Method for detecting and controlling air-fuel ratio in internal combustion engine

A method for detecting and controlling the air-fuel ratio of a multicylinder internal combustion engine through an output of a single air-fuel ratio sensor installed at a confluence point of the exhaust system of the engine. The detection response delay is assumed to be a first-order lag and a state variable model is established. Further, the air-fuel ratio at the confluence point is assumed to be a sum of the products of the past firing histories of the each cylinder of the engine and a second state variable model is established. An observer is then designed to observe the internal state of the second model and the air-fuel ratio at the individual cylinders are estimated from the output of the observer. The deadbeat control is carried out by calculating a ratio between the estimated air-fuel ratio and a target air-fuel ratio. The calculated ratio is multiplied to a correction value at a preceding control cycle earlier by a number corresponding to the number of the engine cylinders.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
This invention relates to a method for detecting and controlling the 
air-fuel ratio in an internal combustion engine, more particularly to a 
method for detecting the air-fuel ratio in a multiple cylinder internal 
combustion engine accurately and controlling to a target air-fuel ratio 
with good convergence. 
2. Description of the Prior Art 
It is a common practice to install a single air-fuel ratio sensor 
constituted as an oxygen concentration detector in the exhaust system of a 
multiple cylinder internal combustion engine and feedback the detected 
value for regulating the amount of fuel supplied to a target air-fuel 
ratio. A system of this type is taught by Japanese Patent Publication No. 
Sho 59(1984)-101562, for example. 
In the system, in order to improve the detection accuracy, a time lag 
counted from a reference timing (a first cylinder's TDC position) and 
required for the exhaust gas flowing out of the individual cylinders to 
reach the air-fuel ratio sensor is predetermined in advance in response to 
the operating condition of the engine. And taking the predetermined time 
lag into consideration, the air-fuel ratio is detected for the individual 
cylinders and is feedback controlled to a target value. However, since the 
air-fuel ratio sensor constituted as an oxygen detector is arranged to 
detect the air-fuel ratio through a generated electromotive force caused 
by a chemical reaction which occurs when an element of the oxygen detector 
comes into contact with the exhaust gas, the sensor can not respond 
immediately and there is a delay in detecting the air-fuel ratio after the 
exhaust gas has reached the sensor. This means that, until the delay has 
been solved, the air-fuel ratio of the burnt mixture could not be detected 
precisely and hence the accurately and excellent convergence could not be 
expected in the air-fuel ratio feedback control. 
SUMMARY OF THE INVENTION 
An object of the invention is therefore to provide a method for detecting 
the air-fuel ratio in an internal combustion engine in which the detection 
response lag in the air-fuel ratio sensor is precisely estimated to 
accurately obtain the air-fuel ratio of the mixture actually burnt such 
that the air-fuel ratio feedback control can, if desired, be conducted in 
a manner excellent in accuracy and convergence. 
Further, when a single air-fuel ratio sensor is installed at or downstream 
of an confluence point (the exhaust manifold joint) of a multicylinder 
engine such as having four or six cylinders, the output of the sensor 
represents a mixture of the values at all cylinders. This makes it hard to 
obtain the actual air-fuel ratio at the individual cylinders and then 
makes it difficult to converge it to a target ratio properly. Thus, some 
cylinders could be supplied with a lean mixture whereas others a rich 
mixture, thereby degrading emission characteristics. 
Although this can be solved by providing the sensor for the individual 
cylinders, the arrangement will necessarily be expensive and what is more, 
brings another problem on sensor's service life. For, it is not 
advantageous to install many air-fuel ratio sensors in the exhaust system 
to suffer them from a hot ambient temperature. The prior art system aimed 
to solve the problem. However, since the air-fuel ratio at the confluence 
point of the exhaust system is a mixture of those at the individual 
cylinders as was explained, the prior art system leaves much to be 
improved, such as in its detection accuracy. 
Another object of the invention is therefore to provide a method for 
estimating the air-fuel ratio in a multicylinder internal combustion 
engine in which the air-fuel ratios of the individual cylinders are 
precisely estimated from the output of a single air-fuel ratio sensor 
installed at or downstream of an exhaust gas confluence point in the 
exhaust system of the engine. 
Further object of the invention is to provide a similar method for 
estimating the air-fuel ratio in a multicylinder internal combustion 
engine in which the air-fuel ratio of each cylinder is precisely estimated 
from the output of a single air-fuel ratio sensor installed at or 
downstream of an exhaust gas confluence point in the exhaust system of the 
engine such that the air-fuel ratios at the individual cylinders are 
feedback controlled to a target ratio in a manner excellent in accuracy 
and convergence. 
Furthermore, in the air-fuel ratio control, the air-fuel ratios at the 
individual cylinders are usually PID-controlled based on their deviation 
from the target value. With this method, however, the convergence on the 
target values is often less than satisfactory. This is because cost and 
durability considerations normally make it impossible to install a 
plurality of air-fuel ratio sensors for detecting the air-fuel ratios at 
the individual cylinders, as stated before. The air-fuel ratios at the 
individual cylinders therefore have to be estimated from the output of a 
single sensor installed in the exhaust system. Since this makes it 
impossible to ascertain the air-fuel ratios at the individual cylinders 
with high precision, the feedback gain has to be kept down in order to 
prevent hunting. The control convergence is therefore not so satisfactory 
than expected. 
Still further object of the invention is therefore to provide a method for 
controlling the air-fuel ratio in a multicylinder internal combustion 
engine wherein the air-fuel ratios at the individual cylinders of the 
engine can be accurately separated and extracted from the output of a 
single air-fuel ratio sensor installed at or downstream of an exhaust gas 
confluence point of the exhaust system and the so-obtained air-fuel ratios 
can be used for conducting the control, what is called the "deadbeat 
control", for immediately converging the air-fuel ratio at each cylinder 
to the target ratio with deadbeat response. 
A technique to immediately converge the air-fuel ratio to the target 
air-fuel ratio is in no ways limited to the multicylinder engine in which 
a single air-fuel ratio sensor is used. 
Yet still further object of the invention is therefore to provide a method 
for controlling the air-fuel ratio in an internal combustion engine which 
is more generally applicable even to an arrangement in which the air-fuel 
ratios are detected by sensors installed at the individual cylinders, 
wherein the deadbeat control is conducted for immediately converging the 
air-fuel ratio at each cylinder on the target air-fuel ratio during the 
next control cycle. 
For realizing these objects, the present invention provides a method for 
detecting the air-fuel ratio of a mixture supplied to an internal 
combustion engine through an output of an air-fuel ratio sensor, 
comprising deeming a detection response lag of the sensor as a first-order 
lag to establish a state variable model, obtaining a state equation 
describing the behavior of the state variable model, discretizing the 
state equation for period delta T to obtain a transfer function, and 
obtaining an inverse transfer function of the transfer function and 
multiplying it to the output of the sensor to estimate the air-fuel ratio 
of the mixture supplied to the engine.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
FIG. 1 is an overall schematic view of an internal combustion engine 
air-fuel ratio detection and control system, in hardware construction, for 
carrying out the method of this invention. Reference numeral 10 in this 
figure designates an internal combustion engine having four cylinders. Air 
drawn in through an air cleaner 14 mounted on the far end of an air intake 
path 12 is supplied to first to fourth cylinders through an air intake 
manifold 18 while the flow thereof is adjusted by a throttle valve 16. An 
injector 20 for injecting fuel is installed in the vicinity of the intake 
valve (not shown) of each cylinder. As is well known, the amount of fuel 
injected by each injector 20 for each intake stroke of the associated 
piston in a cylinder is controlled by control unit 42 and may be varied 
from cylinder-to-cylinder and stroke-by-stroke. The injected fuel mixes 
with the intake air to form an air-fuel mixture that is ignited in the 
associated cylinder by a spark plug (not shown). The resulting combustion 
of the air-fuel mixture drives down a piston (not shown). The exhaust gas 
produced by the combustion is discharged through an exhaust valve (not 
shown) into an exhaust manifold 22, from where it passes through an 
exhaust pipe 24 to a three-way catalytic converter 26 where it is removed 
of noxious components before being discharged to the exterior. In 
addition, the air intake path 12 is bypassed by a bypass 28 provided 
therein in the vicinity of the throttle valve 16. 
A crank-angle sensor 34 for detecting the piston crank angles is provided 
in a distributor (not shown) of the internal combustion engine 10, a 
throttle position sensor 36 is provided for detecting the degree of 
opening of the throttle valve 16, and a manifold absolute pressure sensor 
38 is provided for detecting the pressure of the intake air downstream of 
the throttle valve 16 as an absolute pressure. An air-fuel ratio sensor 40 
constituted as an oxygen concentration detector is provided at the exhaust 
pipe 24 in the exhaust system at a point downstream of the exhaust 
manifold 22 and upstream of the three-way catalytic converter 26, where it 
detects the air-fuel ratio of the exhaust gas. The outputs of these 
sensors are sent to a control unit 42. 
Details of the control unit 42 are shown in the block diagram of FIG. 2. 
The output of the air-fuel ratio sensor 40 is received by a detection 
circuit 46 of the control unit 42, where it is subjected to appropriate 
linearization processing to obtain an air-fuel ratio (A/F) characterized 
in that it varies linearly with the oxygen concentration of the exhaust 
gas over a broad range extending from the lean side to the rich side. As 
this air-fuel ratio is explained in detail in the applicant's earlier 
Japanese patent application (Japanese Patent Application No. Hei 
3(1991)-169456), it will not be explained further here. Hereinafter in 
this explanation, the air-fuel ratio sensor will be referred to as an "LAF 
sensor" (the name is derived from its characteristics in which the 
air-fuel ratio can be detected linearly). The output of the detection 
circuit 46 is forwarded through an A/D (analog/digital) converter 48 to a 
microcomputer comprising a CPU (central processing unit) 50, a ROM 
(read-only memory) 52 and a RAM (random access memory) 54 and is stored in 
the RAM 54. Similarly, the analog outputs of the throttle position sensor 
36 and the manifold absolute pressure sensor 38 are input to the 
microcomputer through a level converter 56, a multiplexer 58 and a second 
A/D converter 60, while the output of the crank-angle sensor 34 is shaped 
by a pulse generator 62 and has its output value counted by a counter 64, 
the result of the count being input to the microcomputer. In accordance 
with commands stored in the ROM 52, the CPU 50 of the microcomputer uses 
the detected values to compute an air-fuel ratio feedback control value, 
drives the injectors 20 of the respective cylinders via a driver 66 and 
drives a solenoid valve 70 via a second driver 68 for controlling the 
amount of secondary air passing through the bypass 28. 
The operation of this control system will now be explained. 
For high-accuracy separation and extraction of the air-fuel ratios of the 
individual cylinders from the output of a single air-fuel ratio sensor 
installed at or downstream of an exhaust gas confluence point in the 
exhaust system of a multiple cylinder engine, it is first necessary to 
accurately ascertain the detection response delay of the air-fuel ratio 
sensor. The solid line curve in FIG. 3, the figure being the result of 
simulation which will be explained at a later stage, shows the air-fuel 
ratio sensor response carried out in a one-cylinder internal combustion 
engine when the amount of intake air was presumed to be maintained 
constant and the amount of fuel supplied was presumed to be varied 
stepwise as illustrated by dashed lines. As can be seen in this figure, 
when the air-fuel ratio is varied stepwise, the LAF sensor output lags 
behind the input value. Since this lag is caused by a chemical reaction as 
was mentioned earlier, however, it is difficult to analyze precisely. The 
inventors therefore used simulation to model this delay as a first-order 
lag. For this they built the model shown in FIG. 4. Here, if we define 
LAF: LAF sensor output and A/F: input air-fuel ratio, the state equation 
can be written as 
EQU L.ANG.F(t)=.alpha.LAF(t)-.alpha.A/F(t) (1) 
When the state equation is discretized in the discrete-time series for 
period delta T, we get 
EQU LAF(k+1)=.alpha.LAF(k)+(1-.alpha.)A/F(k) (2) 
Here: 
EQU .alpha.=1+.alpha..increment.T+(1/2!).alpha..sup.2 .increment.T.sup.2 
+(1/3!).alpha..sup.3 .increment.T.sup.3 +(1/4!).alpha..sup.4 
.increment.T.sup.4 
Equation (2) is represented as a block diagram in FIG. 5. 
Therefore, Equation (2) can be used to obtain the actual air-fuel ratio 
from the sensor output. That is to say, since Equation (2) can be 
rewritten as Equation (3), the value at time k-1 can be calculated back 
from the value at time k as shown by Equation (4). 
EQU A/F(k)={LAF(k+1)-.alpha.LAF(k)}/(1-.alpha.) (3) 
EQU A/F(k-1)={LAF(k)-.alpha.LAF(k-1)}/(1-.alpha.) (4) 
Specifically, use of Z transformation to express Equation (2) as a transfer 
function gives Equation (5), and a real-time estimate of the air-fuel 
ratio in the preceding cycle can be thus obtained by multiplying the 
sensor output LAF of the current cycle by its inverse transfer function. 
FIG. 6 is a block diagram of the real-time A/F estimator. 
EQU t(z)=(1-.alpha.) (5) 
Although, as was mentioned earlier, the response delay of the LAF sensor is 
caused by a chemical reaction and is therefore difficult to analyze, there 
was ascertained to be a correlation between the response delay and the 
engine speed. Therefore, the coefficient of the transfer function is 
varied relative to appropriately set graduations in the engine speed. As a 
result, the accuracy of the estimated air-fuel ratio value can be enhanced 
by using a different A/F estimator, i.e. a different inverse transfer 
function coefficient, for each prescribed graduation in engine speed. 
The simulation results regarding the foregoing will be explained with 
reference to FIG. 3. As mentioned earlier, FIG. 3 shows the sensor's 
actual output obtained when graduated air-fuel ratios are input as 
illustrated by dashed lines. And, broken lines (dotted lines) indicate the 
output of the model (shown in FIG. 5) obtained when the stepwise air-fuel 
ratio is input. In this figure, the sensor's actual output and the model's 
output are seen to be substantially in agreement. The foregoing can be 
taken to verify the validity of the model simulating the sensor response 
delay as a first-order lag. FIG. 7 shows the result of the same simulation 
where the air-fuel ratio is estimated by multiplying the sensor actual 
output value by the inverse transfer function. From this figure, the 
air-fuel ratio at time Ta, for example, can be estimated to be 13.2:1, not 
12.5:1. (The small ups and downs in the estimated air-fuel ratio are the 
result of fine variation in the detected sensor output.) 
The separation and extraction of the air-fuel ratios of the individual 
cylinders using the air-fuel ratio estimated in the foregoing manner will 
now be explained. 
As was explained earlier, when a single air-fuel ratio sensor is installed 
at or downstream of an exhaust gas confluence point of the exhaust system 
of a multiple cylinder internal combustion engine, the output of the 
sensor represents a mixture of the values at all of the cylinders. Since 
this makes it hard to obtain the actual air-fuel ratio at the individual 
cylinders, it was not up to now possible to control the air-fuel ratios at 
the individual cylinders precisely. As the air-fuel mixture therefore 
became lean at some cylinders and rich at others, the quality of the 
exhaust emissions was degraded. While this problem can be overcome by 
installing a separate sensor for each cylinder, this increases costs to an 
unacceptable level and also gives rise to problems regarding sensor 
durability. Now, by modeling the sensor detection response delay as a 
first-order lag, the inventors have made it possible to use the method 
explained in the following to ascertain with high accuracy the air-fuel 
ratios at the individual cylinders of a multiple cylinder (in the 
embodiment, a four-cylinder) internal combustion engine employing only a 
single air-fuel ratio sensor installed at or downstream of a confluence 
point of the exhaust system. The method will now be explained in detail. 
The inventors first established the internal combustion engine exhaust 
system model shown in FIG. 8 (hereinafter called the "exhaust gas model"). 
The discretization sampling time in the exhaust gas model was made the 
same as the TDC (top dead center) period (0.02 sec at an engine speed of 
1,500 rpm). And, as F (fuel) was selected as the controlled variable in 
the exhaust gas model, the term fuel-air ratio F/A was used instead of the 
air-fuel ratio A/F in the figure. However, for ease of understanding, the 
word "air-fuel ratio" will still be used in the following except that the 
use of the words might cause confusion. 
The inventors then assumed the air-fuel ratio at the confluence point of 
the exhaust system to be an average weighted to reflect the time-based 
contribution of the air-fuel ratios of the individual cylinders. This made 
it possible to express the air-fuel ratio at the confluence point at time 
k in the manner of Equation (6). 
##EQU1## 
More specifically, the air-fuel ratio at the confluence point can be 
expressed as the sum of the products of the past firing histories of the 
respective cylinders and weights C (for example, 40% for the cylinder that 
fired most recently, 30% for the one before that, and so on). It must be 
noted, however, that the state in which the exhaust gases from the 
individual cylinders mix at the confluence point varies with the engine 
operating condition. For example, since the TDC period is long in the 
low-speed region of the engine, the degree of mixing of the exhaust gases 
from the different cylinders is lower than in the high-speed region. On 
the other hand, during high-load operation, since the back pressure and 
the exhaust gas discharge pressure are fundamentally larger, the degree of 
mixing of the exhaust gases from the different cylinders is lower than 
during low-load operation. When the degree of mixing of the exhaust gases 
from the different cylinders is low, it becomes necessary to increase the 
weight of the cylinder that fired most recently. In the invention 
therefore the weight C is varied according to the engine operation 
condition. This is achieved by appropriately preparing look-up tables for 
the weights C relative to the engine speed and the engine load as 
parameters and retrieving the weight C for the current operating condition 
from the tables. Incidentally, the #n in the equation indicates the 
cylinder number, and the firing order of the cylinders is defined as 1, 3, 
4, 2. The air-fuel ratio here, correctly the fuel-air ratio (F/A), is the 
estimated value obtained by correcting for the response delay. 
Based on the aforesaid assumptions, the state equation of the exhaust gas 
model can be written as 
##EQU2## 
Further, if the air-fuel ratio at the confluence point is defined as y(k), 
the output equation can be written as 
##EQU3## 
Since u(k) in this equation cannot be observed, it will still not be 
possible, even if an observer is designed from the equation to observe 
x(k). However, if one defines x(k+1)=x(k-3) on the assumption of a stable 
operating state in which there is no abrupt change in the air-fuel ratio 
from that 4 TDC earlier (i.e., from that of the same cylinder), Equation 
(9) will be obtained. 
##EQU4## 
The simulation results for the exhaust gas model obtained in the foregoing 
manner will now be given. FIG. 9 shows a situation of the simulation in 
which fuel is supplied to three cylinders of a four-cylinder internal 
combustion engine so as to obtain an air-fuel ratio of 14.7:1 and to one 
cylinder so as to obtain an air-fuel ratio of 12.0:1. FIG. 10 shows the 
air-fuel ratio at this time at the confluence point (the position where 
the air-fuel ratio sensor 40 is located in the exhaust pipe 24 in FIG. 1) 
as obtained using the aforesaid exhaust gas model. While FIG. 10 shows 
that a stepped output is obtained, when the response delay of the LAF 
sensor is taken into consideration, the sensor output becomes the smoothed 
wave designated "Model's output adjusted for delay" in FIG. 11. The close 
agreement of the waveforms of the model's output and the sensor's output 
verifies the validity of the exhaust gas model as a model of the exhaust 
gas system of a multiple cylinder internal combustion engine. 
Thus, the problem comes down to one of an ordinary Kalman filter in which 
x(k) is observed in the state equation and the output equation shown in 
Equation (10). When the weighted matrices Q, R are determined as shown in 
Equation (11) and the Riccati's equation is solved, the gain matrix K 
becomes as shown in Equation (12). 
##EQU5## 
Obtaining A-KC from this gives Equation (13). 
##EQU6## 
FIG. 12 shows the configuration of an ordinary observer. Since there is no 
input u(k) in the present model, however, the configuration has only y(k) 
as an input, as shown in FIG. 13. This is expressed mathematically by 
Equation (14). 
##EQU7## 
The system matrix S of the observer whose input is y(k), namely of the 
Kalman filter, is 
##EQU8## 
In the present model, when the ratio of the member of the weight imputation 
R in Riccati's equation to the member of Q is 1:1, the system matrix S of 
the Kalman filter is given as 
##EQU9## 
The waveforms of the simulated air-fuel ratios at the respective cylinders 
are then precisely drawn and the result is input to the exhaust gas model 
to obtain the air-fuel ratio at the confluence point, which is in turn 
input to the observer for verifying the estimation of the air-fuel ratios 
at the individual cylinders. The tendency of the weighted matrix and the 
estimated values is also examined. 
Since Equation (17) applies in the present model, the weighted matrix Q is 
a diagonal matrix whose members are all the same. 
EQU X(k)=[x(k-3) x(k-2) x(k-1) x(k)]' (17) 
What needs to be examined, therefore, are the ratio of the members of Q and 
R. The gains obtained by varying the ratio between the members of Q and R 
are shown in a table of FIG. 14. The simulation model combining the 
observer constituted using these gains with the exhaust gas model is shown 
in FIG. 15. In addition, the results of the computation using this model 
when values 12.0:1, 14.7:1, 14.7:1, 14.7:1 are input as the air-fuel 
ratios at the individual cylinders are as shown in FIG. 16 and the 
observer's estimation error at this time between the target ratio and the 
estimated ratio is as shown in a table of FIG. 17. The results of the 
computation using this model when the air-fuel ratios were independently 
varied within the ranges of 12.0.+-.0.2:1, 14.7.+-.0.2:1, 14.7.+-.0.2:1, 
14.7.+-.0.2:1 (for noise simulation) are shown in FIG. 18 and the 
observer's estimation error at this time is as shown in a table of FIG. 
19. In each of FIGS. 16 and 18, (a) to (e) have the following meanings: 
(a) Air-fuel ratio of the respective cylinders (exhaust gas model input), 
(b) Air-fuel ratio at confluence point (exhaust gas model output), 
(c) Observer output (input indicated by (b)) when Q member : R member=1:10, 
(d) Observer output (input indicated by (b)) when Q member : R member=1:1, 
and 
(e) Observer output (input indicated by (b)) when Q member : R member=10:1. 
It will be noted from FIG. 16 that when the same air-fuel ratio was set for 
all cylinders, the rate of convergence increased with increasing weight of 
Q. However, increasing Q/R to 10 or greater caused substantially no change 
in the convergence. The error (target air-fuel ratio at each 
cylinder--estimated air-fuel ratio at each cylinder) in FIG. 18 in time 
series will be shown in FIG. 20. After converged in the observer there is 
little difference between the case where the ratio of Q member: R member 
is 10:1 and the case where it is 1:1 and, therefore, taking external 
disturbance into account, Q member: R member=1:1 is preferable. Thus the 
observer using the Kalman theory with respect to the input air-fuel ratio 
at the confluence point is able to estimate the individual cylinder 
air-fuel ratios with high precision at the confluence point. (Although the 
weighted matrix was best at Q/R=1-10, it is considered necessary to 
determine it from the response using actual data.) 
FIG. 21 shows the result of simulation in which the estimated air-fuel 
ratios at the individual cylinders obtained by inputting to the observer 
the actual confluence point air-fuel ratio data obtained by multiplying 
the actually measured data by the aforesaid inverse transfer function of 
the A/F estimator. In this figure: 
(a) LAF sensor output, 
(b) Air-fuel ratio at confluence point (real-time A/F estimator's output 
(input to the observer), 
(c) Observer output when Q member:R member=1:10 (input indicated by (b)), 
(d) Observer output when Q member: R member=1:1 (input indicated by (b)), 
and 
(e) Observer output when Q member:R member=10:1 (input indicated by (b)). 
The LAF sensor output measurement conditions were: engine speed=1,500 rpm, 
air intake manifold pressure =-281.9 mmHg, A/F=12.0:1 (#2), 14.7:1 (#1, 
#3, #4). 
Since the true values of the actual input air-fuel ratios were unknown, 
12.0:1, 14.7:1, 14.7:1, 14.7:1 were used as approximate values in the 
simulation. As can be seen from this figure, the observer output varies in 
cycles of 4 TDC and substantially estimates the input air-fuel ratio. 
Moreover, the figure shows that the use of the Kalman filter enables 
convergence in 2 to 8 cycles, depending on how the weighted matrices are 
set. 
Use of the cylinder air-fuel ratios estimated in the foregoing manner for 
controlling the air-fuel ratios to the target value will now be explained. 
An example of this control using the PID technique is shown in the block 
diagram of FIG. 22. Although the illustrated control differs from ordinary 
PID control in the point that it conducts feedback through a 
multiplication term, the control method itself is well known. As shown, it 
suffices to calculate for each cylinder the deviation (1-1/lambda) of the 
actual air-fuel ratio from the target value that results from input Ti 
(injection period) and to feedback the product of this and a corresponding 
gain KLAF so as to obtain the target value. While the method is well 
known, its ability to provide control for adjusting the air-fuel ratios of 
the individual cylinders to the target value is dependent on the highly 
accurate detection of the air-ratios of the individual cylinders made 
possible by the invention as described in the foregoing. 
Since the need to prevent hunting in the aforesaid PID control makes it 
impossible to set the feedback gain too high, however, the control 
convergence is not as good as might be desired. FIGS. 23-27 show 
simulation results indicating the response of the PID control of FIG. 22. 
FIG. 23 shows the air-fuel ratio output characteristics when the input 
air-fuel ratio was fixed (21.0:1), FIG. 24 the characteristics of the 
corresponding feedback gain KLAF, FIG. 25 other input air/fuel ratio 
characteristics, FIG. 26 the air-fuel ratio output characteristics at this 
time, and FIG. 27 the characteristics of the corresponding gain KLAF. As 
is clear from FIG. 26, the convergence is by no means rapid. 
Therefore, an explanation will now be made with regard to the deadbeat 
control which enables immediate convergence on the target value with 
deadbeat response. 
Consideration will be given to feedback wherein, as a fundamental policy, 
convergence on the target air-fuel ratio W is achieved by correcting the 
input u(k) using the ratio W/x circumflex (k) between the 
observer-estimated air-fuel ratio x circumflex (k) and the target air-fuel 
ratio W. In the model of FIG. 15, when feedback control of the individual 
cylinders is conducted using as the gain .alpha.(k) the result of 
accumulating the ratios of the observer-estimated air-fuel ratio x 
circumflex (k) and the target air-fuel ratio W, we get what is shown in 
FIG. 28. Assuming the input at this time to be u(k), it holds that 
EQU x(k)=.alpha.(k).multidot.u(k) (18) 
EQU .alpha.(k)=.alpha.(k-4).multidot.W/x(k-4) (19) 
From Equation 18, it follows that 
##EQU10## 
and from Equation (19), that 
##EQU11## 
Therefore, when u(k)/u(k-4).about.1, K.fwdarw..infin., if 
x(k-4).fwdarw.x(k-4), then when k.fwdarw..infin., it should follow that 
x(k).fwdarw.W. 
Expressed in general terms, this becomes Current output]=[Current 
input].times.[Target value]/[Current estimated output value].times. 
[Preceding correction value for specific control cycle] 
In this case, "Preceding correction value for specific control cycle" means 
the output four control cycles (TDC) earlier, i.e. for the output for the 
same cylinder (in a four-cylinder engine). However, when this gain was 
actually used in feedback simulation, the control did not stabilize. 
If the value two times earlier is used for introducing a delay into the 
cumulative calculation of the gain .alpha.(k), the result is as shown in 
FIG. 29. At this time it holds that 
EQU .alpha.(k)=.alpha.(k-8).multidot.W/x(k-4) (22) 
Making the same calculation without a delay gives 
##EQU12## 
and the control stabilized. 
This will be explained with reference to FIG. 30. The air-fuel ratio, x 
circumflex (k) estimated (by the observer) for the specific cylinder are 
the results obtained by control using the correction value .alpha.(k) for 
that cycle. Therefore, in calculating the correction value, since the 
estimated air-fuel ratio is that for a number of times earlier, it is 
necessary to check what the gain value was at that time. In this sense, 
and as shown in FIG. 30, the observer output four times earlier (one time 
earlier, if viewed in terms of the first cylinder) is the estimated first 
cylinder air-fuel ratio 8 times earlier (the time before last). Thus since 
the next control gain is calculated from the control gain 8 times earlier 
and the result (estimated value) obtained by the control using this gain, 
the timing conforms and convergence on the target value is achieved. FIG. 
31 shows the result of this simulation. (It will be noted that control was 
more stable than in the case of no delay shown at the top of FIG. 31. In 
this figure, the solid lines show the results for feedback control and the 
broken lines the results for no feedback control.) FIG. 32 is a block 
diagram of this model (which is obtained by adding a feedback control 
system to the model of FIG. 15). FIGS. 33 to 37 show the results of 
simulation using this model. In will be noted from FIG. 36 that the 
convergence is markedly better than that in PID control. 
Further study regarding delay led to the conclusion that still better 
control can be achieved as shown in FIG. 38, namely, by using the control 
gain 12 times earlier (three times earlier, if viewed in terms of the 
first cylinder). Specifically, in light of the fact that the delays in the 
engine, in particular the delay in the appearance of the control results, 
could be accurately expressed and that the amount of delay contingent on 
the number of cylinders and the amount of delay contingent on the number 
of combustion cycles were clarified, it was concluded that, in view of 
"the delay in the appearance of the control results + the time for one 
combustion stroke + the sampling delay + the time for observer estimation 
+ the gain allocation," it is preferable to use the gain 12 cycles earlier 
(three times earlier, if viewed in terms of the first cylinder) as the 
cumulative gain. This is shown in FIG. 39. For reference, the gains to be 
used for 3, 5, 6 and 12 cylinder engines are shown in FIGS. 40 to 43. 
Next, the feedback control model of FIG. 32 was supplied with ideal input 
for confirming convergence of the air-fuel ratios of the individual 
cylinders on the target value. The effect of the observer weighted matrix 
was also examined. 
The computation results obtained when the individual cylinder air-fuel 
ratios input to the feedback model of FIG. 32 were 12.0:1, 14.7:1, 14.7:1, 
14.7:1 and those obtained when, to simulate imaginary noise, the air-fuel 
ratios input for the individual cylinders were varied within the ranges of 
12.0.+-.0.2:1, 14.7.+-.0.2:1, 14.7.+-.0.2:1, 14.7.+-.0.2:1 are shown in 
FIGS. 44 and 45. In these Figures, (a) to (c) have the following meanings: 
(a) Control results when Q member: R member=1:10, 
(b) Control results when Q member: R member=1:1, 
(c) Control results when Q member: R member=10:1, where the computation was 
made for a target air-fuel ratio of 14.7:1 and the members of the observer 
weighted matrix were such that Q:R=1:10, 1:1, 10:1. The control error 
under these conditions is shown in tables of FIG. 46 and 47. As can be 
seen in FIG. 44, with respect to a fixed ideal input, the rate at which 
the air-fuel ratios at the individual cylinders converge on the target 
value increases with increasing observer convergence weight. As shown in 
FIG. 45, when the air-fuel ratios at the individual cylinders do not 
stabilize, the convergence deteriorates in proportion as the feedback is 
late. 
From FIG. 28 on, the feedback control was conducted with the input air-fuel 
ratio for each cylinder multiplied by the control gain. This was only for 
the purpose of simulation, however, and in actuality the feed back control 
is conducted as shown in FIG. 22. Namely, the gain is calculated as a 
multiplication term for the fuel injection period pulse Ti. 
While the foregoing embodiment controls the air-fuel ratios at the 
individual cylinders to the target value on the basis of estimated values 
of the actual air-fuel ratios at the individual cylinders obtained using 
only a single air-fuel ratio sensor, the embodiment is not limited to this 
arrangement and can also be applied to the case where the deadbeat control 
for achieving the target values is conducted on the basis of the actual 
air-fuel ratios at the individual cylinders detected using a plurality of 
air-fuel ratio sensors installed at the individual cylinders. 
The present invention has thus been shown and described with reference to 
the specific embodiments. However, it should be noted that the present 
invention is in no way limited to the details of the described 
arrangements, changes and modifications may be made without departing from 
the scope of the appended claims.