Fuel injection system of an internal combustion engine

A fuel injecting amount of an internal combustion engine is calculated utilizing equations determined from a physical model describing a behavior of fuel in the engine. The fuel injection system includes estimation means in which estimation values fw and fv of the adhering fuel amount and the vapor fuel amount respectively are calculated based on: a product .lambda.r.multidot.m of the detected fuel/air ratio and the detected air amount; a division Vf/.omega. of fuel evaporating amount by the engine speed; and a fuel injecting amount q. The fuel injecting amount is calculated in the system based on the division Vf/.omega., the estimated values fw and fv, the product .lambda.r.multidot.m, and a summed up deviation from a target ratio. The coefficients of respective terms are determined by analyzing the physical model by modern control theory. A variation of the invention does not use an air/fuel ratio sensor.

BACKGROUND OF THE INVENTION 
The present invention relates to a fuel injection system of an internal 
combustion engine, in which the amount of fuel injected by a fuel 
injection valve, hereforth referred to as the fuel injecting amount, is 
determined based on a physical model describing a behavior of fuel coming 
into a cylinder of the engine. 
A fuel injection system is disclosed, which determines a fuel injecting 
amount of a fuel injection valve so that an air/fuel ratio of an air/fuel 
mixture supplied to an engine is adjusted to coincide with a target ratio, 
for example, in Published Unexamined Japanese Patent Application No. 
59-196930. The system uses identification that the linear approximation 
holds between a control input and a control output. The control input is 
regarded as a compensation value for compensating a basic fuel injecting 
amount obtained from the rotating speed of an engine and the amount of 
intake air. The control output is regarded as an actual measurement of the 
air/fuel ratio detected by an air/fuel ratio sensor. Using such 
identification provides a physical model for describing dynamic behavior 
of the engine, based on which a control law is designed. The system of 
this known type, based on the linear control theory, is thus constructed 
to determine the fuel injecting amount, utilizing the control law. 
Actually, however, the linear relationship does not hold between the 
control input and the control output. The physical model obtained from a 
simple linear approximation, thus, is allowed to describe the dynamic 
behavior of the engine accurately only in a very limited operating 
condition. For this reason, the conventional systems suppose several 
physical models in several regions of the engine operation in each of 
which the linear approximation can almost hold. Accordingly several 
control laws corresponding to the physical models must be designed in 
respective regions. In the aforementioned system, control laws have to be 
switched depending on the physical model in the respective region of the 
engine operation, resulting in cumbersome control. Switching the control 
law might cause the control at the boundary between the regions to be 
unstable. 
A system of this type uses an approximation by lower order physical model 
for improving responsiveness of the control by reducing calculating time. 
In this method, an approximation error or an error due to the difference 
among individual engines is absorbed by an integral operation. However, in 
the conventional method, the physical model is provided based on 
physically meaningless state variables on the assumption that the linear 
approximation can hold between the control output and control input. Hence 
approximating the physical model by lower order will deteriorate the 
control accuracy because of the increase in the amount of the integral 
term. 
Further, since the above system determines the fuel injecting amount in 
accordance with an actual measurement of an air/fuel ratio detected by an 
air/fuel ratio sensor as the control output, the control cannot be applied 
to an engine with no such sensor. 
SUMMARY OF THE INVENTION 
It is an object of the present invention to provide a fuel injection system 
of an internal combustion engine, which determines the fuel injecting 
amount with great accuracy without switching control laws. 
It is another object of the invention to provide a fuel injection system of 
an internal combustion engine, which adjusts an air/fuel ratio to a target 
ratio without using a sensor for detecting the air/fuel ratio. 
One feature of the present invention is, as shown in FIG. 1A, a fuel 
injection system of an internal combustion engine M2 for determining a 
fuel injecting amount q of a fuel injection valve M4 based on a physical 
model describing a behavior of fuel coming into a cylinder M3 of the 
engine M2. The system utilizes an amount fw of fuel adhering to an inner 
wall of an intake pipe M1 and an amount fv of vapor fuel in the intake 
pipe M1 as state variables. The system comprises: 
an operating state detection means M5 for detecting the rotating speed 
.omega. of the engine M2, an evaporating amount Vf of the fuel adhering to 
the inner wall of the intake pipe M1, fuel/air ratio .lambda. of a mixture 
coming into the cylinder M3, and an amount m of air coming into the 
cylinder M3; 
a dividing means M6 for dividing the evaporating amount Vf by the engine 
speed .omega.; 
an estimation means M7 for estimating the adhering fuel amount fw and the 
vapor fuel amount fv, based on a product .lambda..multidot.m of the 
detected fuel/air ratio .lambda. and the detected air amount m, the 
division Vf/.omega. at the dividing means M6 and the injecting amount q, 
utilizing a first equation determined from the physical model; 
a summing means M8 for summing up a difference 
m.multidot.(.lambda.-.lambda.r) between the product .lambda..multidot.m 
and a product .lambda.r.multidot.m of a preset target fuel/air ratio 
.lambda.r and the air amount m; and 
a fuel injecting amount calculation means M9 for calculating the fuel 
injecting amount q, based on the division Vf/.omega., the estimated 
adhering fuel amount fw, the estimated vapor fuel amount fv, the product 
.lambda.r.multidot.m of the target fuel/air ratio .lambda.r and the air 
amount m and the difference summed at the summing means M8, utilizing a 
second equation determined from the physical model. 
The operating state detection means M5 detects: the rotating speed .omega. 
of the engine M2, i.e., an engine speed; an evaporating amount Vf of the 
fuel adhering to the inner wall of the intake pipe M1; fuel/air ratio 
.lambda. of a mixture coming into the cylinder M3; and an amount m of air 
coming into the cylinder M3. 
A known engine speed sensor can be used for detecting the engine speed 
.omega.. A known air/fuel ratio sensor equipped to an exhaust system of an 
engine which outputs detection signals in accordance with the 
concentration of oxygen in the exhaust gas can be used in the operating 
state detection means M5. 
The evaporating amount Vf can be derived from a known function between a 
saturated vapor pressure Ps of the fuel in the intake pipe M1 and a 
pressure P in the intake pipe M1 (intake pipe pressure). The saturated 
vapor pressure Ps is hardly obtained by a sensor. So the following 
equation (1) is utilized for providing it. The pressure Ps is a function 
of a temperature T of the fuel. The temperature T can be represented by 
either the water temperature of a water jacket of the engine M2, or the 
temperature of a cylinder head adjacent to the intake port. Thus the 
temperature T (.degree.K), either in the water jacket or in the cylinder 
head detected by a temperature sensor is used as the parameter in the 
equation (1): 
EQU Ps=.beta.1.multidot.T.sup.2 -.beta.2.multidot.T+.beta.3 (1) 
where .beta.1, .beta.2, .beta.3 are proper constants. 
First, the saturated vapor pressure Ps is obtained based on temperature 
signals from the sensor at the water jacket or at the cylinder head. Then 
a pressure P in the intake pipe is sensed by a known pressure sensor. The 
fuel evaporating amount Vf is detected by utilizing a predetermined data 
map or a predetermined equation based on the saturated vapor pressure Ps 
and the intake pipe pressure P. Alternatively, since the fuel evaporating 
amount Vf greatly changes dependent on the pressure Ps, it may be obtained 
with approximation from the following equation (1)' using only Ps as the 
parameter: 
EQU Vf=.beta.4.multidot.Ps (1)' 
where .beta.4 is a constant. 
The air amount m coming into the cylinder M3 can be easily obtained, for 
example, from the following equation (2). When the engine speed .omega. is 
constant, the air amount m is approximated by a linear function of the 
pressure P, such as: 
EQU m={.beta..times.(.omega.).multidot.P-.beta.y(.omega.)}/Ti, (2) 
where .beta..times.(.omega.) and .beta.y(.omega.) are coefficients 
depending on the engine speed .omega.. Accordingly the air amount m is 
detected based on the pressure P and the temperature Ti detected by the 
respective known sensors, and the engine speed .omega. detected by the 
aforementioned sensor, utilizing the above equation (2). Also, the air 
amount m may be detected by compensating a basic air amount m by the 
temperature Ti. The basic air amount m is obtained from a predetermined 
map using the pressure P and the engine speed .omega. as parameters. The 
air amount m coming into the cylinder M3 at intake stroke still can be 
estimated based on the amount of the air coming into the intake pipe M1 
detected by a known air flow meter attached upstream of a throttle valve. 
An example of the physical model as the basis of the above inventive 
construction will be described. 
A fuel amount fc coming into the cylinder M3 of the engine M2 is given by 
the following equation (3), using the fuel injecting amount q of the fuel 
injection valve M4, the adhering fuel amount fw and the vapor fuel amount 
fv. 
EQU fc=.alpha.1.multidot.q+.alpha.2.multidot.fw+.alpha.3.multidot.fv(3) 
The above equation is given because the fuel amount fc is considered as the 
sum of a direct influx .alpha.1.multidot.q by the fuel injected from the 
fuel injection valve M4, an indirect influx .alpha.2.multidot.fw spilling 
from the intake pipe M1 to which the injected fuel adheres, and a vapor 
fuel influx .alpha.3.multidot.fv remaining in the intake pipe M1 due to 
evaporation of either the injected fuel or the fuel adhering to the inner 
wall. 
Since the fuel injecting amount q is determined by the control parameter of 
the fuel injection valve M4 (e.g., injection valve opening time), which is 
a known variable, the fuel amount fc can be estimated if the adhering fuel 
amount fw and the vapor fuel amount fv are obtained as hereforth 
explained. 
The adhering fuel amount fw decreases by .alpha.2 at every intake cycle 
caused by the flow into the cylinder M3 at the intake stroke as well as by 
evaporation in the intake pipe M1. Conversely it increases by .alpha.4 
which is a part of the fuel injecting amount q injected from the fuel 
injection valve M4 synchronously with the intake cycle. The amount of the 
fuel evaporating at every intake stroke can be represented as 
.alpha.5.multidot.Vf/.omega.. Thus the adhering fuel amount fw is given by 
the following equation (4): 
EQU fw(k+1)=(1-.alpha.2).multidot.fw(k)+.alpha.4.multidot.q(k)-.alpha.5.multido 
t.Vf(k)/.omega.(k) (4) 
where k is a number of the intake cycle time. 
The vapor fuel amount fv decreases by .alpha.3 at every intake cycle caused 
by the flow into the cylinder M3 at the intake stroke. It increases by 
.alpha.6 due to the evaporation of a part of the fuel injecting amount q. 
It further increases by the evaporation of the adhering fuel. The vapor 
fuel amount fv is given by the following equation (5). 
EQU fv(k+1)=(1-.alpha.3).multidot.fv(k)+.alpha.6.multidot.q(k)+.alpha.5.multido 
t.Vf(k)/.omega.(k) (5) 
A fuel amount fc(k) admitted into the cylinder M3 of the engine M2 is 
represented by the following equation (6) using a fuel/air ratio 
.lambda.(k) which can be detected from the concentration of the oxygen in 
the exhaust gas, and the air amount m(k) coming into the cylinder M3. 
EQU fc(k)=.lambda.(k).multidot.m(k) (6) 
When the coefficients .alpha.1 through .alpha.6 of the respective equations 
are determined by the known method of system identification, a state 
equation (7) and an output equation (8) are obtained as shown below. Both 
equations use the adhering fuel amount and the vapor fuel amount as state 
variables, and are described in a discrete system taking the intake cycle 
of the engine as a sampling cycle. Those equations determine a physical 
model for describing behavior of fuel in the engine. 
##EQU1## 
The estimation means M7 obtains estimations fw and fv of the state 
variables fw and fv, based on: a product .lambda..multidot.m (which 
represents fuel amount coming into the cylinder) of the fuel/air ratio 
.lambda. and the air amount m both of which are detected by the operating 
state detection means M5, the division Vf/.omega.; from the dividing means 
M6, and the fuel injecting amount q of the fuel injection valve M4. Here 
the calculation utilizes the first equation set in accordance with the 
aforementioned physical model. Since the adhering fuel amount fw and the 
vapor fuel amount fv cannot be detected directly by a sensor like the 
engine speed .omega. or the fuel/air ratio .lambda., nor detected even 
indirectly by calculations from detected results of sensors like the fuel 
evaporating amount Vf or the air amount m, they are estimated by the 
estimation means M7. 
The estimation means M7 may have a construction of known observers like 
minimal order observer, identity observer, dead beat observer, linear 
function observer, or adaptive observer. The design methods of the 
observers are explained in detail in "Introduction to Dynamic 
System--Theory, Models and Applications" by David G. Luenberger, John 
Wiley & Sons Inc., N.Y. (1979). 
The fuel injecting amount calculation means M9 calculates the fuel 
injecting amount q of the fuel injection valve M4 based on the division 
Vf/.omega. from the dividing means M6, the estimations fw and fv from the 
estimation means M7, the product .lambda.r.multidot.m of the target 
fuel/air ratio .lambda.r and the air amount m, i.e., target fuel amount 
coming into the cylinder M3, and the sum calculated by the summing means 
M8, utilizing the second equation determined from the physical model. 
The fuel injecting amount calculation means M9 is so constructed to 
calculate the control variable of the servo system compensated for the 
non-linearity. The control variable is a sum of the products as follows: 
the products of the state variables fw and fv estimated by the estimation 
means M7 and coefficients predetermined by the physical model; the product 
of sum of differences added by the difference between the target fuel 
amount .lambda.rm and the measured fuel amount .lambda.m and coefficients 
predetermined by the physical model so as to approach the fuel amount 
.lambda.m to the target amount .lambda.rm under the existence of 
disturbance; and the product of the division Vf/.omega.(k) calculated by 
the dividing means M6 and coefficients predetermined by the physical 
model. 
In the above constructed fuel injection system of the invention, the 
estimation means M7 estimates the state variables fw and fv based on the 
product .lambda..multidot.m of the fuel/air ratio .lambda. and the air 
amount m detected by the operating state detection means M5, the division 
Vf/.omega. calculated by the dividing means M6, and the fuel injecting 
amount q of the fuel injection valve M4, utilizing the first equation 
determined from the physical model. The fuel injecting amount calculation 
means M9 calculates the fuel injecting amount q of the fuel injection 
valve M4 based on the division Vf/.omega. from the dividing means M6, the 
estimations fw and fv from the estimation means M7, the product 
.lambda.r.multidot.m of the target fuel/air ratio .lambda.r and the air 
amount m detected by the operating state detection means M5, and the sum 
calculated by the summing means M8, utilizing the second equation 
determined from the physical model. 
The fuel injection system of the present invention calculates the fuel 
injecting amount in accordance with the control law determined from the 
physical model which describes the fuel behavior in the engine as shown by 
the equations (7) and (8), utilizing the adhering fuel amount and the 
vapor fuel amount as state variables. The fuel injecting amount of the 
engine, thus, is subjected to a feedback control. 
The fuel injection system of an internal combustion engine of this 
invention sets a control law in accordance with a physical model 
describing the fuel behavior in the engine, and is compensated for the non 
linearity in accordance with the division calculated by the dividing means 
M6. Therefore the system allows a single control law to cover the control 
of the fuel injecting amount with great accuracy under wide-ranging 
operating conditions of the engine. Accordingly its construction is 
further simplified and can be expressed in lower order, thereby improving 
responsiveness of the control. 
Another feature of the present invention is, as shown in FIG. 1B, a fuel 
injection system of an internal combustion engine M2 for determining an 
injecting amount q of a fuel injection valve M4 based on a physical model 
describing a behavior of fuel coming into a cylinder M3 of the engine M2 
utilizing an amount fw of fuel adhering to an inner wall of an intake pipe 
M1 and an amount fv of vapor fuel in the intake pipe M1 as state 
variables. The system comprises: 
an operating state detection means M15 for detecting a rotating speed 
.omega. of the engine M2, an evaporating amount Vf of the fuel adhering to 
the inner wall of the intake pipe M1 and an amount m of air coming into 
the cylinder M3; 
a dividing means M16 for dividing the evaporating amount Vf by the engine 
speed .omega.; 
an estimation means M17 for calculating estimation values fw and fv of the 
adhering fuel amount fw and the vapor fuel amount fv, based on the 
division Vf/.omega. at the dividing means M16 and the injecting amount q, 
utilizing a first equation determined from the physical model; and 
a fuel injecting amount calculation means M19 for calculating the fuel 
injecting amount q, based on the division Vf/.omega., the estimation 
values fw and fv, and a product .lambda.r.multidot.m of the detected air 
amount m and a target fuel/air ratio .lambda.r, utilizing a second 
equation determined from the physical model. 
This feature is characterized in that the operating state detection means 
M15 does not detect the fuel/air ratio of the mixture. The estimation 
means M17 estimates fw and fv without utilizing .lambda..multidot.m and 
the fuel injecting amount calculation means M19 calculates the injecting 
amount q without the summed up difference. This system is enabled to 
adjust the air/fuel ratio to the target air/fuel ratio without the sensor 
for detecting the air/fuel ratio, thereby simplifying the construction of 
the system.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
A first embodiment of the present invention will be described with 
reference to the drawings. 
Shown in FIG. 2, an intake pipe 4 admits air through an air cleaner 6. The 
intake pipe 4 is provided with a throttle valve 8 for controlling the flow 
of the admitted air, a surge tank 10 for suppressing pulsation of the 
admitted air, a sensor 12 for detecting a pressure P in the intake pipe 4 
(intake pipe pressure), and a sensor 13 for detecting an intake air 
temperature Ti. 
An exhaust pipe 14 is provided with an oxygen sensor 16 for detecting a 
fuel/air ratio of an air/fuel mixture coming into a cylinder 2a of an 
internal combustion engine 2 in accordance with the concentration of 
oxygen in the exhaust gas, and a three way catalytic converter 18 for 
treating the exhaust gas. Residual HC of the fuel and the combustion 
residues such as CO and NOx in the exhaust gas are converted into harmless 
gases in the three way catalytic converter 18. 
The engine 2 is provided with sensors for detecting operating states 
thereof such as an engine speed sensor 22 for detecting the engine speed 
.omega. in accordance with the rotation of a distributor 20, a crank angle 
sensor 24 for detecting a fuel injecting timing t to the engine 2 in 
accordance with the rotation of the distributor 20, a water temperature 
sensor 26 installed on a water jacket of the engine 2 for detecting a 
cooling water temperature T, and the aforementioned sensors 12, 13, and 
16. The distributor 20 is so constructed to apply high voltage from an 
igniter 28 to spark plugs 29 at a predetermined ignition timing. 
Signals detected by the respective sensors are fed to an electronic control 
circuit 30 constructed as an arithmetic logic circuit including a 
microcomputer to be used for driving a fuel injection valve 32 to control 
the amount of the fuel injected therefrom. 
The electronic control circuit 30 comprises a CPU 40, a ROM 42, a RAM 44, 
an input port 46, and an output port 48. The CPU 40 performs arithmetic 
operations for the fuel injection control in accordance with a 
predetermined control program. The control program and initial data used 
for the operation by the CPU 40 are stored in the ROM 42. The data used 
for the operation are temporarily stored in the RAM 44. The detected 
signals from the respective sensors are received through the input port 
46. A driving signal to the fuel injection valve 32 responding to the 
result operated by the CPU 40 is supplied through the output port 48. The 
electronic control circuit 30 is constructed to perform feedback control 
of a fuel injecting amount q of the fuel injection valve 32 so that the 
fuel/air ratio .lambda. of the mixture coming into the cylinder 2a of the 
engine 2 is adjusted to the target fuel/air ratio .lambda.r set in 
accordance with the respective operating state of the engine 2. 
A control system used for the feedback control will be described with 
reference to a block diagram of FIG. 3, which does not show any hardware 
structure. Actually it is realized as a discrete system by executing a 
series of programs shown in the flowchart of FIG. 4. The control system of 
this embodiment is designed based on a physical model represented by the 
equations (7) and (8). 
Referring to FIG. 3, in the control system of this embodiment, the 
temperature T detected by the sensor 26 is input to a first calculator P1. 
Then a saturated vapor pressure Ps is calculated based on the input 
temperature T, utilizing the equation (1). Further a fuel evaporating 
amount Vf is calculated based on the pressure Ps, utilizing the equation 
(1)'. The fuel evaporating amount Vf is input to a divider P2 to be 
divided by the engine speed .omega. detected by the sensor 22. The 
division Vf/.omega. is input to a multiplier P3 to be multiplied by a 
predetermined coefficient f5. 
The engine speed .omega. detected by the sensor 22 is input to a second 
calculator P4 along with the pressure P detected by the sensor 12, and the 
temperature Ti detected by the sensor 13. The second calculator P4 
calculates an air amount m coming into the cylinder 2a based on the engine 
speed .omega., the pressure P, and the temperature Ti, utilizing equation 
(2). The calculated result is output to both a first multiplier P5 and a 
second multiplier P6. At the first multiplier P5, a fuel/air ratio 
.lambda. of the mixture coming into the cylinder 2a detected by the oxygen 
sensor 16 is multiplied by the air amount m calculated by the second 
calculator P4, resulting in the actual fuel amount .lambda..multidot.m 
coming into the cylinder 2a. 
At the second multiplier P6, a target fuel/air ratio .lambda.r determined 
in accordance with the load imposed on the engine 2 is multiplied by the 
air amount m calculated by the second calculator P4, resulting in a 
calculated required fuel amount .lambda.r.multidot.m (target fuel amount) 
to come into the cylinder 2a. The target fuel amount .lambda.r.multidot.m 
calculated by the multiplier P6 is input to a multiplier P7 to be 
multiplied by a predetermined coefficient f4. 
The products of the first and the second multipliers P5 and P6 are input to 
a difference operating portion P8 where the difference of the products 
m.multidot.(.lambda.-.lambda.r) is calculated. The difference is summed up 
at a summing portion P10, which is further multiplied by a predetermined 
coefficient f3 at a multiplier P9. 
The actual fuel amount .lambda..multidot.m calculated by the first 
multiplier P5 and the division Vf/.omega. calculated by the divider P2 are 
output to an observer P11. The observer P11 is so constructed to estimate 
the adhering fuel amount fw and the vapor fuel amount fv based on the 
actual fuel amount .lambda..multidot.m, division Vf/.omega. from the 
divider P2, the fuel injecting amount q of the fuel injection valve 32, 
and the adhering fuel amount fw and the vapor fuel amount fv which are 
estimated in the previous execution of the same routine, utilizing a 
predetermined equation. The obtained estimations fw and fv are multiplied 
by coefficients f1 and f2 at multipliers P12 and P13, respectively. 
The products obtained from the multipliers P12 and P13, along with the 
products from other multipliers P4, P7 and P10, are added by adders P14 
through P17. Accordingly the fuel injecting amount q of the fuel injection 
valve 32 is determined. 
A design method for the aforementioned control system in FIG. 3 will be 
explained. A design method for the control system of this type is 
described in detail, as for example, in the above-cited reference. 
Therefore the method is described only briefly herein. This embodiment 
uses the Smith-Davison design method. 
The control system of this embodiment is designed based on the 
aforementioned physical model represented by the equations (7) and (8). 
This physical model with non-linearity is linearly approximated. 
If the following equations are provided: 
##EQU2## 
the equations (7) and (8) are represented by the following equations. 
EQU x(k+1)=.PHI..multidot.x(k)+.GAMMA..multidot.q(k)+.pi..multidot.Vf(k)/.omega 
.(k) (15) 
EQU y(k)=.PHI..multidot.x(k) (16) 
Suppose a disturbance W(k) is added to the right side of the equation (15), 
the equations (15) and (16) will be as shown by the following equations 
(15)' and (16)'. Variables at this time are represented by subscript a. 
EQU xa(k+1)=.PHI..multidot.xa(k)+.GAMMA..multidot.qa(k)+.pi..multidot.Vf(k)/.om 
ega.(k)+E.multidot.W(k) (15)' 
EQU ya(k)=.PHI..multidot.xa(k) (16)' 
Suppose y(k)=yr (target value), the equations (15) and (16) are represented 
by the following equations (15)" and (16)". 
EQU xr=.PHI..multidot.xr+.GAMMA..multidot.qr+.pi..multidot.Vf(k)/.omega.(k)(15) 
" 
EQU yr=.PHI..multidot.xr (16)" 
From the above equations (15)", (15)" and (16)', (16)", the equations (17) 
and (18) are obtained. 
EQU xa(k+1)-xr=.PHI..multidot.(xa(k)-xr)+.GAMMA..multidot.(qa(k)-qr)+E.multidot 
.W(k) (17) 
EQU ya(k)-yr=.PHI..multidot.(xa(k)-xr) (18) 
Suppose .DELTA.W(k)=W(k)-W(k-1)=0, on the assumption that disturbance W 
changes in a stepwise fashion in the equation (17), the equations (17)' 
and (18)' are obtained from the equations (17) and (18). 
EQU .DELTA.(xa(k+1)-xr)=.PHI..multidot..DELTA.(xa(k)-xr)+.GAMMA..multidot..DELT 
A.(qa(k)-qr) (17)' 
EQU .DELTA.(ya(k)-yr)=.PHI..multidot..DELTA.(xa(k)-xr) (18)' 
Therefore, the above equations (17)' and (18)' entail a state equation 
which is linearly approximated and extended to a servo system as shown by 
the following equation (19). 
##EQU3## 
The above equation (19) is rewritten to the following equation (20). 
EQU .delta.X(k+1)=Pa.multidot..delta.X(k)+Ga.multidot.u(k) (20) 
A quadratic criterion function in the discrete system can be represented as 
follows. 
##EQU4## 
With weighted parameter matrices Q and R selected, the input .delta.u (k) 
for minimizing the quadratic criterion function J is obtained from the 
next equation (22). 
EQU .delta.u(k)=F.multidot..delta.X(k) (22) 
The optimal feedback gain F in the equation (19), thus, is determined by: 
EQU F=-(R+Ga.sup.T .multidot.M.multidot.Ga).sup.-1 .multidot.Ga.sup.T 
.multidot.M.multidot.Pa (23) 
where M is a regular symmetric matrix satisfying a discrete Ricacci 
equation shown by 
EQU M=Pa.sup.T .multidot.M.multidot.Pa+Q-(Pa.sup.T 
.multidot.M.multidot.Ga).multidot.(R+Ga.sup.T 
.multidot.M.multidot.Ga).sup.-1 .multidot.(Ga.sup.T 
.multidot.M.multidot.Pa) (24) 
Hence .DELTA.(qa(k)-qr) is given by: 
##EQU5## 
where F is [F1 F2]. 
With the above equation (25) integrated, qa(k)-qr is given by 
##EQU6## 
When control is performed according to equation (26) under the condition of 
equations (15)" and (16)", i.e., y(k)=yr, the following equation (27) is 
given. 
EQU qr=F1.multidot.xr.multidot.F1.multidot.xa(0)+ya(0) (27) 
Then substituting the equation (27) into the equation (15)" provides the 
following equation. 
EQU xr-[(.PHI.+.GAMMA.F1].multidot.xr+.GAMMA..multidot.(-F1.multidot.xa(0)+qa(0 
))+.pi..multidot.Vf(k)/.omega.(k) (28) 
Suppose xa(k+1)=x(k) (k.fwdarw..infin.), the following equations are 
obtained. 
EQU xr(k)=[I-.PHI.-.GAMMA..multidot.F1].sup.-1 
.multidot..GAMMA..multidot.(-F1.multidot.xa(0)+qa(0))+.PHI..multidot.[I-.P 
HI.-.GAMMA..multidot.F1].sup.-1 .multidot..pi.Vf(k)/.omega.(k)(29) 
EQU Yr(k)=.PHI..multidot.[I-.PHI.-.GAMMA..multidot.F1].sup.-1 
.multidot..GAMMA..multidot.(-F1.multidot.xa(0)+qa(0))+.PHI..multidot.[I-.P 
HI.-.GAMMA..multidot.F1].sup.-1 
.multidot..GAMMA..multidot.Vf(k)/.omega.(k)(30) 
Therefore the following equation is provided. 
##EQU7## 
In the equation (31), substituting the following equations (32) and (33) 
into the equation (26) provides the equation (34). 
##EQU8## 
Substituting the equations (9) and (10) into the equation (34) provides 
##EQU9## 
Accordingly the control system shown in FIG. 3 is designed. The equation 
(36) corresponds to the second equation for calculating the fuel injecting 
amount. 
The observer P11 is so constructed to estimate the adhering fuel amount fw 
and the vapor fuel amount fv in the equation (36) since they cannot be 
directly measured. Gopinath design method or the like is known for the 
design method of the observer of this type, which is described in detail 
by the cited "Basic System Theory". Here the minimal order observer is 
adopted. 
If the following equation (37) is provided, the aforementioned equation 
(15) is rewritten to the equation (38) as below. 
##EQU10## 
The generalized system of the observer for the physical model represented 
by the above equations (38) and (16) is determined as the following 
equation (39). 
EQU x(k+1)=A.multidot.x(k)+B.multidot.y(k)+J.multidot.u(k) (39) 
Therefore the observer P11 of this first embodiment can be designed as the 
following equation (40), by which the adhering fuel amount fw and the 
vapor fuel amount fv are estimated. 
##EQU11## 
The fuel injection control executed by the electronic control circuit 30 
will be described referring to a flowchart of FIG. 4. The variables used 
in the current processing will be hereinafter represented by subscript 
(k). 
The process for fuel injection control begins with the start of the engine 
2, and is repeatedly carried out during the operation of the engine 2. 
When the process is initiated, step 100 is executed where the variables of 
both the adhering fuel amount estimation fwo and the vapor fuel amount 
estimation fvo, and the fuel injecting amount q are initialized. At step 
110, the integral value Sm.lambda. of the difference between the actual 
fuel amount .lambda.m and the target fuel amount .lambda.rm is set at 0. 
At step 120, the fuel/air ratio .lambda.(k), the pressure P(k), the intake 
air temperature Ti(k), the engine speed .omega.(k), and the fuel 
temperature T(k) are calculated based on the output signals from the 
respective sensors. 
At step 130, the target fuel/air ratio .lambda.r responding to the load 
imposed on the engine 2 is calculated based on the pressure P(k) and the 
engine speed .omega.(k) obtained at step 120. At this step 130, the target 
fuel/air ratio .lambda.r is so set that an air excess rate of the air fuel 
mixture becomes 1, i.e., .lambda.r is set at the stoichiometric air/fuel 
ratio. In case of engine operation with heavy load, the target fuel/air 
ratio .lambda.r is set to the richer side so as to increase the output of 
the engine by increasing the fuel amount more than usual. In case of 
engine operation with light load, it is set to the leaner side so as to 
reduce the fuel consumption by decreasing the fuel amount less than usual. 
After the target fuel/air ratio .lambda.r(k) is set at step 130, the 
control proceeds to step 140. The process at this step 140 is executed as 
the second calculator P4 in which the air amount m(k) coming into the 
cylinder 2a is calculated based on the pressure P(k), the intake air 
temperature Ti(k), and the engine speed .omega.(k) which are obtained at 
step 120, utilizing either the equation (2) or a predetermined data map 
representing such relation of equation (2). 
The control further proceeds to step 150 where the process is executed as 
the first calculator P1 and the divider P2. At this step 150, the fuel 
evaporating amount Vf obtained based on the fuel temperature T(k) is 
divided by the engine speed .omega.(k) to calculate the evaporating amount 
Vfw(k), i.e., Vf(k)/.omega.(k), between cycle to cycle of the intake 
stroke. In this embodiment, the saturated vapor pressure Ps(k) is obtained 
from the equation (1) or a predetermined data map, and the pressure Ps(k) 
is used for calculating the evaporating fuel amount Vf based on the 
equation (1)'. Since the evaporating fuel amount Vf also changes dependent 
on the pressure P, it may be calculated based on the saturated vapor 
pressure Ps(k) obtained from the equation (1) and the pressure P(k) 
obtained at step 120. 
The process at following step 160 is executed as the first multiplier P5 
where the fuel/air ratio .lambda.(k) obtained at step 120 is multiplied by 
the air amount m(k) obtained at step 150 to calculate the actual fuel 
amount .lambda.m(k) that has come into the cylinder 2a at the previous 
intake stroke. Then the control proceeds to step 170 where the process is 
executed as the observer P11. At step 170, the estimations of the adhering 
fuel amount fw(k) and the vapor fuel amount fv(k) are provided based on 
the actual fuel amount .lambda.m(k) at step 160, the fuel injecting amount 
q obtained in the previous execution of the same routine, the evaporating 
amount Vfw(k) at step 150, and estimations of the adhering fuel amount fwo 
and the vapor fuel amount fvo obtained in the previous execution of the 
same routine, utilizing the equation (40). 
The process at step 180 is executed as the second multiplier P6. At this 
step 180, the target fuel amount .lambda.rm(k) coming into the cylinder 2a 
is calculated by multiplying the target fuel/air ratio .lambda.r(k) set at 
step 130 by the air amount m(k) obtained at step 140. The control further 
proceeds to step 190 where the fuel injecting amount q is calculated based 
on the integral value Sm.lambda. of the difference between the actual fuel 
amount .lambda.m and the target fuel amount .lambda.rm, estimations fw(k) 
and fv(k) obtained at step 170, the target fuel amount .lambda.rm(k) at 
step 180, and the evaporating amount Vfw(k) at step 150, utilizing 
equation (36). 
At step 200, the fuel injection control is executed by opening the fuel 
injection valve 32 during the period corresponding to the fuel injecting 
amount q(k) obtained at step 190 at the fuel injection timing determined 
based on the detection signal from the crank angle sensor 24. 
When the fuel supply to the engine 2 is terminated after the execution of 
the fuel injection control at step 200, the control proceeds to step 210 
where the process is executed as the summing portion P10. At step 210, the 
difference between the actual fuel injection amount .lambda.m(k) obtained 
at step 160 and the target fuel injection amount .lambda.rm(k) at step 180 
are added to the integral value Sm.lambda.(k) obtained in the previous 
execution of the same routine to obtain an integral value Sm.lambda.(k). 
The control proceeds to step 220 where the estimations fw(k) and fv(k) 
obtained at step 170 are set as the values fwo and fvo used for providing 
estimations of the adhering fuel amount fw and the vapor fuel amount fv at 
next processing. The program then returns to step 120 again. 
In the fuel injection system of this embodiment, the control law is set 
based on the physical model describing the behavior of fuel in the engine 
2. Accordingly the behavior which varies responsive to the temperature of 
the intake pipe of the engine 2, i.e., warming-up state of the engine 2, 
can be compensated for its non-linearity by Vfw (Vf/.omega.), resulting in 
the fuel injection control covered by a single control law. This will 
eliminate cumbersome processing such as switching from one control law to 
another in accordance with the operating state of the engine, thereby 
simplifying the control system. 
Since the system utilizes the physical model enabled to describe the 
behavior of fuel with high accuracy, it can perform the control without 
being influenced by disturbances in spite of the control law with lower 
order, thus improving the control accuracy. 
The state variables estimated at the observer are the adhering fuel amount 
and the vapor fuel amount. Therefore, an abnormality of the system can be 
detected by determining whether they are estimated accurately by the 
observer. 
In the above embodiment, the control system is designed based on the 
physical model represented by equations (7) and (8) on the assumption that 
all the fuel evaporating from the inner wall of the intake pipe is to be 
the vapor fuel. However, some part of the fuel evaporating at the intake 
stroke of the engine (1/4 of the total evaporating amount 
.alpha.5.multidot.Vf/w between an intake cycle to the next intake cycle in 
a 4-cycle engine) may not remain inside the intake pipe as the vapor fuel. 
Instead, it may directly flow into the cylinder of the engine. For the 
case, the equations (5) and (6) are rewritten to the following equations 
(50) and (51). 
EQU fv(k+1)=(1-.alpha.3).multidot.fv(k)+.alpha.6.multidot.q(k)+3.multidot..alph 
a.5.multidot.Vf(k)/4.multidot..omega.(k) (50) 
EQU fc(k)=.lambda.(k).multidot.m(k)+.alpha.5.multidot.Vf(k)/4.multidot..omega.( 
k) (51) 
The physical model is modified as the following equations (52) and (53): 
##EQU12## 
where .alpha.7=.alpha.5.multidot.3/4 and .alpha.8=.alpha.5/4. The control 
system can also be designed by this physical model. 
In this case, the control system can be designed in the same manner as the 
above embodiment by the following equations. 
##EQU13## 
Since the equations (52), (53) can be represented as the aforementioned 
equations (15), (16), the state equation which is linearly approximated 
and extended to the servo system shown by the equation (19) is obtained in 
the same manner as the above embodiment. Then the equation (34) is derived 
from solving the Ricacci equation. Substituting the equations (54) and 
(55) into the equation (34) provides the following equation (60). 
##EQU14## 
Then the control system can be designed, which is the same as the above 
embodiment shown in FIG. 3. 
The observer P11 shown in FIG. 3 is also designed based on the equation 
(40) in the same manner as the above embodiment. 
In the above embodiment, estimations fw and fv of the adhering fuel amount 
fw and the vapor fuel amount fv obtained by the observer P11 are used as 
it is for the control. However in case of the engine operation with light 
load, at low engine speed, and at a high cooling water temperature of 
80.degree. C. or more, the adhering fuel amount fw might be estimated as 
negative due to an increase in the evaporating amount Vf/.omega. 
calculated at every intake stroke. In practice, since the adhering fuel 
amount fw can not become negative, such estimation would disturb the 
stable control. 
The processes executed by steps 171 and 172 shown in FIG. 5 are required 
for solving the aforementioned problem. At those steps, after the amount 
fw is estimated at step 170 shown in FIG. 4, it is determined whether the 
estimated value fw is negative. If the value is determined to be negative, 
it is set at 0. 
A second embodiment will be described, which corresponds to the second 
feature of the present invention shown in FIG. 1B. 
The schematic diagram illustrating the internal combustion engine 2 and its 
peripheral equipments applied to this embodiment are shown in FIG. 6. The 
construction of them, however, is different from that of the first 
embodiment shown in FIG. 2 only in that the oxygen sensor (air/fuel ratio 
sensor) of the exhaust pipe 14 is excluded. Accordingly this embodiment is 
different from the first one in that the fuel/air ratio .lambda. is not 
used in the control to be described hereinafter. 
The control system of the second embodiment is represented by the block 
diagram of FIG. 7. As shown in FIG. 7, the control system is not provided 
with the first multiplier P5, adder P8, summing portion P10, multiplier 
P9, and adder P14 shown in FIG. 3. The observer P31 is constructed to 
calculate estimations fw and fv without using the fuel/air ratio .lambda.. 
Since the other parts of the construction are the same as those of the 
first embodiment, the numerals designating the identical parts will be 
added by 20. 
The design method of the control system of FIG. 7 will be described. 
If the following equations are provided: 
##EQU15## 
the equations (7) and (8) are represented by the following equations, 
respectively. 
EQU x(k+1)=.PHI..multidot.x(k)+.GAMMA..multidot.u(k)+E.multidot.w(k)(79) 
EQU y(k)=.theta..multidot.x(k)+.LAMBDA..multidot.u(k) (80) 
In case of steady state with y(k)=yr (target value), supposing u(k)=ur and 
x(k)=xr entails that the equations (79) and (80) are represented by the 
following equations (79)' and (80)'. 
EQU xr=.PHI..multidot.xr+.GAMMA..multidot.ur+E.multidot.w(k) (79)' 
EQU yr=.theta..multidot.xr+.LAMBDA..multidot.ur (80)' 
From the above equations (79), (79)', and (80), (80)', the following 
equations are derived. 
EQU x(k+1)-xr=.PHI..multidot.(x(k)-xr)+.GAMMA..multidot.(u(k)-ur)(81) 
EQU y(k)-yr=.theta.(x(k)-xr)+.LAMBDA..multidot.(u(k)-ur) (82) 
If the following equations are provided: 
EQU X(k)=x(k)-xr (83) 
EQU U(k)=u(k)-ur (84) 
EQU Y(k)=y(k)-yr-.LAMBDA..multidot.(u(k)-ur) (85) 
the equations (81) and (82) become as follows. 
EQU X(k+1)=.PHI..multidot.X(k)+.GAMMA..multidot.U(k) (86) 
EQU Y(k)=.theta..multidot.X(k) (87) 
In the above equations (86) and (87), supposing X(k).fwdarw.0 entails 
Y(k)=0. Also supposing u(k).fwdarw.ur entails y(k).fwdarw.yr. The next 
step is to design the optimal regulator of the above equation (86) can be 
designed. That is, the optimal regulation is obtained as shown in the 
following equation (88), by solving discrete Ricacci equation. 
EQU U(k)=F.multidot.X(k) (88) 
The equation (88) is transformed into the following equation (89) utilizing 
the equations (83) and (84). 
EQU u(k)=F.multidot.x(k)-F.multidot.xr+ur (89) 
If xr and ur in equations (79)' and (80)' are given by the following 
equation (90), the above equation (79) is determined to provide u(k). 
##EQU16## 
In this embodiment, the above equation (90) is rewritten as the following 
equation (91) from the equations (70) through (78). 
##EQU17## 
Thus, the values xr and ur (i.e., fwr, fvr and qr) are obtained as 
follows. 
EQU fwr=.beta.11.multidot.Vf(k)/.omega.(k)+.beta.12.multidot.{.lambda.r.multido 
t.m(k)-(1-.alpha.4-.alpha.6).multidot.u(k)} (92) 
EQU fvr=.beta.21.multidot.Vf(k)/.omega.(k)+.beta.22.multidot.{.lambda.r.multido 
t.m(k)-(1-.alpha.4-.alpha.6).multidot.u(k)} (93) 
EQU qr=.beta.21.multidot.Vf(k)/.omega.(k)+.beta.23.multidot.{.lambda.r.multidot 
.m(k)-(1-.alpha.4-.alpha.6).multidot.u(k)} (94) 
where .alpha.11 through .alpha.23 are constants. 
The following equation (95) is obtained from the equation (89) using 
coefficients f1, f2, f4, and f5. 
EQU u(k)=f1.multidot.fw(k)+f2.multidot.fv(k)+f4.multidot.m(k).lambda.r+f5.multi 
dot.Vf(k)/.omega.(k) (95) 
In this way, the control system shown in FIG. 7 can be designed. 
The equation (95) corresponds to the second equation in the fuel injecting 
amount calculation means M19 for obtaining the fuel injecting amount. 
The observer P31 is so constructed to estimate the adhering fuel amount fw 
and the vapor fuel amount fv utilized in the equation (95) since they 
cannot be directly measured. Gopinath design method or the like is known 
for the design method of the observer of this type. This embodiment cannot 
use the conventional observer because the air/fuel ratio .lambda. of the 
mixture which is actually supplied to the engine 2 cannot be measured. 
However, the equation (7) which describes the behavior of fuel in the 
engine 2 provides the amounts fw and fv without the actual value of 
.lambda.. The reason is as follows. 
The second and third terms of the right side of the equation (7) be 
calculated because q(k) is derived from the electronic control circuit 30 
as the control parameter, Vf(k) is detected by the saturated vapor 
pressure Ps from the cooling water temperature T from the sensor 26, and 
the intake pipe pressure P from the sensor 12, and further the engine 
speed .omega.(k) is detected by the engine speed sensor 22. If the 
following equations (96) and (97) are provided, the equation (98) is 
obtained as below. 
##EQU18## 
The equation (98) is stable because 1-.alpha.2&lt;1 and 1-.alpha.3&lt;3 1. 
Therefore, .delta.w(k) and .delta.v(k).fwdarw.0, i.e., fw(k).fwdarw.fw(k), 
and fv(k).fwdarw.fv(k). If appropriate initial values are provided for 
fw(k) and fv(k), they can be estimated by utilizing the equation (7). 
In this embodiment, the observer P31 is so constructed to estimate the 
adhering fuel amount fw and the vapor fuel amount fv by utilizing the 
equation (7). Even if the disturbance brings such conditions as 
fw(k).noteq.fw, and fv(k).noteq.fv, the equation (95) will provide u(k) 
(i.e., fuel injecting amount q(k)) with no problem, since the fw(k) and 
fv(k) follow fw(k) and fv(k), 
The fuel injection control executed by the electronic control circuit 30 in 
this second embodiment will be described referring to a flowchart of FIG. 
8. Hereinafter, the variables used for the current processing will be 
represented by subscript (k). 
The process for fuel injection control begins with the start of the engine 
2, and is repeatedly carried out during the operation of the engine 2. 
When the process is initiated, step 300 is executed where the variables of 
the adhering fuel amount estimation fwo and the vapor fuel amount 
estimation fvo, and the fuel injecting amount q are initialized. At step 
310, intake pipe pressure P(k), intake air temperature Ti(k), engine speed 
.omega.(k), and cooling water temperature T(k) are obtained based on the 
output signals from the respective sensors. Then the control proceeds to 
step 320 where a target fuel/air ratio .lambda.r responding to the load 
imposed on the engine 2 is calculated based on the P(k) and .omega.(k) 
obtained at step 310. At this step 320, the target fuel/air ratio 
.lambda.r is so set that an air excess rate of the air fuel mixture 
becomes 1, i.e., at the stoichiometric air/fuel ratio. In case of the 
engine operation with heavy load, the target fuel/air ratio .lambda.r is 
set to the richer side to increase the output of the engine by increasing 
the fuel amount more than usual. Conversely in case of the engine 
operation with light load, it is set to the leaner side to reduce the fuel 
consumption by decreasing the fuel amount less than usual. 
After the target fuel/air ratio .lambda.r(k) is set at step 320, the 
control proceeds to step 330. The process at step 330 is executed as the 
second calculator P24 in which an air amount m(k) coming into the cylinder 
2a is calculated based on P(k), Ti(k), and .omega.(k) obtained at step 
320, utilizing either the equation (2) or a predetermined data map. 
The process at the following step 340 is executed as the first calculator 
P21 and the divider P22. At this step 340, the fuel evaporating amount Vf 
obtained based on T(k) and P(k) at step 310 is divided by the engine speed 
.omega.(k) to calculate the evaporating amount Vfw(k), i.e., 
Vf(k)/.omega.(k) between an intake cycle to the next intake cycle. 
The process at step 350 is executed as the observer P31 in which 
estimations of the adhering fuel amount fw(k) and the vapor fuel amount 
fv(k) are provided based on the evaporating amount Vfw(k) at step 340, the 
fuel injecting amount q obtained in the previous execution of the same 
routine, and estimations fwo, fvo obtained in the previous execution of 
the same routine, utilizing the following equation (99) which is derived 
from the equation (7). 
##EQU19## 
The process at step 360 is executed as the multiplier P26. There, the 
target fuel amount .lambda.rm(k) coming into the cylinder 2a is calculated 
by multiplying the target fuel/air ratio .lambda.r/(k) set at step 320 by 
the air amount m(k) at step 330. The control proceeds to step 370 where 
the fuel injecting amount q(k) is calculated based on the estimations 
fw(k), fv(k) obtained at step 350, the target fuel amount .lambda.rm(k) at 
step 360, and the evaporating amount Vfw(k) at step 340, utilizing the 
equation (95). 
At step 380, the fuel injection is executed by opening the fuel injection 
valve 32 during the period corresponding to the fuel injecting amount q(k) 
determined at step 370 at the fuel injection timing determined based on 
the detection signal from the crank angle sensor 24. 
When the fuel supply to the engine 2 is terminated after the execution of 
the fuel injection at step 380, the control proceeds to step 390. At step 
390, the estimation fw(k) and fv(k) obtained at step 350 are set as the 
values of the adhering fuel amount fwo and the vapor fuel amount fvo used 
for providing estimations fw and fv at next processing. Then the program 
returns to step 310 again. 
In the fuel injection system of this embodiment, the control law is set 
based on the physical model describing the behavior of fuel in the engine 
2. The behavior which varies responsive to the temperature of the intake 
pipe in the engine 2, i.e., warming-up state of the engine, can be 
compensated for its non-linearity by Vfw, i.e., Vf/.omega.. Accordingly 
the fuel injection control is covered by a single control law. This will 
eliminate the cumbersome processing such as switching from one control law 
to another in accordance with the operating state of the engine, thereby 
simplifying the control system. 
The fuel/air ratio can be adjusted to the target ratio without using a 
sensor for detecting the fuel/air ratio .lambda. of the mixture actually 
supplied to the engine 2, thereby simplifying the construction of the 
device. 
The state variables estimated at the observer are the adhering fuel amount 
and the vapor fuel amount. Therefore, an abnormality of the system can be 
detected by determining whether they are estimated accurately by the 
observer. 
The control system of this embodiment is designed based on the physical 
model represented by the equations (7) and (8) on the assumption that all 
the fuel evaporating from the inner wall of the intake pipe would be the 
vapor fuel. However, some part of the evaporating fuel at the intake 
stroke of the engine (1/4 of the total evaporating amount 
.alpha.5.multidot.Vf/.omega. between an intake cycle to the next intake 
cycle in a 4-cycle engine) may not remain inside the intake pipe as the 
vapor fuel. Instead it may directly flow into the cylinder of the engine. 
Thus the equations (5) and (6) are rewritten to the equations (100) and 
(101) as follows. 
EQU fv(k+1)=(1-.alpha.3).multidot.fv(k)+.alpha.6.multidot.q(k)+3.multidot..alph 
a.5.multidot.Vf(k)/4.multidot..omega.(k) (100) 
EQU fc(k)=.lambda.(k).multidot.m(k)+.alpha.5.multidot.Vf(k)/4.multidot..omega.( 
k) (101) 
The physical model is modified as the following equations (102) and (103): 
##EQU20## 
where .alpha.7=.alpha.5.multidot.3/4 and .alpha.8=.alpha.5/4. The control 
system can be designed from this physical model. 
In this embodiment, the observer P31 is designed by using the equation (7). 
A known observer may be available in which the state variables are 
estimated on the assumption that the fuel/air ratio .lambda. is controlled 
to coincide with the target fuel/air ratio .lambda.r. 
In case a minimal order observer is designed from the equation (7), the 
following equation is given. 
##EQU21## 
This observer cannot be directly applied to the device which does not 
detect the fuel/air ratio .lambda.. However, the adhering fuel amount fw 
and the vapor fuel amount fv can be estimated by making the second term of 
the equation (104) as B.lambda.rm(k) on the assumption that the fuel/air 
ratio .lambda. is adjusted to the target ratio .lambda.r by the fuel 
injection control.