1. Introduction
Step-ratio automatic transmissions (AT), equipped with torque converter, planetary gear sets, and wet clutches/brakes, are commonly found in conventional but also electrified powertrains [
1,
2]. A large number of AT gears (up to 10 for passenger vehicles, nowadays) introduces new types of shifts, such as multi-step and double-transition shifts, which are more difficult to control [
2,
3]. This places increased requirements on gear shift control system in terms of improving shift responsiveness, comfort, and energy efficiency [
3].
Current AT shift control systems predominantly rely upon open-loop-generated control input profiles, whose parameters can effectively be determined by using off-line control parameter optimization (CPO) methods previously applied to conventional single-transition shifts [
4,
5] and more demanding double-transition shifts [
6,
7]. The control input profiles themselves can be designed based on the insights gained through control trajectory optimization (CTO), as presented in References [
8,
9] based on the examples of single- and double-transition shifts, respectively. Although CPO is faster and more straightforward than CTO, it still requires significant computational time to find optimal parameters, which motivates the development of new, simplified optimization approaches.
The emphasis of AT upshift controller design is usually on the inertia phase, where the main, oncoming (ONC) clutch torque control action is supplemented by the engine torque reduction action implemented through a fuel cut or spark retardation control channel [
1] in order to reduce the uncomfortable output shaft torque overshoot. Since open-loop shift control is rarely robust to AT parameter variations, it is usually supplemented by closed-loop control actions aimed at tracking the desired ONC clutch slip speed or engine speed reference to ensure shift duration [
1]. Different formulations of closed-loop control systems are proposed in the literature, such as PID control [
10], robust control [
11], and LPV control [
12]. The drawback of these methods is that the ONC clutch control action does not directly account for (and anticipate) the open-loop engine control action. For better overall performance, more advanced, multi-input/multi-output (MIMO) closed-loop control strategies can be applied, which are typically based on linear quadratic regulators (LQR) [
13,
14,
15] and model predictive controllers (MPC) [
16,
17]. Although the advanced MIMO methods offer performance improvements over open-loop and basic closed-loop control strategies, they may be computationally demanding and require full-state feedback that may not be available in application.
During the upshift inertia phase, the ONC clutch and engine torque control inputs may be augmented by the off-going (OFG) clutch torque input. The benefit of using OFG clutch during the inertia phase, i.e., introducing a controlled tie-up of the two clutches, is the suppression of output shaft torque oscillation and/or the prevention of the engine flair at the inertia phase start, as confirmed by bond-graph analysis in Reference [
18] and experimentally in Reference [
19]. It is demonstrated through CTO in Reference [
8] and CPO in Reference [
20] that using the OFG clutch either reduces shift time or improves shift comfort at the expense of somewhat increased clutch energy loss. LQR-based MIMO control involving the OFG clutch during the inertia phase is considered in References [
14,
15].
This paper expands on the work presented in Reference [
20] by proposing a simplified control parameter optimization approach for the upshift inertia phase based on an analytical, static model of shift objectives. This approach is further employed in order to formulate a practical, static model-based model predictive control (S-MPC) law. The S-MPC law utilizes the ONC clutch slip speed feedback signal and commands the ONC clutch torque capacity based on the static model prediction of the open-loop engine and OFG clutch control actions. This constitutes the following main contributions of the paper: (i) a numerically efficient shift control parameter optimization method based on analytical static model of optimization objectives and (ii) a corresponding, simple-to-implement, computationally inexpensive, and robust model predictive control strategy.
The remainder of this paper is organized as follows.
Section 2 describes a control-oriented powertrain dynamics model including a 10-speed AT submodel.
Section 3 overviews the genetic algorithm-based control parameter optimization results for 1–3 upshift based on Reference [
20]. The static model-based optimization method is presented in
Section 4, while the corresponding simulation analysis results are given in
Section 5. The S-MPC strategy is proposed and verified in
Section 6. Concluding remarks are given in
Section 7.
2. Powertrain Dynamics Model
The mechanical scheme of the considered powertrain and the corresponding control-oriented bond graph are shown in
Figure 1 [
21,
22]. A source-of-torque element is used to model the spark ignition engine, where the dependence of torque
τe on the accelerator pedal position
pth and the engine speed
ωe is established through an experimentally recorded engine map. The engine rotational dynamics are described by:
where
Iei is the total, engine and impeller moment of inertia, Δ
τec denotes the engine torque cutting control input, and
τi is the impeller torque.
The torque converter is represented by nonlinear static maps
Rτ(
Rω) and
Ki(
Rω), where
Rω =
ωt/ωi =
ωt/ωe is the speed ratio, and
Rτ =
τt/
τi and
Ki = ωi/
are the torque ratio and the capacity factor, respectively. This gives the following torque converter static model (see Reference [
22] and references therein):
The considered 10-speed AT gearbox [
23] includes four planetary gear sets, four clutches, and two brakes. A full, sixth-order mathematical model of the considered 10-speed AT [
23] gearbox is derived in References [
24,
25]. When accounting for locked and open clutch/brake states for a given single-transition shift, the following reduced-order model can be obtained, which is represented by the bond graph shown in
Figure 2 [
25,
26]:
where the gearbox input speed
ωis =
ωt and the output speed
ωos constitutes the state vector, while the off-going (OFG) clutch torque
τOFG, the oncoming (ONC) clutch torque
τONC, the gearbox input torque
τis =
τt and the output torque
τos form the input vector. The inertia matrix
Ared contains equivalent gearbox input and output inertia
Iin and
Iout, respectively, as well as the cross-coupling inertia
Iio. The input matrix
Bred contains the equivalent input/output gear ratios
i1,
g1 and
i2,
g2 of the OFG and ONC clutch torque transfer paths (
Figure 2), respectively. It should be mentioned that the bond graph in
Figure 2 is applicable to another common type of AT, which is dual clutch transmission (DCT), with the main being difference that there is no inertia coupling effect (nor the torque converter in
Figure 1a). Thus, the optimization and control methods proposed in this paper are applicable to DCTs, as well.
The AT model further simplifies in the case of a locked torque converter, where
ωis =
ωi =
ωe and
τis =
τi =
τe − Δ
τec hold. The engine rotational dynamics given by Equation (1) become redundant [
25], and the engine-side inertia
Iei is lumped to the torque converter turbine inertia
Iin given in the gearbox state Equation (3a).
A clutch friction model is used to determine the clutch torques
τj,
j ∈ {OFG, ONC}, in Equation (3a) based on the clutch slip speeds
ωj given by Equation (3b). Here, the Karnopp friction model [
27] is used, which is described as [
22]:
where
τhj is the commanded clutch torque capacity,
= 1 is the normalized Coulomb friction torque, and
τstick,j is the locked clutch element hold torque that is obtained from slipping clutch and input/output torques [
22]. The clutch actuator dynamics are not considered (except in a control system robustness test in
Section 6) for the sake of simplicity and in order to establish benchmark performance in a straightforward manner.
The differential reduces the gearbox output speed (
ωdo =
ωos/
id) and provides equal left and right halfshaft torque
τhs =
id τos/2 (
Figure 1). The halfshafts are modeled by an equivalent compliant shaft (
Figure 1, [
21,
22]). Details on the driveline and longitudinal dynamics submodels are given in Reference [
7]. Note that a linearized, adhesion-region tire model is used, and that the zero amount of road slope and a constant rolling resistance are assumed.
7. Conclusions
Replacing the powertrain dynamics model and related optimization objectives with simplified, static expressions for inertia phase duration, energy loss, and inertia bump objectives, as functions of control input profile parameters, facilitates control parameter optimization and control performance algebraic analyses. The objective expressions have been obtained for two characteristic control scenarios: (i) constant control inputs that includes oncoming (ONC) and off-going (OFG) clutch torque, and engine torque reduction; and (ii) a piecewise linear profile of OFG clutch torque, while keeping the ONC clutch and engine torque reduction constant.
Algebraic analysis was conducted based on the assumptions of constant output shaft speed, linear ONC clutch slip speed profile, constant control inputs, and locked torque converter. The analysis has shown that the inertia phase duration can be reduced by boosting the engine torque reduction, increasing the OFG clutch torque, and raising the allowable level of inertia bump. When applying the OFG clutch control input, there is an inertia bump level above which the clutch energy loss increases with shortening of the shift, thus switching the Pareto frontier from 2D to 3D shape (all the three objectives become conflicting). The above observations have been illustrated by static objective model-based numerical optimization results. They both agree with the genetic algorithm-based control parameter optimization (CPO) benchmark results, which showed that by using the OFG clutch control, the inertia phase duration could be reduced by 20% at the expense of clutch energy loss rise of 4%.
The insights gained by applying the simplified off-line optimization approach have been employed for the derivation of an on-line static model-based predictive control (S-MPC) strategy. The S-MPC strategy commands the oncoming clutch torque capacity on the shrinking, inertia phase horizon based on two actions: (i) an oncoming clutch slip-speed feedback term designed to achieve the target inertia phase duration, and (ii) anticipation of open-loop mean-value control actions of engine torque reduction and off-going clutch torque. The oncoming clutch torque capacity command is further modified by a control signal difference dead zone element in order to suppress chattering effects. The simulation verification results have indicated that the S-MPC strategy can provide performance comparable to the CPO benchmark. For certain faster shift designs, S-MPC outperforms CPO in terms of reduced RMS jerk, owing to its ability to freely shape the ONC clutch torque profile, as opposed to the piecewise linear open-loop profile used in CPO. A robustness analysis has shown that in the case of AT parameter variations the S-MPC system maintains the control performance (in particular the shift time) close to that of the nominal system, i.e., within ±1% compared to ±10% in the CPO case. This is because of the inherent ONC clutch slip speed feedback loop incorporated in the S-MPC law. Similarly, the S-MPC system is rather insensitive to unmodeled clutch actuator dynamics.
The main advantage of the proposed control strategy lies in its ability to provide MPC functionality while being simple and practical to implement and tune. Since the S-MPC strategy relies upon the existing ONC clutch slip speed feedback in order to determine the ONC clutch torque demand, it can readily substitute the existing closed-loop controllers. From the tuning perspective, the calibration engineer only needs to set the desired shift duration, while the time-varying feedback gain is automatically determined by the strategy. However, the strategy still relies on calibration or off-line-optimization of the open-loop OFG clutch torque demand and engine control input, and as such, it is strictly speaking sub-optimal. Therefore, the future work will mainly be directed towards the development of more general, on-line-implementable, multi-input MPC strategies. They would be aimed at reproducing the CPO benchmark performance while incorporating inherent feedback paths through all control channels. The herein presented and future MPC strategies are aimed to be extended to torque phase control, as well.