Patent Publication Number: US-2002011067-A1

Title: Air-fuel ratio control system for engine with in-catalyst state compensation

Description:
CROSS REFERENCE TO RELATED APPLICATION  
       [0001] This application is based on and incorporates herein by reference Japanese Patent Applications No. 2000-189733 filed Jun. 20, 2000 and No. 2000-303146 filed Oct. 3, 2000.  
       BACKGROUND OF THE INVENTION  
       [0002] The present invention relates to an air-fuel ratio control system for an internal combustion engine that controls an air-fuel ratio in accordance with an in-catalyst state.  
       [0003] Recently, an automobile is provided with a three-way catalyst in an exhaust pipe and an air-fuel ratio sensor on the upstream side of the catalyst. The amount of fuel injection is feedback-controlled so that the air-fuel ratio of exhaust gas is controlled within the purification window (around stoichiometric air-fuel ratio) based on the output from the air-fuel ratio sensor, so that the exhaust gas is purified efficiently.  
       [0004] The catalyst functions to subject lean component (NOx, O 2 ) and rich components (HC, H 2 ) in exhaust gas to oxidation/reduction reaction to thereby convert it to harmless neutral gas components (CO 2 , H 2 O, N 2 ), and functions additionally to adsorb or occlude non-reacted lean components and rich components temporarily in the catalyst. The catalyst purifies exhaust gas by both ways of the oxidation/reduction reaction and adsorption function. The state in which the air-fuel ratio of exhaust gas that enters catalyst deviates from the stoichiometric air-fuel ratio to the rich side or lean side continues for a while depending on the engine operation condition. As a result of such continuous deviation of the air-fuel ratio, the amount of adsorbed lean component and amount of adsorbed rich components in catalyst increase up to the saturation. This saturation results in low adsorption capability of the catalyst and low exhaust gas purification ratio of the catalyst.  
       SUMMARY OF THE INVENTION  
       [0005] The present invention has an object to provide an air-fuel ratio control system for an internal combustion engine that functions to improve the exhaust gas purification ratio by controlling an air-fuel ratio properly in accordance with an in-catalyst state.  
       [0006] According to the present invention, an air-fuel ratio control system for an internal combustion engine detects an air-fuel ratio of exhaust gas upstream or downstream a catalyst, detects air flow taken into an internal combustion engine, calculates in-catalyst state amount based on the detected air-fuel ratio of exhaust gas and air flow, and corrects a fuel injection amount so that the deviation between the in-catalyst state amount and a target in-catalyst state amount is reduced. Thereby, the internal combustion engine is controlled so that the adsorption capability of the catalyst is maintained high, and the exhaust gas purification ratio is improved.  
     
    
    
     BRIEF DESCRIPTION OF THE DRAWING  
     [0007] The above and other objects, features and advantages of the present invention will become more apparent from the following detailed description made with reference to the accompanying drawings. In the drawings:  
     [0008]FIG. 1 is a schematic diagram showing an air-fuel ratio control system in accordance with a first embodiment of the present invention;  
     [0009]FIG. 2 is a flow chart showing a target Φ calculation program executed in the first embodiment;  
     [0010]FIG. 3 is a flow chart showing a target Φ calculation program executed in the first embodiment;  
     [0011]FIG. 4 is a graph showing the relation between guard values OSmin and OSmax corresponding to the calculated in-catalyst state amount OS value and the air flow Q;  
     [0012]FIG. 5 is a time chart showing an exemplary behavior of the excess fuel factor and the in-catalyst state amount OS during the transient operation in the first embodiment;  
     [0013]FIG. 6 is a flow chart showing a method of calculating control parameters A 1 , A 2 , B 1 , B 2 , and B 3  by means of approximate differentiation executed in the first embodiment;  
     [0014]FIG. 7 is a schematic diagram showing an air-fuel ratio control system in accordance with a second embodiment of the present invention;  
     [0015]FIG. 8 is a flow chart showing a target Φ calculation program executed in the second embodiment;  
     [0016]FIG. 9 is a flow chart showing a ΔΦrefA calculation program executed in the second embodiment; and  
     [0017]FIG. 10 is a time chart showing the control characteristic of the second embodiment in comparison with that of the first embodiment. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT  
     [0018] [First Embodiment] 
     [0019] The first embodiment of the present invention will be described hereinafter with reference to FIG. 1 to FIG. 5.  
     [0020] Referring first to FIG. 1, an air cleaner  13  is provided at the uppermost portion of an intake pipe  12  of an internal combustion engine  11 , and an airflow meter  14  for detecting the intake air flow is provided at the downstream side of the air cleaner  13 . A throttle valve  15  and a throttle valve opening sensor  16  for detecting the throttle opening angle are provided on the downstream side of the air flow meter  14 .  
     [0021] Furthermore, a surge tank  17  is provided on the downstream side of the throttle valve  15 . The surge tank  17  is provided with an intake pipe pressure sensor  18  for detecting the intake pipe pressure. Furthermore, the surge tank  17  is provided with an intake manifold  19  for introducing air to each cylinder of the engine  11 . A fuel injection valve  20  is provided for injecting fuel near the intake port of the intake manifold  19  of each cylinder.  
     [0022] A catalyst  22  such as a three-way catalyst for reducing noxious components (CO, HC, NOx) in exhaust gas is disposed on the middle of an exhaust pipe  21  of the engine  11 . An air-fuel(A/F) ratio sensor  23  such as a linear A/F sensor for detecting the air-fuel ratio of exhaust gas is provided on the upstream side of the catalyst  22 . Furthermore, a cooling water temperature sensor  24  for detecting the cooling water temperature and a crank angle sensor  25  are mounted on the cylinder block of the engine  11 .  
     [0023] Sensor outputs of these sensors are supplied to an electronic control circuit (ECU)  26 . The ECU  26  comprises a microcomputer and executes a target Φ calculation program stored in a built-in ROM (memory) as shown in FIG. 2 and FIG. 3 to thereby calculate the in-catalyst state amount OS, and calculates a target excess fuel factor Φref on the upstream side of the catalyst  22  depending on the in-catalyst state amount OS. Herein, the term “excess fuel factor” means the inverse of the excess air factor λ (Φ=1/λ), and the excess air factor λ means the ratio of the actual air-fuel ratio to the stoichiometric air-fuel ratio.  
     [0024] λ=actual air-fuel ratio/stoichiometric air-fuel ratio  
     [0025] Φ=stoichiometric air-fuel ratio/actual air-fuel ratio  
     [0026] The ECU  26  feedback-corrects the fuel injection amount so that the deviation between the target excess fuel factor calculated by use of the target Φ calculation program shown in FIG. 2 and FIG. 3 and the actual excess fuel factor is reduced to a small value to thereby control the in-catalyst state amount OS around the target in-catalyst state amount OSref.  
     [0027] The target Φ calculation program shown in FIG. 2 and FIG. 3 is activated every predetermined crank angle (for example, every 180 degrees CA) or every predetermined time interval, and calculates the current in-catalyst state amount OS (i) as described hereinunder at step  101 . At first, the ECU  26  calculates the variation ΔOS(i) of the in-catalyst state amount based on the deviation (Φref) between the actual excess fuel factor detected by means of the air-fuel ratio sensor  23  on the upstream side of the catalyst  22  and the target excess fuel factor Φref and air flow Q entering the catalyst  22  per unit time. 
     Δ OS ( i )=(Φref)× Q ( i−d )  (1) 
     [0028] Herein, the previous air flow Q(i−d) derived a lag (delay) time d earlier from the present time point i is used as the air flow Q with consideration of the lag time d from the fuel injection to detection of the excess fuel factor of exhaust gas. At that time, a fixed time value may be used as the lag time d for simplification of the arithmetic processing, but a varying time value that varies depending on the air flow Q may be used as the lag time d. Because the air flow rate becomes faster and the actual lag time d becomes shorter as the air flow Q becomes larger, the lag time d may be set so as to be shorter as the air flow Q is larger.  
     [0029] The variation ΔOS(i) of in-catalyst state amount calculated by use of the above equation (1) is added to the previous in-catalyst state amount value OS(i−1) to thereby obtain the current in-catalyst state amount OS(i). 
       OS ( i )=Δ OS ( i )+ OS ( i− 1)  (2) 
     [0030] The routine proceeds to step  102  thereafter, and the calculated in-catalyst state amount OS(i) value is subjected to guard (limit) processing by use of guard values OSmin and OSmax corresponding to the lean side/rich side saturated adsorption amount of the catalyst  22 . For example, if the calculated in-catalyst state amount OS(i) value lies within the range from the guard value OSmax to the guard value OSmin (Osmin≦OS(i)≦OSmax), the calculated in-catalyst state amount OS(i) value is used as it is. On the other hand, if the calculated in-catalyst state amount value exceeds the guard value OSmin (or OSmax), the calculated in-catalyst state amount OS(i) value is replaced with the guard value OSmin (or OSmax), that is, OS(i)=OSmin (or OS(i)=OSmax).  
     [0031] At that time, a fixed value may be used as the guard values OSmin and OSmax for simplification of the arithmetic processing. However, because of the saturated adsorption characteristic that the saturated adsorption amount value of the catalyst  22  is smaller as the exhaust gas flow rate flowing in the catalyst is faster (the air flow Q is increased), the guard values OSmin and OSmax may be varied according to a map corresponding to the air flow Q. In this case, it is desirable to set the guard values OSmin and OSmax so as to be smaller as the air flow Q value is larger. By setting the guard values OSmin and OSmax as above, the guard values OSmin and OSmax that conform to the actual saturated adsorption characteristic of the catalyst  22  can be set.  
     [0032] The routine proceeds to step  103  thereafter, and the deviation OSerror between the target in-catalyst state amount OSref and the current actual in-catalyst state amount OS(i) is calculated. 
       Os error= Os ref− OS ( i ) 
     [0033] A proportional gain kp, integral gain ki, and differential gain kd of a P-I-D controller are set according to a data map at the next step  104 . At that time, kp, ki and kd may be set variable depending on the engine operation condition such as intake air flow Q and intake pipe pressure P.  
     [0034] The routine proceeds to step  105  thereafter, control parameters A 1 , A 2 , B 1 , B 2 , and B 3  of the P-I-D controller are calculated according to the following equations by use of gains kp, ki, and kd, and operation interval dt (for example, the time required for rotation of 180 degrees CA). 
     A 1 =1 
     A 2 =0 
       B   1 = kp ·(1 +dt/ki+kd/dt ) 
       B   2 = kp ·(1+2 ·kd/dt ) 
     
       B 
       3 
       =kp·kd/dt 
     
     [0035] The routine proceeds to step  106  shown in FIG. 3 thereafter, and the target excess fuel factor Φref at this time is calculated as described hereinunder by use of the above parameters A 1 , A 2 , B 1 , B 2 , and B 3 , in-catalyst state amount deviation OSerror, and previous excess fuel factor Φref. At first, the target excess fuel factor correction value ΔΦref is calculated according to the following equation.  
                                      Δ φ ref   =B1 · OSerror(i) − B2 · OSerror(i-1)           +B3 · OSerror(i-2) + A1 · Φ ref(i-1)           −A2 · Φ ref(i-2)                  
 
     [0036] The target excess fuel factor correction value ΔΦref is added to the base value “1” to obtain the target excess fuel factor Φref on the upstream side of the catalyst  22 , and the program ends. 
     Φref=1+ΔΦref 
     [0037] Next, the exemplary behavior of the excess fuel factor and the in-catalyst state amount OS is described with reference to FIG. 5. For example, when the excess fuel factor on the upstream side of the catalyst  22  varies to the rich side, the rich component is adsorbed in the catalyst  22  to increase the in-catalyst state amount OS. However, after the in-catalyst state amount OS reaches the guard value OSmax on the rich side (saturated adsorbed amount on the rich side), the catalyst  22  cannot adsorb the rich component further. The calculated in-catalyst state amount OS value is subjected to guard processing by use of the guard value OSmax and the calculated in-catalyst state amount OS value is kept at the guard value OSmax. The expansion of error of the calculated in-catalyst state amount OS value is limited thereby when the catalyst  22  is saturated.  
     [0038] When the excess fuel factor varies to the lean side thereafter, because the rich component sorbed in the catalyst  22  is consumed due to oxidation/reduction reaction with the lean component in exhaust gas, the in-catalyst state amount OS begins to decrease. The in-catalyst state amount OS returns around the target in-catalyst state amount OSref thereby. As a result, the adsorption capacity of the catalyst  22  is maintained in good condition and the exhaust gas purification ratio is improved.  
     [0039] Because the in-catalyst state amount OS is calculated based on the excess fuel factor of exhaust gas and the air flow Q in the first embodiment and the position (intake pipe  12 ) where the air flow Q is detected is located far from the position (exhaust pipe  21 ) where the excess fuel factor of exhaust gas is detected, the time when air passes through the air flow detection position (intake pipe  12 ) is earlier than the time when the air reaches to the excess fuel factor detection position by the lag time during which the air passes through the air flow detection position, mixes with injected fuel, burns, and reaches to the excess fuel factor detection position. The in-catalyst state amount OS cannot be calculated correctly therefore by use of the excess fuel factor Φ(i) and the air flow Q(i) that are detected at the same time during transitional operation while the air flow Q is varying.  
     [0040] To solve such problem, the previous air flow Q(i−d) derived the lag time d earlier than the present time point i is used as the air flow to be used for calculation of the in-catalyst state amount OS in the first embodiment with consideration of the lag time d corresponding the time period from fuel injection to detection of the excess fuel factor of exhaust gas. The temporal deviation between the time when the air flow Q used for calculation of the in-catalyst state amount OS is detected and the time when the excess fuel factor used for calculation of the in-catalyst state amount OS is detected is corrected thereby, and the in-catalyst state amount OS can be calculated accurately even during the transitional operation while the air flow Q is varying.  
     [0041] The lag time from fuel injection to detection of the excess fuel factor of exhaust gas is taken into consideration in the first embodiment, but the previous air flow Q(i−d′) derived a lag time d′ earlier than the present time point i may be used with consideration of the lag time d′ corresponding to the time period from the time when air passes through the air flow detection position to the time when the air reaches to the excess fuel factor detection position. The important point is that at least the previous air flow derived earlier by the lag time corresponding to the time period from the time of fuel injection to the time when the excess fuel factor of exhaust gas is detected is used.  
     [0042] Furthermore, the control parameters A 1 , A 2 , B 2 , B 2 , and B 3  are calculated by use of the P-I-D controller in the first embodiment, but the control parameters A 1 , A 2 , B 1 , B 2 , and B 3  are calculated by use of approximate differentiation instead of the P-I-D controller in another embodiment of the present invention shown in FIG. 6. Other processing may be the same as the processing of the steps shown in FIG. 2 and FIG. 3. In the case that the control parameters A 1 , A 2 , B 1 , B 2 , and B 3  are calculated by use of approximate differentiation as described above, the same effect as obtained in the above-mentioned embodiment is also obtained.  
     [0043] The air-fuel ratio sensor  23  is provided on only the upstream side of the catalyst  22  in the exemplary system configuration shown in FIG. 1, but the present embodiment may be applied to a system in which the air-fuel ratio sensor is provided on the upstream side and the downstream side of the catalyst  22 . In this case, a system configuration in which the in-catalyst state amount is calculated based on the excess fuel factor and air flow that are detected by means of an air-fuel ratio sensor provided on the downstream side of the catalyst and the target excess factor Φ on the upstream side of the catalyst is calculated so that the deviation between the in-catalyst state amount and the target in-catalyst state amount is decreased. At that time, at least the previous air flow derived earlier by a lag time is preferably used as the air flow used for calculation of the in-catalyst state amount with consideration of the lag time from the time of fuel injection to the time of excess fuel factor detection of exhaust gas on the downstream side of the catalyst.  
     [0044] Though the excess fuel factor is used as the air-fuel ratio information in the above embodiments, as a matter of course the excess air factor Φ or air-fuel ratio A/F may be used instead of the excess fuel factor.  
     [0045] A gas concentration sensor for detecting the gas concentration of exhaust gas may be used instead of the air-fuel ratio sensor  23 . In this case, the in-catalyst state amount is calculated based on the detected gas concentration of exhaust gas and the detected air flow, and the fuel injection amount may be corrected so that the deviation between the in-catalyst state amount and the target in-catalyst state amount is decreased.  
     [0046] [Second Embodiment] 
     [0047] Next, the second embodiment of the present invention will be described hereinafter with reference to FIG. 7 to FIG. 10. In the second embodiment, as shown in FIG. 7, air-fuel ratio sensors  23  and  27  are provided both on the upstream side and downstream side of the catalyst  23 . A linear A/F sensor for generating the linear air-fuel ratio signal corresponding to the air-fuel ratio of exhaust gas as used in the first embodiment is used as the air-fuel ratio sensor on the upstream side (upstream side sensor)  23 . An oxygen sensor for generating the output voltage that is inverted depending on rich/lean ratio of the air-fuel ratio of exhaust gas is used as the air-fuel ratio sensor on the downstream side (downstream side sensor)  27 . As a matter of course, a linear A/F sensor as used as the upstream side sensor  23  may be used as the downstream side sensor  27 . Other system configuration is the same as that of the first embodiment.  
     [0048] The ECU  26  feedback-corrects the fuel injection amount so that the deviation between the target excess fuel factor Φref calculated by means of the target Φ calculation program shown in FIG. 8 and the actual fuel excess factor Φ is reduced to a small value to thereby control the in-catalyst state amount OS around the target in-catalyst state amount OSref.  
     [0049] The target Φ calculation program shown in FIG. 8 is activated every predetermined crank angle or every predetermined time interval. At step  201 , whether the output of the upstream side sensor  23  (air-fuel ratio of exhaust gas entering the catalyst  22 ) is rich or lean is determined.  
     [0050] If the output of the upstream side sensor  23  is determined to be rich, the routine proceeds to step  202 , and the in-catalyst state amount variation ΔOS(i) between the in-catalyst state amount at the time of the previous arithmetic operation and that at the time of this arithmetic operation is calculated according to the following equation. 
     Δ OS ( i )= kr ×(Φ−Φref)× Q ( i−d ) 
     [0051] Wherein kr denotes a weighing factor, Φ denotes an actual excess fuel factor detected by means of the upstream side sensor  23 , Φref denotes a target excess fuel factor, and Q(i−d) denotes a previous air flow at the time earlier than the present time i by the lag time d.  
     [0052] Herein the weighing factor kr is set corresponding to the output of the downstream side sensor  27  that is used as the substitute information of the actual in-catalyst state amount (air-fuel ratio of exhaust gas flowing out from the catalyst  22 ) as described hereinunder.  
     [0053] (1) The weighing factor kr is set to a predetermined value smaller than 1 when the output of the downstream sensor  27  (actual in-catalyst state amount) is rich. The predetermined value may be a fixed value that has been set previously, or may be set corresponding to the output of the downstream sensor  27  according to a data map or an equation.  
     [0054] (2) The weighing factor kr is set to a predetermined value larger than 1 when the output of the downstream side sensor  27  (actual in-catalyst state amount) is lean. The predetermined value may be a fixed value that has been set previously, or may be set corresponding to the output of the downstream sensor  27  according to a map or an equation.  
     [0055] (3) The weighing factor kr is set to 1 when the output of the downstream side sensor  27  (actual in-catalyst state amount) is stoichiometric. The output of the downstream side sensor  27  corresponding to kr=1 may have some range and kr is set to 1 when the output of the downstream side sensor  27  (actual in-catalyst state amount) is approximately stoichiometric.  
     [0056] On the other hand, if the output of the upstream side sensor  23  is determined to be lean at step  201 , the routine proceeds to step  203 , the in-catalyst state amount variation ΔOS(i) between the in-catalyst state amount at the time of previous arithmetic operation and that at the time of this arithmetic operation is calculated by use of the weighing factor k 1  according to the following equation. 
     Δ OS ( i )= k 1   ·(Φ−Φref)× Q ( i−d ) 
     [0057] Herein, the weighing factor k 1  is set corresponding to the output of the downstream side sensor  27  that is used as the substitute information of the actual in-catalyst state amount (air-fuel ratio of exhaust gas flowing out from the catalyst  22 ) as described hereinunder.  
     [0058] (1) The weighing factor k 1  is set to a predetermined value larger than 1 when the output of the downstream sensor  27  (actual in-catalyst state amount) is rich. The predetermined value may be a fixed value that has been set previously, or may be set corresponding to the output of the downstream sensor  27  according to a map or an equation.  
     [0059] (2) The weighing factor k 1  is set to a predetermined value smaller than 1 when the output of the downstream side sensor  27  (actual in-catalyst state amount) is lean. The predetermined value may be a fixed value that has been set previously, or may be set corresponding to the output of the downstream sensor  27  according to a map or an equation.  
     [0060] (3) The weighing factor k 1  is set to 1 when the output of the downstream side sensor  27  (actual in-catalyst state amount) is stoichiometric. The output of the downstream side sensor  27  corresponding to k 1 =1 may have some range and k 1  is set to 1 when the output of the downstream side sensor  27  (actual in-catalyst state amount) is approximately stoichiometric.  
     [0061] The steps  202  and  203  thus render the parameters (weighing factors kr and k 1 ) of the in-catalyst state amount variation ΔOS(i) arithmetic expression variable corresponding to the output of the downstream sensor  27 . Furthermore, the steps  202  and  203  also calculates the in-catalyst state amount variation Δ OS(i) with correction by use of the parameters (weighing factors kr and k 1 ).  
     [0062] The in-catalyst state amount variation ΔOS(i) is calculated at steps  202  or  203 , the routine proceeds to step  204 , and the in-catalyst state amount variation ΔOS(i) is added to the previously calculated in-catalyst state amount value OS(i−1) to thereby obtain the current in-catalyst state amount OS(i). 
       OS ( i )=Δ OS ( i )+ OS ( i− 1) 
     [0063] The routine proceeds to step  205  thereafter, and the calculated in-catalyst state amount OS(i) value is subjected to guard processing by use of the guard values OSmin and OSmax corresponding to lean side/rich side saturated adsorption amount of the catalyst  22 , respectively.  
     [0064] The routine proceeds to step  206  thereafter, and the deviation OSerror between the target in-catalyst state amount OSref and the current in-catalyst state amount OS(i) is calculated. 
       Os error= Os ref− OS (i) 
     [0065] The proportional gain, integral gain, and differential gain of the P-I-D controller are calculated according to a data map at the next step  207 , the routine proceeds to step  208  thereafter, and the control parameters A 1 , A 2 , B 1 , B 2 , and B 3  of the P-I-D controller are calculated in the same manner as used in the first embodiment by use of the gains.  
     [0066] The routine proceeds to step  209  thereafter, and the target excess fuel factor correction value ΔΦrefB is calculated by use of the control parameters A 1 , A 2 , B 1 , B 2 , and B 3  and the previous target excess fuel factor Φref according to the following equation.  
               ΔΦ                 refB     =                  B1   ·     OSerror        (   i   )         -     B2   ·     OSerror        (     i   -   1     )         +                                B3   ·     OSerror        (     i   -   2     )         +       A1   ·   Φ                     ref        (     i   -   1     )         -                                A2   ·   Φ                     ref        (     i   -   2     )                             
 
     [0067] ΔΦrefA calculation program shown in FIG. 9, which will be described hereinafter, is executed at next step  210  to thereby calculate the target excess fuel factor correction value ΔΦrefA by means of sub-feedback. The routine proceeds to step  211  thereafter, the two target excess fuel factor correction values ΔΦrefB and ΔΦrefA are added to the base value “1” to thereby set the target excess fuel factor Φref on the upstream side of the catalyst  22 , and the program is brought to an end. 
     Φref=1+ΔΦref A +Δref B   
     [0068] On the other hand, when ΔΦrefA calculation program shown in FIG. 9 is activated at the above step  210 , at first the control parameters ki and kp are calculated by means of sub-feedback according to the following equation in the above-mentioned step  210  corresponding to the deviation OSerror(i) between the target in-catalyst state amount OSref and the current in-catalyst state amount OS(i). 
       Ki=kis×OS error( i ) 
       Kp=kps×OS error( i ) 
     [0069] Wherein kis and kps denote the respective base values of the control parameters ki and kp.  
     [0070] The routine proceeds to step  302  thereafter, and whether the current output of the downstream side sensor  27  (actual in-catalyst state amount) is rich or lean is determined. If the output of the downstream side sensor  27  is determined to be rich, the routine proceeds to step  303 , and whether the output was also rich or not previously is determined. If the output was also rich previously, the routine proceeds to step  304 , and the control parameter ki is added to the previous target excess fuel factor correction value ΔΦrefA(i−1) to thereby obtain the current target excess fuel factor correction value ΔΦref(i). 
     ΔΦref A ( i )=ΔΦref A ( i− 1)+ ki   
     [0071] Otherwise, if the output of the downstream side sensor  27  was in the lean side previously and is inverted to the rich side this time, the routine proceeds to step  305 , and the control parameter kp is added to the previous target excess fuel factor correction value ΔΦrefA(i−1) to thereby obtain the current target excess fuel factor correction value ΔΦrefA(i). 
     ΔΦref A ( i )=ΔΦref A ( i− 1)+ kp   
     [0072] On the other hand, if the current output of the downstream side sensor  27  (actual in-catalyst state amount) is determined to be lean at step  302 , the routine proceeds to step  306 , and whether the output was also lean previously or not is determined. If the output was lean previously and is also lean currently, the routine proceeds to step  308 , and the control parameter ki is added to the previous target excess fuel factor correction value ΔΦrefA(i−1) to thereby obtain the current target excess fuel factor correction value ΔΦrefA(i). 
     ΔΦref A ( i )=ΔΦref A ( i− 1)+ ki   
     [0073] On the other hand, if the output of the downstream side sensor  27  was in the rich side previously and is inverted to the lean side this time, the routine proceeds to step  307 , and the control parameter kp is added to the previous target excess fuel factor correction value ΔΦrefA(i−1) to thereby obtain the current target excess fuel factor correction value ΔΦrefA(i). 
     ΔΦref A ( i )=ΔΦref A ( i− 1)+ kp   
     [0074] As described above, the target excess fuel factor correction value ΔΦrefA(i) is calculated in any of the steps  304 ,  305 ,  307 , and  308 , the routine proceeds to step  309 , the target excess fuel factor correction value ΔΦrefA(i) is subjected to guard processing by use of a proper guard value, and the program is brought to an end.  
     [0075] The transitional control characteristic of the second embodiment will be described with reference to a time chart shown in FIG. 10. The time chart shown in FIG. 10 shows the control characteristic of the second embodiment (#2) in comparison with the control characteristic of the first embodiment (#1).  
     [0076] In the second embodiment, the parameter (weighing factors kr and k 1 ) of the arithmetic expression of the in-catalyst state amount variation ΔOS(i) is rendered variable corresponding to the output of the downstream side sensor  27  (actual in-catalyst state amount) as a result of attention that the output of the downstream side sensor  27  disposed on the downstream side of the catalyst  22  (air-fuel ratio of exhaust gas flowing out from the catalyst  22 ) varies following the actual in-catalyst state amount. Thereby, the calculated in-catalyst state amount OS(i) value can be corrected successively corresponding to the output of the downstream side sensor  27  (actual in-catalyst state amount).  
     [0077] As a result, in the second embodiment, the calculation error (estimation error) of the in-catalyst state amount OS(i) can be reduced to a value smaller than that in the case of the first embodiment, and the high speed response air-fuel ratio control that follows the actual in-catalyst state amount is actualized. Thereby, the excess fuel factor on the upstream side of the catalyst and the output of the downstream side sensor  27  (actual in-catalyst state amount) are converged soon to the stoichiometric value, and the exhaust gas purification ratio can be maintained stable even in the transient condition.  
     [0078] Furthermore, in the case of the second embodiment, because the sub-feedback control parameters ki and kp that are served for reflecting the air-fuel ratio of the downstream side of the catalyst  22  (output of the downstream side sensor  27 ) on the target excess fuel factor Φref (target in-catalyst state amount OSref) is rendered variable corresponding to the output of the downstream side sensor  27  (actual in-catalyst state amount), the target excess fuel factor Φref (target in-catalyst state amount OSref) can be varied at high speed corresponding to the output of the downstream side sensor  27  (actual in-catalyst state amount).