Nonlinear model-predictive driving control of ICE vehicles equipped with CVTs via relaxation of switching behaviours

In this study, a design method of driving controller based on nonlinear model predictive control (NMPC) is presented for internal-combustion-engine (ICE) vehicles equipped with continuously variable transmissions (CVTs). The clutch-engaged and -released modes and engine-on and -off modes are considered, along with the engine-braking effect. The switching characteristics of the vehicle system dynamics are relaxed into continuous transitions. This allows for the use of standard NMPC software packages to solve the NMPC problem. Numerical examples indicate that the designed controller is capable of achieving real-time driving control based on NMPC.


Introduction
The continuous advancement of vehicular technologies aimed at enhancing safety, comfort and ecofriendliness is a critical objective within mobility society.A strategy for accomplishing this involves the control of vehicle systems to augment these desired characteristics.For instance, unstable vehicle motions (such as spinning out) during steering manoeuvres can frequently be mitigated via suitable adjustment of the braking force on each wheel [1].This strategy drives the development of advanced driver-assistance systems (ADASs) and, more broadly, lays the groundwork for automated driving concepts.These technologies have garnered significant interest in recent years [2,3].
Automated driving systems, along with certain ADASs, modulate the vehicle propulsion/braking force and steering angle to enhance desirable attributes.This modulation process is referred to as driving control.Optimal control presents a viable strategy for the development of driving control systems because the enhancement of desired properties often naturally translates into a minimization problem for a cost functional.In practical scenarios, offline (i.e.open-loop) optimal control is not feasible as the entire control duration is not always pre-determined, and road conditions, such as surrounding vehicles and pedestrians, are subject to real-time changes.Hence, in recent years there has been a growing interest in driving control based on nonlinear model predictive control (NMPC) [4][5][6].
Vehicles generally encompass a range of switched components.For instance, internal combustion engine (ICE) vehicles (ICEVs) are equipped with clutches.
When engaged, the clutch allows the engine's output torque to be transferred to the wheels, enabling vehicle propulsion or retardation via engine braking.Conversely, when the clutch is disengaged, a physical disconnect occurs between the engine and the vehicle, thereby interrupting the torque transfer.This disengagement occurs not just during gear changes but also during coasting periods when there is no need for engine braking or fuel consumption.This type of an operation, termed as free-wheeling technology, has been recognized as a significant contributor to fuelefficient driving [7,8].
Implementing NMPC-based driving control in vehicles with switched components necessitates solving a hybrid optimal control problem (HOCP) at each sampling interval.Over the years, various numerical solution methods for HOCP, such as dynamic programming, indirect methods and direct methods, have been proposed [9].These methods have their respective advantages and disadvantages, and as such, the development of solution methods for HOCP is still an active area of research (e.g.[9,10]).The challenge intensifies during online implementation, as the optimization computation must terminate within each sampling period (typically 1-100 ms) on an onboard computer with limited computational capabilities.This requirement generally makes it unfeasible to apply the (offline) solution methods of HOCP, rendering the NMPC-based driving control of vehicles with switched components a complex task.The difficulty of this task has resulted in very few reported results on the topic.In [11,12], NMPC-based driving controllers were designed using offline solution methods, in which the resulting computation time was not sufficiently fast for real-time implementation.
In this study, we aim at designing an NMPC-based driving controller for an ICEV equipped with a continuously variable transmission (CVT).The CVT ratio (i.e. the transmission ratio of the CVT) is assumed to be pre-scheduled according to two ratio maps.The switched transitions between clutch-engaged andreleased modes, as well as engine-on and -off modes, are considered.To accommodate the switched dynamics of the vehicle system, these switched transitions are transformed into continuous changes.This transformation allows the HOCP to be converted into a continuous optimal control problem.Over the past few decades, NMPC problems based on continuous optimal control problems have been the subject of extensive study.This has led to the development of a variety of efficient NMPC algorithms, which have been distributed as software packages, such as ParNMPC [13,14] and C/GMRES [15,16].This suggests that the proposed controller, derived through this method of relaxation, has a significant potential for implementation across a broad range of practical scenarios.The methodology for proposed modelling and control design, facilitated through this process of relaxation, exhibits the following characteristics.
• Unlike previous studies, such as [4][5][6]11,12], the engine-braking effect due to energy losses, like pumping loss and mechanical friction loss, is incorporated into the vehicle model and controller design.• In contrast to existing works [10,17], the engine on/off command is not independently controlled but is tied to the vehicle propulsion-force command, without loss of generality (Section 2.5).This approach reduces the number of control variables without increasing the number of state variables, potentially lessening the computational burden of real-time optimization.• Prior knowledge of the switching structure, such as the mode sequence, switch timing and the number of switches, is not required in the proposed method.This stands in contrast to many offline HOCPs, which often require such knowledge (e.g.[10]).• The proposed method of relaxation is intuitive and straightforward: it replaces switching signals with sigmoid functions and adds a penalty term to the cost functional to prevent the sigmoid functions from constantly taking intermediate values.Despite its simplicity, the numerical examples in this paper show that the proposed controller functions effectively (Section 5).
The designed driving controller is evaluated using numerical examples.The results indicate that the controller is capable of achieving real-time driving control based on NMPC.
Preliminary results were presented in the authors' conference presentations [18,19].In these preliminary studies, the clutch and engine-braking effect were not considered, whereas these factors exist in all ICEVs and have crucial effects on driving.From a driving-control perspective, adding these factors produces two driving modes, namely coasting and engine braking.Although this makes the control problem more involved, it is properly handled in this paper.Moreover, the realtime computability was not verified in the preliminary studies.
The rest of this paper is organized as follows.In Section 2, we describe the architecture of objective vehicle and develop its dynamical model.Its switching behaviours are relaxed in Section 3. The NMPC-based driving control problem is formulated in Section 4. The designed controller is tested on numerical examples in Section 5.In Section 6, the conclusion of the study is presented.

Notation
In the following sections, R represents the set of real numbers.Symbols of the form F * (where " * " is a wild card) signify forces acting on the wheels, and f * designates an intermediate function used to define F * , expressed in terms of force.Positive values of these quantities indicate vehicle propulsion.The symbol f, when without a subscript, represents the vector field of the state equation, not forces, as is common practice.The subscripts "c ", "e ", and "a" correspond to the clutch, engine and alternator, respectively.Function arguments may frequently be omitted if it does not result in confusion.

Overall description
The target vehicle is an ICEV equipped with a CVT.Its architecture is depicted in Figure 1.In this section, a vehicle model is developed involving a binary control and switching signal in their original form.These discrete and switching characteristics will be relaxed in Section 3.
The vehicle dynamics are modelled as a state equation of the form where t denotes the time parameter, x denotes a twodimensional state vector, u denotes a three-dimensional input vector, and f denotes a mapping The vectors x and u are defined as follows: where p [m] denotes the travelled distance along the road of interest, v [m/s] denotes the (longitudinal) vehicle speed, u c denotes the clutch engage/release command defined in Section 2.4, u e denotes the vehicle propulsion-force command defined in Section 2.5, and u a denotes the electricity generation rate of alternator defined in Section 2.7.The function f will be detailed in the following sections.
The initial condition is given as follows: where p 0 and v 0 denote the given initial states.
The instantaneous fuel consumption of the engine, denoted by W f [cc/s], will be modelled in Section 2.8, as well as the fuel-consumption equivalent of the electrical power generated by the alternator, denoted by W a [cc/s].These quantities appear in the cost functional (Section 4) as opposed to that in the state equation.

Vehicle motion
The vehicle motion is modelled as follows: where m [kg] denotes the vehicle mass, and F r , F a , F e [N] denote the forces acting on the wheels, which represent the driving resistance, braking force due to the electricity (re)generation of the alternator, and the propulsion/braking force delivered from the engine, respectively.These forces will be defined in the rest of this section.A constraint is imposed as follows: with pre-specified positive constants v min and v max .

Clutch
The clutch engaged/released status is defined as This is essentially a binary control input.To apply the relaxation method that will be derived in Section 3, let us transcribe the clutch status as by introducing a continuous input variable u c ∈ R.
The switching characteristic of s c will be relaxed in Section 3.
A box constraint is imposed as with arbitrary constants u min e 0 and u max e 0. This prevents u c from being blown away to −∞ or ∞.

Engine
The engine on/off status is defined as follows: The driveline losses arising in the interval from the clutch to the wheels is assumed to be negligible as typically done in driving control [20,21].Hence, the engine output torque is directly transferred from the engine output shaft to the wheels.When the clutch is released, the engine is assumed to be kept off.When the clutch is engaged, the value of the vehicle propulsion/braking force F e [N] from engine is classified according to Table 1.The case F e >0 arises when the engine output torque is positive, indicating that the theoretical engine torque surpasses the energy losses within the engine.The case F e = 0 arises when the engine output torque is zero, implying that the theoretical engine torque is just sufficient to offset the energy losses.Both these cases occur only in engine-on mode (σ e = 1), and they are respectively labelled as "Propulsion" and "Balance".It should be noted that the case F e <0, where the theoretical engine torque is insufficient to cancel out the energy losses, is excluded in engine-on mode because this situation is typically avoided in commercial vehicles.In engine-off mode (σ e = 0), only F e = f eb occurs, where f eb (<0) [N] represents the braking force resulting from the engine-braking effect.This case is labelled as "Engine braking".The engine braking force is modelled as follows:  where τ eb (<0) [Nm] denotes the engine-braking torque defined in Figure 2 where ε e <0 and k>1 denote design parameters, and F max e [N] denotes the maximum vehicle-propulsion force from the engine, which is pre-specified.The shape of these functions is illustrated in Figure 3.It should be noted that f e is a class-C k−1 function.Using these functions, σ e and F e can be described as follows: ) The above descriptions can be explained as follows.
When the clutch is released (s c = 0), we have σ e = 0 and F e = 0 as expected.Now, let us consider the case in which the clutch is engaged (s c = 1): when 0 ≤ u e ≤ 1, the result is σ e = 1 and F e = F max e u k e , corresponding to the "Propulsion" scenario; when ε e ≤ u e ≤ 0, the result is σ e = 1 and F e = 0, corresponding to the "Balance" scenario; when u e <ε e , the result is σ e = 0 and F e = f eb , corresponding to the "Engine braking" scenario.From these observations, it is evident that σ e and F e can be controlled by manipulating the single and continuous control input u e , without any loss of generality.Moreover, the vehicle propulsion power delivered from the engine is defined as Some NMPC software (e.g.AutoGenU [15,16] and ParNMPC [13,14]) do not accept if-then descriptions, such as (8), in their problem formulations.To apply the proposed method through the NMPC software, one can use the following Softplus-type [22] function instead of f e : where a SP >0 denotes a design parameter.The shape of this function is illustrated in Figure 4.This function is of class-C ω and for a sufficiently large a SP : The function fe plays a role similar to f e in the proposed method.
A box constraint is imposed as where u min e denotes an arbitrary constant sufficiently smaller than ε e .The minimum value u min e is set to prevent u e from being blown away to −∞.The switching characteristic of s e will be relaxed in Section 3. It should be noted that the right-hand sides of ( 8), ( 11), ( 9) and ( 10) are explicitly described in the state variables and input variables.Hence, we obtained the explicit expressions of σ e (u), F e (x, u), and P e (x, u). (13)

Remark 2.1:
We can consider developing a simpler model of the form with the constraint u th e ≥ 0, where u th e denotes the input variable representing the vehicle propulsion force produced by the theoretical engine torque and f el (<0) denotes the braking force produced by the energy losses in the engine such as heat-transfer loss, incompletecombustion loss, pumping loss and friction loss.It should be noted that f el coincides with f eb (enginebraking effect) in the engine-off mode (i.e. when u th e = 0 in this case).For the sake of simple discussion, let us consider the case in which σ c = 1.The formulation provided in (14) appears to be more beneficial than (10) because of its non-switching structure.However, this formulation is less practical than (10) for the following reasons: modelling the braking force f el (ω e , u th e , . . . ) in the engine-on mode is difficult even through experiments; this formulation allows F e to take a negative value in the engine-on mode (when 0<u th e < f el (ω e , u th e , . . . ) ), which should be avoided as explained above.Therefore, we use model (10) in this study.

Transmission
The engine speed ω e and vehicle speed v are related by the following equation: where the speed ratio G [rad/m] is composed of the CVT ratio, gear ratio of final drive and wheel radius.Since the latter two quantities are (regarded as) constant, we refer to G as "CVT ratio" below.As stated in the introduction, the CVT ratio G is assumed to be pre-scheduled according to two ratio maps, which are prescribed.These maps are switched depending on the clutch status σ c and engine status σ e , as follows: where G α and G o denote the prescribed ratio maps provided for each engine mode.Note that the CVT ratio G is not a control input.The map G α is defined as where ω α (P e ) denotes the function defined in Figure 5, which corresponds to the target engine speed in the engine-on mode.The map G o is defined as where ω o (v) denotes the function defined in Figure 6, which corresponds to the target engine speed in the engine-off mode.
The engine speed is constrained by where ω max e [rad/s] denotes the maximum engine speed, which is pre-specified.

Alternator
The braking force F a (≤ 0) [N] due to the electricity (re)generation of the alternator is modelled as  where f max a (≤ 0) [N] denotes the maximum possible braking force due to the electricity (re)generation, and u a ∈ R denotes the continuous input variable representing the (re)generation ratio, which is constrained by The force f max a is represented as where P max a (≥ 0) [W] denotes the maximum possible power generated by the alternator defined in Figure 7.
It should be noted from ( 18), ( 21) and ( 19) that we obtained the explicit expression of F a (x, u).

Fuel consumption and electricity generation
The instantaneous fuel consumption W f (≥ 0) [cc/s] of the engine is modelled as where w f (>0) [cc/s] denotes the instantaneous fuel consumption in the engine-on mode defined for P e ≥ 0, which is defined in Figure 8.The fuel-consumption equivalent of the electrical power (re)generation by the alternator is modelled as where H [J/mL] denotes the lower heating value of petrol.It should be noted that W a denotes either zero or negative because the alternator works only for (re)generation.Furthermore, from ( 13), ( 18), (22), and (23), we obtained the explicit expressions of W f (x, u) and W a (x, u).The total consumption of the fuel equivalents in the vehicle system is described as

Driving resistance
The driving resistance force F r (<0) [N] is modelled as where C d denotes the drag coefficient, A s [m 2 ] denotes the maximum vehicle cross-section area, ρ a [kg/m 3 ] denotes the air density, μ r denotes the rolling resistance coefficient, g [m/s 2 ] denotes the acceleration of gravity, and θ [rad] denotes the road slope.The first term represents air drag, second term denotes the rolling resistance, and third term denotes the gravitational force.
The road slope information θ(p) is assumed to be available via vehicle-to-infrastructure communication.Obviously, we obtained the explicit expression of F r (x).

Preliminary
The switching signals found in the vehicle model are s e and s c , each defined by Equations ( 7) and ( 4), respectively.To relax these switching characteristics, the sigmoid function is introduced: where ν denotes a variable, and ε and α>0 denote design parameters which correspond to the horizontal translation and slope of the function, respectively.It should be noted that this is a smooth function.

Relaxation of engine mode switching
To relax the switching characteristic of s e , we approximate s e by the sigmoid function as s e (u e ) ≈ se (u e ) := sgm(ε e , α e ; u e ), (24) with the design parameters ε e <0 and α e >0.The shape of s e and se is illustrated in Figure 9.Although ( 24) is a common relaxation method, it may lead to the situation that se keeps taking an intermediate value such as 0.5, which must be avoided.To resolve this problem, let us introduce a penalty term denoted by L e (u e ), which satisfies L e (u e ) ≈ 0, se (u e ) ≈ 1 or se (u e ) ≈ 0, This term is added to the cost functional (Section 4).In this study, we put where parameter β e denotes the scaling coefficient of u e -axis, in which the scaling is centred at u e = ε e .The shape of s e , se , L e , and f e is illustrated in Figure 9. Owing to the penalty term, we can expect that the region of "intermediate values" (hatched in the figure) will be avoided or quickly passed through.It should be noted that this penalty term differs from the time derivative ds e dt , which vanishes even when the value of se is steady in the "intermediate" zone.

Relaxation of clutch mode switching
The relaxation procedure provided in the previous section also applies to the clutch.According to the procedure, we approximate s c as with the design parameter α c >0.The penalty term is defined as with the scaling coefficient β c .

Remark 3.1:
The signals u e , u a , and s e may take arbitrary values when s c (= σ c ) = 0 because they do not affect the vehicle dynamics as indicated in ( 9), ( 10), ( 16), ( 19), (22), and (23).Furthermore, u c and u e might not be unique for certain values of s c and s e , respectively, as suggested by definitions ( 4) and ( 8).These ambiguities could lead to non-unique optima in the NMPC problem to be formulated.Details of this issue will not be delved into as it falls outside the scope of this paper.However, it is expected that the proposed relaxation method could reduce these redundancies, by its approximation nature.For instance, the values of sc (u c ) for u c = −1 and −0.5 show slight differences, in contrast to the fact that s c (−1) = s c (−0.5) = 0.Moreover, since sc (u c ) = 0 holds even when the clutch is released (u c 0), the values of u e , u a , and s e exert slight influences on the NMPC problem (refer to, for example, Equations 10 and 23).These properties may aid in reducing the redundancies while approximating the switching characteristics.It is worth noting that, even in this scenario, the effects of u e , u a , and s e on vehicle behaviours are practically negligible.

NMPC formulation
In what follows, the relaxed vehicle model will be discussed.Let us define the cost functional as The first term is the instantaneous fuel economy that encourages eco-driving, second term denotes speed tracking, third term denotes the ride comfort, and the last two terms denote the penalty terms defined in (25) and ( 26).These terms are weighted by positive constants r W , r v , r F e , r s e , and r s c .Note that the first term reduces the instantaneous fuel consumption per distance; therefore, it does not encourage the vehicle to stop.
Since there is no hybrid characteristic in this problem, we can use standard (non-hybrid) NMPC algorithms.

Settings
The road slope is set as Figure 10.The parameters of vehicle model and NMPC formulation are set as Table 2, where t denotes the sampling period.
We use ParNMPC [13,14] ver.1903-1 with MAT-LAB R2020b on a laptop computer (Intel R Core TM i7-1065G7 CPU @1.30GHz/1.50GHz; 16.0GB RAM) to execute the proposed NMPC computation.Since ParN-MPC does not allow if-then rules in its problem formulation, we select (11) in the proposed method.At this stage, the plant model is the relaxed one that is identical to the one used in the controller design.
We pose two control objectives: fuel-economic speed tracking and eco-driving as follows.

Results and discussion: fuel-economic speed tracking
We set the weight coefficients as r W = 1 × 10 −6 , r v = 2 × 10 5 , r F e = 1 × 10 −6 , which represent the fuel-economic (and comfortable) speed tracking.The weights for penalty terms are set as r s e = 1 × 10 −2 and r s c = 2 × 10 −2 .The design parameters of the penalty terms are set as β e = 2 and β c = 0.8.We obtained the result shown in Figures 11-14.
We can observe that signals sc and se took only values of approximately 0 or 1.The results confirm that the  proposed relaxation method accurately approximates the behaviour of the switching signals s c and s e .
Initially, the vehicle decelerated to achieve the target speed by utilizing the engine braking force.This was accomplished by adjusting u c and u e so that sc (= σ c ) ≈ 1 and se ≈ 0 would hold, leading to σ e ≈ 0 and F e ≈ f eb (refer to (10)).This operation corresponds to the engine braking mode (refer to Table 1).Throughout this deceleration period, the alternator was fully activated (u a ≈ 1) to enhance deceleration.The CVT ratio was set to G ≈ G o as required (refer to Equation 16).Consequently, no fuel was consumed and the alternator was able to regenerate power.
Once the vehicle speed attained its target value, the vehicle transitioned to the propulsion mode (Table 1) to maintain the target speed.This mode was initiated by adjusting the input u e to make se ≈ 1 and σ e ≈ 1 hold and F e positive.Consequently, the CVT ratio switched to G ≈ G α as needed.Fuel was consumed to generate the propulsion force.Generally, higher power generation by the alternator leads to greater resistance, which in turn increases fuel consumption for vehicle cruising.The generation ratio observed in this example, approximately 0.5, was a result of balancing this trade-off.Even as the vehicle climbed uphill during this propulsion period, the vehicle speed remained at the target value due to appropriate manipulation of the propulsion force.
The computation time at each sampling instant was significantly shorter than the sampling period.Moreover, all the constraints were satisfied.These results suggest that the proposed method successfully achieves real-time NMPC while managing the switching characteristics.

Results and discussion: eco-driving
We set the weight coefficients as r W = 1 × 10 −6 , r v = 0 and r F e = 1 × 10 −6 , which represent the (comfortable) eco-driving.The weights for penalty terms are set as r s e = 1 × 10 −2 and r s c = 8 × 10 −3 .The design parameters of the penalty terms are set as β e = 1 and β c = 0.8.We obtained the result shown in Figures 15-18.
Similar to the speed-tracking scenario discussed in the previous section, the proposed relaxation method worked effectively to approximate the behaviour of switching signals.
Initially, the vehicle glided, i.e. the clutch was released, to minimize fuel consumption while maintaining the vehicle speed as high as possible.This gliding was achieved by manipulating u c so that sc (= σ c ) ≈ 0. It is important to note that σ e ≈ 0 held automatically by Equation (9).During this gliding period, fuel consumption ceased (refer to Equation 22).The propulsion/braking force from the engine was not exerted on the vehicle, i.e.F e ≈ 0 (see Equation 10).These    behaviours contribute to reducing the first and third terms of the cost functional (27).Regeneration by the alternator was automatically halted (see Equation 23).The CVT ratio was adjusted to G ≈ 0 as required (refer to Equation 16).Note that the effects of u e , u a and se were almost negligible during this period (see Remark 3.1).
Once the vehicle speed decreased, the vehicle switched to the propulsion mode for cruising.The clutch became engaged (s c ≈ 1), the engine state turned σ e ≈ 1, and F e was made positive.Accordingly, the CVT ratio was adjusted to G ≈ G α as required.Fuel was consumed to generate the propulsion force.The alternator's generation ratio was managed to balance the trade-off, as demonstrated in the speed-tracking example.F e responded to changes in road slope.
The computation time at each sampling instant was significantly shorter than the sampling period.Moreover, all constraints satisfied.These results suggest that the proposed method successfully achieves real-time NMPC while managing the switching characteristics.

Comparative results
We compared the proposed method with the existing methods of HOCP relaxation.
A typical method of HOCP relaxation is embedding [9].However, this method is not applicable to the present problem because the control-dependent switching mechanism involved in (7) cannot be handled by the embedding method.This highlights an advantage of the proposed method.
Another typical method is to only replace the switching signals s c and s e with sigmoid functions without adding the penalty terms to the cost functional.This is done by setting r s c = r s e = 0 in (27).We conducted an NMPC simulation for the fuel-economic speed tracking problem with the same weight coefficients as given in Section 5.2, except that r s c = r s e = 0, and obtained the result shown in Figure 19.Similar to the result of the proposed method shown in Figure 11, the vehicle followed the target speed.However, the switching signal sc constantly took intermediate values during the period after about t = 10 [s].Subsequently, the CVT ratio G constantly took improper values far from G α and G o .These behaviours are not acceptable for a solution to the present problem.In contrast, the proposed method provided proper results with sc ≈ 0 or 1 as desired.These contrastive results demonstrate the effectiveness of the proposed method.

Conclusion
In this study, an NMPC-based driving controller was proposed for ICEVs equipped with CVTs.The clutchengaged and -released modes and engine-on andoff modes were considered along with engine-braking effect.The switching characteristics of the vehicle system dynamics were relaxed into continuous transitions, allowing for the use of standard NMPC software packages to solve the NMPC problem.The numerical examples demonstrated that the designed controller successfully achieved real-time NMPC-based driving control.
Further, as a future work, the proposed controller can be examined on the original (unrelaxed) plant.Further verifications on more accurate simulators and experiments on real-life vehicles are desired for practical applications.Strategies for determining the design parameters of the sigmoid functions and penalty terms as well as the weight coefficients ought to be investigated.Further reduction of the NMPC computational burden would contribute to real-time implementations.These issues would be considered in future studies.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was supported by JSPS KAKENHI Grant Number JP21K14187.The authors report there are no competing interests to declare.

Figure 3 .
Figure 3. Shape of s e (u e ) and f e (u e ).The corresponding modes when the clutch is engaged are described at the bottom.

Figure 4 .
Figure 4. Shape of fe (u e ) when k, a SP and F max e are set asTable 2 of Section 5.

Figure 7 .
Figure 7. Maximum possible power P max a (ω e ) generated by the alternator.

Figure 8 .
Figure 8. Instantaneous fuel consumption w f (P e ) in the engine-on mode.

Figure 9 .
Figure 9. Shape of s e (u e ), se (u e ), L e (u e ), and f e (u e ).

Figure 10 .
Figure10.Road slope.For reference, the altitude was produced by numerically integrating the road slope.

Figure 11 .
Figure 11.Main result of the fuel-economic speed-tracking problem.The grey dashed lines denote the constraints on respective quantities.The red and green dash-dotted lines denote G α and G o , respectively.The black dotted line denotes the target speed.

Figure 12 .
Figure 12.Time histories of the control inputs in the fueleconomic speed-tracking problem.The grey dashed lines denote the constraints on respective quantities.The black dashdotted line denotes u e = ε e .

Figure 13 .
Figure 13.Time histories of the fuel consumption and (re)generation power in the fuel-economic speed-tracking problem.

Figure 14 .
Figure 14.Time history of the computation time at each sampling instant in the fuel-economic speed-tracking problem.The grey dashed line denotes the sampling period.

Figure 15 .
Figure 15.Main result of the eco-driving problem.The grey dashed lines denote the constraints on respective quantities.The red and green dash-dotted lines denote G α and G o , respectively.

Figure 16 .
Figure 16.Time histories of the control inputs in the ecodriving problem.The grey dashed lines denote the constraints on respective quantities.The black dash-dotted line denotes u e = ε e .

Figure 17 .
Figure 17.Time histories of the fuel consumption and (re)generation power in the eco-driving problem.

Figure 18 .
Figure 18.Time history of the computation time at each sampling instant in the eco-driving problem.The grey dashed line denotes the sampling period.

Figure 19 .
Figure 19.Main result of the fuel-economic speed-tracking problem without the penalty terms.The grey dashed lines denote the constraints on respective quantities.The red and green dash-dotted lines denote G α and o , respectively.The black dotted line denotes the target speed.

Table 1 .
Modes of vehicle propulsion/braking force from engine when the clutch is engaged.
, and G o [rad/m] and ω o [rad/s] denote the transmission ratio and engine speed, respectively, in the engine-off mode.The explicit descriptions of functions G o (v) and ω o (v) will be provided in Section 2.6.It should be noted that the righthand side of (6) is expressed as an explicit function of v. Furthermore, function τ eb (ω o ) can be obtained via an engine dynamometer test with the engine off.To model the engine on/off status σ e and vehicle propulsion/braking force F e according to Table1, let us introduce a continuous input variable u e ∈ (−∞, 1]

Table 2 .
Parameters of vehicle model and NMPC formulation.