**1. Introduction**

In recent years, an ever-increasing number of researchers have focused their efforts on the substantial challenge of powertrain electrification in an attempt to deal with new government requirements that aim at reducing pollutant emissions and fuel consumption [1]. In this context, the Research and Development (R&D) departments of several automobile companies are engaged in the development of innovative powertrain solutions to replace vehicles with fuel motorization. A slow and progressive transition from internal combustion engine vehicles (ICEVs) to zero emissions vehicles (ZEVs), battery electric vehicles (BEVs), and fuel cell electric vehicles (FCEVs) is occurring, along with the development of hybrid electric vehicles (HEVs) and plug-in hybrid electric vehicles (PHEVs).

As proposed in [2], several possible topological architectures can be implemented in HEVs, where two or more sources of power/energy are combined to achieve the required power necessary to propel the vehicle. HEV vehicles combine the ICE power with an electric traction motor, which is powered by an energy storage device generally known as battery pack. The most common configurations adopted to couple different power sources can be summarized as parallel hybrid [3], series hybrid [4] or combined series–parallel architectures, with a mechanical power-split [5–8] or innovative magnetic split [9]. Different control strategies have been investigated and proposed to achieve optimal energy management [10] between different power sources. HEVs combine the advantages of electric motor drives—having quick acceleration—with good ICE performance at constant speeds. On the other hand, vehicle drivability [11] is highly affected, as during rapid torque transients—due to drivers' abrupt accelerations—noise and vibrations [12] can occur during engine start/stop conditions [13] in mild hybrid electric vehicles.

Academic Editors: Vincenzo Di Dio, Antonella Castellano and Marco Cammalleri

Received: 22 February 2023 Revised: 8 March 2023 Accepted: 9 March 2023 Published: 14 March 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Besides this aspect, sudden driver torque requests can generate an increase in NOx emissions, as investigated in [14] during tip-in/tip-out manoeuvres. Hence, control strategies should act in a multiphysics way, also considering—besides optimal energy management—NOx and CO2 emissions [15] and, more generally, the reduction of pollutant emissions into the environment [16].

The aim of this paper is to address the topics of both drivability enhancement and the reduction of pollutant emissions; thus, the development of a vehicle model that is able to predict and highlight all these aspects is required—as carried out in [17], where Gear Shift Patterns were optimized to fulfil multiple constraints in terms of customer requirements such as drivability, NVH performance, emissions, and fuel consumption. These last two aspects were considered in [18] to evaluate fuel savings and NOx benefits when predictive control is applied to a mild hybrid truck, using dynamic programming with backtracking. Conversely, [19] proposed a novel real-time Energy Management Strategy (EMS) that was integrated with a model to use physical considerations to estimate energy consumption during gearshifts in Dual-Clutch Transmissions (DCT).

The development of a transmission model with a high level of accuracy is necessary to describe the dynamic behaviour of the whole system well, although its complexity can be reduced with appropriate simplifications—as carried out in [20] with a hybrid model of a vehicle driveline that was partially lumped and partially distributed to investigate vibrational phenomena inside the transmissions known as "shuffle" and "clonk". A methodology for the evaluation of clunking noises during gearshifts was also proposed in [21] using a nonlinear lumped parameter model of DCT, developed in an Amesim environment, while a Matlab/Simulink environment was adopted in [22] for the evaluation of NVH performance in the DCT transmission of a C-segment passenger car. Coming back to hybrid vehicles, mathematical models of both conventional and mild hybrid powertrain were developed in [23] through the integration of a conventional manual transmission-equipped powertrain and an electric motor as a secondary power source, in order to study the performance of partial power-on gear shifts through the implementation of torque hole-filling by the electric motor during gearshifts.

Besides the modelling of mechanical components, an appropriate tyre model can have relevant effects on simulated dynamics, as they have also a considerable influence on damping effects inside the powertrain. [24] demonstrated, using appropriate identification techniques aimed at describing tyre torsional dynamics in the frequency range 10–50 Hz, that damping is more accurately represented if modelled as hysteretic and not as viscous when the tyre load is low. Conversely, [25] proposed three different tyre models, e.g., modelled as a simple torsional spring, adopting a linear slip model, or assuming a fixed relaxation length; these last two models introduced nonlinearity, as an instantaneous slip ratio was considered. The relaxation length was estimated from the steady state tyre properties in [26] and included wheel load and slip dependencies. The adoption of this parameter is necessary for the description of interactions between tyres and roads, as all the other chassis components influence vehicle dynamic properties through tyre contact forces and torques.

A lumped LuGre tyre ground contact model was modelled in [27] and in [28], where the developed dynamic friction tyre model was able to accurately capture the transient behaviour of the friction force observed during transitions between braking and acceleration phases. Other researchers have instead used software for tyre dynamics to compute the forces acting on the tyre and hence on the steel wheel for other purposes, such as fatigue assessment [29].

The parallel mild hybrid architecture of a heavy commercial vehicle developed for this research study is similar to the one adopted in [30], but with the improvement of implementing three different control strategies for the direct control of vehicle acceleration, vehicle speed control, and torque control, with superposed active damping of drive train oscillations. The torque distribution between the two power sources is managed as a different task from the regulation task (oscillation damping), adopting a specific control

algorithm that does not require the implementation of an optimisation problem. Moreover, as regards the linear model used for the design of the controller, an accurate tyre model with a fixed relaxation length is adopted—thus decoupling the kinematic behaviour of the driving rear tyres from the free rolling condition, as carried out in [25].

The developed model should also be able to reduce drivetrain vibrations; different works addressing this aim can be found in literature. In [31], a control for vibration reduction was developed for hybrid vehicles and experimentally validated, with two different controllers that reduced both the vibrations due to engine torque ripple and ones related to the driveline. Conversely, the root locus technique [32] has been adopted for antijerk control designs capable of preserving the driveline from oscillations. To improve NVH performance, a magnetic differential for vehicle drivetrains was developed in [33], while a controller for driving the engine—in order to actively damp driveline oscillations—was proposed in [34] to improve drivability and passenger comfort.

The present paper aims to propose an accurate nonlinear vehicle model and a modelbased torque allocation control to achieve both improved vehicle drivability and pollutant emissions performance. The controller is designed in two steps with different aims: highlevel logic was used in order to aim at providing a total powertrain torque able to satisfy the desired drivability performance in terms of longitudinal acceleration dynamics, through the integration of a direct feedforward term with a feedback contribution designed based on the Linear Quadratic Regulator (LQR) theory [35–37]; meanwhile, the low-level control allocation is designed to split the torque between the engine and the electric motor for emissions reduction purposes. The main contribution of the paper is to:


The paper is organised as follows: in Section 2 the nonlinear truck model—developed in a Matlab/Simulink environment—is shown and its characteristic dynamic equations are described, while in Section 3, the torque allocation control—based on the driver's intentions—is designed to reduce driveline oscillations and improve emissions performance. In Section 4, the controller performance is investigated to analyse the dynamic response of the powertrain and to evaluate the reduction of engine emissions. Finally, the conclusions of the work are discussed in Section 5.

#### **2. Hybrid Vehicle Powertrain Layout**

In this paper, a P0 mild parallel hybrid electric powertrain with a diesel engine was adopted. In this system architecture, better known as a belted alternator starter (BAS) [38,39], the internal combustion engine (ICE) and the electric motor (EM) are combined by means of a belt and work together to supply the necessary power to the vehicle. The EM is not able to propel the vehicle by itself; hence, it helps the ICE when extra power is required, behaving as a power booster [10]. With this configuration, an efficient engine start/stop functionality was ensured as well as the storage of energy in the battery during regenerative braking. Then, the power unit system was connected to the wheels through a gearbox and a differential.

The implementation of a parallel mild hybrid allowed us to benefit from the relevant advantages of the cooperation of EM and ICE, such as emissions reductions in case of slower transient behaviour in the ICE—better known as phlegmatization [40]—while the EM covered possible aggressive driver torque requests thanks to its high bandwidth—typical of electrical machines.

#### *2.1. Matlab/Simulink Nonlinear Truck Model*

The truck, shown in Figure 1, was modelled by means of a block-oriented approach [41,42], using a customized library of driveline components and adopting a user-friendly graphical interface—as shown in Figure 2. For each system component, it is necessary to define the dynamic laws by means of mechanical equilibrium principles, defining dynamic constants and constraints, and imposing the initial conditions. The input/output quantities of each component depend on the component topology. The Simulink model was grouped into subsystems, i.e., the engine, clutch damper, gearbox, differential, and wheels (2 driving and 2 driven), and a longitudinal truck vehicle model with longitudinal load transfer computation.

**Figure 1.** Transmission scheme of the parallel hybrid truck vehicle.

**Figure 2.** Nonlinear truck vehicle model in Matlab/Simulink environment.

Moreover, the model included the bi-linear elastic and damping behaviour of the torsional damper, the linear characteristic of the left and right-half shafts, a transient, nonlinear model for the tyres, and a Pacejka magic formulation [43] for road–tyre interactions.

Inside both the front and rear-tyre Simulink blocks, a wheel sensor was modelled, in order to apply a time delay and the effects of signal quantization to the simulated-wheel angular speed signal, to better reproduce experimental conditions.

#### *2.2. Dynamic Equations*

All the dynamic equations reported in this section are written in agreement with the free body diagrams of the drivetrain components and of the vehicle, which are reported in Figure 3.

**Figure 3.** Free body diagrams of the main driveline components.

#### 2.2.1. Power Source: ICE and EM

In the parallel hybrid configuration analysed, the ICE and the electric motor were connected by means of a belt, with a certain transmission ratio *τ<sup>b</sup>* = *ωEM*/*ωICE*; hence, the total driving torque *Te*, assuming 100% belt transmission efficiency, is given by:

$$T\_{\mathfrak{e}} = T\_{ICE} + T\_{EM} \,\mathfrak{r}\_{\mathfrak{b}} \tag{1}$$

The maximum torque of both the ICE and EM are reported in Figure 4. It is evident that the EM characteristic is symmetrical with respect to zero in case of traction or regenerative braking.

**Figure 4.** Torque characteristics of ICE and EM.

The acceleration of the engine .. *θ<sup>e</sup>* was obtained from the equilibrium between the driving torque *Te*, and the resistive torque of the primary shaft *Tp*, according to:

$$
\ddot{\theta}\_{\varepsilon} = \frac{T\_{\varepsilon} - T\_p}{f\_{\varepsilon}} \tag{2}
$$

where *Je* is the equivalent moment of inertia of the power source, so it depends on the ICE and EM inertias, which are related by the transmission ratio of the belted system as follows:

$$J\_e = J\_{ICE} + J\_{EM} \,\, \tau\_b^2 \tag{3}$$

#### 2.2.2. Transmission: From Clutch Damper to Differential

The torque at the primary shaft *Tp* was obtained from the bi-linear clutch damper behaviour—whose nonlinear characteristics are reported in Figure 5—according to Equation (4):

$$T\_p = T\_{cd} = k\_{cd} \left(\theta\_c - \theta\_p\right) + c\_{cd} (\dot{\theta}\_c - \dot{\theta}\_p) \tag{4}$$

where *kcd* and *ccd* are the torsional stiffness and the torsional damping of the clutch damper, respectively.

**Figure 5.** Elastic torque and viscous damping of the clutch damper.

The angular speed of the gearbox primary shaft . *θ<sup>p</sup>* is related to the secondary shaft angular speed . *θ<sup>s</sup>* by the kinematic relationship:

$$
\dot{\theta}\_p = \dot{\theta}\_s \ \tau\_d \tag{5}
$$

where *τ<sup>d</sup>* is the global driveline transmission ratio, depending on the actual gearbox gear ratio and the differential gear ratio, while the available torque at the secondary shaft—considering driveline efficiency *ηd*—is given by:

$$T\_s = \eta\_d \text{ } \tau\_d \text{ } T\_p = 2 \text{ } T\_{ds, \text{L}/R} = 2[k\_s(\theta\_s - \theta\_R) + c\_s(\dot{\theta}\_s - \dot{\theta}\_R)] \tag{6}$$

As reported in Equation (6), the drive shaft torque *Tds*,*L*/*<sup>R</sup>* was lumped using a viscous damper *cs* and a torsional spring *ks*. Instead, the torque *TG* = *Ts* − *Tp* corresponds to the torque supported by the gearbox constraints. Finally, torque dissipations due to transmission bearings [44] were modelled, but not reported in the FBDs of Figure 3.

### 2.2.3. Vehicle and Wheels

The vehicle model considers the longitudinal dynamics and load transfer caused by the vehicle acceleration *ax*, which is computed from the equilibrium equation—reported in Equation (7)—of the forces acting on the vehicle, i.e., the longitudinal forces on the tyres *Fx*,*FL*, *Fx*,*FR*, *Fx*,*RL*, *Fx*,*RR*, where the subscripts *FL*, *FR*, *RL*, *RR* refer to the front-left, front-right, rear-left and rear-right wheels, respectively, the aerodynamic force *Fa* and the resistance force *Fi* due to the road slope angle *α*.

$$a\_x = \frac{\sum\_{n}^{4} F\_{\mathbf{x},n} - F\_a - F\_i}{M\_v} \tag{7}$$

The term ∑<sup>4</sup> *<sup>n</sup> Fx*,*n*—where the subscript *n* refers to the *FL*, *FR*, *RL*, *RR* wheels—sums all the longitudinal forces on the four wheels, while the two terms *Fa* and *Fi* are as described by Equation (8):

$$\begin{cases} F\_a = 0.5 \,\rho \,\text{S}\_v \,\text{C}\_d \, V^2\\\ F\_i = M\_v \,\text{g} \, \text{sin}\,\alpha \end{cases} \tag{8}$$

Regarding the aerodynamic terms, *ρ* is the air density, *Sv* is the vehicle front area, *Cd* is the aerodynamic drag coefficient, and *V* is the vehicle speed. Moreover, each tyre is described by an additional degree of freedom, and since vehicle lateral dynamics are not taken into account, the angular accelerations .. *<sup>θ</sup><sup>F</sup>* and .. *θ<sup>R</sup>* are the same for the two wheels at the front and rear axles, and are computed according to Equation (9):

$$\begin{cases} \begin{array}{l} \ddot{\theta}\_F = \frac{-T\_{b,F} - F\_{x,F}R\_{\nu} - T\_{r,F}}{f\_{\nu,F}}\\ \ddot{\theta}\_R = \frac{T\_{\nu r} - T\_{b,R} - F\_{x,R}R\_{\nu r} - T\_{r,R}}{f\_{w,R}} \end{array} \tag{9}$$

where *Tw* = *Tds*,*L*/*<sup>R</sup>* is the wheel torque coming from the half shaft—which participates only for the rear wheels—*Tb*,*<sup>F</sup>* and *Tb*,*<sup>R</sup>* are the braking torques, and *Tr*,*<sup>n</sup>* (*Tr*,*<sup>F</sup>* for the front and *Tr*,*<sup>R</sup>* for the rear) is the rolling resistance torque, which has a quadratic relationship to the wheels' rotational speed *θn*, according to Equation (10):

$$T\_{r,n} = F\_{z,n} \, R\_w \, (f + \mathcal{K} \dot{\theta}\_n^2) \tag{10}$$

The terms *f* and *K* are the constant and quadratic coefficients of the rolling resistance and *Fz*,*<sup>n</sup>* is the vertical load on each wheel, which is computed considering the static load distribution between the two axles—by means of parameter *γ*—and the dynamic load transfer due to vehicle longitudinal acceleration, using Equation (11):

$$\begin{cases} \, \, \_{F\_{z,F}} = \frac{m \, \_{\frac{\alpha}{2}} \gamma}{\frac{2}{2}} - \frac{m \, h\_G \, a\_x}{2 \, L} \\ \, \_{F\_{z,R}} = \frac{m \, \_{\frac{\alpha}{2}} (1 - \gamma)}{\frac{2}{2}} + \frac{m \, h\_G \, a\_x}{2 \, L} \end{cases} \tag{11}$$

where *L* is the vehicle wheelbase and *hG* is the height of the vehicle's centre of mass, whereas *Fz*,*<sup>F</sup>* and *Fz*,*<sup>R</sup>* indicate the vertical load on the front and rear wheels, respectively. For each wheel, the longitudinal tyre slip is computed using Equation (12):

$$
\sigma = \frac{\omega R\_w - V}{\omega R\_w} \tag{12}
$$

For sake of simplicity, the tyre model assumes a constant relaxation length *Lt*, even if—for more accurate results—it should be considered dependent on the vertical load and longitudinal slip [26]. Introducing the relaxation length, it is possible to correct the tyre slip, obtaining a transient slip for each tyre. Finally, the Pacejka formulation was adopted in the tyre model to evaluate the longitudinal force *Fx*,*<sup>n</sup>* on each tyre, as a function of the different constant Pacejka parameters and three main dynamic quantities:

$$F\_{\mathbf{x},n} = f(F\_{\mathbf{z},n}, \mu, \sigma) \tag{13}$$

where *μ* is the tyre–road friction coefficient.

#### 2.2.4. NOx Modelling

As proposed in [14], the NOx emissions were modelled considering the transient dynamics of the ICE. It has been experimentally proven that, in case of sudden acceleration by the driver, there is a spike in NOx emissions—which is expected to be 80% higher than the steady state value. Thus, in the Simulink model, a transfer function of the second order was adopted with a variable gain with respect to the engine torque *TICE*. The NOx emissions were estimated using the transfer function reported in Equation (14):

$$NO\_X(s) = G(T\_{ICE}(s)) \ H(s) T\_c(s) \tag{14}$$

where *G*(*TICE*(*s*)) is a variable gain between the steady state NOx and the engine torque, while the second order transfer function *H*(*s*) is defined as:

$$H(s) = \frac{\omega\_n^2}{s^2 + 2\zeta\omega\_n + \omega\_n^2} \tag{15}$$

where *ω<sup>n</sup>* and *ζ* have been properly tuned to reproduce the transient dynamics relationship between the NOx emissions and the engine torque proposed by [14]. The NOx model here adopted takes into account only the engine-out emissions, without considering an after-treatment system device.

#### **3. Closed Loop Control System**

In this paper, the torque allocation control logic was designed to improve the vehicle drivability, to damp driveline oscillations, and to reduce engine emissions. The schematic diagram of the control strategy is reported in Figure 6. The total torque requested by the high-level part of the control logic *Te* was obtained using the FeedForward (FF) contribution *Te*,*FF*, supported by the FeedBack (FB) intervention *Te*,*FB*. The total torque was then elaborated by the control allocation strategy (low level) to split the contribution between the engine *TICE* and the electric motor *TEM* that were the input for the nonlinear model described in the previous section.

**Figure 6.** Driver control allocation control scheme.

#### *3.1. High-Level Control Strategy*

The gas pedal position (GPP) imposed by the driver was converted into a requested torque *Tdr* through the steady-state engine map in Figure 4 and then filtered out with a cut-off frequency of 2 Hz to elaborate the FF contribution *Te*,*FF*.

The FB terms were designed based on a five-state linearised model, whose state space representation was defined according to the methodology presented in [11], where the effect of the torsional damper and load transfers was neglected with respect to the nonlinear model:

$$\begin{Bmatrix} \dot{x} \end{Bmatrix} = [A] \{x\} + [B\_1] v + [H] \tag{16}$$

where *v* = *Te* is the control output and {*x*} is the state vector, composed as follows:

$$\begin{bmatrix} \mathbf{x} \end{bmatrix} = \begin{bmatrix} \mathbf{x}\_1 & \mathbf{x}\_2 & \mathbf{x}\_3 & \mathbf{x}\_4 & \mathbf{x}\_5 \end{bmatrix}^T = \begin{bmatrix} \frac{\theta\_p}{\mathbf{x}\_d} - \theta\_R & \dot{\theta}\_R & \dot{\theta}\_p & \dot{\theta}\_F & T\_t \end{bmatrix}^T \tag{17}$$

where *Tt* = (*Fx*,*RL* + *Fx*,*RR*) *Rw* is the steady-state driven wheel torque. Matrices [*A*], [*B*], and [*H*] are described in [11]. The coordinate system of the state-space in Equation (16) is converted from the state vector {*x*} of the error dynamics representation, as follows:

$$\begin{Bmatrix} \dot{\varepsilon} \end{Bmatrix} = [A] \begin{Bmatrix} \varepsilon \end{Bmatrix} + [B\_1] \begin{Bmatrix} v' \end{Bmatrix} \tag{18}$$

where {*e*} = *xRef* <sup>−</sup> {*x*} and *<sup>v</sup>* <sup>=</sup> *Te*,*FF* <sup>−</sup> *<sup>v</sup>*, with *xRef* represents the reference state vector, which is elaborated from the driver requested torque *Tdr*:

$$\begin{Bmatrix} \mathbf{x}\_{Rcf} \end{Bmatrix} = \begin{bmatrix} \frac{T\_{dr}\mathbf{r}\_d}{k\_i} & \dot{\boldsymbol{\theta}}\_{Rcf} & \dot{\boldsymbol{\theta}}\_{Rcf}\mathbf{r}\_d & \dot{\boldsymbol{\theta}}\_{Rcf} & T\_{dr}\mathbf{r}\_d \end{bmatrix}^T \tag{19}$$

The reference wheel speed . *θRef* is obtained by integrating the reference acceleration .. *θRef* , defined based on the driver's intention (*Tdr* and *τd*), and the measured wheel speed . *θF*,*m*:

$$\ddot{\theta}\_{Rcf} = \frac{T\_{dr}\eta\_d \tau\_d - 0.5 \,\rho \,\text{S}\_v \,\text{C}\_d \left(\dot{\theta}\_{F,m} R\_w\right)^2 R\_w - M\_v g \left(f + K \,\dot{\theta}\_{F,w}^2\right) R\_w}{\left(f\_{w,F} + f\_{w,R} + M\_v R\_w^2 + f\_c \eta\_d \tau\_d^2\right)}\tag{20}$$

The measured wheel speed . *θF*,*m*, as mentioned in Section 2, was obtained using simulated sensor dynamics that introduced a communication time delay and the digitalization of the continuous measurement signal (zero order hold technique).

The FB torque contribution was then calculated as a full state feedback law:

$$w' = -\left[K\_{FB}\right]\{e\}\tag{21}$$

where the feedback gains, saved in [*KFB*], are elaborated based on the LQR design—as proposed in [35,36]. The LQR control allowed the asymptotical stability of the closed loop system and the minimisation of the quadratic performance functional *J*:

$$J = \int\_0^\infty (\{e\}^T [Q] \{e\} + r v'^2) dt \tag{22}$$

where [*Q*] = *diag*(1*<sup>e</sup>* − 9 1 1/*τ*<sup>2</sup> *<sup>d</sup>* 1 1*e* − 9) is a diagonal weighted matrix where the elements relative to angular speeds are emphasized with respect to the other states.

Finally, the feedback matrix [*KFB*] was obtained as:

$$\left[K\_{FB}\right] = r^{-1} \left[B\_1\right]^T \left[P\right] \tag{23}$$

where *P* is the solution of the algebraic Riccati equation.

The total requested torque was then formulated as:

$$T\_{\varepsilon} = \upsilon = T\_{\varepsilon, FF} - \upsilon' = T\_{\varepsilon, FF} + \left[K\_{FB}\right]\{\varepsilon\} \tag{24}$$

To implement a full state feedback controller, a Kalman filter was also designed to estimate the state vector *x*ˆ, based on the inputs of the nonlinear vehicle model *v* and on its measured vector *y*.

$$\{\mathcal{Y}\} = \begin{bmatrix} \omega\_{\mathfrak{e}} & \omega\_{\mathcal{F}} \end{bmatrix} = [\mathbb{C}] \{\mathfrak{x}\} \tag{25}$$

where [*C*] = 00100; 00010 is the output matrix.

#### *3.2. Control Allocation*

The total torque *Te* was then split between the ICE and the EM by the control allocation strategy. In this paper, the ICE provided the main torque contribution and the EM only intervened to compensate for the dynamic limits of the engine, e.g., when the driver requested a sudden torque through the GPP. The scheme of the control allocation strategy is reported in Figure 7.

**Figure 7.** Control effort distribution between the ICE and EM, according to the dynamic limits of the engine.

As explained by several authors [14–16], the dynamic performance of the ICE is usually limited to reduce the engine emissions that may occur in presence of extremely dynamic torque requests. To accomplish the same task, a rate limiter was introduced into the control allocation strategy to saturate the maximum torque rate of the engine. Therefore, the torque of the EM, *TEM*, is given by the difference between the total torque request *Te* and the output torque requested to the ICE. As indicated in the scheme, the EM torque request is limited by a rate limiter, which was set at a higher level with respect to the ICE rate limiter.

In order to obtain the full benefits achieved by the methodology proposed in this paper, the energy stored in the battery should be sufficient to guarantee the requested activation of the electric motor. This energy management strategy is responsible for storing energy in the battery when the engine is more efficient, with lower polluting emissions. To exploit the benefits of this methodology, a more comprehensive design for the energy management system is required, but this is out of the scope of this paper.

#### **4. Results**

The torque allocation control designed in the previous section was finally applied to the nonlinear model described in Section 2 in order to evaluate its reference tracking performance, torque split efficacy between the ICE and the EM, and its influence on NOx emissions.

#### *4.1. Reference Tracking Performance*

The main purpose of the torque allocation control is to satisfy drivers' requests in terms of vehicle longitudinal acceleration. The driver's intention, expressed in terms of a reference longitudinal acceleration *ax*,*ref* <sup>=</sup> .. *θRef Rw*, is converted into a total torque request *Te*, which is obtained by the sum of the FF and FB contributions. The simulation results obtained during a tip-in manoeuvre—i.e., an instantaneous wheel torque request—are shown in Figures 8 and 9, where the influence of the two controller contributions is highlighted.

**Figure 8.** Vehicle state tracking performance during a tip-in manoeuvre with only the FF torque contribution (**a**) and with FF + FB integration (**b**): longitudinal acceleration (top left), front wheels, rear wheels, and primary shaft speeds (top right), rear axle deformation (down left) and torque *Tt*.

**Figure 9.** Control torque request (top), jerk (centre) and weighted states errors (down) during a tip-in manoeuvre with only the FF torque contribution (**a**) and with FF + FB integration (**b**).

The control, with and without the activation of the FB contribution, provided a satisfactory reference tracking performance for the speed states. However, the presence of the FB contribution improved the transient vehicle response, with an evident oscillation reduction for the whole set of states. This also implies a considerable improvement in vehicle acceleration performance and in the consequent amplitude of the corresponding jerk (i.e., derivative of the vehicle acceleration), with beneficial implications for vehicle comfort. The advantages provided by the FB integration were paid off with a peak torque request of about 20% higher than that of controller with only the FF term activated.

The FB intervention also represented a fundamental contribution to the rejection of external disturbances that may affect controller tracking performance. Indeed, the FF formulation was based on the simplified linear model, and did not include motion resistance increments due to sudden road slopes. The influence of step disturbances on the road slope is analysed in Figure 10.

**Figure 10.** Control torque request (**a**), road slope (**b**), longitudinal acceleration (**c**) and weighted states errors (**d**) during a tip-in manoeuvre with the FF + FB controller mode.

The application of the torque requested by only the FF term would compromise the desired performance of the torque controller if external disturbances or uncertain/unmodelled dynamics affected the vehicle dynamics. Indeed, the FF formulation did not include the influence of the road slope, since this is a quantity that is difficult to measure or estimate—thus not representing a suitable input for feeding into a control logic. For this reason, the intervention of the FB term is essential to compensate for the higher vehicle motion resistance by applying an additional torque contribution that is able to bring the longitudinal acceleration back to the reference value; this was automatically achieved

with an acceptable worsening of state errors and without any information related to the road profile.

The effect of the FB contribution was, however, influenced by the design of the LQR cost function and the selection of the weights *qi* and *r*. The optimal control theory [35,36] suggests that the ratio *qi*/*r* is responsible for the trade-off between reference tracking performance and energy consumption. A sensitivity analysis of the weight *r* on the torque controller performance is shown in Figure 11.

For low *r* values, the FB contribution aims at improving the reference tracking performance by reducing state errors and the amplitude of the vehicle acceleration and jerk oscillations. This advantage is obtained with a transient peak power request, i.e., the product *Te* . *θ<sup>e</sup>* of the engine and the electric motor. Vice versa, an increment of *r* would save more energy by reducing the total power request, but with an evident decline in terms of vehicle performance and comfort.

#### *4.2. Control Allocation Evaluation*

The previous section focused attention on the performance of the FF + FB controller in order to ascertain a reference total torque *Te* that is able to dampen oscillations in vehicle acceleration, even in the presence of external disturbances, e.g., a road slope. However, the torque controller is not designed to take into account the powertrain's limits in terms of engine/electric motor torque saturation and dynamic constraints. Indeed, the ICE is usually operated at low frequency dynamics to reduce combustion emissions when a steep torque is requested, e.g., during a tip-in manoeuvre. The presence of a redundant driving source, i.e., the EM, allows the ICE's limitations to be overcome in terms of combustion emissions without compromising the vehicle's drivability. To evaluate the influence of ICE dynamics on the torque controller performance, a sensitivity analysis of the ICE rate limiter (see the scheme in Figure 7) is reported via the simulation results shown in Figures 12 and 13.

**Figure 12.** Torque split during a tip-in manoeuvre with only the ICE (**a**) and with a hybrid configuration (**b**) for different constraints on the ICE torque slope saturation.

**Figure 13.** *Cont*.

**Figure 13.** Longitudinal acceleration during a tip-in manoeuvre with only the ICE (**a**) and with a hybrid configuration (**b**) for different constraints on the ICE torque slope saturation.

The results presented in Figure 13a show how the dynamics of vehicle acceleration are affected by the rate limitation on the ICE-requested torque *TICE*. The lower the rate constraint on the *TICE*, the higher the overshoot and the rising time of the acceleration. These results are well explained, as the rate saturation on the torque requested by the ICE drastically influences the total torque *Te*—see Figure 12a—demanded by the controller, which acts in order to reduce the states errors and so the acceleration error. The presence of an electric machine, with a typically faster response and lower emission issues than a combustion engine, allows the ICE to operate at lower frequency dynamics—thus achieving the desired performance in terms of vehicle acceleration, as shown in Figure 13b. The performance of the hybrid configuration is totally unaffected by limitations on the engine torque rate.

An important consequence of the results presented in Figures 12 and 13 is that the redundancy of power sources can also produce effective advantages in terms of the NOx emitted by the ICE. Indeed, assuming that the correlation between NOx emissions and the ICE output torque is expressed by Equation (15) [14], the benefits of reduced ICE dynamics are reported in Figure 14.

**Figure 14.** NOx emissions during a tip-in manoeuvre with the hybrid configuration for different constraints on the ICE torque slope saturation.

The hybrid configuration is then able to achieve the desired drivability performance, with a reduction of almost 80% in the transient NOx overshoot peak for the case of a 100 Nm/s torque rate saturation.

#### **5. Conclusions**

The paper presents a methodology for designing a torque allocation control strategy aiming at improving vehicle performance and NOx emissions. The results obtained can be summarized in the following final remarks:


The results confirmed the methodology's efficacy for vehicle drivability performance and engine-out emissions. However, there is still potential scope for exploring the impact of the torque control strategy on other important features related to hybrid powertrains, e.g., energy management performance, clutch engagement, and regenerative braking benefits, which are reserved for future activities.

**Author Contributions:** Conceptualization, L.D., A.T., E.G. and M.V.; methodology, L.D., A.T., E.G. and M.V.; software, L.D., A.T., E.G. and M.V.; validation, L.D., A.T., E.G. and M.V.; formal analysis, L.D., A.T., E.G. and M.V.; investigation, L.D., A.T., E.G. and M.V.; writing—original draft preparation, L.D., A.T., E.G. and M.V.; writing—review and editing, L.D., A.T., E.G. and M.V.; visualization, L.D., A.T., E.G. and M.V.; supervision, E.G. and M.V. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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