- Email: [email protected]
A Method to Extend Inverse Dynamic Simulation of Powertrains with Additional Dynamics
Copyright © IFAC Advances in Automotive Control Salerno, Italy, 2004
ELSEVIER
IFAC PUBUCATIONS www.elsevier.comllocatelifac
A METHOD TO EXTEND INVERSE DYNAMIC SIMULATION OF POWERTRAINS WITH ADDITIONAL DYNAMICS Anders Froberg' Lars Nielsen'
• Linkopings Universitet, SE-581 83 Linkoping, Sweden email: [[email protected]. [email protected]
Abstract: Inverse dynamic powertrain simulation, like in Advisor or the QSStoolbox, has proven to be an efficient and successful approach to simulate vehicles during drive cycles. The approach L-.; based on back-calculation of accelerations and torques from the prescribed velocities in the drive cycle, and the differentiation requirements in this simulation process limits the possibility to include additional states in the powertrain models. The main objective here L-.; to extend the simulation with additional dynamics like e.g. mean vdlue modeL<; of the engine. This is achieved using stable inversion of nonlinear systems that can handle such additional dynamics. Computer algebra can be used to perform the necessary model transformations. A key step in obtaining sufficient differentiation properties is to smooth the drive cycle using a kernel with interpretation as an implicit driver model. The proposed method is demonstrated using :Mathematica for model transformation and Matlab for simulation. Copyright © 2004 IFAC Keywords: Fuel consumption, Engine modeling, Driver Model, Vehicle simulation.
1. INTRODUCTION
ertrain modeLe; including for example mean value engine models. A major contribution of this paper Le; to propose a new methodology that uses stable inversion of nonlinear systems that can handle such additional dynamics. Such stable inversion has been studied in non-linear control (Devasia et al., 1996; Hunt and Meyer, 1997; Isidori, 1995), but will here be utilized for simulation. The method requires model transformations that can be handled by computer algebra.
Simulation of longitudinal vehicle models is a commonly used tool for powertrain design, powertrain optimization, and, design of powertrain control strategies. There are two common ways to do this, inverse dynamic, and forward dynamic simulation, e.g. using Matlab/ Simulink or similar tools. The inverse dynamic simulation Le; implemented in well known tooL<; such as Advisor (Wipke et al., 1999) and the QSS-Toolbox (Guzzella and Amstutz , 1999), and these tooL<; show very good efficiency in simulation time. 1Jore background on these methods are given in the next section, but the reason for their success is that these tools use vehicle speed and acceleration to calculate required torques and speeds backwards through the powertrain. Note that this is a type of system inversion, which is a key observation to e>..1:end these methods to more general pow-
The paper is organized as follows. First some background to inverse dynamic simulation is given. In Section 3 a general procedure for nonlinear system inversion is discussed. These methods are then demonstrated on an example model of a powertrain in Sections 4 and 5. The powertrain model includes engine dynamics. Drive cycle tracking for forward dynamic and inverse dynamic simulation is discussed in Section 6. There, a new
643
2.1 Proposed
implicit driver model for inverse dynamic models is presented, and the overall concept is collected in Section 7. Simulatioll.'i of the presented inverse powertrain model are presented in Section 8, and in the last section the conclusions are drawn.
It would of course be of considerable value to be able to extend the tim~efficient quasi-static simulation with important dynamics without significantly loosing simulation performance. In this paper we study the use of stable inversion of non-linear systems (Devasia et al., 1996; Hunt and Meyer, 1997) as a mean to simulate dynamic systems. This inverse dynamic simulation is quite similar to the quasi-static approach, but extends its use to also include more dynamics in addition to the stationary maps. As we will see, there will still be explicit tracking as in Figure 1, but this will require drive-cycle smoothing which has a natural interpretation as an implicit driver model.
2. BACKGROl.JND Inverse dynamic simulation is implemented in well known tools such as Advisor (Wipke et al. , 1999) and the QSS-Toolbox (GuzzeIla and Amstutz, 1999). These tools use vehicle speed and acceleration to calculate required torques and speeds backwards through the powertrain. Finally fuel flow can be calculated. The computational scheme is depicted in Figure 1. The models mainly
I Cycle
H H HTransm·H Vehicle
Wheel
Engine
Exten.~ion
3. NONLINEAR SYSTEM INVERSION
I
It is assumed that the vehicle models can be written on the following form,
Fig. 1. Schematic depiction of the computational scheme in inverse dynamic simulation as in Advisor and the QSS-Toolbox. The simulated states are calculated back-wards from the drive cycle , and drive cycle tracking is thus explicit. No driver model is used.
x(t)
= f(x(t)) + g(x(t))u(t)
yet) = h(x(t) ) where yet) is vehicle speed. If one wants to compute the inputs required for the vehicle to follow a desired speed trajectory Yd(t) , given from a drive cycle, one way is to solve
consists of static equations and maps of the components efficiency. In each simulation step speed and acceleration are considered to be COll.'itant. Because of this, these models are called quasistatic models. A major advantage of this method is that simulation time is so low that it can be used in design exploration and optimization loops.
Xd(t) = f(Xd(t))
+ g(Xd(t))Ud(t)
(1)
Yd(t) = h(Xd(t)) for Xd(t), the state trajectory, and Ud(t), the input needed to produce it. Thi'i can be accomplished by using the methods of stable inversion of non linear systems presented in (Devasia et al., 1996; Hunt and Meyer, 1997). There it is also described how non minimum-phase systems can be simulated in a stable way by making a non-causal implementation. This is no restriction in drive cycle simulations since the whole drive cycle is known before hand.
In the forward dynamic simulation, differential equations are solved using, e.g., throttle position (or fuel flow ) as input, and vehicle speed as output. These kinds of models also require a controller, a driver model, to track a given speed trajectory (drive cycle). The computational scheme is depicted in Figure 2. The differential equations
The procedure for inversion is as follows: first find the relative degree r of the system, see (Isidori, 1995). Then by differentiating the output r times we get
Fig. 2. Schematic depiction of the computational scheme in forward dynamic simulation as in e.g. Matlabj Simulink. Drive cycle tracking is implicit since it is obtained using an explicit driver model.
y(r)(t) = Ujh(x(t) ) + LgL'j-lh(x(t))u(t)
(2)
The following change of coordinates is done: (~T , rtf= 'Ij;( x)
(3)
where ~ is Y and the r - 1 first derivatives thereof. To fill out the new state basis, 1] is chosen in a way so that 'Ij; forms a change of coordinates, which means that 1] is the state vector of the internal dynamics or zero dynamics of the system. Now the following definition is made:
that then have to be solved typically gives an order of magnitude longer simulation times for drive cycle simulations than what is typical for Advisor or QSS. On the other hand, more effects can be included making the modeling more accurate than in usual quasi-static simulation.
644
cx(~ , 1]) = Lfh(1jJ -1(~ , 1]»
(4)
{3(~ , 1]) = LgL,/1 h(1jJ-1 (~ , 1]»
(5)
41fT BM t;jJ = Vd
= mJQhv1]th
1 M t;?
FM t;jJ =
u(t ) = {3( ~ (t ), 1](t» -1 (y (r ) (t ) - cx (~ (t ), 1](t» ) (6 )
~aux (0.464 + 0.0072 (S;c
t
S
)1fb/ . 105
--s
41fT) (0 .075)0.5 +0.0215 Vd
and the original state trajectories can be found from the inverse coordinate change
x (t ) = 1jJ -1(~(t) , 1](t»
(16)
Vd
9
From (2), the signal u(t) can be calculated from yet) and the r first derivatives thereof as
(15 )
PMEP= (Pc - Pi )
(7)
(20)
(21 )
4. POWERTRAIN MODEL
The friction model (17) is according to (Soltic, 2000 ) and the normalized air ma.'iS charge (20) is according to (Hendricks et al., 1996) As can be seen, the model has two states, intake manifold pressure and vehicle speed. The fuel mass of Equation (20) is in simulations replaced with a fixed v-Mue at idle.
A basic powertrain model including a two-state model of a naturally a.-;pirated SI engine ",ill be used as an example to demonstrate vehicle simulation using stable inversion. The engine model is a mean value model with two states, intake manifold pressure Pi and vehicle speed v. Altogether this is a simple example of a standard model of a powertrain that it is desirable to be able to simulate efficiently as discussed in Section 2.
4.1 Resulting Powertrain Model By combining equations (8) - (21) and assuming constant exhaust manifold pressure we get a system on the form
All variables in the following are explained in the nomenclature in Section 10. The equations are either standard as described in {Kiencke and Nielsen, 2000 ), or a reference is given.
v = Cl v 2 + C2Vl.S + C3Pi + C4
The interaction between the environment and the vehicle is modeled as 1
= mv + 2CdPAv2 + m9er
Pi = CsV + Gt;VPi + C7mat
(8 )
= clxi + c2 x i's + C3X2 + C4 = h (X) (24 ) X2 = C5Xl + C6X1X2 + C7U = h ex ) + 92 (X)U Y=Xl = hex )
(10) The transmission is modeled as an ideal gear with constant efficiency
5. INVERSE VERSION OF THE POWERTRAIN MODEL In this section it will be shown how the method of stable inversion described in Section 3 can be applied to the powertrain model described in the previous section.
(11)
(12) The engine is modeled as a rotating mass ]wc =
'1' -
'1~
By computing the Lie derivatives of hex ) in the directions f (x ) and g(x ) it L-; seen that Lgh(x ) = 0 and LgLJh(x) = C3C7 # 0, \:Ix. As described in (hidori, 1995; Devasia et al., 1996) this means that the system (24) has relative degree r = 2. By differentiating yet ) two times , equation (2) becomes
(13)
where, '1 ', is the driving torque on the crankshaft. The driving torque, '1', comes from the following mean value engine model
BMEP = IMEPg
-
FMEP - PMEP
(23 )
Xl
(9)
wc=iww
(22)
Introducing Xl = V, X2 = Pi , U = mat and y = v leads to the state space form
The wheel is modeled as a stiff rolling wheel without slip
Tw=Fr
(18) (19)
In the coming sections this method will be demonstrated on an example of an extended powertrain model.
F
(17)
(14)
where
fi(t)
645
= L}h(x(t» + LgLJh(x(t» u (t )
(25)
6.1 Smoothed drive cycle
Since the relative degree of the system (24) equals the dimension of the system, there are no zero dynamics and the change of coordinates (3) becomes f, = 1j;(x ) ,
The method in Section 3 requires that the desired speed Yd (t ) is r (the relative degree of the system) times continuously differentiable, and r is two in our example. The New European Drive Cycle L<; piecewise linear and can thus not be continllollsly differentiated two times as required. Therefore the drive cycle used as input to the inverse model has to be modified so that the derivatives are bounded, and this is done as follows.
(26)
(27) Equation (4) and (5) becomes
To achieve a Yd(t ) that is r times continuously differentiable, any given drive cycle Yc (t ) can be convolved with an r times continuously differentiable kernel get ) which gives
a(f,) = L~h (1j; -l (f,» = ( 1.8C2~~· 8 + 2C16 )6 + C3C56 {3(f,) = L g L f h(1j;-l (f,» = C3 C7
(28) (29)
Yd (t) = C1
where Ci are the model parameters in Equations (22)- (23).
get ) =
(30)
X2 (t ) = (E.2 - cIE.r - c2E.P - Ci)/C3
(32)
T)Yc (T) dT
(33)
,ItI :::;
a;
(34 )
otherwi se;
See Figure 4 for a plot of get ) where the parameter value is a = 1.
The original state trajectories can be found from the inverse coordinate change x(t ) = 1j; -l ( ~(t)
(31 )
{e5:~" 0,
by using (26) and (27).
Xl (t ) = 6
-00
where C = J get ) dt . A good choice for the convolution kernel, get) , L<; to use the definition
Then Ud (t ) in Equation (1) is calculated from Yd(t ), Yd (t ) and fid (t ) as
Ud (t ) = {3(~ (t» -l (iid (t ) - a(f,(t »)
100 get -
6. DRIVE CYCLE TRACKING Fig. 4. The convolution kernel used to smooth the drive cycle. (a=I )
A drive cycle is a velocity profile and the New European Drive Cycle (NEDC) in Figure 3 L<; a typical example. When extending the inverse dynamic simulation with more dynamics, higher derivatives of the speed have to be calculated. This can be seen in our example in Equation (30) where also the second derivative of Yd is needed.
A nice property of this choice is that g (T) (t ) E Coo, '
1100
'~r---------------------~---------
y~T) (t) = C ' 20
-00
g (T) (t - r )yc(r ) dr
(35)
and can thus be calculated without numerical differentiation.
' 00
Another way of achieving an r times differentiable speed trajectory, could be to use a low pass filter of relative degree r . This would however only give asymptotic tracking of linear segments, whereas (33) gives exact tracking already at distance a from corners.
d~ired
,-
, ,
I .
°O~--~~-----_--~-~ ~--~~-----'~ ~----~ 'm
6.2 Implicit driver model
Tmo(s]
Fig. 3. The New European Drive Cycle is piecewise linear.
(l'I~DC)
The smoothing of the drive cycle Yc(t ) corresponds to the driver model in the forward simulation. By
646
8. SThfULATIONS
changing the parameter a in the kerneL different driver behaviors can be simulated. The parameter a is the look-ahead time for the driver, and also the time it takes to adjust to a new speed. This implicit driver model is only depending on the desired tracking behavior of the drive cycle and not on the vehicle model. In forward simulation, the driver model may have to be changed when a parameter in the vehicle model is changed, if the tracking behavior is to be preserved. In Figure 5 the behavior of a driver with 5 seconds look-ahead time is shown in the ECE (first part of the NEDC) drive cycle.
The example in Section 4 has been simulated to demonstrate the feasibility of the new extended inverse dynamic concept. The main point is to demonstrate simulation of Equation (30), to show inverse dynamic simulation including engine dynamics. Fuel consumption is chosen as output in the same way as in Advisor and the QSSToolbox. With an a..'>Sumption of stoichiometric air fuel ratio, calculating fuel flow io; equivalent of calculating the airflow. The model input, the air flow past throttle, and the model output, vehicle speed, can be seen in Figure 7. The corresponding forward simulation has also been done, using Equations (22)- (23) together with a driver model implemented a..<; a PI-controller. The result is similar a..<; expected and is just presented as an additional validation. In (Froberg and Nielsen, 2004), it is shown that the simulation time for forward dynamic simulation is typically
/
40
!
I I
r-----\
I
\
2 - 10....
: ,'\ I \ / !
\
/
ou~~ o
~
~
\
~
,
~
'0. ,
0
•
/
\~~' '--~-'~~ ,~
,~
,~
,~
,~
Time (s)
Fig. 5. The ECE cycle and the tracking behavior of an implicit driver model with a look-ahead time of 5 seconds.
~1 7. NEW EXTENDED CONCEPT
1000
600
200
1200
i rme(s] 1~' r-----------------------------~----'
The overall new scheme can be illustrated as in FigllTe 6. The "Driver" block in the scheme contains the convolution with get ). Note the extension to the quasi-static approach depicted in Figure 1, and note also how the "Driver" block relates to the corresponding block in the forward scheme depicted in Figure 2.
120
100
o
Fig. 6. Schematic depiction of the new extended inverse dynamic concept that is an extension of inverse dynamic simulation as in Advisor and the QSS-Toolbox. The simulated states are calculated back-wards from the drive cycle. The drive cycle tracking is explicit, whereas the driver model is implicit. In standard forward dynamic simulation, as in Figure 2, the driver model is e>q)licit whereas tracking is implicit.
200
400
I!IOO Time (s1
eoo
1000
1200
Fig. 7. Control signal (airflow past throttle) and vehicle speed in the NEDC cycle. Inverse (thick) and forward mode. The main point is to demonstrate inverse simulation of a powertrain model that includes engine dynamics. Mathematica was used for the model transformations described in Section 3, and Equation (34) was used as the implicit driver model.
647
an order of magnitude longer than what is typical for inverse dynamic simulation.
S '1' '1~
Ti '1 'w v
9. CONCLUSIONS Inverse dynamic powertrain simulation, like in Advisor or the QSS-toolbox, has proven to be an efficient and successful approach to simulate vehicles during drive cycles. The main objective here was to extend the model class, possible to simulate lL.<;ing this approach, with additional dynamics like e.g. mean value models of the engine. This is achieved using a framework based on stable inversion of nonlinear systems. A key step to obtain the differentiation properties needed is to smooth the drive cycle using a kernel with interpretation as an implicit driver model. A powertrain model presented in Section 4, that includes engine manifold dynamics as the additional dynamics, was used as a demonstrating example, and the new proposed method successfully handled it.
Vd
Vi Yi 11th 11trn
>. ~au:r 7rb/
p Ww Wc
REFERENCES
ACKNOWLEDGlvIENTS The Swedi<;h Energy Agency and the Swedish Foundation for Strategic Research are gratefully acknowledged for their funding.
10. NOMENCLATURE
A
(~)s B BMEP Cd
c.,.
F
FMEP 9
IMEPg i
J m mac
mat
m,
nr
Pc Pi PMJ:JP
R Si
Engine stroke Crank shaft torque Engine (flywheel) torque Intake manifold temperature Wheel torque Vehicle speed Engine displacement volume Intake manifold volume y-intercept of the normalized air charge Thermal efficiency Transmission efficiency Normalized air fuel ratio Friction coefficient Boost layout Air density Wheel rotational speed Engine rotational speed
Vehicle cross sectional area Stoichiometric air fuel ratio Engine bore Brake mean effective pressure Air drag coefficient Rolling resil>"tance coefficient Wheel traction force Friction mean effective pressure Gravitational acceleration Gross indicated mean effective pressure Gear ratio Engine inertia Vehicle mass Engine air charge Air mass flow past throttle Engine fuel charge Revolutions per engine cycle Exhaust manifold pressure Intake manifold pressure Pump mean affective pressure Fuel heating value Wheel radius Gas constant Slope of the normalized air charge
648
Devasia, S., D. Chen and B. Paden (1996). Nonlinear inversion-based output tracking. IEEE Transactions on Automatic Control 41, 930942. Fr6berg, A. and L Nielsen (2004). Dynamic vehicle simulation -forward, inverse and new mi'Ced possibilities for optimized design and control. SAE Technical Paper SerieB, 2004-01-1619. Guzzella, L. and A. AIIll>"tutz (1999). Ca.e tools for quasi-static mode ling and optimization of hybrid powertrarn.<;. IEEE Tran.~actions on Vehicular Technology 48, 1762-1769. Hendricks, E., A. Chevalier, M. Jensen, S. C. Sorenson, D. Trumpy and J. Asik (1996). :Modelling of the intake manifold filling dynamics. SAE Technical Paper Series SP1149, 1-25. Hunt, L. R. and G. Meyer (1997). Stable inversion for nonlinear systems. Automatica 33, 15491554. Isidori, A. (1995). Nonlinear Control Systems. Springer-Verlag. Kiencke, U. and L. Nielsen (2000) . .4.utomotive Control System~. Springer-Verlag. Soltic, P. (2000). Part-Load Optimized SI Engine System~. PhD thesis. Swiss Federal Institute of Technology, Zurich. Wipke, K. B., M. R. Cuddy and S. D. Burch (1999). Advisor 2.1: A user-friendly advanced powertrain simulation using a combined backward/forward approach .. IEEE Trnnsaction.~ on Vehicular Technology 48, 1751-1761.
ELSEVIER
IFAC PUBUCATIONS www.elsevier.comllocatelifac
A METHOD TO EXTEND INVERSE DYNAMIC SIMULATION OF POWERTRAINS WITH ADDITIONAL DYNAMICS Anders Froberg' Lars Nielsen'
• Linkopings Universitet, SE-581 83 Linkoping, Sweden email: [[email protected]. [email protected]
Abstract: Inverse dynamic powertrain simulation, like in Advisor or the QSStoolbox, has proven to be an efficient and successful approach to simulate vehicles during drive cycles. The approach L-.; based on back-calculation of accelerations and torques from the prescribed velocities in the drive cycle, and the differentiation requirements in this simulation process limits the possibility to include additional states in the powertrain models. The main objective here L-.; to extend the simulation with additional dynamics like e.g. mean vdlue modeL<; of the engine. This is achieved using stable inversion of nonlinear systems that can handle such additional dynamics. Computer algebra can be used to perform the necessary model transformations. A key step in obtaining sufficient differentiation properties is to smooth the drive cycle using a kernel with interpretation as an implicit driver model. The proposed method is demonstrated using :Mathematica for model transformation and Matlab for simulation. Copyright © 2004 IFAC Keywords: Fuel consumption, Engine modeling, Driver Model, Vehicle simulation.
1. INTRODUCTION
ertrain modeLe; including for example mean value engine models. A major contribution of this paper Le; to propose a new methodology that uses stable inversion of nonlinear systems that can handle such additional dynamics. Such stable inversion has been studied in non-linear control (Devasia et al., 1996; Hunt and Meyer, 1997; Isidori, 1995), but will here be utilized for simulation. The method requires model transformations that can be handled by computer algebra.
Simulation of longitudinal vehicle models is a commonly used tool for powertrain design, powertrain optimization, and, design of powertrain control strategies. There are two common ways to do this, inverse dynamic, and forward dynamic simulation, e.g. using Matlab/ Simulink or similar tools. The inverse dynamic simulation Le; implemented in well known tooL<; such as Advisor (Wipke et al., 1999) and the QSS-Toolbox (Guzzella and Amstutz , 1999), and these tooL<; show very good efficiency in simulation time. 1Jore background on these methods are given in the next section, but the reason for their success is that these tools use vehicle speed and acceleration to calculate required torques and speeds backwards through the powertrain. Note that this is a type of system inversion, which is a key observation to e>..1:end these methods to more general pow-
The paper is organized as follows. First some background to inverse dynamic simulation is given. In Section 3 a general procedure for nonlinear system inversion is discussed. These methods are then demonstrated on an example model of a powertrain in Sections 4 and 5. The powertrain model includes engine dynamics. Drive cycle tracking for forward dynamic and inverse dynamic simulation is discussed in Section 6. There, a new
643
2.1 Proposed
implicit driver model for inverse dynamic models is presented, and the overall concept is collected in Section 7. Simulatioll.'i of the presented inverse powertrain model are presented in Section 8, and in the last section the conclusions are drawn.
It would of course be of considerable value to be able to extend the tim~efficient quasi-static simulation with important dynamics without significantly loosing simulation performance. In this paper we study the use of stable inversion of non-linear systems (Devasia et al., 1996; Hunt and Meyer, 1997) as a mean to simulate dynamic systems. This inverse dynamic simulation is quite similar to the quasi-static approach, but extends its use to also include more dynamics in addition to the stationary maps. As we will see, there will still be explicit tracking as in Figure 1, but this will require drive-cycle smoothing which has a natural interpretation as an implicit driver model.
2. BACKGROl.JND Inverse dynamic simulation is implemented in well known tools such as Advisor (Wipke et al. , 1999) and the QSS-Toolbox (GuzzeIla and Amstutz, 1999). These tools use vehicle speed and acceleration to calculate required torques and speeds backwards through the powertrain. Finally fuel flow can be calculated. The computational scheme is depicted in Figure 1. The models mainly
I Cycle
H H HTransm·H Vehicle
Wheel
Engine
Exten.~ion
3. NONLINEAR SYSTEM INVERSION
I
It is assumed that the vehicle models can be written on the following form,
Fig. 1. Schematic depiction of the computational scheme in inverse dynamic simulation as in Advisor and the QSS-Toolbox. The simulated states are calculated back-wards from the drive cycle , and drive cycle tracking is thus explicit. No driver model is used.
x(t)
= f(x(t)) + g(x(t))u(t)
yet) = h(x(t) ) where yet) is vehicle speed. If one wants to compute the inputs required for the vehicle to follow a desired speed trajectory Yd(t) , given from a drive cycle, one way is to solve
consists of static equations and maps of the components efficiency. In each simulation step speed and acceleration are considered to be COll.'itant. Because of this, these models are called quasistatic models. A major advantage of this method is that simulation time is so low that it can be used in design exploration and optimization loops.
Xd(t) = f(Xd(t))
+ g(Xd(t))Ud(t)
(1)
Yd(t) = h(Xd(t)) for Xd(t), the state trajectory, and Ud(t), the input needed to produce it. Thi'i can be accomplished by using the methods of stable inversion of non linear systems presented in (Devasia et al., 1996; Hunt and Meyer, 1997). There it is also described how non minimum-phase systems can be simulated in a stable way by making a non-causal implementation. This is no restriction in drive cycle simulations since the whole drive cycle is known before hand.
In the forward dynamic simulation, differential equations are solved using, e.g., throttle position (or fuel flow ) as input, and vehicle speed as output. These kinds of models also require a controller, a driver model, to track a given speed trajectory (drive cycle). The computational scheme is depicted in Figure 2. The differential equations
The procedure for inversion is as follows: first find the relative degree r of the system, see (Isidori, 1995). Then by differentiating the output r times we get
Fig. 2. Schematic depiction of the computational scheme in forward dynamic simulation as in e.g. Matlabj Simulink. Drive cycle tracking is implicit since it is obtained using an explicit driver model.
y(r)(t) = Ujh(x(t) ) + LgL'j-lh(x(t))u(t)
(2)
The following change of coordinates is done: (~T , rtf= 'Ij;( x)
(3)
where ~ is Y and the r - 1 first derivatives thereof. To fill out the new state basis, 1] is chosen in a way so that 'Ij; forms a change of coordinates, which means that 1] is the state vector of the internal dynamics or zero dynamics of the system. Now the following definition is made:
that then have to be solved typically gives an order of magnitude longer simulation times for drive cycle simulations than what is typical for Advisor or QSS. On the other hand, more effects can be included making the modeling more accurate than in usual quasi-static simulation.
644
cx(~ , 1]) = Lfh(1jJ -1(~ , 1]»
(4)
{3(~ , 1]) = LgL,/1 h(1jJ-1 (~ , 1]»
(5)
41fT BM t;jJ = Vd
= mJQhv1]th
1 M t;?
FM t;jJ =
u(t ) = {3( ~ (t ), 1](t» -1 (y (r ) (t ) - cx (~ (t ), 1](t» ) (6 )
~aux (0.464 + 0.0072 (S;c
t
S
)1fb/ . 105
--s
41fT) (0 .075)0.5 +0.0215 Vd
and the original state trajectories can be found from the inverse coordinate change
x (t ) = 1jJ -1(~(t) , 1](t»
(16)
Vd
9
From (2), the signal u(t) can be calculated from yet) and the r first derivatives thereof as
(15 )
PMEP= (Pc - Pi )
(7)
(20)
(21 )
4. POWERTRAIN MODEL
The friction model (17) is according to (Soltic, 2000 ) and the normalized air ma.'iS charge (20) is according to (Hendricks et al., 1996) As can be seen, the model has two states, intake manifold pressure and vehicle speed. The fuel mass of Equation (20) is in simulations replaced with a fixed v-Mue at idle.
A basic powertrain model including a two-state model of a naturally a.-;pirated SI engine ",ill be used as an example to demonstrate vehicle simulation using stable inversion. The engine model is a mean value model with two states, intake manifold pressure Pi and vehicle speed v. Altogether this is a simple example of a standard model of a powertrain that it is desirable to be able to simulate efficiently as discussed in Section 2.
4.1 Resulting Powertrain Model By combining equations (8) - (21) and assuming constant exhaust manifold pressure we get a system on the form
All variables in the following are explained in the nomenclature in Section 10. The equations are either standard as described in {Kiencke and Nielsen, 2000 ), or a reference is given.
v = Cl v 2 + C2Vl.S + C3Pi + C4
The interaction between the environment and the vehicle is modeled as 1
= mv + 2CdPAv2 + m9er
Pi = CsV + Gt;VPi + C7mat
(8 )
= clxi + c2 x i's + C3X2 + C4 = h (X) (24 ) X2 = C5Xl + C6X1X2 + C7U = h ex ) + 92 (X)U Y=Xl = hex )
(10) The transmission is modeled as an ideal gear with constant efficiency
5. INVERSE VERSION OF THE POWERTRAIN MODEL In this section it will be shown how the method of stable inversion described in Section 3 can be applied to the powertrain model described in the previous section.
(11)
(12) The engine is modeled as a rotating mass ]wc =
'1' -
'1~
By computing the Lie derivatives of hex ) in the directions f (x ) and g(x ) it L-; seen that Lgh(x ) = 0 and LgLJh(x) = C3C7 # 0, \:Ix. As described in (hidori, 1995; Devasia et al., 1996) this means that the system (24) has relative degree r = 2. By differentiating yet ) two times , equation (2) becomes
(13)
where, '1 ', is the driving torque on the crankshaft. The driving torque, '1', comes from the following mean value engine model
BMEP = IMEPg
-
FMEP - PMEP
(23 )
Xl
(9)
wc=iww
(22)
Introducing Xl = V, X2 = Pi , U = mat and y = v leads to the state space form
The wheel is modeled as a stiff rolling wheel without slip
Tw=Fr
(18) (19)
In the coming sections this method will be demonstrated on an example of an extended powertrain model.
F
(17)
(14)
where
fi(t)
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= L}h(x(t» + LgLJh(x(t» u (t )
(25)
6.1 Smoothed drive cycle
Since the relative degree of the system (24) equals the dimension of the system, there are no zero dynamics and the change of coordinates (3) becomes f, = 1j;(x ) ,
The method in Section 3 requires that the desired speed Yd (t ) is r (the relative degree of the system) times continuously differentiable, and r is two in our example. The New European Drive Cycle L<; piecewise linear and can thus not be continllollsly differentiated two times as required. Therefore the drive cycle used as input to the inverse model has to be modified so that the derivatives are bounded, and this is done as follows.
(26)
(27) Equation (4) and (5) becomes
To achieve a Yd(t ) that is r times continuously differentiable, any given drive cycle Yc (t ) can be convolved with an r times continuously differentiable kernel get ) which gives
a(f,) = L~h (1j; -l (f,» = ( 1.8C2~~· 8 + 2C16 )6 + C3C56 {3(f,) = L g L f h(1j;-l (f,» = C3 C7
(28) (29)
Yd (t) = C1
where Ci are the model parameters in Equations (22)- (23).
get ) =
(30)
X2 (t ) = (E.2 - cIE.r - c2E.P - Ci)/C3
(32)
T)Yc (T) dT
(33)
,ItI :::;
a;
(34 )
otherwi se;
See Figure 4 for a plot of get ) where the parameter value is a = 1.
The original state trajectories can be found from the inverse coordinate change x(t ) = 1j; -l ( ~(t)
(31 )
{e5:~" 0,
by using (26) and (27).
Xl (t ) = 6
-00
where C = J get ) dt . A good choice for the convolution kernel, get) , L<; to use the definition
Then Ud (t ) in Equation (1) is calculated from Yd(t ), Yd (t ) and fid (t ) as
Ud (t ) = {3(~ (t» -l (iid (t ) - a(f,(t »)
100 get -
6. DRIVE CYCLE TRACKING Fig. 4. The convolution kernel used to smooth the drive cycle. (a=I )
A drive cycle is a velocity profile and the New European Drive Cycle (NEDC) in Figure 3 L<; a typical example. When extending the inverse dynamic simulation with more dynamics, higher derivatives of the speed have to be calculated. This can be seen in our example in Equation (30) where also the second derivative of Yd is needed.
A nice property of this choice is that g (T) (t ) E Coo, '
1100
'~r---------------------~---------
y~T) (t) = C ' 20
-00
g (T) (t - r )yc(r ) dr
(35)
and can thus be calculated without numerical differentiation.
' 00
Another way of achieving an r times differentiable speed trajectory, could be to use a low pass filter of relative degree r . This would however only give asymptotic tracking of linear segments, whereas (33) gives exact tracking already at distance a from corners.
d~ired
,-
, ,
I .
°O~--~~-----_--~-~ ~--~~-----'~ ~----~ 'm
6.2 Implicit driver model
Tmo(s]
Fig. 3. The New European Drive Cycle is piecewise linear.
(l'I~DC)
The smoothing of the drive cycle Yc(t ) corresponds to the driver model in the forward simulation. By
646
8. SThfULATIONS
changing the parameter a in the kerneL different driver behaviors can be simulated. The parameter a is the look-ahead time for the driver, and also the time it takes to adjust to a new speed. This implicit driver model is only depending on the desired tracking behavior of the drive cycle and not on the vehicle model. In forward simulation, the driver model may have to be changed when a parameter in the vehicle model is changed, if the tracking behavior is to be preserved. In Figure 5 the behavior of a driver with 5 seconds look-ahead time is shown in the ECE (first part of the NEDC) drive cycle.
The example in Section 4 has been simulated to demonstrate the feasibility of the new extended inverse dynamic concept. The main point is to demonstrate simulation of Equation (30), to show inverse dynamic simulation including engine dynamics. Fuel consumption is chosen as output in the same way as in Advisor and the QSSToolbox. With an a..'>Sumption of stoichiometric air fuel ratio, calculating fuel flow io; equivalent of calculating the airflow. The model input, the air flow past throttle, and the model output, vehicle speed, can be seen in Figure 7. The corresponding forward simulation has also been done, using Equations (22)- (23) together with a driver model implemented a..<; a PI-controller. The result is similar a..<; expected and is just presented as an additional validation. In (Froberg and Nielsen, 2004), it is shown that the simulation time for forward dynamic simulation is typically
/
40
!
I I
r-----\
I
\
2 - 10....
: ,'\ I \ / !
\
/
ou~~ o
~
~
\
~
,
~
'0. ,
0
•
/
\~~' '--~-'~~ ,~
,~
,~
,~
,~
Time (s)
Fig. 5. The ECE cycle and the tracking behavior of an implicit driver model with a look-ahead time of 5 seconds.
~1 7. NEW EXTENDED CONCEPT
1000
600
200
1200
i rme(s] 1~' r-----------------------------~----'
The overall new scheme can be illustrated as in FigllTe 6. The "Driver" block in the scheme contains the convolution with get ). Note the extension to the quasi-static approach depicted in Figure 1, and note also how the "Driver" block relates to the corresponding block in the forward scheme depicted in Figure 2.
120
100
o
Fig. 6. Schematic depiction of the new extended inverse dynamic concept that is an extension of inverse dynamic simulation as in Advisor and the QSS-Toolbox. The simulated states are calculated back-wards from the drive cycle. The drive cycle tracking is explicit, whereas the driver model is implicit. In standard forward dynamic simulation, as in Figure 2, the driver model is e>q)licit whereas tracking is implicit.
200
400
I!IOO Time (s1
eoo
1000
1200
Fig. 7. Control signal (airflow past throttle) and vehicle speed in the NEDC cycle. Inverse (thick) and forward mode. The main point is to demonstrate inverse simulation of a powertrain model that includes engine dynamics. Mathematica was used for the model transformations described in Section 3, and Equation (34) was used as the implicit driver model.
647
an order of magnitude longer than what is typical for inverse dynamic simulation.
S '1' '1~
Ti '1 'w v
9. CONCLUSIONS Inverse dynamic powertrain simulation, like in Advisor or the QSS-toolbox, has proven to be an efficient and successful approach to simulate vehicles during drive cycles. The main objective here was to extend the model class, possible to simulate lL.<;ing this approach, with additional dynamics like e.g. mean value models of the engine. This is achieved using a framework based on stable inversion of nonlinear systems. A key step to obtain the differentiation properties needed is to smooth the drive cycle using a kernel with interpretation as an implicit driver model. A powertrain model presented in Section 4, that includes engine manifold dynamics as the additional dynamics, was used as a demonstrating example, and the new proposed method successfully handled it.
Vd
Vi Yi 11th 11trn
>. ~au:r 7rb/
p Ww Wc
REFERENCES
ACKNOWLEDGlvIENTS The Swedi<;h Energy Agency and the Swedish Foundation for Strategic Research are gratefully acknowledged for their funding.
10. NOMENCLATURE
A
(~)s B BMEP Cd
c.,.
F
FMEP 9
IMEPg i
J m mac
mat
m,
nr
Pc Pi PMJ:JP
R Si
Engine stroke Crank shaft torque Engine (flywheel) torque Intake manifold temperature Wheel torque Vehicle speed Engine displacement volume Intake manifold volume y-intercept of the normalized air charge Thermal efficiency Transmission efficiency Normalized air fuel ratio Friction coefficient Boost layout Air density Wheel rotational speed Engine rotational speed
Vehicle cross sectional area Stoichiometric air fuel ratio Engine bore Brake mean effective pressure Air drag coefficient Rolling resil>"tance coefficient Wheel traction force Friction mean effective pressure Gravitational acceleration Gross indicated mean effective pressure Gear ratio Engine inertia Vehicle mass Engine air charge Air mass flow past throttle Engine fuel charge Revolutions per engine cycle Exhaust manifold pressure Intake manifold pressure Pump mean affective pressure Fuel heating value Wheel radius Gas constant Slope of the normalized air charge
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Devasia, S., D. Chen and B. Paden (1996). Nonlinear inversion-based output tracking. IEEE Transactions on Automatic Control 41, 930942. Fr6berg, A. and L Nielsen (2004). Dynamic vehicle simulation -forward, inverse and new mi'Ced possibilities for optimized design and control. SAE Technical Paper SerieB, 2004-01-1619. Guzzella, L. and A. AIIll>"tutz (1999). Ca.e tools for quasi-static mode ling and optimization of hybrid powertrarn.<;. IEEE Tran.~actions on Vehicular Technology 48, 1762-1769. Hendricks, E., A. Chevalier, M. Jensen, S. C. Sorenson, D. Trumpy and J. Asik (1996). :Modelling of the intake manifold filling dynamics. SAE Technical Paper Series SP1149, 1-25. Hunt, L. R. and G. Meyer (1997). Stable inversion for nonlinear systems. Automatica 33, 15491554. Isidori, A. (1995). Nonlinear Control Systems. Springer-Verlag. Kiencke, U. and L. Nielsen (2000) . .4.utomotive Control System~. Springer-Verlag. Soltic, P. (2000). Part-Load Optimized SI Engine System~. PhD thesis. Swiss Federal Institute of Technology, Zurich. Wipke, K. B., M. R. Cuddy and S. D. Burch (1999). Advisor 2.1: A user-friendly advanced powertrain simulation using a combined backward/forward approach .. IEEE Trnnsaction.~ on Vehicular Technology 48, 1751-1761.