Abstract:
A method for determining a condition of a catalyst disposed in an exhaust of an engine. The method includes sensing a common property of both the exhaust upstream and downstream of the catalyst; taking samples of such upstream and downstream sensed property over a period of time; accumulating the samples over the period of time; determining statistical characteristics of the sensed common property of the upstream and downstream common property; comparing the determined statistical characteristics of the sensed upstream property with the determined statistical characteristics of the downstream property to determine whether catalyst was in a proper operating condition during such period of time. With such method statistical characteristics are determined from samples as they are obtained and then once obtained, processed to determine the condition of the catalyst. Such method thereby reduces memory or data storage requirements and also reduces computational requirements.

Description:
RELATED APPLICATIONS 
     This application relates to co-pending patent application Ser. No. 09/828,020 filed Apr. 7, 2001, assigned to the same assignee as the present patent application, the entire subject matter thereof being incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     This invention relates to engine catalyst monitoring system and methods. 
     BACKGROUND 
     As is known in the art, Government regulations place strict emissions standards on automobiles. Furthermore, to ensure low emissions throughout vehicle life, automobiles must be self-diagnosing; i.e., failures in any emissions related component must be reported to the driver by illuminating a malfunction indicator light (MIL), for example. The catalytic converter (herein referred to as the catalyst), being one of the most critical emission control components on an automobile, falls under these rules. The ability of the catalyst to convert emissions into benign compounds must be monitored during engine operation. 
     Traditionally, three way catalysts (TWCs) are diagnosed based on oxygen storage since it correlates with hydrocarbon (HC) and NO x  , conversion efficiency. To measure oxygen storage, oxygen sensors are used to detect the oxygen concentration upstream and downstream of the catalyst. By analyzing the differences in the upstream and downstream concentrations of oxygen, the oxygen storage of the catalyst, and therefore its conversion efficiency, can be inferred. 
     A variety of methods have been developed and patented to analyze the oxygen sensor signals for catalyst diagnosis. Features of the sensor signals are converted into metrics, and diagnosis is performed using these metrics. Some metrics include number of rich/lean switches, amplitudes, slopes, length of line, and step responses. As vehicle emissions levels proceed from Low Emission Vehicle (LEV) to Ultra-low Emission Vehicle (ULEV) to Super Ultr-Low Emission Vehicle/Partial Zero Emission Vehicle (SULEV/PZEV), however, all these metrics have increased difficulty distinguishing between good and failed TWCs, leading to unnecessary warranty costs. Furthermore, new types of TWC materials and designs—conditioning catalysts and low oxygen storage catalysts, for example—create challenges for existing catalyst monitor techniques. 
     In summary, the automotive industry faces the problem of diagnosing new types of catalysts with more accuracy using modest computing resources and new technology has been created to meet this challenge. 
     SUMMARY 
     In accordance with the present invention, a method is provided for determining a condition of a catalyst disposed in an exhaust of an engine. The method includes sensing a common property of both the exhaust upstream and downstream of the catalyst. Samples of such upstream and downstream sensed property are taken over a period of time. The taken samples are accumulated over the period of time. Statistical characteristics of the sensed common property of the upstream and downstream common property are determined. The determined statistical characteristics of the sensed upstream property are compared with the determined statistical characteristics of the downstream property to determine whether catalyst was in a proper operating condition during such period of time. 
     With such method statistical characteristics are determined from samples as they are obtained and then once obtained, processed to determine the condition of the catalyst. Such method thereby reduces memory or data storage requirements and also reduces computational requirements. 
     In one embodiment, the common property is oxygen content in the exhaust. 
     In one embodiment one of the statistical characteristics is the mean of the samples of the upstream property and the mean of the downstream property. 
     In one embodiment another one of the statistical characteristics is the mean of the square of the samples of the upstream property and the square of the mean of the downstream property. 
     In one embodiment, another one of the statistical characteristic is the variance of the samples of the upstream property and the variance of the samples of the downstream property. 
     In one embodiment, the variance of the samples of the upstream property is determined by calculating the average of the squares of such upstream samples minus the square of the mean of such upstream samples and wherein the variance of the samples of the downstream property is determined by calculating the average of the squares of such downstream samples minus the square of the mean of such downstream samples. 
     In accordance with the invention, a method is provided for determining a condition of a catalyst disposed in an exhaust of an engine. The method includes providing oxygen sensors in the exhaust upstream and downstream of the catalyst; estimating the mean value of the upstream and downstream sensor signals, m f  and m r  respectively, in accordance with:            m   f     =         1   N            ∑     t   =   1     N                         v   f          (   t   )                     and                   m   r           =       1   N            ∑     t   =   1     N                       v   r          (   t   )               ,              respectively                          
     where v f (t) and v r (t) are the voltages of the front and rear oxygen sensor respectively at time t, and N is the number of samples over the period of time; estimating the variances and correlation coefficient, denoted by s f   2 , s r   2 , and r respectively, in accordance with:          s   f   2     =           1   N            ∑     t   =   1     N                     (       (       v   f          (   t   )       )     2     )         -       (       1   N            ∑     t   =   1     N                       v   f          (   t   )           )       2        
                
            s   r   2     =           1   N            ∑     t   =   1     N                     (       (       v   r          (   t   )       )     2     )         -       (       1   N            ∑     t   =   1     N                       v   r          (   t   )           )       2        
                
              rs   f          s   r       =         1   N            ∑     t   =   1     N                         v   f          (     t   -   T     )              v   r          (   t   )             -       (       1   N            ∑     t   =   1     N                       v   f          (     t   -   T     )           )          (       1   N            ∑     t   =   1     N                       v   r          (   t   )           )                                        
     where T is a positive constant; and determining the condition of the catalyst by comparing the differences between the estimated means of the upstream and downstream sensor signals, the difference in the estimated variances of the upstream and downstream sensor signals, and the estimated correlation coefficient. 
     The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. 
    
    
     DESCRIPTION OF DRAWINGS 
     FIG. 1 is a block diagram of an engine system having a processor for determining the condition of a catalyst used in the exhaust of such engine; and 
     FIGS. 2-6 are flow diagrams of process used by the processor of the system of FIG. 1 to determine the condition of the catalyst. 
    
    
     Like reference symbols in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     Referring now to FIG. 1, a block diagram illustrating an engine control system  10  for a representative internal combustion engine with conditioning catalyst monitor according to one embodiment of the present invention is shown in FIG.  1 . System  10  preferably includes an internal combustion engine  12  having a plurality of cylinders, represented by cylinder  14 . In one preferred embodiment, engine  12  includes eight cylinders arranged in a “V” configuration having two cylinder banks with four cylinders each. 
     As one of ordinary skill in the art will appreciate, system  10  includes various sensors and actuators to effect control of the engine. One or more sensors or actuators may be provided for each cylinder  14 , or a single sensor or actuator may be provided for the engine. For example, each cylinder  14  may include four actuators which operate corresponding intake and exhaust valves, while only including a single engine coolant temperature sensor. 
     System  10  preferably includes a controller  16  having a microprocessor  28  in communication with various computer-readable storage media, indicated generally by reference numeral  20 . The computer readable storage media preferably include a read-only memory (ROM)  22 , a random-access memory (RAM)  24 , and a keep-alive memory (KAM)  26 . As known by those of ordinary skill in the art, KAM  26  is used to store various operating variables while controller  16  is powered down but is connected to the vehicle battery. Computer-readable storage media  20  may be implemented using any of a number of known memory devices such as PROMs, EPROMs, EEPROMs, flash memory, or any other electric, magnetic, optical, or combination memory device capable of storing data, some of which represents executable instructions, used by microprocessor  28  in controlling the engine. Microprocessor  28  communicates with the various sensors and actuators via an input/output (I/O) interface  32 . Of course, the present invention could utilize more than one physical controller, such as controller  16 , to provide engine/vehicle control depending upon the particular application. 
     In operation, air passes through intake  34  where it may be distributed to the plurality of cylinders via an intake manifold, indicated generally by reference numeral  36 . System  10  preferably includes a mass airflow sensor  38  which provides a corresponding signal (MAF) to controller  16  indicative of the mass airflow. If no mass airflow sensor is present, a mass airflow value may be inferred from various engine operating parameters. A throttle valve  40  may be used to modulate the airflow through intake  34  during certain operating modes. Throttle valve  40  is preferably electronically controlled by an appropriate actuator  42  based on a corresponding throttle position signal generated by controller  16 . A throttle position sensor provides a feedback signal (TP) indicative of the actual position of throttle valve  40  to controller  16  to implement closed loop control of throttle valve  40 . 
     As illustrated in FIG. 1, a manifold absolute pressure sensor  46  may be used to provide a signal (MAP) indicative of the manifold pressure to controller  16 . Air passing through intake  34  enters the combustion chambers or cylinders  14  through appropriate control of one or more intake valves. The intake and exhaust valves may be controlled directly or indirectly by controller  16  along with ignition timing (spark) and fuel to selectively activate/deactivate one or more cylinders  12  to provide variable displacement operation. A fuel injector  48  injects an appropriate quantity of fuel in one or more injection events for the current operating mode based on a signal (FPW) generated by controller  16  processed by an appropriate driver. Control of the fuel injection events is generally based on the position of the pistons within respective cylinders  14 . Position information is acquired by an appropriate crankshaft sensor which provides a position signal (PIP) indicative of crankshaft rotational position. At the appropriate time during the combustion cycle, controller  16  generates a spark signal (SA) which is processed by ignition system  58  to control spark plug  60  and initiate combustion within an associated cylinder  14 . 
     Controller  16  (or a camshaft arrangement) controls one or more exhaust valves to exhaust the combusted air/fuel mixture of activated or running cylinders through an associated exhaust manifold, indicated generally by reference numeral  27 . Depending upon the particular engine configuration, one or more exhaust manifolds may be used. In one preferred embodiment, engine  12  includes an exhaust manifold  27  associated with each bank of cylinders as illustrated in FIG.  1 . 
     Monitoring sensors  62 A and  62 B are preferably associated with a bank of cylinders and provide a signal (EGO) indicative of the oxygen content of the exhaust gases to controller  16 . Monitoring sensors  62 A and  62 B are preferably exhaust gas oxygen sensors. The present invention is independent of the particular type of exhaust gas oxygen sensor utilized, which may depend on the particular application. In one embodiment, heated exhaust gas oxygen sensors (HEGO) are used for monitoring and feedback control as described below. Of course, various other types of air/fuel ratio sensors/indicators may be used such as a universal exhaust gas oxygen sensor (UEGO), for example. 
     The exhaust gas passes through the exhaust manifolds  27  and through associated catalysts  64 A and  64 B which act as mechanical and chemical filters by straightening the exhaust gas flow and acting as catalysts for conversion of a portion of the exhaust gases. Catalysts  64 A and  64 B are positioned upstream relative to control exhaust gas oxygen sensors  66 A and  66 B, respectively. Control sensors  66 A and  66 B may be used during closed loop control of the air/fuel ratio during certain modes of engine operation. The filtering provided by catalysts  64 A and  64 B reduces the contaminants contacting control sensors  66 A and  66 B. 
     According to one embodiment of the present invention, signals from monitoring sensor  62 A and control sensor  66 A are sampled at a predetermined interval and adjusted to reduce the effect of any difference in signal amplitudes and mean values on a subsequent comparison of the signals. The adjusted signals are then compared to determine operational efficiency of the conditioning catalyst based on the relative similarity or differences between the signal shapes. As the conversion efficiency of conditioning catalyst  64 A decreases, the adjusted signal shapes become similar provided the time delay between sensors is considered. The relative similarity or difference between signal shapes may be determined using a correlation coefficient, for example, as illustrated and described below and in the above-referenced co-pending patent application, the entire subject matter thereof being incorporated into this patent application by reference. 
     After passing through conditioning catalysts  64 A and  66 B, exhaust gases flow through an associated close-coupled catalyst  68 A,  68   b , respectively, and are combined prior to flowing through a main underbody catalyst  70 . 
     A temperature sensor  72  may be provided to monitor the temperature of a catalyst within emission control device or underbody catalyst  70 , depending upon the particular application. Alternatively, the temperature may be estimated using an appropriate temperature model based on various other sensed engine/vehicle parameters which may include mass airflow, manifold absolute pressure or load, engine speed, air temperature, engine coolant temperature, and/or engine oil temperature, for example. A representative temperature model could be developed to determine catalyst temperature for any one of the emission control devices  64 A,  64 B,  68 A,  68 B and/or  70  using various sensed and estimated engine operating parameters as described in U.S. Pat. No. 5,956,941, for example. 
     It should first be noted that catalyst  64 A is monitored in response to signals from upstream and downstream sensors  62 A and  66 A, respectively, in the same manner in which catalyst  64 B is monitored in response to signals from upstream and downstream sensors  62 B and  66 B, respectively. Thus, we will consider the signals from sensors  62 A and  66 A in evaluating the condition of catalyst  64 A, understanding the processing of signals from sensors  62 B and  66 B is the same to thereby indicated the condition of catalyst  64 B. Thus, as noted above, the two sensors  62 A,  66 A, one upstream and one downstream of the catalyst  64 A, detect the oxygen concentration in the exhaust passing through such catalyst  64 A. When the engine operates in closed loop fuel control, the upstream concentration of oxygen oscillates around stoichiometry, but due to the oxygen storage of the catalyst, the downstream concentration has a different time history. 
     These differences are quantified based on the first and second order statistics of the two signals, produced by the upstream sensor, or front  62 A and the downstream or rear sensor  66 B, respectively. In other words, the mean value and variance of each signal are computed by the processor  16 , along with the correlation coefficient between them. The diagnostic decision uses all five of these statistical quantities. If the two signals have the same mean value, and the same variance, then they are quite similar. If in addition the correlation coefficient is near +1, then the two signals are statistically the same to an engineering approximation. If the pre- and post-catalyst oxygen sensor signals are the same then the oxygen storage of the catalyst  64 A is depleted, and the conversion efficiency is degraded. The calculation methodology is efficient from both a computational and memory standpoint. Furthermore, a key computational issue is solved by a novel application of statistical identities to be described. 
     More particularly, in accordance with the invention, several novel features allowing diagnosis of a TWC, here catalyst  64 A. Because this algorithm is based on a detailed statistical analysis of the sensor signals, highly accurate decisions can be made, a critical feature at the SULEV/PZEV emissions standards. Furthermore, a simple computer, here processor  16 , can execute the algorithm quickly due to the modest computational and memory requirements. 
     Computing Statistical Quantities 
     The mean value of the front and rear sensor signals, m f  and m r  respectively, (i.e., the mean of the signals produced by the sensors  62 A and  66 A, respectively) are estimated by the well-known expressions given in Equation 1 and Equation 2.                m   f     =       1   N            ∑     t   =   1     N                       v   f          (   t   )                   Equation                 1                                              m   r     =       1   N            ∑     t   =   1     N                       v   r          (   t   )                   Equation                 2                                 
     where v f (t) and v r (t) are the voltages of the front and rear oxygen sensors  62 A and  66 A, respectively at time t, and N is the number of samples. The following notion is used: 
     
       
         
               
               
               
             
           
               
                   
                   
               
             
             
               
                   
                 s f , (s r ) 
                 Standard deviation of the front 
               
               
                   
                   
                 (rear) 
               
               
                   
                   
                 HEGO voltage 
               
               
                   
                 s f   2 , (s r   2 ) 
                 Variance of the front (rear) 
               
               
                   
                   
                 HEGO voltage 
               
               
                   
                 R 
                 Correlation coefficient 
               
               
                   
                   
                 between 
               
               
                   
                   
                 front and rear HEGO voltages 
               
               
                   
                   
               
             
          
         
       
     
     Ignoring some theoretical subtleties, the variances and correlation coefficient, denoted by s f   2 , s r   2 , and r respectively, are usually estimated by Equation 3, Equation 4, and Equation 5.                s   f   2     =       1   N            ∑     i   =   1     N                       (         v   f          (   i   )       -     m   f       )     2                 Equation                 3                                              s   r   2     =       1   N            ∑     i   =   1     N                       (         v   r          (   i   )       -     m   r       )     2                 Equation                 4                                                 rs   f          s   r       =       1   N            ∑     i   =   1     N                       (         v   r          (   i   )       -     m   r       )          (         v   f          (   i   )       -     m   f       )                   Equation                 5                                 
     Using this method to calculate the variances and correlation coefficient, however, creates a problem for implementation in the engine control computer, here processor  16 . To execute these calculations, the mean value must already have been determined, but the mean value is not known until all the data has been collected. Therefore, implementing these equations as given would require storing all the past voltages from both oxygen sensors. Once all the data is collected, the mean value can be estimated, and the variances can be estimated. If these sensors are sampled at 20 Hz in some applications, storing all these values would require about 50 kilobytes of RAM, an unreasonable requirement. 
     This invention includes a novel method to avoid the storage requirement. One key idea behind this calculation is the following identity:                s   f   2     =         1   N            ∑     t   =   1     N                     (           v   f          (   t   )       2     -     2          v   f          (   t   )            m   f       +     m   f   2       )         =         1   N            ∑     t   =   1     N                     (       (       v   f          (   t   )       )     2     )         -       (       1   N            ∑     t   =   1     N                       v   f          (   t   )           )     2                 Equation                 6                                
     This form of the equation shows that the variance of the data is equal to the average of the squares of the data minus the square of the average. By using this equation, each data point can be processed as it is acquired. Instead of requiring N memory registers to store all samples of the data, only four memory registers (i.e., accumulators) are required in the processor  16 : one, ACCUM 1 , to accumulate the sum of the voltage v f (t), one, ACCUM 2 , to accumulate the sum of the voltages v r (t), one, ACCUM 3 , to accumulate the sum of the squares of the voltage V f (t) and one, ACCUM 4 , to accumulate the sum of the squares of the voltage v r (t). 
     Another expansion, parallel to Equation 6, exists for s r   2 . 
     The product rs f s r , is expanded as shown in Equation 7.                  rs   f          s   r       =         1   N            ∑     t   =   1     N                         v   f          (   t   )              v   r          (   t   )             -       (       1   N            ∑     t   =   1     N                       v   f          (   t   )           )          (       1   N            ∑     t   =   1     N                       v   r          (   t   )           )                 Equation                 7                                
     Thus, fifth accumulator ACCUM 5  is provided in processor  16  to store the products V f (t)*V r (t) as measurements from the oxygen sensors  62 A,  66 A are made. As will be described in more detail below in connection with FIGS. 2 through 6, the values (i.e., the samples from the sensors  62 A and  66 A) can be immediately processed and intermediate values stored in the accumulators ACCUM 1 -ACCUM 5  within the processor  16 . Once all the data has been collected, the mean values have been computed, and the variances have been computed, then Equation 7 can be solved for r. 
     Finally, it should be noted that a sixth accumulator ACCUM 0  is used to keep track of the number of samples taken and thus stores N. 
     Time Delay and Oxygen Storage 
     The signals of the two oxygen sensors  62 A,  66 A have a time delay between them. This time delay arises from two sources: the time it takes the exhaust gases to travel from the first sensor  62 A to the second sensor  66 A and the oxygen storage capacity of the catalyst  64 A. As the oxygen storage degrades, the time delay shrinks. Likewise, as the speed and load of the engine change, the time delay shrinks. While the mean values and variances are insensitive to the time delay between the signals, the correlation coefficient is a strong function of the delay. The correlation coefficient as shown in Equation Equation 5 and Equation 7 is calculated with a time delay of zero; in these equations the processed voltages are both from the same time instant. Another alternative is to perform the correlation calculation at many different time delays and then use the maximum value of the correlation. Such a calculation, however, would require extensive computation and memory. 
     A third alternative is to set a fixed time delay off-line: find the time delay corresponding to a threshold catalyst system and calibrate this as a constant delay, T, for all calculations. Many methods exist to determine an appropriate value for T. In one embodiment a catalyst which has borderline emissions effectiveness is provided. This catalyst is then inserted between two oxygen sensors. A step change in the oxygen content of the exhaust, or a step change in air/fuel ratio, injected at the input of the upstream sensor. The length of time it takes for the step change to occur in the downstream sensor is the above-mentioned time delay T. 
     Using a constant value for T is shown mathematically in Equation 8 below. To eliminate the influence of speed and load on the time delay, the monitor should be run at specified speed load points such that either the impact is averaged out or this portion of the time delay is constant. In addition to reducing the computation time and memory requirements, selecting a fixed time delay in this embodiment improves the sensitivity of the correlation analysis. The improved accuracy arises from the fact that a fresh catalyst system is correlated using the time delay for a threshold system. The “wrong” time delay is used, and the resulting correlation coefficient is not the maximum possible for those two signals. As the catalyst ages, the actual time delay approaches the calibrated time delay, while simultaneously the shapes of the two signals become more similar. Both of these trends cause the correlation coefficient to increase. This single calibrated time delay is a key feature of this correlation method described in the above-referenced co-pending patent application                  rs   f          s   r       =         1   N            ∑     t   =   1     N                         v   f          (     t   -   T     )              v   r          (   t   )             -       (       1   N            ∑     t   =   1     N                       v   f          (     t   -   T     )           )          (       1   N            ∑     t   =   1     N                       v   r          (   t   )           )                 Equation                 8                                
     Using a pre-calculated time delay, T, requires a simple FIFO buffer memory in the KAM  24  to keep track of old values of one signal. If the sample rate is 20 Hz, then the buffer must hold 20T samples. Typically, T is around 2 seconds. 
     Diagnosis Using Statistical Quantities 
     In general, these five statistical quantities measure different aspects of two signals, here the two signals produced by the sensors  62 A and  66 A. If the two signals share the same mean value and the same variance, they are quite similar. If in addition the correlation coefficient between the two is near unity, then the two signals are nearly the same, statistically speaking. If the oxygen sensors  62 A and  66 A see the same signal, the oxygen storage capacity of the catalyst  64 A is depleted, and the conversion efficiency of the catalyst is low. 
     The most important quantity for diagnosis is the correlation coefficient, r. This number measures the similarity of the two waveforms. If the catalyst has large oxygen storage, then the rear oxygen sensor  66 A does not see the same oxygen concentration as the front sensor  62 A; the two sensor signals will have different shapes. If the catalyst  64 A has less oxygen storage, more “break-through” occurs and the two sensors  62 A and  66 A see more similar concentrations of oxygen; in this case the two signals have similar shape. Furthermore, the correlation coefficient is highly sensitive to the time delay or phase lag between the two signals, and this time delay is also a strong function of the oxygen storage in the catalyst. Therefore, a low correlation indicates a fresh catalyst, while a high correlation indicates a degraded catalyst. 
     The diagnostic rules flow from an understanding of these interpretations, as well as an understanding of how the fueling strategy functions. In general, these five statistical quantities measure different aspects of two signals, here the two signals produced by the sensors  62 A and  66 A. If the two signals share the same mean value and the same variance, they are quite similar. If in addition the correlation coefficient between the two is near unity, then the two signals are nearly the same, statistically speaking. If the oxygen sensors  62 A and  66 A see the same signal, the oxygen storage capacity of the catalyst  64 A is depleted, and the conversion efficiency of the catalyst is low. 
     
       
         
               
             
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Interpretation of Statistical Quantities 
               
             
          
           
               
                 Statistical Quantity 
                 Meaning 
               
               
                   
               
               
                 Mean, m f  and m r   
                 Average value of the signal 
               
               
                 Variance, s f  and s r   
                 Gives a measure of the average 
               
               
                   
                 amplitude or spread of the signal, RMS 
               
               
                   
                 value 
               
               
                 Correlation Coefficient, r 
                 Compares “shapes” while ignoring 
               
               
                   
                 mean value and amplitude, bounded 
               
               
                   
                 between +1 and −1 
               
               
                   
               
             
          
         
       
     
     While the correlation coefficient lies at the heart of catalyst diagnostics, the other four quantities, mean values and variances, also provide information about the system. The variances provide a check against a possible failure of the correlation coefficient. The correlation coefficient ignores amplitudes and mean values and only compares shapes. Therefore, small amplitude of the upstream oxygen sensor signal with shape similar to the control HEGO signal would falsely indicate a failed catalyst. By comparing the variances and mean values of the signals, however, this case can be easily detected. Since the variance of a signal is a measure of its amplitude, catalyst failure should only be declared when the variance of the front and rear sensors are roughly equal. Requiring the mean values to be equal adds further robustness. 
     This logic is summarized in Table 2 below. By utilizing all five pieces of information about the signals, the catalyst diagnosis is more accurate and robust than if only a single piece of information is used. While Table 2 indicates that the checks are made against unity, clearly this number can be a calibrated constant. 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Diagnostic Inference Table for Catalyst 
               
             
          
           
               
                   
                   
                 Inferred Catalyst 
               
               
                   
                 Statistical Quantities 
                 Condition 
               
               
                   
                   
               
               
                   
                 (r = 1) and (s r /s f  = 1) and (m r  = 
                 Catalyst Failed 
               
               
                   
                 m f ) 
               
               
                   
                 (r small) or (s r /s f  small) or 
                 Catalyst OK, other failure 
               
               
                   
                 (m r  ≠ m f ) 
                 possible 
               
               
                   
                   
               
             
          
         
       
     
     This method of monitoring TWCs can be used for a variety of different applications, including the daunting task of monitoring a TWC at PZEV levels. Other applications for this technology include: low oxygen storage catalysts; lean burn applications; ACCRO (zoned catalysts); and LEVII NO x  monitoring, for example. 
     Referring now to FIGS. 2 through 6 flow diagrams are shown which execute code stored in the processor  16 . Thus, referring to FIG. 2, the catalyst monitor begins at  300 . When the closed loop fuelling is being controlled, i.e.,  302 , the upstream concentration of oxygen, as monitored by sensor  62 A (FIG. 1) oscillates around stoichiometry. Once operating around stoichiometry, variable in the process are initialized in Step  304  as shown in FIG.  3 . i.e., the accumulators ACCUM 0 -ACCUM 5  for the mean, standard deviation and correlation are set to zero. In Step  306  samples of V f (t) are stored for T second (T is the same as the above-described time delay T.) In Step  308 , the sensor voltages V f  and V r  are sampled. The sampled voltages V f (t) and V r (t) are fed to accumulators ACCUM 1  and ACCUM 2 , respectively as shown in FIG.  4 . The sampled voltages V f (t) and V r (t) are each squared to form V f (t) 2  and V r (t) 2 , respectively, and such squared voltages V f (t) 2  and V r (t) 2  are fed to accumulators ACCUM 3  and ACCUM 4 , respectively as shown in FIG.  4 . Further, the product of the sampled voltages V f (t−T), taken from the FIFO buffer in KAM  24 , and V r (t) (i.e., V f (t−T)*V r (t) are fed to ACCUM 5 , as shown in FIG.  4 . 
     Once the accumulation is performed for the T the predetermined number of samples N (Step  312 ), the statistics described above in connection with Equations 6 and 7 are performed by the processor  16  (Step  314 ), as shown in FIG.  5 . The number of samples can be chosen by many methods. In this embodiment, for example, the number of samples, N, is selected to ensure acceptable estimates of all quantities. That is, after this data acquisition process (Step  312 ), the statistics are calculated in accordance with the equations presented above (Step  314 ). Thus, reference is made to FIG.  5 . As shown therein, the accumulated voltage in ACCUM 1  is divided by the number of samples and the result is stored in the RAM  24  (FIG. 1) at a specified location herein designated as location A. Thus, location A stores the mean of Vf(t), m f , as shown in FIG.  5 . 
     The accumulated voltage in ACCUM 2  is divided by the number of samples (i.e., N stored in ACCUM 0 ) and the result is stored in the RAM  24  (FIG. 1) at a specified location herein designated as location B. Thus, location B stores the mean of Vr(t), m r , as shown in FIG.  5 . 
     The accumulated voltage in ACCUM 3  is divided by the number of samples and this has subtracted from it the square of the value at location A of RAM  24  (i.e., m f   2 ). The result is the variance of Vf(t), i.e., S f . The result S f  is stored in the RAM  24  (FIG. 1) at a specified location herein designated as location C, as shown in FIG.  5 . 
     The accumulated voltage in ACCUM 4  is divided by the number of samples and this has subtracted from it the square of the value at location B of RAM  24  (i.e., m r   2 ). The result is the variance of Vr(t), i.e., S r . The result S r  is stored in the RAM  24  (FIG. 1) at a specified location herein designated as location D, as shown in FIG.  5 . 
     The accumulated voltage in ACCUM 5  is divided by the number of samples and this has subtracted from it the product of the value stored in location A of RAM  24  (i.e., m f ) and the value stored in location B of RAM  24  (i.e., m r ) to thereby form rS f S r  in accordance with Equation 8 above, i.e., the square of the value at location B of RAM  24  (i.e., m r   2 ). The result, rS f S r  is stored in RAM  24  at location E. 
     To determine r, the result rS f S r  stored in location E is divided by the square root of the product of the data, S f , stored in location C and the data, S r , stored in location D. 
     Having the statistics from Step  314 , the diagnosis described above in Table 2 are performed in Step  316  as shown in FIG.  6 . Thus, in Step  700  a determination is made whether the difference in the mean of V f (t) and the mean of V(t) r  is less than some a priori established threshold thr_m. If greater than the threshold the catalyst is determined to be OK (Step  705 ); otherwise a test is made in Step  702  to determine if the difference in the variances of Vf(t) and V(t) f  is less than some predetermined threshold, thr_s. If the difference is greater than the threshold the catalyst is determined to be OK (Step  705 ); if not, the deviation of the correlation coefficient r between the front sensor  62 A V f (t) and rear sensor  66 A from 1 is compared with a predetermined threshold thr_r (Step  703 ). If it is greater than this threshold the catalyst is OK (Step  705 ); otherwise, the operator of the vehicle is advised that the catalyst has failed (Step  704 ). 
     A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. Accordingly, other embodiments are within the scope of the following claims.