and tracking/drivability, i.e., depending on the driving mode different weightings among the terms of the cost function in Eq. (35) can be selected. At the end of the control design procedure, the computed control gains are stored in the memory of the vehicle TCU. Since typically the preferred engagement time for an upshift is very short (i.e., 0.25 s for the speciﬁc application), only a few computations of the gains are required and the resultant look-up table will occupy a small space in the TCU memory. As an alternative to the look-up table-based approach, it is noted that the proposed technique is more efﬁcient than backward integration, so the on-line computation of the gains could be achievable using the processing power of the controller. With the on-line approach, the only off-line requirement is the computation of the state transition matrix as a function of the gear number, which can be stored in the TCU memory.

Fig. 6. Simpliﬁed schematic of the proposed controller with the auxiliary half-shaft torque estimator.

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

365

3.3. Control system implementation and half-shaft torque estimation Fig. 6 is the schematic of the overall control structure. The main look-up table includes the sets of piecewise constant gains of the controller (variable in the time domain), which are functions of the only ﬁnal gear ratio. In fact, as outlined by Eq. (49), the computation of Stn is based on the state transition matrix, which is constructed from the vehicle and transmission physical parameters (such as the gear ratios, inertias, mass of the vehicle, etc.…). One of the inputs of the feedback controller is represented by the half-shaft torque, Ths, which cannot be directly measured on the car, and is therefore estimated with a Kalman ﬁlter. The main inputs of the Kalman ﬁlter are transmission speed, wheel speed, clutch torque and EM torque, which are either directly measured by sensors or estimated on-line with algorithms already present on production vehicles. Eq. (50) includes the model formulation adopted for the Kalman Filter design. 2 6 6 6 4

θ_ d € θd θ_ w €θ w

2

0 6 −K hs 6 7 6J 7 6 eq;sim 7¼6 5 6 0 4 K hs J v;l 3

1 −C hs J eq;sim 0 C hs J v;l

0 K hs J eq;sim 0 −K hs J v;l

3 2 0 0 C hs 72 θd 3 7 6 τg τd _ 7 7 6 6 J eq;sim 76 θd 7 þ 6 J eq;sim 4 5 6 1 7 7 θw 4 0 −C hs 5 θ_ w 0 J v;l

3 0 τ EM τd 7 7 T cl J eq;sim 7 7 T 0 5 EM 0

ð50Þ

At each time step the discretized version of Eq. (50) is used to compute the a-priori estimate of the states. The detailed formulation of the Kalman Filter, implemented according to the theory in [22], is omitted for brevity. In the ﬁlter the half-shaft torque is calculated starting from transmission output speed and wheel speed. Because of the measurement feedback, the estimator provides satisfactory performance even when approximated values of the half-shaft torsional compliance are used. 4. Results and discussion This section provides a simulation-based analysis of the beneﬁts of the proposed optimal controller, by using the validated non-linear model discussed in Section 2. Fig. 7 shows a gearshift from gear 2 to gear 3 at 70% of engine torque demand and zero EM torque demand from the energy management system. The desired clutch engagement time is assumed to be tf = 0.25 s. The controller activation interval is divided into ﬁve equally spaced time steps, which results in a gain look-up table with a size of only 8 kb. In the speciﬁc simulation of Fig. 7, in order to evaluate the controller capability of reducing the clutch energy losses, the cost function of the controller is set to have a relatively heavy penalty on the clutch dissipation energy, and moderate-to-low values of the weights on the

Fig. 7. Simulation of an upshift from gear 2 to gear 3 at 70% of engine torque demand and 0% of electric motor torque demand from the energy management system, with an energy efﬁciency-oriented tuning of the proposed controller (the vertical lines show the activation interval of the clutch engagement controller).

366

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

Fig. 8. Simulation of an upshift from gear 2 to gear 3 at 70% of engine torque demand and 0% of electric motor torque demand from the energy management system, with a tracking performance-oriented tuning of the proposed controller (the vertical lines show the activation interval of the clutch engagement controller).

reference tracking errors and their integrals. The clutch and engine torque penalties are empirically chosen from simulation results, to achieve desirable performance. The selected weighting matrices are reported in the Appendix. The actuation torque values in the simulation model are constrained between the allowable upper and lower bounds of the physical prototype. This is essential especially when the clutch energy dissipation is highly penalized, as extreme tunings of the controller could try to generate an unrealistic engine braking torque to accomplish the engagement. In Fig. 7 the clutch engagement controller is activated at roughly t = 0.75 s and is terminated slightly before t = 1 s, i.e., when the slip speed is within a speciﬁed boundary for a sufﬁciently long time interval. Throughout the engagement, the controller keeps the engine torque at its minimum possible value and clutch actuation is signiﬁcantly delayed. Despite the major deviations of the actual trajectories from the reference values, the engagement is accomplished at the prescribed ﬁnal time, because of the ﬁnal time penalties in the cost function formulation. Moreover, within the same time frame the clutch torque reaches a level corresponding to the driver demand. As in this case the derivative of the slip speed at the clutch engagement is relatively high, the vehicle acceleration is characterized by oscillations after the engagement. However, based on the experience of Oerlikon Graziano SpA (the industrial company involved in this study) in subjective assessments of the gearshift quality and correlation with objective measurement and simulation results, the oscillations of the acceleration proﬁle for the energy-oriented set-up of the controller are still acceptable for the user, as the gearshift is accomplished in a very limited time interval.

Fig. 9. Comparison of the three analyzed controllers in terms of energy dissipation within the clutch, for different values of the initial gear number, at engine torque demands of 70% (top) and 100% (bottom), with an electric motor torque demand from the energy management system of 0%.

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

367

In Fig. 8 the same maneuver of Fig. 7 is repeated, this time with a nil penalty on the clutch energy dissipation within the cost function for control system design, while the weights on the tracking errors and their integrals are signiﬁcantly increased to enforce the tracking performance (tracking performance-oriented set-up of the controller). Fig. 8 shows signiﬁcant differences for gearshift performance compared with Fig. 7, i.e., the controller continuously tracks the reference trajectories and is capable of handling the extensive model non-linearities, such as those related to the engine torque characteristic as a function of speed and tire dynamic behavior. Since clutch energy dissipation is neglected within this tracking-oriented tuning, the energy dissipated by the clutch is 2432 J, against the 1232 J of the previous case. In Fig. 9 the optimal controller with the energy-oriented set-up is compared with the baseline controller (i.e., the conventional gearshift controller presented in Section 2) and the clutch engagement controller formulation of [16], for a selection of gearshifts with engine torque demands of 70% (top) and 100% (bottom). The controller in [16] is realized with two PID controllers aiming at tracking predeﬁned engine and clutch speed proﬁles, together with a feedforward contribution related to engine speed. In all of the gearshifts the optimal controller provides lower energy dissipation than the other controllers (i.e., average reduction of 1014 J with respect to the PID controllers in [16] and 1127 J with respect to the baseline controller). However, for high gear numbers the difference between the energy dissipation of the baseline controller and that of the optimal controller is reduced because in the speciﬁc tests the engine operates within a relatively lower speed region, with smaller initial clutch slip speed values. Also, at low engine speed the amount of available engine braking torque is limited, which forces the optimal controller to rely mainly on the clutch torque in order to achieve the timely engagement of the clutch. 5. Conclusion The activity presented in the paper leads to the following conclusions: • Hybrid electric drivetrains based on the integration of an electric motor drive into a conventional automated manual transmission layout allow combined mechanical simplicity, cost effectiveness and torque-ﬁll capability for gearshifts executed on the engine side of the drivetrain, while providing the energy efﬁciency beneﬁts typical of electriﬁed architectures. • A simple formulation of the electric motor torque demand during gearshifts was assessed through simulations and experiments, and provided the expected torque-ﬁll performance. • The formulation of a novel optimal controller for the clutch re-engagement phase was discussed in detail. The approach is based on the augmentation of the reference trajectories, resulting into a homogeneous two-point-boundary-value-problem. The problem was solved through the off-line computation of the optimal sets of feedback gains, starting from the system state transition matrix. The controller, based on a small-size look-up table suitable for storage in the transmission control unit, allows consideration of advanced cost functions for the control of the clutch re-engagement process, without penalties in terms of computational load. • The novel controller permits different set-ups, either focused on tracking performance (hence, on smooth vehicle acceleration proﬁle) or energy efﬁciency (hence, on clutch energy dissipation reduction). The controller set-ups (e.g., variable depending on the selected driving mode) can be easily tuned by transmission engineers without signiﬁcant experience of advanced control theory, through the weights used in the controller cost function. • The performance of the controller was assessed against that of two more conventional clutch engagement controllers, by using the experimentally validated simulation model of the novel drivetrain layout. In the energy efﬁciency-oriented set-up, the proposed controller allows signiﬁcant reductions of clutch energy dissipation. Further studies will focus on: i) integration of the novel controller with the energy management system of the hybrid electric drivetrain concept; ii) analysis of the effect of further simpliﬁcations of the model equations used for control system design; iii) vehicle validation of the algorithm; and iv) optimal controller application to other transmission system layouts, such as dualclutch transmissions. Acknowledgements The authors would like to thank Oerlikon Graziano SpA for the ﬁnancial and technical support provided throughout the research project. Appendix A A.1. Matrices of the reference trajectory dynamics C¼

C1 024

2 0 60 042 ; C1 ¼ 6 40 C2 0

1 0 0 0

0 1 0 0

3 0 07 7; C2 ¼ 0 15 0 0

1 0

ðA:1Þ

368

A.M. Gavgani et al. / Mechanism and Machine Theory 105 (2016) 352–368

A.2. Penalty matrices used in the simulations Energy efﬁciency-oriented tuning

2

100000 6 0 Q ¼6 4 0 0

2 0 3 60 0 0 0 6 6 5 0 0 7 7; N ¼ 6 0 60 0 50 0 5 6 40 0 0 40 0

3 4300 3 2 70 0 0 0 0 7 7 6 7 0 9900 0 0 7 100 0 0 7 7 ;S ¼ 6 ;R ¼ 4 0 0 800 0 5 0 300 0 7 7 5 0 0 0 900 0 0

ðA:2Þ

Tracking performance-oriented tuning

2 6 Q ¼6 4

100000 0 0 50 0 0 0 0

2

0 60 0 0 6 6 7 0 0 7; N ¼ 6 0 60 5 500000 0 6 40 0 300000 0 3

3 0 2 70 0 07 7 7 6 0 9900 100 0 07 ;S ¼ 6 ;R ¼ 4 0 0 0 300 07 7 5 0 0 0 0

3 0 0 0 0 7 7 800 0 5 0 900

ðA:3Þ

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

E. Galvagno, M. Velardocchia, A. Vigliani, Dynamic and kinematic model of a dual clutch transmission, Mech. Mach. Theory 46 (6) (2011) 794–805. E. Galvagno, M. Velardocchia, A. Vigliani, A model for a flywheel automatic assisted manual transmission, Mech. Mach. Theory 44 (6) (2009) 1294–1305. F. Garofalo, L. Glielmo, L. Iannelli, F. Vasca, Smooth engagement for automotive dry clutch, 40th IEEE Conference on Decision and Control, 2011. A. Serrarens, A. Dassen, M. Steinbuch, Simulation and control of an automotive dry clutch, American Control Conference, 2004. P. Dolcini, C.C. de Wit, H. Becahrt, Lurch avoidance strategy and its implementation in AMT vehicles, Mechatronics 18 (5) (2008) 289–300. L. Glielmo, F. Vasca, Optimal control of dry clutch engagement, SAE Technical Paper 2000-01-0837, 2000. T. Jin, P. Li, G. Zhu, Optimal decoupled control for dry clutch engagement, American Control Conference, 2013. B. Tapes, L. Kasac, J. Deur, Optimal control of automated transmission engagement process, IEEE International Conference on Control Applications, 2012. X. Song, Z. Sun, X. Yang, G. Zhu, Modelling, control, and hardware-in-the-loop simulation of an automated manual transmission, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 224 (2) (2010) 143–160. A. Haj-Fraj, F. Pfeiffer, Optimal control of gear shift operations in automatic transmissions, J. Frankl. Inst. 338 (2) (2001) 371–390. X. Lu, B. Wang, B. Gao, H. Chen, Model predictive control of AMT clutch during start-up process, Control and Decision Conference, 2011. A.V.D. Heijden, A. Serrarens, M. Camlibel, H. Nijmeijer, Hybrid optimal control of dry clutch engagement, Int. J. Control. 80 (11) (2007) 1717–1728. A. Bemporad, F. Borrelli, L. Glielmo, F. Vasca, Hybrid control of dry clutch engagement, European Control Conference, 2001. T. Lu, F. Dai, J. Zhang, M. Wu, Optimal control of dry clutch engagement based on the driver's starting intentions, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 226 (8) (2012) 1048–1057. S.S. Vaddi, P.K. Menon, G.D. Sweriduk, Multi-stepping solution to linear two boundary problems in missile integrated control, AIAA Guidance, Navigation and Control Conference, 2005. L. Glielmo, L. Iannelli, F. Vasca, Speed control for automated manual transmission with dry clutch, 43rd IEEE Conference in Decision and Control, 2004. A. Mehdizadeh Gavgani, T. Bingham, A. Sorniotti, J. Doherty, C. Cavallino, M. Fracchia, A parallel hybrid electric drivetrain layout with torque-fill capability, SAE International Journal of Passenger Cars – Mechanical Systems 8 (2) (2015) 767–778. H.B. Pacejka, Tire and Vehicle Dynamics, Elsevier, 2005. G. Genta, G. Morello, The Automotive Chassis Vol. 1: Components Design, Springer, 2009. S. Abuasaker, A. Sorniotti, Drivability analysis of heavy goods vehicles, SAE Int. J. Commer. Veh. 3 (1) (2010) 195–215. A.E. Bryson, Applied Optimal Control: Optimization, Estimation and Control, CRC Press, 1975. C.K. Chui, G. Chen, Kalman Filtering with Real-Time Applications, Springer, 1991.

Copyright © 2022 COEK.INFO. All rights reserved.