Efficient Drive Cycle Simulation

1442 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 3, MAY 2008

Efficient Drive Cycle SimulationAnders Fröberg and Lars Nielsen

Abstract—Drive cycle simulations of longitudinal vehicle modelsare important aids for the design and analysis of power trains, andtools currently on the market mainly use two different methods forsuch simulations: the forward dynamic and quasi-static inversesimulations. Here, a known theory for the stable inversion ofnonlinear systems is used to combine the fast simulation times ofthe quasi-static inverse simulation with the ability of the forwarddynamic simulation to include transient dynamics. The stableinversion technique and a new implicit driver model togetherform a new concept: inverse dynamic simulation. This techniqueis demonstrated to be feasible for vehicle propulsion simulationand specifically for three power train applications that includeimportant dynamics that cannot be handled using quasi-staticinverse simulation. The extensions are engine dynamics, drivelinedynamics, and gas flow dynamics for diesel engines, which are alsoselected to represent important properties, such as zero dynamics,resonances, and nonminimum-phase systems. It is shown thatinverse dynamic simulation is easy to set up, gives short simulationtimes, and gives consistent results for design space exploration.This makes inverse dynamic simulation a suitable method to usefor drive cycle simulation, particularly in situations requiringmany simulations, such as optimization over design space, powertrain configuration optimization, or the development of powertrain control strategies.

Index Terms—Driver models, inverse simulation, power trainmodeling.

I. INTRODUCTION

THERE are a number of important uses of drive cyclesimulation, and among all of its applications, there are

many where simulation time is important. It has often been usedin concept studies [1]–[3], examples of which are optimizationover a design space of parameters [2], [4], optimization ofpower train configuration [5], [6], and design of power traincontrol systems, [7], [8]. Drive cycle simulation is also used incontrollers such as model predictive cruise controllers [9].

There are two main ways to perform longitudinal vehiclesimulation: 1) quasi-static inverse simulation and 2) forwarddynamic simulation. The quasi-static inverse simulation usesvehicle speed and acceleration to calculate the required torquesand speeds backward through the driveline. Fuel flow is thencalculated. Fig. 1 shows a typical computational scheme of thequasi-static inverse simulation. No driver model is used, andthe drive cycle tracking is explicit. In currently existing tools,following inverse simulation, the quasi-static approach [10],

Manuscript received July 5, 2006; revised March 30, 2007, June 25, 2007,and July 23, 2007. This work was supported by the Swedish Energy Agency.The review of this paper was coordinated by Prof. A. Emadi.

The authors are with the Division of Vehicular Systems, Department ofElectrical Engineering, Linköping University, 581 83 Linköping, Sweden(e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2007.907310

[11] is taken. Speed v(t) and acceleration v(t) are approxi-mated by v(t) = (v(kh + h) + v(kh))/2 and v(t) = (v(kh +h) − v(kh))/h, respectively. This way, only static equations aresolved when the input is computed, and a major advantage ofthis method is the low simulation time. On the other hand, inforward dynamic simulation, differential equations are solvedusing, e.g., throttle position or fuel flow as input and vehiclespeed as output. Given an initial value of the vehicle’s states andthe input, the system is numerically integrated to compute thespeed trajectory. This type of simulation also requires a drivermodel, i.e., a controller, to track a given speed trajectory (drivecycle), as depicted in Fig. 1. For forward dynamic simulation,drive cycle tracking is implicit since it is obtained using anexplicit driver model, and it is straightforward to include addi-tional significant dynamics. However, the differential equationsthat have to be solved typically give simulation times that are anorder of magnitude longer for drive cycle simulation than thosefor quasi-static simulation.

Because of the importance of drive cycle simulation, it isnatural that there are several tools for the simulation of longitu-dinal vehicle models currently on the market. Advisor [11] andQuasi-Static Simulation toolbox (QSS-TB) [10] are examplesof quasi-static inverse tools, and PSAT [12], Capsim [13], andV-Elph [14] are examples of forward dynamic simulation tools.All these tools use Matlab/Simulink.

A. Goal of this Paper

It would, of course, be of considerable value to be ableto extend time-efficient quasi-static simulation with importantdynamics without significantly losing simulation performance.The goal of this paper is to find a simulation method withsuch properties. The guiding principles in this developmenthave been to make extensions to the quasi-static method thatare sufficiently general to include the dynamics of the powertrain, such that important transients can be captured in thesimulation. Quasi-static inverse simulation uses only one state,i.e., vehicle speed v(t), so that the control is determined byu(t) = f(v(t), v(t)). If more dynamics needs to be added tothe models, then the inverse simulation strategy has to beextended. More states z have to be included, and they have tobe obtainable from the velocity profile, which means that higherderivatives of the speed or the drive cycle may be needed. Thiscan be formally written as u(t) = F (v(t), v(t), z(t), z(t)) =F(v(t), v(t), v(t), . . .), where either additional states or higherderivatives may be used. Such simulation, which is here-after called inverse dynamic simulation, is the main topic ofthis paper.

Given the main objective of this paper, i.e., combiningthe good properties of the quasi-static and forward dynamic

0018-9545/$25.00 © 2008 IEEE

FRÖBERG AND NIELSEN: EFFICIENT DRIVE CYCLE SIMULATION 1443

Fig. 1. Schematic depiction of different computational schemes, also illustrating the goal of this paper.

simulations (see Fig. 1), there are a number of ramificationsof the problem that have to be considered, and the rest of thispaper is outlined as follows: In Section II, inversion of nonlinearsystems with added dynamics is described. Since drive cycles,in general, are nonsmooth, tracking within limits is an issue indrive cycle simulations. This, together with driver models, ispresented in Section III. In Section IV, the proposed method ofinverse dynamic simulation is demonstrated to be feasible onpower train models. This includes additional dynamics that hassignificant influence on fuel consumption and emissions, andinternal variables are also captured. In Section V, simulationperformance measures such as simulation time, setup effort,and consistency in design space exploration are discussed, andinverse dynamic simulation is compared to forward dynamicsimulation.

II. INVERSION OF NONLINEAR SYSTEMS

Inverse dynamic simulation entails the computation of gen-erating variables, such as fuel flow and engine torque, whenthe output, i.e., the velocity profile, is given, which meansthat it is a problem of system inversion. This section willpresent the theory to be used, whereas the practical use of thetheory will be demonstrated on important power train models inSection IV.

Various domain-specific solutions to system inversion havebeen developed, e.g., in rigid body dynamics [15]–[17]. Areview of methods for inverse dynamic simulation of nonlinearsystems in aerospace applications is given in [18], where amethod based on numerical differentiation, followed by alge-braic inversion, is presented and an application of that methodis presented in [19]. This method is limited in that it uses a Eulerapproximation of the derivatives. Another way of performingsystem inversion is the use of a tool for noncausal simulationsuch as Dymola, where the method is based on a structuralmanipulation of equations. However, only minimum-phase sys-tems are treated, and numerical differentiation of the inputs isused [20].

A different possibility utilized in this paper is the use of thetheory of stable inversion of nonlinear systems. Drive cyclesneed not be continuously differentiable, but for physical powertrains having finite forces and torques, simulation signals willbe smooth, and it turns out that it is not a limitation to assumesmoothness. Therefore, the method described in [21] and [22]can be fruitfully adapted. A related theory is presented in [23].The rest of this section presents the method by showing howa system of ordinary differential equations is manipulated toperform an inverse dynamic simulation of it. The method han-dles minimum- as well as nonminimum-phase systems, and inthe next section, it will be shown how numerical differentiationof the inputs is avoided. A large class of systems that can besimulated in the forward dynamic way can thereby also besimulated in the inverse dynamic way.

A. Stable Inversion of Nonlinear Systems

A short review of the inversion method used will be givenhere. Let u(t) be the inputs, y(t) be the outputs, and x(t) be thestates of a given system. Suppose that the system can be writtenin input-affine form as

x(t) = f (x(t)) + g (x(t)) u(t) (1)

y(t) =h (x(t)) . (2)

Assume that the number of inputs q is equal to the numberof outputs. This is not a crucial assumption, as demonstrated,for example, in [24], where redundancy is used to cast theproblem in this formulation. Consider also a system wherevariables with suffix d corresponds to a system where variablesare smooth, corresponding to the tracking of a velocity profilewithin tracking limits. Given a desired output yd(t) that issmooth, the problem is solving

xd(t) = f (xd(t)) + g (xd(t)) ud(t) (3)

yd(t) =h (xd(t)) (4)


for input ud(t) and, possibly, also states xd(t). With the notionbeing smooth, here, it is meant that a signal is sufficientlymany times continuously differentiable. From here on, it will beassumed that all functions are smooth such that the necessaryderivatives exist and can be computed. It will also be assumedthat f(0) = 0 and h(0) = 0. This can always be achieved forsystems such as (3) and (4) by a simple change of coordinates.

Let L be the standard notation for Lie derivativesaccording to

Lfh(x) =∑

i

fi(x)∂

∂xih(x) (5)

Lrfh(x) =Lf

(Lr−1

f h(x))

. (6)

If, for all 1 ≤ i, j ≤ q, for all k < ri − 1, and for all x in aneighborhood of x0

LgjLk

fhi(x) = 0 (7)

and the q × q matrix

β(x) =

Lg1Lr1−1f h1(x) . . . Lgq

Lr1−1f h1(x)

Lg1Lr2−1f h2(x) . . . Lgq

Lr2−1f h2(x)

.... . .

...Lg1L

rq−1f hq(x) . . . Lgq

Lrq−1f hq(x)

(8)

is nonsingular, then the system is said to have a vector relativedegree r = (r1, r2, . . . , rq) at point x0, i.e., the relative degreeof the system is the number of times one has to differentiate theoutputs for at least one input to explicitly appear.

The first step in the inversion procedure is the computationof the relative degree. The next step is to partially linearize thesystem. This is done by differentiating yi(t) until at least oneuj(t) explicitly appears. Define ξi

k = y(k−1)i (t) for i = 1, . . . , q

and k = 1, . . . , ri, and let

ξ(t) =(ξ11(t), ξ1

2(t), . . . , ξ1r1

(t),

ξ21(t), . . . , ξ2

r2(t), . . . , ξq

rq(t)

)T

=(y1(t), y1(t), . . . , y

(r1−1)1 (t),

y2(t), . . . , y(r2−1)2 (t), . . . , y(rq−1)

q (t))T

. (9)

Now, the change of coordinates can be defined. Since y is afunction of x, it can be written as

(ξT , ηT )T = ψ(x) (10)

where η are the variables needed, and the only constraint inthe choice of η is that the Jacobian matrix of ψ(x) must benonsingular at x0 so that it is a local coordinate transformationin a neighborhood of x0. It is always possible to find such

a ψ(x) [23]. With this choice of coordinates, the followingsystem, which is partially linear in ξ, is achieved:

ξi1(t)=ξi

2(t)...ξiri−1(t)=ξi

ri(t),

ξiri

(t)=αi (ξ(t), η(t))+βi (ξ(t), η(t)) u(t)

for i=1, . . . , q

η(t)=sa (ξ(t), η(t))+sb (ξ(t), η(t)) u(t) (11)

where vector α and matrix β are given by

α(ξ, η) =Lrfh

(ψ−1(ξ, η)

)(12)

β(ξ, η) =LgLr−1f h

(ψ−1(ξ, η)

)(13)

and functions sa(ξ, η) and sb(ξ, η) are given by the choice of η.

Denote y(r) = (y(r1)1 , . . . , y

(rq)q ). Then, it can be seen in the

first part of (11) that y(r)(t) = αi(ξ(t), η(t)) + βi(ξ(t),η(t))u(t). By the definition of relative degree, β(ξ, η) is non-singular. Given a desired output trajectory yd(t), the requiredcontrol input ud(t) can be calculated as

ud(t) = βi (ξ(t), η(t))−1(y(r)d (t) − αi (ξ(t), η(t))

). (14)

Here, it is seen that, to calculate the required input for thesystem to follow the prescribed trajectory, not only yd(t) butalso the trajectories of η(t) and the r first derivatives of yd(t)have to be known. If the state trajectories producing the desiredoutput are of interest, they can be calculated from the inversecoordinate change

x(t) = ψ−1 (ξ(t), η(t)) . (15)

How the zero dynamics is to be solved depends on thesystem being studied. There are three classes of systems thatcan be written in the form of (11). The first class includes allsystems where the relative degree is equal to the dimension ofthe system, which means that there are no zero dynamics. Thesecond class includes all systems with stable zero dynamics,i.e., minimum-phase systems. The third class includes systemswith unstable zero dynamics, i.e., nonminimum-phase systems.For the class of systems without zero dynamics, (14) becomes asystem of static equations where the required input can be cal-culated from the desired output and its derivatives. In the caseof stable zero dynamics, the procedure is also straightforward.Substitute (14) in (11), i.e.,

η(t) = sa (ξ(t), η(t)) + sb (ξ(t), η(t)) βi (ξ(t), η(t))−1

×(y(r)d (t) − αi (ξ(t), η(t))

)≡ s (η(t), Yd(t)) (16)

where

Yd(t) =(y1(t), y1(t), . . . , y

(r1)1 (t), y2(t), . . .

y(r2)2 (t), . . . , y(rq)

q (t))

. (17)


Choose appropriate initial values for η(t), and solve the systemof differential equations (16) to find the trajectories of the zerodynamics.

In the case of unstable zero dynamics, however, there is nosuch straightforward way since (16) cannot then be integratedas an initial value problem. It is not possible to solve theunstable zero dynamics in the general case, but in drive cyclesimulations, the desired output trajectory is known beforehand.This gives the possibility of computing a noncausal solutionof (16) and still receiving a stable result. In [21], a Picard-like iteration is used to find the zero dynamics of a nonlinearnonminimum-phase system. To illustrate that method, a linearsystem is first studied, i.e.,

η(t) = Aη(t) + Bu(t), η(±∞) = 0 (18)

where A has no eigenvalues on the jω-axis. For such systems,it is possible to find a similarity transformation η = T η, whereT is invertible, that brings the system on a form( ˙η1

˙η2

)=

(An 00 Ap

)(η1

η2

)+

(B1

B2

)u, η(±∞) = 0

(19)

where An has all the eigenvalues in the left half-plane, and Ap

has all the eigenvalues in the right half-plane. The solution tothe boundary value problem (19) is

η(t) =

∞∫−∞

φ(t − τ)Bu(τ)dτ (20)

where

φ(t) =(

1(t)eAnt 00 −1(−t)eApt

)(21)

and 1(t) is the unit step function. Solving for the first statesη1(t) in (19) is easy. The system is stable, and the solution

η1 =

∞∫0

eAnτ B1u(t − τ)dτ (22)

can be computed with numerical solvers for ordinary differen-tial equations (ODEs). If the input signal is known beforehand,the solution to the last states η2(t) can be computed in a stableway by reversing time as

η2 =

0∫−∞

−eApτ B2u(t − τ)dτ (23)

or a noncausal numerical solution can be used. Hence, a stablesimulation of the system can be done. For a nonlinear systemsuch as (16), separation of the system into stable and unstableparts as in (19) cannot be done, in general. The approachof Devasia et al. [21] is writing the nonlinear zero dynamics(16) as

s(η, Yd) = Aη + (s(η, Yd) − Aη) (24)

Fig. 2. NEDC.

which can be interpreted such that the nonlinearities are seenas disturbances to a linear system. In the preceding equation,A gives a linear approximation of s(·), which is typically∂s/∂η|(0,0). The iteration method simulates the linear system,with the disturbances (nonlinearities) as inputs, according to

ηm+1(t) =

∞∫−∞

φ(t − τ) [s(ηm(τ), Yd(τ) − Aηm(τ)] dτ (25)

where φ(t) is the state transition matrix for the linear system. Ifa change of coordinates that brings the linearization in the formof (19) is applied, the system becomes

˙η(t) = T−1AT η(t) + T−1 (s (T η(t), Yd(t)) − AT η(t)) (26)

and iteration (25) can be separated in the form of (19), i.e.,

(η1,m+1(t)η2,m+1(t)

)=

∞∫−∞

φ(t − τ)T−1

× [s(T ηm(τ), Yd(τ) − AT ηm(τ)] dτ (27)

where φ(t) is defined by (21). For this iteration to converge toa solution of (25), there are restrictions on the area of attractionthat has to be handled [21].

III. TRACKING PERFORMANCE AND DRIVER MODELS

A drive cycle is a speed profile where speed is given as afunction of time (see, e.g., Fig. 2), and as can be seen in thefigure, a drive cycle cannot be expected to be continuouslydifferentiable. This means that, for models including additionaldynamics, it is impossible to exactly follow or track a drivecycle; hence, the speed has to be controlled to track the drivecycle in a desired way. When drive cycles are used for emissionlegislations, the cycle is tracked by a human driver within cer-tain limits. It is therefore logical, also in simulation, to designdriver models to track cycles within defined limits. As will beshown later, results in, for example, fuel consumption can differbetween two different velocity trajectories that are both withinprescribed limits (see Fig. 3), and it is thus important to studyhow a drive cycle is tracked and not only that the tracking iswithin limits.

To achieve good drive cycle tracking, more or less sophisti-cated driver models are needed, and here, driver models for both


Fig. 3. Two different simulations of NEDC. Both are within tracking limitsbut result in different fuel consumptions.

forward and inverse dynamic simulations will be discussed.To compare tracking and behavior, different driver models aretested on the New European Drive Cycle [(NEDC) see Fig. 2].

A. Tracking in Forward Dynamic Simulation

In forward dynamic simulation, a driver model acts as acontroller to track the reference speed trajectory, and as shownin Fig. 1, the driver model compares the drive cycle with theactual speed and decides how to control the inputs, e.g., theaccelerator pedal, brake pedal, clutch, and gear selector, tofollow the prescribed drive cycle within defined limits. A drivermodel can be as simple as a proportional–integral–derivative(PID) controller [25] or a more sophisticated controller withprediction abilities [26]. The parameters of the driver modeldetermine driver alertness, which, in this type of simulation,results in tightness of the cycle tracking. Fig. 4 shows anexample of tracking behavior for forward driver models.

B. Enabling Tracking in Inverse Dynamic Simulation

In inverse dynamic simulation, the reference speed trajectoryis exactly followed if the physics of the model allows it,provided that the reference speed trajectory is continuouslydifferentiable as many times as the relative degree of thesystem, as mentioned in [27] and [28] or as shown in (14)(see also [23]). If it is not sufficiently differentiable, then thetrajectory has to be smoothed to create a reference speed thatthe simulated model can follow with bounded states and inputs[27]. This smoothing of the drive cycle is done in real life by atest driver but is done here by mathematical smoothing, whichwill be given the interpretation of an implicit driver model.There are several possibilities in smoothing the drive cycle,and the shape of the smoothed speed profile is decided by thebehavior of the driver model. One way is to filter the drive cyclewith a standard linear low-pass filter with a relative degree ofat least the same order as the system. This will ensure that thefiltered drive cycle is sufficiently many times differentiable for(14) to be solvable [23]. Using a noncausal linear filter givesthe driver look-ahead properties. Another way to smooth drive

cycle vc(t) is to compute the desired trajectory vd(t) as theconvolution

vd(t) =1C

∞∫−∞

g(t − τ)vc(τ)dτ. (28)

A good choice for convolution kernel g(t) (see Fig. 5) is to usethe definition

g(t) =

{e

−a2

a2−t2 , |t| ≤ a0, otherwise

(29)

where parameter a is a tracking time constant. Linear filteringof the drive cycle gives only asymptotically exact tracking,whereas the use of (29), due to compact support, gives exacttracking on large parts of the drive cycle, as shown in Fig. 4. Asmaller a gives “tighter” tracking, which corresponds to a moreaggressive or alert driver behavior.

Since the choice (29) of convolution kernel g(t) has compactsupport and is infinitely many times continuously differen-tiable, the resulting trajectory vd(t) is infinitely many timescontinuously differentiable. Moreover, the calculation of thederivatives does not require numerical differentiation. Instead,the derivatives are calculated using the following convolution:

v(r)d (t) =

∞∫−∞

g(r)(t − τ)vc(τ)dτ (30)

where the different derivatives g(r) are obtained by analyticaldifferentiation of (29). Fig. 5 shows some examples with param-eter a = 1.

Driver model. The new driver model for inverse dynamicsimulation is defined by (28)–(30).

Inverse Dynamic Simulation. The driver model previouslypresented, together with the inversion procedure presented inSection II, is an extension to quasi-static simulation and formsa new concept of inverse dynamic vehicle simulation (Fig. 1).

Thus, the smoothing of the drive cycle is interpreted as animplicit driver model, where the tracking behavior is specifiedin the velocity domain and is independent of the vehicle. Infact, this way of specifying tracking behavior can be argued tobe closer to human behavior when performing drive cycle testson a chassis dynamometer since the tracking of a human driveris not specified by controller parameters but rather in terms oftracking smoothness.

IV. TYPICAL POWER TRAIN MODEL EXAMPLES

When using drive cycle simulation, it may be desired tocapture the behavior in transients to achieve sufficient precisionin the processes generating fuel consumption and emissions.Therefore, it is necessary to not only use quasi-static simula-tions, such as Advisor [11] or the QSS-TB [10], but also includedominant dynamics.

In this section, three of the most important dynamics [26]that capture such transients will be presented, and the feasibilityof the inversion procedure presented in Section II will be


Fig. 4. Driver behavior for (left) forward dynamic simulation and (right) inverse dynamic simulation for a part of the NEDC.

Fig. 5. Convolution kernel and its first derivatives (normalized curves).

demonstrated on these models. One example is the dynamics ofthe air filling in the intake manifold when using throttle positionas input to the power train. Another example is resonances inthe driveline, which are mainly caused by torsion in the drive-shafts. These are modeled in Section IV-A. The third examplecomes from diesel engines with exhaust gas recycling (EGR),where the gas flow in the engine is important for emissionmodeling. This flow is modeled in Section IV-B. The modelsare only briefly presented here; for a more detailed presentation,see [29].

Even though the examples in this section are motivated bythe application, they each impose interesting and illustrativemathematical characteristics. The first example introduces in-take manifold pressure to illustrate the extension with one extrastate, compared to quasi-static modeling. The second exam-ple additionally includes driveshaft torsion, which includes aresonant system and zero dynamics. The third example withgas flow for diesel engines includes the nonminimum phase,i.e., unstable zero dynamics. The model examples will beused for the comparison of different simulation methods inSection V.

A. Power Train With Intake Dynamics andDriveline Dynamics

This section will describe the first two examples, and it willbe done by first giving the most complex example, which is apower train with both intake pressure dynamics and driveshaftflexibility. Then, the model will be specialized to excludedriveshaft flexibility.

Fig. 6. Components of the power train model.

The main dynamic variable of a vehicle is vehicle speed v,which is the only one used in quasi-static models. Includingdriveshaft torsion θd decouples vehicle speed v and enginespeed ωe. In addition, the dynamics of intake manifold pres-sure pi is included. The model (Fig. 6) is more thoroughlydescribed by Fröberg [29] and will be referred to as the flexiblepower train model FP. The tractive input is effective throttle areaAeff . From ambient pressure pa and intake manifold pressurepi, the flow past the throttle is calculated via a flow restrictionmodel Ψ. Then, the gas flow into the cylinders is calculated,giving the gross work. Subtracting pump work and frictiongives the output torque from the engine. The rest of the drivelineis modeled as standard inertias, torsion springs, dampers, andgears, giving the driving force on the wheels. This driving forcereduced by air resistance and rolling resistance gives the vehiclespeed. When needed, a brake torque proportional to the driverinput is applied to the wheels, and the simulation is handled asin [24]. During the tractive part of the drive cycle, the modelhas input u = Aeff , states x = [v, ωe, θd, pi]T , and outputy = v. The model parameters are lumped into coefficients ci.Let functions fi, gi, and h be defined by the following modelequations:

v = c1v2 + c2v + c3ωe + c4θd + c5 ≡ f1(x) (31)

ωe = c6v + c7ωe + c8ω1.8e + c9θd + c10pi + c11

≡ f2(x) (32)

θd = c12v + c13ωe ≡ f3(x) (33)

pi = c14ωe + c15ωepi + c16Ψ(

pi

pa

)Aeff

≡ f4(x) + g4(x)u (34)

y = v ≡ h(x). (35)

The relative degree of this system is r = 3. Hence, the zerodynamics and the inverse dynamic model have an order of one.


Fig. 7. Velocity of a simulation of the power train model with flexibledriveshaft from Section IV-A.

The coordinate change (10) is chosen as

(ξ1, ξ2, ξ3, η)T = (v, v, v, pi)T . (36)

Using (14) in the zero dynamics gives

η = f4

(ψ−1(ξ, η)

)+ g4

(ψ−1(ξ, η)

)β−1

(v(3)d − α

)(37)

using (12) and (13), which are

α =L3fh (38)

β =LgL2fh. (39)

The zero dynamics [see (37)] are stable for this model and,hence, can be solved as an initial value problem. It is seen thatonly output vd(t) and the first three derivatives thereof have tobe known when solving for the zero dynamics and the resultingrequired input (14), i.e.,

Aeff,d = β−1(ξ, η)(v(3)d − α(ξ, η)

). (40)

Simulations of this model are presented in Figs. 7 and 8, whichshow both forward dynamic and inverse dynamic simulations.The results are comparable, which demonstrates the feasibilityof the proposed new method. A more detailed evaluation is donein Section V.Power Train With Intake Dynamics: The stiff power train

SP includes the dynamics of intake manifold pressure pi butnot the driveshaft torsion. This simplified version of the model(31)–(35) with stiff driveshafts is written as

v = c1v2 + c2v

1.8 + c3pi + c4 ≡ f1(x) (41)

pi = c5v + c6vpi + c7Ψ(

pi

pa

)Aeff ≡ f4(x) + g4(x)u (42)

y = v ≡ h(x). (43)

The inversion of this system follows the same procedure as thatfor the previous example. The system’s relative degree is r = 2,giving an inverse model that consists of static equations.

Fig. 8. Effective throttle area of a simulation of the power train model withflexible driveshaft from Section IV-A.

Fig. 9. Schematic of a diesel engine with EGR and VGT. The figure is adaptedfrom [30].

B. Gas Flow Control in a Diesel Engine

As mentioned earlier, nonminimum-phase systems may bechallenging, and one important power train component that hasthis property is the gas flow for a diesel engine with EGR andvariable geometry turbine [(VGT), see Fig. 9]. Such a systemis modeled by Wahlström [30], and the simplified version thatis used here is detailed by Fröberg [29] and will be referred tohere as “DE.” The control variables are EGR valve uegr and thearea of the VGT Avgt. These two inputs are used to control themass flow and its composition into the cylinder. Here, the fuelflow and engine speed are considered to be model parameters.The outputs are normalized air fuel ratio λ and EGR ratio xegr.This is a nonminimum-phase system for many operating points,and intuitively, it can be reasoned as follows: When the VGT isclosed, initially, the flow past the turbine will decrease, whichresults in a decrease of compressor flow and a decrease of λ.However, there will be a buildup of pressure in the exhaustmanifold; after a while, this will result in a higher flow pastthe turbine, speeding up the turbo, and, later, will result in anincrease of fresh air flow, which means an increase in λ. Themodel has four states x = [pi, pe, ωt, Aegr]T , i.e., the intake


Fig. 10. (Solid) Specified output trajectories for λ and xegr. (Dashed) Corresponding forward dynamic simulation is also shown.

manifold pressure, exhaust manifold pressure, turbine speed,and EGR valve area. As previously mentioned, coefficients ci

are lumped model parameters, and functions fi, gi, and h aredefined in the model equations that are written as

pi = c1pi + c2ωt + c3AegrpeΨ(

pi

pe

)≡ f1(x) (44)

pe = c4pi + c5AegrpeΨ(

pi

pe

)+ c6

+ c7pe

√1 − c8

pc9e

Avgt ≡ f2(x) + g2(x)u (45)

ωt = c10pc11i + c12 +

(c13 +

c14

pc15e

) √1 − c8

pc9e

pe

ωtAvgt

≡ f3(x) + g3(x)u (46)

Aegr = c16Aegr − c16uegr ≡ f4(x) + g4(x)u. (47)

The outputs are y = [λ, xegr]T , i.e.,

λ = c17ωt ≡ h1(x) (48)

xegr =c18AegrpeΨ

(pi

pe

)c19ωt + c18AegrpeΨ

(pi

pe

) ≡ h2(x). (49)

Computing the Lie derivatives for this system gives the vec-tor relative degree as r = (1, 1), which means that the systemzero dynamics has an order of two. As previously mentioned,it is assumed in the inversion procedure that f(0) = 0 andh(0) = 0, which is easily achieved for this system by a simplecoordinate change, where the origin is moved to a stationarypoint, such as pi = pi − pi,stat. When this is done, the coordi-nate change to ξ and η can be chosen as

(ξ1, ξ2, η1, η2)T = (λ, xegr, pi, pe)T . (50)

Using (14), the zero dynamics can be written as

η1 = f1

(ψ−1(ξ, η)

)(51)

η2 = f2

(ψ−1(ξ, η)

)+ g2

(ψ−1(ξ, η)

)× β(ξ, η)−1

(ξ − α(ξ, η)

)(52)

where

α =Lfh (53)β =Lgh. (54)

For the stationary point pi = 2.06 · 105 Pa, pe = 2.41 · 105 Pa,ωt = 7001 rad/s, and Aegr = 1 · 10−4 m2, this zero dynamicsis linearized, i.e.,

η = Aη (55)

and the eigenvalues of A become σ1 = −6.8 and σ2 = 15.8;thus, clearly, the zero dynamics are unstable. The iterativetechnique described by (27) is used to solve for the trajectoriesof the zero, dynamics. No measures such as multiple lineariza-tions, are used to optimize performance. Having obtained thetrajectories, the required input is calculated as in (14), whichbecomes(

uegr,d

Avgt,d

)= β−1(ξ, η)

(( ˙λd

˙xegr,d

)− α(ξ, η)

). (56)

Forward dynamic simulations of the diesel engine are non-trivial [31] since it is hard to design a controller for thenonminimum-phase multivariable system. Instead, the proce-dure here is, first, to specify desired outputs and the drivecycle, then to perform an inverse simulation, and, finally, to usethe resulting inputs as inputs to the forward dynamic simula-tion. Figs. 10 and 11 show simulations of the model, whichdemonstrates feasibility since the results are almost identicalfor forward and inverse simulations.

V. SIMULATION USABILITY AND PERFORMANCE

The extended concept of the inverse dynamic vehicle simu-lation summarized in Sections II and III-B was demonstratedin the previous section to be feasible. However, as has beenmentioned, the goal of this paper is increased performance bycombining the advantages of forward dynamic simulation andquasi-static inverse simulation in a new method, i.e., inversedynamic simulation (Fig. 1). This section is devoted to acomparison with standard forward simulation, realizing that,when performing a simulation, there are several properties thatare important. This concerns the setup effort, consequences oftracking performance, consistency when scanning a parameterset, and, of course, simulation time, which are the topics of thefollowing sections.


Fig. 11. Required inputs from an inverse dynamic simulation of the diesel engine with desired outputs according to Fig. 10.

A. Simulations

The simulation study for investigating properties and formaking comparisons has been set up in the following way.The power train models from Section IV are used, with pa-rameter values corresponding to a typical personal sedan car,except for the diesel example, which is a typical heavy diesel.Detailed parameter values can be found in [29]. Regardingdriver models, the forward model is controlled with a drivermodel implemented as a PID controller that is connected tothe vehicle model, as depicted in Fig. 1. The PID parametersare Kp = 8 · 10−5, Ki = 1 · 10−6, and Kd = 0. The inversedynamic simulation (see Fig. 1) uses a driver model, as de-scribed by (28) and (29), with parameter a = 5. To get com-parable tracking, the parameter setting of the driver modelshas been done as follows: First, an inverse dynamic drivermodel that has a suitable tracking behavior has been designed.Then, a forward dynamic PID-type driver model has beendesigned such that it has comparable response time as theinverse driver model. The models are simulated in the NEDC(Fig. 2).

B. Simulation Setup Effort

It was seen in Section IV-A that both the inverse and forwarddynamic simulations qualitatively produced the same result.However, it can be seen in Fig. 8 that the forward dynamicsimulation in that example has an oscillating behavior in thethrottle position. This stems from the driver model since, forthis example, it is not sufficient with a single PID controller dueto the nonlinearities in the system. It may take quite some effortto design a driver model that adapts to the vehicle’s behaviorin different operating points. On the other hand, it is seen thatthe inverse dynamic simulation has a smooth control that betterresembles the behavior of a real driver. Thus, the complexity ofdesigning a driver model may be a drawback for the forwarddynamic simulation considering setup effort.

It is straightforward to implement a forward dynamic modelin a simulation tool such as Matlab/Simulink, whereas aninverse dynamic simulation requires manipulations with thedifferential equations describing the system before it can besimulated. These manipulations are, however systematic, asdescribed in Section II, and can be automatized. The systematicconversion reduces the need to build dedicated inverse modelssuch as those used in, e.g., Advisor. This way, the new simula-

tion concept is a bridge that ties forward dynamic simulations,e.g., PSAT, to quasi-static inverse simulations, e.g., Advisor, inthe sense that common model databases can be used.

C. Consistency for Parameter Exploration

An important use of drive cycle simulation is for design spaceexploration such as optimizing gear ratios of the transmission,optimizing the size of the engine for minimal fuel consumption,or tuning control system parameters. Then, it is important tohave consistency when some parameters vary between simula-tions so that correct conclusions can be drawn.

A first complication is that tracking may vary, as alreadydescribed in Section III and illustrated in Fig. 3. Using themodel from Section IV-A, it is seen in Figs. 7 and 8 thatboth the forward and inverse dynamic simulations producequalitatively comparable results. In [29], an experiment withparameter changes in this model is done by keeping the driverparameters; however, the engine size is changed from 2.3 to1.6 L, and the vehicle mass is changed from 1700 to 1100 kg.It turns out that the tracking behavior in the forward simu-lations is different in the two cases but is, by construction,exactly the same for the inverse dynamic simulations. Fur-thermore, even though all the simulations are within trackinglimits, the different behaviors of the forward simulation giveoscillations in the lightweight-vehicle case. This shows thatit can be difficult to run a set of forward dynamic simula-tions and get consistent fuel consumption calculations withoutretuning the driver model when the vehicle parameters arechanged.

An even more important point will now be demonstrated.The same model is used again but, now, with a stiff driveline.It is simulated to study the fuel consumption in the NEDCdriving cycle. The fuel consumption is taken as the integral ofthe maximum of the fuel flow given by the model and an idleflow. Simulations are performed both with the inverse dynamicmethod and the forward dynamic method. Keeping the samedriver models, the simulations are performed with two differentwheel radii, i.e., 0.3 and 0.45 m. The results are shown inTable I. For the forward case, the order between the twocases are reversed (8.619 < 8.634 compared to 7.315 > 7.293),which means that the order in fuel consumption betweenthe driver models is not preserved when the wheel radius ischanged. This indicates that it can be difficult to evaluate


TABLE ICONSISTENCY IN SIMULATION WITH CHANGE IN WHEEL RADIUS

simulations where model parameters change if the trackingcharacteristics are not preserved. For the inverse case, the orderis consistent (8.595 < 8.627 and 7.265 < 7.271).

The preceding examples show effects in forward dynamicsimulation that deteriorate the usefulness of the set of sim-ulations as a design space exploration. However, for inversedynamic simulation, due to the computational scheme withthe implicit driver model, the behavior within tracking limitsis consistent. The conclusion is that inverse dynamic simula-tion has an advantage regarding consistency in design spaceexploration.

D. Simulation Time

Recall that the main goal was to extend quasi-static inversesimulation with additional dynamics while still keeping thesimulation time low.

To study the simulation speed of the inverse and forwarddynamic simulations, the three models from Section IV willbe used again in a more elaborate study that includes morecases than that in the previous sections. Aside from the drivecycle NEDC, FTP 75 has also been used. Furthermore, differentdriver models have been used, as summarized in Table III.Twenty-six combinations of vehicle models, driver models,and drive cycles have been simulated, and they are listed inthe presentation of the results in Table II. The “SP” modelsare the models from Section IV-A with a stiff driveshaft andengine dynamics. The “FP” model is the model of a power trainwith flexibility in the driveshaft from Section IV-A. The dieselengine model “DE” is the one with unstable zero dynamicsdescribed in Section IV-B. One iteration has been used to findthe zero dynamics of the inverse diesel engine model. Thenumber of integration steps in the inverse simulation of thediesel engine is the sum of the steps for the simulations ofthe zero dynamics and for finding the required inputs.1) Comparison of the Approaches: The inverse dynamic

simulations are set up to resemble automatic conversion fromthe corresponding forward model using only the conversiondescribed in Section II. This means that there is additionalpotential for speedup, such as the choice of coordinates, orto see if all or at least some of the variables in the inversecoordinate change can be analytically solved.

For each case in both the inverse and forward simulations,the fastest solver has been chosen, and the resulting choice islisted in a column of the result presentation in Table II.2) Timing Results: The main result that can be seen in

Table II is that the inverse dynamic simulation has a con-siderable speed advantage compared to the forward dynamic

TABLE IISIMULATION TIMES

TABLE IIIDRIVER MODEL PARAMETERS

simulation. This holds for all three power train applicationsthat have different added dynamics. Going a bit more intodetail, there are a number of interesting observations to be madein the comparisons. There are mainly two things that affectthe simulation time: 1) the number of steps required in theintegration and 2) the number of calculations in each step. InTable II, it is seen that the inverse dynamic simulation takesat least an order of magnitude fewer steps than the forwarddynamic simulation.

One reason for the difference in the number of integrationsteps is related to the characteristics of the vehicle model.The number of steps in the integration mainly depends on theeigenvalues of the simulated system, where large eigenvalues,i.e., far away from the imaginary axis, will require small stepsand, thus, more steps. This means that, when the eigenvaluesof the zero dynamics are smaller than the eigenvalues for theforward dynamic model, the inverse dynamic simulation willrequire fewer steps than the forward simulation, and for manysystems, this is the case. For example, in the diesel enginemodel, the eigenvalues of the forward model in the linearizationpoint are −18.2, −12.7, −5.00, and −1.26, and for the zerodynamics, the eigenvalues are 15.8 and −6.8. For the stiffpower train model, there are no zero dynamics, so the inversemodel is correct in each sample point, independent of howfew the steps that were taken. In addition, for the flexible


power train model, the zero dynamics, stemming from thedriveshaft flexibility, are slower than, e.g., the intake pressuredynamics.

Another reason for the advantage in simulation time is thecalculations needed in each integration step. If there is a needto numerically solve the inverse coordinate change online, thenumber of calculations in each step can increase for the inversedynamic models. However, since the dimension of the zerodynamics is smaller than the number of states in the forwarddynamic models, the number of calculations in each step arefewer in most cases.

A further reason for the relatively slow simulations of theforward models of the basic power train and the flexible powertrain is the need for extra dynamics in the form of a controller,i.e., the driver model. Typically, such a controller may have tentimes faster dynamics than the controlled system. Plots fromsuch simulations presented in Table II have been presented inSection IV. In those plots, the forward dynamic simulation isunnecessarily oscillating in some cases due to the driver model,but this is not the main reason for the long simulation times.The trends in simulation time reported in Table III are still validfor other tunings of the forward driver models.

VI. CONCLUSION

The importance and characteristics, e.g., regarding trackingand repeatability, of drive cycle simulations have been pre-sented. Particularly aiming for repeated drive cycle simulations,the new implicit driver model presented in Section III-B, to-gether with the inversion procedure presented in Section II,forms a new concept of inverse dynamic vehicle simulation.

The use of the new driver model is a good way of specifyingtracking behavior. Compared to specifying tracking using theforward driver model, it is closer to human behavior whenperforming drive cycle tests on a chassis dynamometer sinceit is not specified by controller parameters but rather in terms oftracking smoothness.

In Section IV, three power train applications that includedimportant dynamics that cannot be handled using quasi-staticinverse simulation were presented. The extensions were enginedynamics, driveline dynamics, and gas flow dynamics arounddiesel engines. These three cases also represented interestingmathematical properties such as zero dynamics, resonances,and nonminimum-phase systems, i.e., unstable zero dynamics.The feasibility of inverse dynamic simulation of these systemswas shown.

In addition to feasibility, more importantly, good charac-teristics were also demonstrated regarding consistency andsimulation speed. It was found that specifying parameter ain the interpolating driver model is simpler than retuning theforward driver model (PID controller) for each model parametersetting, which is necessary to achieve consistency in designspace exploration. Simulation comparisons demonstrated thatthe new method has good performance with faster simulations,compared to the standard forward simulation. This means thatthe goals of Section I and Fig. 1 are achieved. Inverse dy-namic simulation was favorably compared to forward dynamicsimulation regarding simulation setup effort, simulation time,

and consistency for parameter exploration. This makes inversedynamic simulation a suitable method to use for drive cyclesimulation, particularly in situations requiring many simula-tions, such as optimization over design space, power trainconfiguration optimization, or the development of power traincontrol strategies.

REFERENCES

[1] R. P. Wolfson and J. H. Gower, “The role of computer modeling and sim-ulation in electric and hybrid vehicle research and development,” IEEETrans. Veh. Technol., vol. VT-32, no. 1, pp. 62–73, Feb. 1983.

[2] L. Guzella and A. Sciaretta, Vehicle Propulsion Systems. New York:Springer-Verlag, 2005.

[3] S. Onoda and A. Emadi, “PSIM-based modeling of automotive power sys-tems: Conventional, electric, and hybrid electric vehicles,” IEEE Trans.Veh. Technol., vol. 53, no. 2, pp. 390–400, Mar. 2004.

[4] S. M. Lukic and A. Emadi, “Effects of drivetrain hybridization on fueleconomy and dynamic performance of parallel hybrid electric vehicles,”IEEE Trans. Veh. Technol., vol. 53, no. 2, pp. 385–389, Mar. 2004.

[5] T. Sandberg, “Heavy truck modeling for fuel consumption simulations andmeasurements,” Licentiate thesis, Linköping Inst. Technol., Linköping,Sweden, 2001.

[6] A. I. Antoniou, J. Komyathy, J. Bench, and A. Emadi, “Modeling andsimulation of various hybrid-electric configurations of the high-mobilitymultipurpose wheeled vehicle (HMMWV),” IEEE Trans. Veh. Technol.,vol. 56, no. 2, pp. 459–465, Mar. 2007.

[7] M. Back, S. Terwen, and V. Krebs, “Predictive powertrain control for hy-brid electric vehicles,” in Proc. 1st IFAC Symp. Adv. Automotive Control,2004, pp. 121–126.

[8] A. Sciaretta, L. Guzella, and C. H. Onder, “On the power split controlof parallel hybrid vehicles: From global optimization towards real-timecontrol,” Automatisierungstechnik, vol. 51, no. 5, pp. 195–203, 2003.

[9] A. Fröberg, L. Nielsen, L.-G. Hedström, and M. Pettersson, “Controllinggear engagement and disengagement on heavy trucks for minimization offuel consumption,” in Proc. IFAC World Congr., Prague, Czech Republic,2005.

[10] L. Guzzella and A. Amstutz, “CAE tools for quasi-static modeling andoptimization of hybrid powertrains,” IEEE Trans. Veh. Technol., vol. 48,no. 6, pp. 1762–1769, Nov. 1999.

[11] K. B. Wipke, M. R. Cuddy, and S. D. Burch, “Advisor 2.1: A user-friendlyadvanced powertrain simulation using a combined backward/forwardapproach,” IEEE Trans. Veh. Technol., vol. 48, no. 6, pp. 1751–1761,Nov. 1999.

[12] A. Rousseau, P. Sharer, and F. Besnier, “Feasibility of reusable vehiclemodeling: Application to hybrid vehicle,” SAE Technical Paper SeriesSP-1826, 2004. 2004-01-1618.

[13] Center for automotive propulsion simulation. [Online]. Available:http://www.capsim.se

[14] K. L. Butler, M. Ehsani, and P. Kamath, “A Matlab-based modeling andsimulation package for electric and hybrid electric vehicle design,” IEEETrans. Veh. Technol., vol. 48, no. 6, pp. 1770–1778, Nov. 1999.

[15] G. Rodriguez, “Kalman filtering, smoothing, and recursive robot armforward and inverse dynamics,” IEEE J. Robot. Autom., vol. RA-3, no. 6,pp. 624–639, Dec. 1987.

[16] C. Wampler, “Manipulator inverse kinematic solutions based on vectorformulations and damped least-squares methods,” IEEE Trans. Syst.,Man, Cybern., vol. SMC-16, no. 1, pp. 93–101, Jan. 1986.

[17] G. Wood, Simulating Mechanical Systems in Simulink With SimMechan-ics, 2003. [Online]. Available: www.mathworks.com

[18] D. Murray-Smith, “The inverse simulation approach: A focused review ofmethods and applications,” Math. Comput. Simul., vol. 53, no. 4, pp. 239–247, Oct. 2000.

[19] D. G. Thomson and R. Bradley, “The principles and practical applicationof helicopter inverse simulation,” Simul. Pract. Theory, vol. 6, no. 1,pp. 47–70, Jan. 1998.

[20] J. Bals, G. Hofer, A. Pfeiffer, and C. Schallert, “Object-oriented inversemodelling of multi-domain aircraft equipment systems and assessmentwith modelica,” in Proc. 3rd Int. Modelica Conf., Linköping, Sweden,Nov. 2003.

[21] S. Devasia, D. Chen, and B. Paden, “Nonlinear inversion-based outputtracking,” IEEE Trans. Autom. Control, vol. 41, no. 7, pp. 930–942,Jul. 1996.

[22] L. R. Hunt and G. Meyer, “Stable inversion for nonlinear systems,” Auto-matica, vol. 33, no. 8, pp. 1549–1554, Aug. 1997.


[23] A. Isidori, Nonlinear Control Systems. New York: Springer-Verlag,1995.

[24] A. Fröberg, “Inverse dynamic simulation of non-quadratic MIMO power-train models: Application to hybrid vehicles,” in Proc. IEEE VehiclePower Propuls. Conf., Windsor, U.K., 2006, pp. 1–6.

[25] L. Eriksson, “Simulation of a vehicle in longitudinal motion with clutchlock and clutch release,” in Proc. 3rd IFAC Workshop Adv. AutomotiveControl, 2001, pp. 65–70.

[26] U. Kiencke and L. Nielsen, Automotive Control Systems, 2nd ed.New York: Springer-Verlag, 2005.

[27] A. Fröberg and L. Nielsen, “A method to extend inverse dynamic simu-lation of powertrains with additional dynamics,” in Proc. 1st IFAC Symp.Adv. Automotive Control, Apr. 2004, pp. 662–667.

[28] A. Fröberg and L. Nielsen, “Dynamic vehicle simulation: Forward, in-verse and new mixed possibilities for optimized design and control,” SAETechnical Paper Series SP-1826, 2004. 2004-01-1619.

[29] A. Fröberg, “Extending the inverse vehicle propulsion simulation conceptto improve simulation performance,” Licentiate thesis, Linköping Inst.Technol., Linköping, Sweden, 2005.

[30] J. Wahlström, “Control of EGR and VGT for emission control and pump-ing work minimization in diesel engines,” Licentiate thesis, LinköpingInst. Technol., Linköping, Sweden, 2006.

[31] J. Wahlström, L. Eriksson, L. Nielsen, and M. Pettersson, “PID controllersand their tuning for EGR and VGT control in diesel engines,” in Proc.IFAC World Congr., Prague, Czech Republic, 2005.

Anders Fröberg received the M.S. degree in appliedphysics and electrical engineering from LinköpingUniversity, Linköping, Sweden, in 2002. He is cur-rently working toward the Ph.D. degree at the Divi-sion of Vehicular Systems, Department of ElectricalEngineering at Linköping University.

His research interests include control and simula-tion of vehicle propulsion.

Lars Nielsen was born in Sweden in 1955. Hereceived the M.Sc. degree in engineering physics andthe Ph.D. degree in automatic control from LundInstitute of Technology, Lund, Sweden, in 1979 and1985, respectively.

Since 1992, he has been a Professor with theDivision of Vehicular Systems, Department of Elec-trical Engineering, Linköping University, Linköping,Sweden, where he holds the Sten Gustafsson Chair.His research interests include automotive modeling,control, and diagnosis.

Documents

Efficient Drive Cycle Simulation