3.3.1 General Remarks
The pressure of the gases in the cylinder provides the most direct signal avail- able for engine control purposes. Since the interpretation of this variable is a well-known tool in ICE research and development, numerous control algo- rithms based on that information have been proposed (see, e.g., [150], [151], [183] [212]). However, the equipment used in those experiments is far too ex- pensive to be included in series production engines. Recently, new in-cylinder pressure sensors have been developed that are sufficiently robust and rea- sonably priced such that their inclusion in series production vehicles appears feasible (see e.g., [95] and [205]).
Since full process calculations are not yet possible in the ECU, algorithms requiring a reduced amount of computational burden are of high interest. In [140], such simplified correlations useful for a control-oriented cylinder pres- sure interpretation have been derived. The highlights of that approach are reviewed in this section.
The pressure in the cylinder is determined by the movement of the piston and the corresponding compression of the air/fuel mixture inside the cylin- der, the heat release caused by the combustion of this mixture, and the heat transfer from the gas to the cylinder walls. The position of the piston can be described by
s(φ) =rcs· l+rcs
rcs − s l2
r2cs −sin2φ−cosφ
!
(3.64) wheres(φ) is the distance of the piston from its TDC position,φis the crank angle (φ= 0 at TDC),rcs is the radius of the crankshaft, i.e., one-half of the
stroke, and l is the length of the connecting rod. A good approximation of (3.64) is given by
s(φ)≈rcs·
1−cosφ+ rcs
2·lsin2φ
(3.65) The equations describing the piston speed and the piston acceleration, in particular, are easier to formulate using (3.65). Denoting the compression ratio by ε and the displaced volume byVd, the following equation holds for the cylinder volumeVc
Vc(φ) =Vd· 1
ε−1 + s(φ) 2·rcs
(3.66)
3.3.2 Estimation of Burned-Mass Fraction
In “forward” thermodynamic process calculations, the development of com- bustion is often described with the burnt-mass fractionxB(φ) defined as
xB(φ) = mϕb(φ)
mϕt (3.67)
wheremϕb(φ) is the mass of burnt fuel at crank angleφandmϕt is the total burnt fuel at the end of combustion. For rich conditions, the total mass of burnt fuel must be calculated with the available mass air, thusmϕt=λ·mϕ
wheremϕis the fuel mass in the cylinder at IVC. Under lean conditions,mϕt
is simplymϕ.
The burnt-mass fraction (3.67) is usually parameterized using the well- known Vibe function, [208] and Appendix C
xB(φ) = 1−ec·(
φ−φs−∆φd
∆φbd )mv+1
(3.68) where the parameter∆φd indicates the spark delay,∆φbd represents the du- ration of the combustion,c≈log(0.001) indicates that the end of combustion is assumed atxB = 0.999, andmvis used to shape the resulting curve for the burnt-mass fraction.
Note that if a symmetric burnt-mass fraction is acceptable, an even simpler approximation can be used as well
xB(φ) = sin2
φ−(φs+∆φd)
∆φbd ·π 2
(3.69) Using a heat release function, combined with a heat-transfer approxima- tion and the thermodynamic conservation laws, the pressure inside the cylin- der can be estimated [97]. This information can then be used to compute in- dicated efficiencies or, combined with a friction model such as that expressed by (2.112), to compute brake efficiencies.
In [49], functions were derived that describe the variations of the three Vibe parameters of equation (3.68) for different operating conditions. This approach allows a control-oriented description of the combustion processes and, thus, for instance, the determination of the engine inputs for optimal fuel consumption [156]. More details are given in Appendix C.
If the cylinder pressure is measured, the burnt-mass fraction can be deter- mined by an “inverse” thermodynamic process calculation. However, this is computationally quite demanding, some details can be found in Appendix C.
Therefore, an approximation that was first described in [171] is used here. The primary points of that approach are visualized in Fig. 3.21.
Fig. 3.21.Estimation of the burn rate.
The procedure starts with the selection of a pressure/volume pair, pcomp
andVcomp, during the compression phase (before the combustion starts) and a pressure/volume pair,pexpandVexp, during the expansion phase (after the combustion is completed). Extrapolating these two pairs to the minimum volume value using a polytropic approximation yields the pressure difference indicated byain Fig. 3.21. The actual pressure/volume data during the com- bustion is extrapolated in the same way, yielding the pressure difference b.
The mass of burnt fuel is then estimated by the ratio of these two pressure differences
xB(φ)≈ b a ≈
pc(φ)·V
c(φ) VT DC
n(φ)
−pcomp·V
comp
VT DC
ncomp
pexp· V
exp
VT DC
nexp
−pcomp·V
comp
VT DC
ncomp (3.70)
n(φ) =
ncomp for ∂V∂φc(φ) ≤0 nexp for ∂V∂φc(φ) >0
Choosing the reference points to be symmetric with respect to TDC (say at 80◦before and after11TDC) simplifies the computations. In fact, as illustrated in Fig. 3.22, in this case the burn rate can be estimated using the expression
xB(φ)≈
pc(φ)·V
c(φ) VT DC
npol
−pc1
∆pc (3.71)
To further simplify the computations, a constant value ofnpol= 1.31 can be used for the polytropic exponent.
Fig. 3.22. Pressure/Volume (pV) diagram using a double-logarithmic representa- tion including the reference points chosen for the symmetric case.
3.3.3 Cylinder Charge Estimation Introduction
A typical application for the method introduced in the last section is the estimation of the 50% burnt-mass position. Of course, the cylinder pressure trace can be used for various other purposes. Two straightforward possibilities are the detection of knock and misfiring or the identification of the actual engine cycle.
A computationally more demanding use of the pressure trace, which is based on physical first principles, is the estimation of the fresh air, fuel, and burnt-gas mass in the cylinder. This application is discussed in this section.
Assuming each cylinder to be equipped with a pressure sensor, this approach offers the possibility of cylinder-individual control, resulting in a more uniform load and air/fuel ratio distribution. Moreover, if the estimation results are sufficiently accurate, the air mass sensor can be left out.
11Note that the combustion must be completed at that point. This is the case for most regular combustion systems, but not necessarily for new, “exotic” engines.
The total mass in the cylinder, defined bymtot=mβ+mϕ+mbg, is fixed as soon as the intake valve has closed. The variablembg=mrg+megr is the total mass of burnt gas, i.e., the sum of the residual gas and the externally recirculated exhaust gas. Choosing a reference position during the compression stroke (indicated by an asterisk in Fig. 3.22), and using the measured cylinder pressurepc1 and the known cylinder volume (3.66), the ideal gas law
ˆ
mtot= pc1·Vc1
R·ϑc1
(3.72) yields an estimation ofmtotbyassuming the temperatureϑc1at the same reference point to have a value of approximately 470◦ K. Note that (3.72) is valid only because the gas constantRof the air/fuel mixture and burnt gases is quite similar (a value of 287J/kgK is recommended). Once the total mass is known, the amount of burnt gas can be estimated if the mass of aspirated mixture is estimated with any of the methods presented in either Sect. 2.3 or Sect. 3.2.2, or with the approaches shown below.
The estimation ofmtot using (3.72) is straightforward but does not yield sufficiently precise results that would allow for the omission of the air mass flow sensor. To achieve this goal, additional information must be obtained by analyzing the complete pressure trajectorypc(φ). However, in this case some information on the heat release during the combustion must be taken into account as well.
Basic Model
Using the lower heating valueHl of the fuel and introducing the parameter Cc (“completeness of combustion”) that indicates how much of the fuel is actually burnt until the exhaust valve starts to open, the following equation describes the total heat release
Qc =mϕ·Hl·Cc =mtot· Hl
1 +λ·σ0 ·(1−xbg)·Cc (3.73) At ideal conditions, i.e., formbg= 0,λ= 1, andCc= 1, the heat released is
Qc,ideal=mtot· Hl
1 +σ0
= pc1·Vc1
R·ϑc1 · Hl
1 +σ0
(3.74) Thus, a new variableηc, referred to as thecylinder charge efficiency, can be defined as follows
ηc= Qc
Qc,ideal
= (1−xbg)·Cc· 1 +σ0
1 +λσ0
(3.75) For a given value ofmtot, this parameter quantifies the ratio between the ac- tual and the maximum theoretically possible heat released in the combustion.
The relationship between heat release and pressure in the cylinder is an- alyzed next. For the two reference positions indicated above, the pressure difference
∆pc=pc2−pc1 (3.76)
can be calculated. For the relevant operating range of a six-cylinder 3.2-liter SI engine, the pressure difference as a function of the combustion energy is depicted in Fig. 3.23. The air/fuel ratio is set to stoichiometry, and the 50%
burnt-mass position,φ50, is always kept near its optimum value of approxi- mately 8◦ crank angle after TDC.
Fig. 3.23. Pressure difference as a function of the combustion energy. Engine op- erating rangen= 1000−4500rpmandpme= 0−6bar. SI engine,λ= 1,φ50≈8◦ crank angle after TDC.
As expected, an almost linear dependency can be observed. Note that this is just another confirmation of the remarks made in Sect. 2.5.1. In fact, the parameter e in the Willans approximation shown in Fig. 2.29 can be interpreted as the “indicated” (internal) efficiency that describes the ratio between combustion energy and pressure difference that eventually results in mechanical work on the piston.
Improved Model
The deviations of the single pressure difference point from a purely linear dependency occur mainly for two reasons:
• Variations in the engine speed influence the time available for the combus- tion and heat exchange with the cylinder walls.
• Variations in the combustion center, represented by the 50% burnt-mass position, influence the conversion of heat to mechanical work and, hence, the heat balance.
These observations lead to the following parametrization of the pressure difference as a function of engine speed and heat-release center
∆pc =k(ωe, φ50)·Qc
=k(ωe, φ50)·ηc·Qc,ideal (3.77)
=k(ωe, φ50)·ηc·pc1·Vc1
R·ϑc1 · Hl
1 +σ0
To investigate these influences, the effects of varying the 50% burnt-mass position and the engine speed can be estimated using thermodynamic process simulations. Assuming an ideal combustion withηc = 1, the factork(ωe, φ50) can be estimated by using (3.78), because all other terms on the right-hand side of this equation can be assumed to be constant
∆pc·ϑc1
pc1 =k(ωe, φ50)·Vc1
R · Hl
1 +σ0 (3.78)
Fig. 3.24. Numerical value of ∆pp c
c1ϑc1 for simulated variations in engine speed n and 50% burned-mass positionφ50.
Figure 3.24 shows the calculated variation for three different total mass values (representing idle, half-load, and full-load conditions). Obviously, the time available for the combustion is an important parameter. Accordingly, in this figure the inverse engine speed is used as an independent variable. A bilinear parametrization is proposed here
k(φ50, ωe)·Vc1
R · Hl
1 +σ0
= X
j∈corners
ϑc1·∆pc
pc1
j
·µj (3.79)
where µj=
1− |φ50,OP −φ50,j| φ50,max−φ50,min
· 1−|ω−OP1 −ωj−1| ω−min1 −ωmax−1
!
(3.80)
where µj are the weighting parameters that allow for the calculation of the value of an arbitrary operating point OP within a rectangle defined by the corner values.
With these preparations, the cylinder charge efficiency can be calculated based on the knowledge ofpc1 andφ50, and an estimation forϑc1
ηc = ∆pc
k(ωe, φ50)· R·ϑc1
pc1·Vc1 ·1 +σ0
Hl (3.81)
Choosingϑc1= 470◦K as a first approximation of the temperature of the cylinder charge at the IVC position, an estimation for the charge efficiency with an accuracy of typically±5% is obtained. If a better estimation of ϑc1
can be obtained from a detailed thermodynamic process calculation, the ap- proximation of the charge efficiency can be improved. As Fig. 3.25 shows, an estimation of (3.81) with an accuracy of±3% is feasible (see [139] for details).
Fig. 3.25.Estimation of the charge efficiency.
Finally, assuming a minimum and a maximum completeness of combustion of 0.93 and 0.98, respectively, the second-order effects of the amount of fuel injected, of the 50% burned-mass position, of the combustion duration, etc., may all be collected in a linear function depending onpc2
Cc = 0.93 +0.05(pc2−pc2,min) pc2,max−pc2,min
(3.82) Reasonable values for a naturally aspirated engine forpc2,max andpc2,minare
pc2,min= 1.5 bar and pc2,max= 15 bar
Now, using (3.75), ifλ is known from sensor information, the burned-gas fraction at IVC can be estimated. The estimated values plotted against the reference values (obtained with a two-zone thermodynamic process calcula- tion) are shown in Fig. 3.26.