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Research ArticleEstimation of Individual Cylinder Air-Fuel Ratio inGasoline Engine with Output Delay
Changhui Wang and Zhiyuan Liu
Department of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, China
Correspondence should be addressed to Changhui Wang; wang changhui@126.com
Received 1 March 2016; Revised 23 June 2016; Accepted 14 July 2016
Academic Editor: Jose A. Somolinos
Copyright Β© 2016 C. Wang and Z. Liu.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The estimation of the individual cylinder air-fuel ratio (AFR) with a single universal exhaust gas oxygen (UEGO) sensor installedin the exhaust pipe is an important issue for the cylinder-to-cylinder AFR balancing control, which can provide high-quality torquegeneration and reduce emissions in multicylinder engine. In this paper, the system dynamic for the gas in exhaust pipe includingthe gasmixing, gas transport, and sensor dynamics is described as an output delay system, and a newmethod using the output delaysystem observer is developed to estimate the individual cylinder AFR. With the AFR at confluence point augmented as a systemstate, an observer for the augmented discrete system with output delay is designed to estimate the AFR at confluence point. Usingthe gas mixingmodel, a method with the designed observer to estimate the individual cylinder AFR is presented.The validity of theproposedmethod is verified by the simulation results from a spark ignition gasoline engine from engine software enDYNAby Tesis.
1. Introduction
Cylinder-to-cylinder air-fuel ratio (AFR) balancing control ininternal combustion engines with multiple cylinders is oneof the technology trends to satisfy the increasingly stringentemission regulations, which can also improve the engineperformance, such as thermal efficiency and fuel economy.The AFR of each cylinder is decided by the aspirated airmass, the injected fuel mass, and the residual gas fromthe prior cycle, in which the combustion stroke of eachcylinder sequentially occurs along the rotational angle of thecrankshaft. Due to the air breathing variability and injectorvariability, there exists AFR imbalance between cylinders,leading to adverse impacts on emission performance usingthe conventional controllers [1, 2].
In order to improve the AFR control accuracy, therehas been a great deal of research that focuses on the AFRcontrol of individual cylinders [3β10]. In fact, the estimationof the individual cylinder AFRwith a single universal exhaustgas oxygen (UEGO) sensor installed in the exhaust pipeis one of the key technology trends for the individualcylinder AFR control. The digital filtering techniques areemployed to extract the AFR imbalance signals from oxygensensor voltage signals in [11], in which the oxygen sensor
voltage signal is processed to determine imbalanced cylinderidentification and AFR cylinder imbalance levels. In [12], amodeling method to estimate the individual fuel-gas ratio isproposed to estimate AFR, which is used for an adaptive gen-eralized predictive control approach to balance the individualcylinder characteristics in the static engine operation mode.A static steady state observer based on the individual cylinderAFR model along the air mass flow, gas mixing, and sensordynamics in an exhaust manifold can be found in [13]. In thediesel engines, a nonlinear observer is proposed to estimatethe individual cylinder AFR [14]. In [15], a PI compensator isdesigned to compensate the cylinder-by-cylinder variations,in which an input observer is proposed to estimate individualcylinder AFR.
However, the transport delay and sensor delay from theexhaust confluence point to UEGO sensor output are ignoredin the proposed observers from the above papers, whichmay reduce the accuracy of the AFR estimation of eachcylinder. In order to improve the individual cylinder AFRestimation accuracy, the system dynamics in the exhaust pipeincluding gas transport and sensor dynamics is describedas an augmented discrete system with output delay in thispaper, in which the AFR at confluence point is augmented as
Hindawi Publishing CorporationJournal of SensorsVolume 2016, Article ID 5908459, 9 pageshttp://dx.doi.org/10.1155/2016/5908459
2 Journal of Sensors
Three way catalyst
Cylinders
Fuel injector
Throttle
Confluence point
UEGO sensor
Exhaust manifoldIntake manifold
Figure 1: Schematic of the 4-cylinder SI gasoline engine.
a system state. Then, an observer for the augmented discretesystem with output delay is designed. With the combinationof the designed observer and the gas mixing model atconfluence point, the method to estimate the individualcylinder AFR is presented. The performance of the proposedmethod is validated against the simulation result from enginesoftware enDYNA provided by Tesis, and a comparison withexistingmethod is given during an urban driving cycle, whichdemonstrates that the proposed method can improve theaccuracy of the individual cylinder AFR estimation.
This paper is organized as follows. In Section 2, the systemdynamics in the exhaust pipe including gas transport andUEGO sensor dynamics is described as an augmented systemwith output delay. In Section 3, an observer for the outputdelay system is designed, and the method to estimate theindividual cylinder AFR is presented. Simulation results fromenDYNA are presented in Section 4, and the conclusions aresummarized in Section 5.
2. Problem Formulation
A schematic diagram of a 4-cylinder spark-ignited (SI)gasoline engine is shown in Figure 1, where the fuel injectorsequipped at the inlet port near to the intake valve are con-trolled individually. The fuel mass burnt in each cylinder isinjected by the corresponding injector, and the fuel injectioncommand is delivered to the injector of each cylinder seriallyalong the crank angle. The AFR of each cylinder is π
π=
πcyli/πair, π = 1, 2, 3, 4, where πcyli is the fuel mass into thecylinder andπair is the air mass into the cylinder.
After combustion, the combusted gas of each cylinder isexhausted into the corresponding runner during the exhauststroke of each cylinder and passes through their runners andconfluence in the public exhaust manifold. Then, the AFRof the mixed gas is measured by a UEGO sensor, and themixed gas runs to the outside passing through the catalyst.The system dynamics in the exhaust pipe includes the gasmixing, gas transport, and sensor dynamics, in which thetransfer function from the confluence point to sensor outputcan be given by [13]
πΊexh (π ) =πsen (π )
ππ (π )
=πβπΏπ
(1 + πmixπ ) (1 + πsenπ ), (1)
where ππis the AFR at the exhaust confluence point, πsen is
the measured AFR of the UEGO sensor, πΏ = πΏmix + πΏsen isthe time delay including the transport delay πΏmix and sensordelay πΏsen, πmix is the time constant of themixing process, andπsen is the sensor time constant.
For an engine with 4 cylinders, the combined UEGOsensor signal is sampled at the exhaust top dead center of eachcylinder, and the sampling period π
π is related to the engine
cycle period as ππ = ππ/4, in which the engine cycle period is
ππ= 120/π
π, where π
πis the engine speed in rpm. Consisting
of the zero-order holder and themixing and sensor dynamics,the discrete formofmodel (1) with the sampling periodπ
π can
be given by [13]
πexh (π§) = π§βπ(1 + π
π§ β 1
π§ β πΌmix+ π
π§ β 1
π§ β πΌsen) , (2)
where
πΌmix = πβππ /πmix ,
πΌsen = πβππ /πsen ,
π =πΏ
ππ
,
π =βπmixπmix β πsen
,
π =πsen
πmix β πsen,
π§ = πππ π .
(3)
Furthermore, (2) can be written as a state-space form withinput delay [15]:
π₯π (π + 1) = π΄ππ₯π (π) + π΅πππ (π β π) ,
πsen (π) = πΆππ₯π (π) ,(4)
Journal of Sensors 3
where
π΄π= (πΌmix 0
0 πΌsen) ,
π΅π= (1
1) ,
πΆπ= (π (πΌmix β 1) π (πΌsen β 1)) ,
π₯π (π) = (
π₯π1 (π)
π₯π2 (π)) ,
π₯π1 (π) =
πsen (π) β π (πΌsen β 1) π₯π2 (π)
π (πΌmix β 1),
π₯π2 (π) =
1
(πΌsen β πΌmix) π (πΌsen β 1)(πΌsenπsen (π)
β πΌmixπΌsenπsen (π β 1)
+ (1 β 2ππΌsen β 2ππΌmix β πΌmix β πΌsen + π + π)
β ππ (π β π) + (ππΌsen + ππΌmix + πΌmixπΌsen)
β ππ (π β π β 1)) .
(5)
Define the new system state as π₯π(π) = π₯
π(π + π), and (4)
with input delay can also be rewritten as the following discretesystem with output delay:
π₯π (π + 1) = π΄ππ₯π (π) + π΅πππ (π) ,
πsen (π) = πΆππ₯π (π β π) .(6)
ππ(π) in (6) is unknown, which can be considered as a
system state due to the small rate of change of ππ(π). Then,
an augmented discrete system with output delay from (6) canbe obtained:
π₯ (π + 1) = π΄π₯ (π) ,
π¦ (π) = πΆπ₯ (π β π) ,
(7)
where
π₯ (π) = (π₯π (π)
ππ (π)) ,
π΄ = (
πΌmix 0 1
0 πΌsen 1
0 0 1
) ,
π¦ (π) = πsen (π) ,
πΆ = (π (πΌmix β 1) π (πΌsen β 1) 0) .
(8)
Equation (7) indicates that the estimation of the AFR ππ
at the exhaust confluence point becomes the state estimationof the discrete system with output delay (7).
3. Observer Design for Discrete System withOutput Delay
The observer for the output delay system (7) is given:
οΏ½οΏ½ (π + 1) = π΄οΏ½οΏ½ (π) + πΏ (π¦ (π) β πΆοΏ½οΏ½ (π β π)) ,
οΏ½οΏ½ (π) = πΆοΏ½οΏ½ (π β π) ,
(9)
where οΏ½οΏ½ β R is the state estimate and πΏ β R3Γ1 isthe feedback gain matrix. The asymptotical stability of theproposed observer (9) is analyzed in the following theorem.
Theorem 1. There exist matrices πΏ, π = ππ > 0, π = ππ β₯ 0,and π = ππ β₯ 0, such that the following linear matrixinequality (LMI) is feasible:
Ξ =(((
(
βπ + π +ππ+π βπ
π+ π π΄
ππ π
π(π΄πβ πΌ)π
βπ +ππ
βπ β ππβ π βπΆ
ππΏππ π
πβπΆππΏππ
ππ΄ βππΏπΆ βπ 0 0
π π 0 βπβ1π 0
π (π΄ β πΌ) βππΏπΆ 0 0 πβ1(π β 2π)
)))
)
< 0; (10)
then observer (9) is asymptotically stable.
Proof. Set the estimation error π(π) = οΏ½οΏ½(π)βπ₯(π), and the errordynamic system between (7) and (9) is obtained:
π (π + 1) = π΄π (π) β πΏπΆπ (π β π) . (11)
Denote π(π) = π(π + 1) β π(π); then we have
π (π β π) = π (π) β
πβ1
β
π=πβπ
π (π) ,
π (π) = (π΄ β πΌ) π (π) β πΏπΆπ (π β π) .
(12)
4 Journal of Sensors
Choose a Lyapunov functional candidate as
π (π) = ππ(π) ππ (π) +
πβ1
β
π=πβπ
ππ(π) ππ (π)
+
0
β
π=βπ+1
πβ1
β
π=πβ1+π
ππ(π) ππ (π) .
(13)
Define Ξπ = π(π + 1) β π(π); then along the solution of (11)and (12) we have
Ξπ (π) = (π (π)
π (π β π))
π
β (βπ + π + π΄
πππ΄ βπ΄
πππΏπΆ
βπΆππΏπππ΄ βπ + πΆ
ππΏπππΏπΆ
)(π (π)
π (π β π))
+ πππ(π) ππ (π) β
πβ1
β
π=πβπ
ππ(π) ππ (π) .
(14)
According to Lemma 1 in [16], for any constantmatrixπ > 0,π, andπ, the following inequality holds:
β
πβ1
β
π=πβπ
ππ(π) ππ (π) β€ (
π (π)
π (π β π))
π
β (ππ+π βπ
π+ π
βπ +ππβππβ π)(
π (π)
π (π β π))
+ π(π (π)
π (π β π))
π
β (ππ
ππ)πβ1(π π)(
π (π)
π (π β π)) .
(15)
With the combination of (14) and (15), we have
Ξπ (π) β€ (π (π)
π (π β π))
π
Ξ©(π (π)
π (π β π)) , (16)
where
Ξ©
= (βπ + π +π
π+π + π΄
πππ΄ + π (π΄ β πΌ)
ππ (π΄ β πΌ) + ππ
ππβ1π βπ
π+ π β π΄
πππΏπΆ β π (π΄ β πΌ)
πππΏπΆ + ππ
ππβ1π
β βπ β ππβ π + πΆ
ππΏπππΏπΆ β ππΆ
ππΏπππΏπΆ + ππ
ππβ1π
) .
(17)
By Schur complement [17], the following LMI (18)guarantees Ξ© < 0, which can guarantee Ξπ(π) < 0 and theasymptotical stability of observer (9):
Ξ =(((
(
βπ + π +ππ+π βπ
π+ π π΄
ππ π
π(π΄πβ πΌ)π
βπ +ππ
βπ β ππβ π βπΆ
ππΏππ π
πβπΆππΏππ
ππ΄ βππΏπΆ βπ 0 0
π π 0 βπβ1π 0
π (π΄ β πΌ) βππΏπΆ 0 0 βπβ1π
)))
)
< 0. (18)
Now, condition (10) guaranteeing (18) must be proved.Defineπ = diag(πΌ, πΌ, πΌ, πΌ, ππβ1), and we have
ππΞ π =
((
(
βπ + π +ππ+π βπ
π+ π π΄
ππ π
π(π΄πβ πΌ) π
βπ +ππ
βπ β ππβ π βπΆ
ππΏππ π
πβπΆππΏππ
ππ΄ βππΏπΆ βπ 0 0
π π 0 βπβ1π 0
π (π΄ β πΌ) βππΏπΆ 0 0 βπβ1ππβ1π
))
)
. (19)
Journal of Sensors 5
enDYNA R4-cylinder SI-engine
Output delay observer (23)
Gas mixing
+
β
e
π1
π2 πc
π3
π4
οΏ½οΏ½1
οΏ½οΏ½2
οΏ½οΏ½3
οΏ½οΏ½4
οΏ½οΏ½sen
πsenGexh(s)
Figure 2: Schematic diagram for the estimation of the individual cylinder AFR.
Because of the fact that (π β π)πβ1(π β π) β₯ 0, we haveβππβ1π β€ π β 2π. Therefore, condition (10) can guarantee
π΀Ππ < 0; then (18) holds.The estimation of the AFR π
πat the exhaust confluence
point can be obtained according to observer (9). In order toobtain the AFR of each cylinder, the relationship between theAFR of each cylinder and the AFR at the exhaust confluencepoint is analyzed in the following.
The combusted gas of each cylinder is discharged intothe corresponding exhaust port and flows to the exhaustconfluence point in the exhaust manifold, in which weassume that exhaust gas mixing in the individual exhaustrunner can be neglected. Hence, the AFR in the exhaustrunner is constant during one engine cycle, and the AFR π
π
at the exhaust confluence point in the πth engine cycle can begiven by [15]
ππ(πππ+ (π β 1) ππ )
=βπ
π=1οΏ½οΏ½ππ(πππ+ (π β 1) ππ ) ππ (π)
β4
π=1οΏ½οΏ½ππ(πππ+ (π β 1) ππ )
+β4
π=1οΏ½οΏ½ππ(πππ+ (π β 1) ππ ) ππ (π β 1)
β4
π=1οΏ½οΏ½ππ(πππ+ (π β 1) ππ )
,
(20)
where οΏ½οΏ½ππis the exhaust air mass flow in the πth exhaust
manifold passing through the confluence point and ππis the
AFR in the πth cylinder. Furthermore, under the assumptionthat airmass aspirated into cylinders in each cycle is constant,exhaust air flowhas the same shape between successive cycles;then a periodic function can be obtained:
πΎππ (π) =
οΏ½οΏ½ππ(πππ+ (π β 1) ππ )
β4
π=1οΏ½οΏ½ππ(πππ+ (π β 1) ππ )
=οΏ½οΏ½ππ((π β 1) ππ )
β4
π=1οΏ½οΏ½ππ((π β 1) ππ )
.
(21)
Therefore, the gas mixing behavior (20) can be rewrittenin the π
π domain as
ππ (π) =
3
β
π=0
πΎ[π][πβπ]
π[πβπ] (π β π) , (22)
where [π] = (π mod 4) + 1. The relationship between the AFRof each cylinder and the AFR at the exhaust confluence pointcan be obtained by (22).
The algorithm to estimate the individual cylinder AFR isas follows: First, the AFR π
πat the exhaust confluence point
is obtained according to observer (9). Then, the individualcylinder AFR can be calculated through (22). With thecombination of (9) and (22), the method for the estimationof each cylinder AFR can be given as follows:
οΏ½οΏ½ (π + 1) = π΄οΏ½οΏ½ (π) + πΏ (π¦ (π) β πΆοΏ½οΏ½ (π β π)) ,
οΏ½οΏ½π (π) = (0 0 1) β οΏ½οΏ½ (π) ,
οΏ½οΏ½[π] (π) =
1
πΎ[π][π]
(β
3
β
π=1
(πΎ[π][πβπ]
β οΏ½οΏ½[πβπ] (π β π)) + οΏ½οΏ½π (π)) .
(23)
4. Simulation Studies
In this section, the simulation study of the estimation of theindividual cylinder AFR is presented in the environment ofa 2.0 L 4-cylinder SI gasoline engine from enDYNA [18, 19].The enDYNA is a professional software tool for the real-timesimulation of internal combustion engines, providing ready-to-usemodels for all common engine types comprising crankangle synchronous combustion, gas path, fuel system, coolingsystem, drivetrain, driver, and soft-ECU.The R4-cylinder SI-engine is an example in enDYNA to simulate a 4-cylinder SIgasoline engine, whose specifications are given in Table 1.Theobserver architecture is illustrated in Figure 2.
6 Journal of Sensors
0.998
1
#1 A
FR
0.998
1
#2 A
FR
0.998
1
#3 A
FR
0.998
1
#4 A
FR
enDYNAEstimated
enDYNAEstimated
enDYNAEstimated
enDYNAEstimated
5 10 15 20 250t (s)
5 10 15 20 250t (s)
5 10 15 20 250t (s)
5 10 15 20 250t (s)
5 10 15 20 250t (s)
0102030
uth
(deg
)
Figure 3: Individual cylinder AFR estimation results.
Table 1: Engine specifications.
Fuel system Direct injectionDisplacement (m3) 0.002Intake manifold volume (m3) 0.004Exhaust manifold volume (m3) 0.0015Max engine speed (rad/s) 785
The parameters of the system dynamics in the exhaustpipe (2) and the gas mixing (22) are presented as follows [15]:
πmix = 0.01 s,
πsen = 0.12 s,
Γ10β3
Γ10β3
Γ10β3
Γ10β3
β1
0
1
#1 er
ror
β1
0
1
#2 er
ror
β1
0
1
#3 er
ror
β1
0
1#4
erro
r
5 10 15 20 250t (s)
5 10 15 20 250t (s)
5 10 15 20 250t (s)
5 10 15 20 250t (s)
Figure 4: Evolution of the estimation error.
π = 1,
ππ= 0.08 s,
ππ = 0.02 s,
Ξ = (
0.55 0.05 0.15 0.25
0.25 0.6 0.05 0.1
0.15 0.25 0.5 0.1
0.05 0.1 0.15 0.7
) .
(24)
Then, the system matrix can be obtained:
π΄ = (
0.1353 0 1
0 0.8465 1
0 0 1
) ,
πΆ = (β0.0786 0.1675 0) .
(25)
According to the inequality (10), the gain matrix can begiven by πΏ = (0.97 3.79 0.73)π.
Here, the input of the throttle angle in enDYNA isdesigned as a step signal presented in Figure 3. Accordingly,the estimation results of the individual cylinder AFR by theproposed method are shown in Figure 3, and the estimationerrors are plotted in Figure 4. When the throttle changes
Journal of Sensors 7Ve
hicle
vel
ocity
50 100 150 2000t (s)
50 100 150 2000t (s)
50 100 150 2000t (s)
50 100 150 2000t (s)
Γ104
0
5
10
0
2000
4000
ne
(r/m
in)
0
20
40uth
(deg
)
0
20
40
60
(km
/h)
pim
(Pa)
Figure 5: Evolution of throttle angle π’th, engine speed ππ, intake
manifold pressure πim, and vehicle velocity under ECE cycle.
suddenly, the estimation errors of the individual cylinderAFR are about 0.06%, and then the steady state errors tendto decay within 0.01%.
In order to verify the effectiveness of the proposedmethod under driving cycle condition, one segment of theurban driving cycle ECE (EconomicCommission for Europe)is used [19], under which the throttle angle π’th, enginespeed π
π, intake manifold pressure πim, and vehicle velocity
are plotted in Figure 5. Accordingly, the comparison of theindividual cylinder AFR estimation between the proposedmethod and the input observer in [15] are presented inFigure 6, and the estimation errors are plotted in Figure 7.Clearly, the error of the proposed method is smaller thanthe input observer when the AFR changes slowly, in whichthe steady state error of the input observer is 0.03%. Whenthe AFR changes severely, there exist fluctuations of the AFRestimation error from both the proposed method and inputobserver in [15]. However, the estimation error from theproposed method is smaller. It is demonstrating that theproposed method considering time delay can improve theaccuracy of the individual cylinder AFR estimation.
enDYNAInput observerTime delay observer
enDYNAInput observerTime delay observer
enDYNAInput observerTime delay observer
enDYNAInput observerTime delay observer
50 100 150 2000t (s)
50 100 150 2000t (s)
50 100 150 2000t (s)
50 100 150 2000t (s)
0.998
0.999
1
#1 A
FR
0.998
0.999
1
1.001
#2 A
FR
0.998
0.999
1
1.001#3
AFR
0.998
0.999
1
1.001
#4 A
FR
Figure 6: Individual cylinder AFR estimation results of the pro-posed method and unknown input observer under ECE cycle.
5. Conclusion
An efficient method for the estimation of the individualcylinder AFR with a single UEGO sensor was developed toimprove the estimation accuracy. The system dynamics inthe exhaust pipe was described as an augmented discretesystem with output delay, in which the AFR at confluencepoint was augmented as a system state and beneficial to beestimated comparing the system with input delay. Then, anobserver for the augmented system with output delay wasdesigned to estimate the AFR at confluence point, whichcan avoid accurately inverting the engine model includingdelays. Using the gas mixing model, a method to estimate
8 Journal of Sensors
Input observerTime delay observer
Input observerTime delay observer
Input observerTime delay observer
Input observerTime delay observer
Γ10β3
Γ10β3
Γ10β3
Γ10β3
β1
0
1
#4 er
ror
50 100 150 2000t (s)
50 100 150 2000t (s)
50 100 150 2000t (s)
50 100 150 2000t (s)
β1
0
1
#1 er
ror
β1
0
1
#2 er
ror
β1
0
1
#3 er
ror
Figure 7: Evolution of the estimation error of the proposed methodand unknown input observer.
the individual cylinder AFR based on the proposed observerwas presented. The performance of the proposed methodwas validated by the simulation data from engine softwareenDYNA provided by Tesis, and a comparison with existingmethod was obtained during ECE cycle, demonstrating thatthe proposed method considering time delay from exhaustgas transport and UEGO sensor dynamics can improve theaccuracy of the individual cylinder AFR estimation.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work was supported by China Automobile IndustryInnovation and Development Joint Fund (no. U1564213).
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