Patent Publication Number: US-2013245919-A1

Title: Low dimensional three way catalyst model for control and diagnostics

Description:
FIELD 
     The present disclosure relates to feedback control of air-fuel ratio in an internal combustion engine. 
     BACKGROUND AND SUMMARY 
     Efficient conversion of exhaust gas emissions in a gasoline engine includes maintaining the catalyst feedgas air-fuel ratio at a narrow window around stoichiometry. However, during actual engine operation, slight excursions away from stoichiometry may occur. To increase the operating window and thus improve emissions performance, catalysts often include ceria to provide a buffer for oxygen storage. To maintain optimal catalyst performance, stored oxygen may be maintained at a desired set point, calibrated based on engine load and temperature, via feedback control of engine air-fuel ratio. 
     However, the inventors herein have recognized an issue with the above approach. Determining the level of stored oxygen in a catalyst typically involves utilization of a physics-based catalyst model that includes a plurality of partial differential equations in one or more dimensions. Such a model may be difficult to implement and may require more processing power than typically available in an engine controller. 
     Thus in one example, the above issue may be at least partly addressed by a method for an engine exhaust system. In one embodiment, the method comprises adjusting a fuel injection amount based on a fractional oxidation state of a catalyst, the fractional oxidation state based on reaction rates of a plurality of exhaust gas species throughout a catalyst longitudinal axis and a set of axially-averaged mass balance and energy balance equations for a fluid phase and a washcoat of the catalyst. 
     For example, the fractional oxidation state may be determined based on a zero-dimensional model represented by a set of ordinary differential equations. The model may track the evolution of a one or more exhaust chemical species through the catalyst. Further, the model also accounts for the diffusion within the washcoat where the reactions take place through the use of effective mass transfer concept. In this way, a simplified model may be used to predict both a total oxygen storage capacity and fractional oxidation state of the catalyst. These may be used in feedback control of the engine air-fuel ratio in order to maintain the fractional oxidation state of the catalyst at a desired amount. Further, catalyst degradation may be indicated if the catalyst activity or the total oxygen storage amount is below a threshold. 
     The present disclosure may offer several advantages. For example, processing resources devoted to the catalyst model may be reduced. Further, emissions control may be improved by maintaining the catalyst at a desired fractional oxidation state. In addition, the evolution of critical exhaust species, such as HC, NOx and CO, or aggregate oxidants and reductants, may be monitored, and if breakthrough is predicted, an operator of the vehicle may be notified and/or additional engine control operations may be undertaken to control the production of the exhaust species. Another advantage of the present approach is that it offers a non-intrusive catalyst monitor for control and diagnostics, which is less dependent on sensor location and hence will be equally applicable to both partial and full volume catalyst systems. 
     The above advantages and other advantages, and features of the present description will be readily apparent from the following Detailed Description when taken alone or in connection with the accompanying drawings. 
     It should be understood that the summary above is provided to introduce in simplified form a selection of concepts that are further described in the detailed description. It is not meant to identify key or essential features of the claimed subject matter, the scope of which is defined uniquely by the claims that follow the detailed description. Furthermore, the claimed subject matter is not limited to implementations that solve any disadvantages noted above or in any part of this disclosure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  schematically shows an example vehicle system. 
         FIG. 2  illustrates a control operation for estimating catalyst gain. 
         FIGS. 3A-3C  schematically show example diagrams of inner and outer loop control strategies. 
         FIG. 4  is a flow chart illustrating an example method for monitoring a catalyst according to an embodiment of the present disclosure. 
         FIG. 5  is a flow chart illustrating an example method for determining an oxidation state of a catalyst according to an embodiment of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     To reduce the breakthrough of emissions, catalysts may utilize oxygen storage material, for example ceria in the form of cerium oxide, to provide buffer for oxygen during rich or lean excursions. The air-fuel ratio entering the catalyst may be controlled such that the oxidation state of the catalyst is maintained at a desired level. In one example model of the present disclosure, the concentration of various exhaust gas species, such as H 2 , CO, NOx, HC, and O 2 , at the inlet through the outlet of the catalyst may be modeled using a simplified low-dimensional model. The model accounts for complex catalyst dynamics, such as diffusion and reaction in the washcoat and catalyst aging, and simplifies the dynamics into a set of axially-averaged model equations. The model equations track the balance of each exhaust species in the fluid phase and in the washcoat of the catalyst. Further, the model compensates for overall energy balance in the fluid phase and the washcoat of the catalyst. 
     In particular, the model may track the change in the concentration of oxidants and reductants in order to determine a fractional oxidation state of the catalyst, which may be used to control the air-fuel ratio of the engine. Further, a catalyst gain may be determined and applied to the model to track a change in total oxygen storage capacity, which may indicate whether or not the catalyst is degraded. Additionally, the concentration of the various exhaust components may be used to predict overall tailpipe emissions.  FIG. 1  shows an example engine including a catalyst and a control system.  FIGS. 2-5  illustrate various control routines that may be carried out by the engine of  FIG. 1 . 
       FIG. 1  shows a schematic depiction of a vehicle system  6 . The vehicle system  6  includes an engine  10  having a plurality of cylinders  30 . The engine  10  includes an intake  23  and an exhaust  25 . The intake  23  includes a throttle  62  fluidly coupled to the engine intake manifold  44  via an intake passage  42 . The exhaust  25  includes an exhaust manifold  48  leading to an exhaust passage  35  that routes exhaust gas to the atmosphere. The exhaust  25  may include one or more emission control devices  70 , which may be mounted in a close-coupled position in the exhaust. One or more emission control devices may include a three-way catalyst, lean NOx trap, diesel or gasoline particulate filter, oxidation catalyst, etc. It can be appreciated that other components may be included in the engine such as a variety of valves and sensors. 
     Engine  10  may receive fuel from a fuel system (not shown) including a fuel tank and one or more pumps for pressurizing fuel delivered to the injectors  66  of engine  10 . While only a single injector  66  is shown, additional injectors are provided for each cylinder. It can be appreciated that the fuel system may be a returnless fuel system, a return fuel system, or various other types of fuel system. The fuel tank may hold a plurality of fuel blends, including fuel with a range of alcohol concentrations, such as various gasoline-ethanol blends, including E10, E85, gasoline, etc., and combinations thereof. 
     The vehicle system  6  may further include control system  14 . Control system  14  is shown receiving information from a plurality of sensors  16  (various examples of which are described herein) and sending control signals to a plurality of actuators  81  (various examples of which are described herein). As one example, sensors  16  may include exhaust gas sensor  126  (such as a linear UEGO sensor) located upstream of the emission control device, temperature sensor  128 , and downstream exhaust gas sensor  129  (such as a binary HEGO sensor). Other sensors such as pressure, temperature, and composition sensors may be coupled to various locations in the vehicle system  6 , as discussed in more detail herein. In one example, an actuator may include a “message center” including an operation display  82  where, in response to an indication of catalyst degradation, a message may be output to a vehicle operator indicating a need to service the emission system, for example. As another example, the actuators may include fuel injector  66 , and throttle  62 . The control system  14  may include a controller  12 . The controller may receive input data from the various sensors, process the input data, and trigger the actuators in response to the processed input data based on instructions or code programmed therein corresponding to one or more routines. Example control routines are described herein with regard to  FIGS. 2-5 . 
     For catalyst diagnostics, various input parameters into a catalyst model may be used. In one embodiment, the input parameters may include catalyst gain, air amount (AM) such as mass airflow rate from MAF sensor, catalyst temperature estimated based on engine operating conditions such as speed, load, etc., HEGO output, and UEGO output. In some embodiments, all the example inputs listed above may be used in the catalyst model. In another embodiment, a HEGO model may be used in series with the catalyst model. In such a model, the model estimated voltage is compared with the measured sensor voltage (e.g., HEGO voltage), and the error computed is then used to update the catalyst activity (a c ). The catalyst activity is used as an indicative of catalyst age for diagnostics. This model-based approach is non-intrusive and less dependent on the HEGO sensor location, making it equally valid for both partial and full volume catalyst. In other embodiments, only a subset of the input parameters may be used, such as catalyst temperature and catalyst gain. 
     The catalyst gain is an on-line estimation of the oxygen storage capacity of the catalyst, which reduces as the catalyst ages, and is illustrated in  FIG. 2 . The example function of  FIG. 2  shows that the catalyst gain is a function of airmass, catalyst temperature, and relative exhaust air-fuel ratio (e.g., lambda). The catalyst gain can be indicative of catalyst conditions, such as an amount of oxygen stored in the catalyst, catalyst conversion efficiency, etc. 
       FIG. 2  illustrates an example function  200  of calculating catalyst gain from UEGO and HEGO sensor inputs. The catalyst gain may be defined as a linear, time-independent system that responds as an impulse to the inputs described above. Determining the catalyst gain relies on transfer functions (TF), which represent the relationship between the inputs and the outputs in the system. The two transfer functions (TF) are shown below in the laplace domain with s being the Laplace operator: 
     
       
         
           
             
               
                 
                   a 
                   
                     s 
                     + 
                     a 
                   
                 
               
               
                 
                   Transfer 
                    
                   
                       
                   
                    
                   function 
                    
                   
                       
                   
                    
                   1 
                    
                   
                       
                   
                    
                   
                     ( 
                     
                       TF 
                        
                       
                           
                       
                        
                       1 
                     
                     ) 
                   
                 
               
             
             
               
                 
                   
                     b 
                      
                     
                       ( 
                       s 
                       ) 
                     
                   
                   
                     conv 
                     ( 
                     
                       
                         [ 
                         xy 
                         ] 
                       
                       , 
                       
                         
                           [ 
                           xz 
                           ] 
                         
                          
                         
                           ( 
                           s 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Transfer 
                    
                   
                       
                   
                    
                   function 
                    
                   
                       
                   
                    
                   2 
                    
                   
                       
                   
                    
                   
                     ( 
                     
                       TF 
                        
                       
                           
                       
                        
                       2 
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Where w=conv(u,v) convolves vectors u and v. Algebraically, convolution is the same operation as multiplying the polynomials whose coefficients are the elements of u and v. 
     Determining the catalyst gain comprises determining the output of TF 1  using input from the HEGO sensor at  210 . This output may be fed into the output of TF 2 , as will be described in more detail below. At  212 , the difference between the UEGO sensor output and lambda (e.g. 1) is determined, and this difference is multiplied by the air mass at  214 . This product is used as the input for TF 2  at  216 . As the catalyst gain may be calculated and updated continually, the output of previous catalyst gain determinations may be fed into the function at  218 . The product of TF 2  and previous catalyst gain may be added to the output of TF 1  at  220 . At  222 , the difference is determined between the input from the HEGO sensor and the product of  220 , and this is multiplied by the output of TF 2  at  224 . To determine the catalyst gain, K, the integral is taken at  226  of the product determined in  224 . 
       FIGS. 3A-3B  are example diagrams depicting inner loop and outer loop control strategies for maintaining air-fuel ratio in an engine. Engine  10  and emission control device  70  of  FIG. 1  are non-limiting examples of engine components which may be monitored and/or controlled using the following control strategies.  FIG. 3A  depicts an example diagram  300  including an inner loop  302  and an outer loop  304 . The inner loop  302  control strategy includes a first air-fuel controller C 1   306 , which supplies a fuel command to the engine  308 . The engine produces exhaust, the oxygen concentration of which is determined by an upstream sensor, such as a UEGO  310 , before reaching a catalyst, such as TWC  312 . The outer loop  304  includes feedback from a downstream oxygen sensor, such as HEGO  314 , which is fed to a second air-fuel controller C 2   316 . Output from a catalyst gain model  318  (see  FIG. 2 ), which receives input from UEGO  310 , engine  308 , and HEGO  314 , is fed into a catalyst model  320  (see  FIG. 5 ). As will be explained in more detail below, the catalyst model determines a total oxygen storage capacity and fractional oxidation state of the catalyst. A difference may be determined between the output of C 2  and the UEGO signal at  322 , which is output as an error signal to the first controller C 1 . 
       FIG. 3B  depicts an example diagram  330  that is similar to the control strategy of diagram  300  of  FIG. 3A , except the catalyst model  320  receives input from a HEGO model  324  rather than the catalyst gain model. HEGO model  324  may be used in series with the catalyst model  320 . The HEGO model  324  compares HEGO voltage as predicted by the catalyst model  320  to measured HEGO voltage. The error computed is then used to update the catalyst activity (a c ). 
       FIG. 3C  depicts an example diagram  340  with a control strategy wherein the catalyst model  320  receives input from both the catalyst gain model  318  and the HEGO model  324 . 
       FIG. 4  is a flow chart illustrating a method  400  for monitoring a catalyst according to an embodiment of the present disclosure. Method  400  may be carried out by an engine control system, such as control system  14  of  FIG. 1 , using feedback from various engine sensors. At  402 , method  400  includes determining catalyst gain. Catalyst gain may be determined according to the process described above with respect to  FIG. 2 . At  404 , the concentration of exhaust species at the inlet of the catalyst is determined. Determining the concentration of the inlet species may include determining the concentration of one or more of O 2 , H 2 O, CO, HC, NOx, H 2 , and CO 2 . The inlet species concentrations may be determined based on one or more of air mass, temperature, air-fuel ratio, engine speed, spark timing, and load. For example, the respective species concentrations may be mapped to air mass, temperature, air-fuel ratio, and engine speed offline, and the concentrations stored in a look-up table in the memory of the control system. 
     At  406 , the catalyst gain and species concentration are input into a catalyst model. In another embodiment, a HEGO model is used to update the catalyst activity in real time instead of catalyst gain. The catalyst model includes a set of axially-averaged ordinary differential equations that calculate, for the longitudinal axis of a catalyst channel, a balance in the fluid phase of the catalyst for each species, a balance in the washcoat of the catalyst for each species, the energy balance of the fluid phase and washcoat, and the oxidation/reduction balance of ceria in the catalyst. At  408 , the total oxygen storage capacity and fractional oxidation state of the catalyst are determined from the catalyst model, which will be explained in greater detail with respect to  FIG. 5  below. At  410 , fuel injection is adjusted to maintain a desired fractional oxidation state. For example, it may be desired to maintain the fractional oxidation state of the catalyst (e.g., the fractional oxidation of ceria within the catalyst) at a desired level, calibrated based on engine load and temperature, for optimal performance, such as 50%. 
     At  412 , it is determined if the total oxygen storage capacity of the catalyst is greater than a threshold. The total oxygen storage capacity of the catalyst is indicative of the state of the catalyst, e.g., a fresh catalyst will have a relatively high oxygen storage capacity while a degraded catalyst will have a relatively low oxygen storage capacity, due to the diminished capacity of the ceria to store oxygen. The total oxygen storage capacity of a fresh catalyst may be determined based on the amount of ceria present in the catalyst during production, or it may be determined during initial operation of the catalyst. The threshold may be a suitable threshold below which the catalyst ceases to effectively control emissions. If the total oxygen storage capacity is greater than the threshold, no degradation is indicated at  414 , and then method  400  returns. If the total oxygen storage capacity is not greater than the threshold, that is if the oxygen storage capacity is less than the threshold, catalyst degradation is indicated  416 , and default action is taken. Default action may include notifying an operator of the vehicle via a malfunction indicator lamp, setting a diagnostic code, and/or adjusting engine operating parameters in order to reduce emissions production. Method  400  then returns. 
       FIG. 5  is a flow chart illustrating a method  500  for determining an oxidation state of a catalyst using a catalyst model. Method  500  may be carried out by engine control system  14  during execution of method  400  of  FIG. 4 . At  502 , the mass balance for the fluid phase of the catalyst for each species is calculated. The mass balance accounts for the transfer of species mass from the fluid phase to the washcoat. The mass balance for the fluid phase may be calculated using the following equation (1): 
     
       
         
           
             
               
                  
                 
                   X 
                   fm 
                 
               
               
                  
                 t 
               
             
             = 
             
               
                 
                   - 
                   
                     
                       〈 
                       u 
                       〉 
                     
                     L 
                   
                 
                  
                 
                   ( 
                   
                     
                       X 
                       fm 
                     
                     - 
                     
                       
                         X 
                         fm 
                         in 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   ) 
                 
               
               - 
               
                 
                   
                     K 
                     mo 
                   
                   
                     R 
                     Ω 
                   
                 
                  
                 
                   ( 
                   
                     
                       X 
                       fm 
                     
                     - 
                     
                       〈 
                       
                         X 
                         wc 
                       
                       〉 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     Where X fm  is the mole fraction of gaseous species in the bulk fluid phase, &lt;X wc &gt; is the mole fraction of the species in the washcoat, R Ω  is the hydraulic radius of the channel, &lt;u&gt; is the average feedgas velocity, L is the length of the catalyst, and K mo  is the mass transfer coefficient between the fluid and the washcoat, defined as: 
     
       
      
       K 
       mo 
       −1 
       =K 
       me 
       −1 
       +K 
       mi 
       −1  
      
     
     Here, k me  and k mi  are the external and internal mass transfer coefficients. 
     At  504 , the mass balance for the washcoat for each species, which accounts for the contribution from the mass transfer from the interface to the bulk washcoat and consumption due to the reaction, is calculated using the following equation (2): 
     
       
         
           
             
               
                 ɛ 
                 w 
               
                
               
                 
                    
                   
                     〈 
                     
                       X 
                       wc 
                     
                     〉 
                   
                 
                 
                    
                   t 
                 
               
             
             = 
             
               
                 
                   1 
                   
                     C 
                     Total 
                   
                 
                  
                 
                   v 
                   T 
                 
                  
                 r 
               
               + 
               
                 
                   
                     K 
                     mo 
                   
                   
                     δ 
                     c 
                   
                 
                  
                 
                   ( 
                   
                     
                       X 
                       fm 
                     
                     - 
                     
                       〈 
                       
                         
                           X 
                           fm 
                         
                         - 
                         
                           〈 
                           
                             X 
                             wc 
                           
                           〉 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     Where r is the reaction rate, ε w  is the porosity of the washcoat, υ represents the stoichiometric matrix, and δ c  is the washcoat thickness. 
     At  506 , the energy balance for the fluid phase is calculated, using the following equation (3): 
     
       
         
           
             
               
                 ρ 
                 f 
               
                
               
                 Cp 
                 f 
               
                
               
                 
                    
                   
                     T 
                     f 
                   
                 
                 
                    
                   t 
                 
               
             
             = 
             
               
                 
                   - 
                   
                     
                       
                         〈 
                         u 
                         〉 
                       
                        
                       
                         ρ 
                         f 
                       
                        
                       
                         Cp 
                         f 
                       
                     
                     L 
                   
                 
                  
                 
                   ( 
                   
                     
                       T 
                       f 
                     
                     - 
                     
                       
                         T 
                         f 
                         in 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   ) 
                 
               
               - 
               
                 
                   h 
                   
                     R 
                     Ω 
                   
                 
                  
                 
                   ( 
                   
                     
                       T 
                       f 
                     
                     - 
                     
                       T 
                       s 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     Where ρ f  is the average density of gas, T f  is the temperature of fluid phase, T f   in  represents the feed inlet temperature, T s  is the temperature of the solid phase, Cp f  is the specific heat capacity, and h is the heat transfer coefficient. 
     At  508 , the energy balance for the washcoat is calculated, using the equation (4): 
     
       
         
           
             
               
                 δ 
                 w 
               
                
               
                 ρ 
                 w 
               
                
               
                 Cp 
                 w 
               
                
               
                 
                    
                   
                     T 
                     s 
                   
                 
                 
                    
                   t 
                 
               
             
             = 
             
               
                 h 
                  
                 
                   ( 
                   
                     
                       T 
                       f 
                     
                     - 
                     
                       T 
                       s 
                     
                   
                   ) 
                 
               
               + 
               
                 
                   δ 
                   c 
                 
                  
                 
                   
                     ∑ 
                     i 
                     Nr 
                   
                    
                   
                     
                       r 
                       i 
                     
                      
                     
                       ( 
                       
                         
                           - 
                           Δ 
                         
                          
                         
                             
                         
                          
                         
                           H 
                           i 
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     Where δ c  is the washcoat thickness and δ w  is the effective wall thickness. 
     At  510 , the rate of oxidation of ceria is calculated using the following equation (5): 
     
       
         
           
             
               
                  
                 θ 
               
               
                  
                 t 
               
             
             = 
             
               
                 1 
                 
                   2 
                    
                   TOSC 
                 
               
                
               
                 ( 
                 
                   
                     R 
                     storage 
                   
                   + 
                   
                     R 
                     release 
                   
                 
                 ) 
               
             
           
         
       
     
     Where θ is the fractional oxidation state of ceria (FOS), 
     
       
         
           
             θ 
             = 
             
               
                 [ 
                 
                   
                     Ce 
                     2 
                   
                    
                   
                     O 
                     4 
                   
                 
                 ] 
               
               
                 
                   2 
                    
                   
                     [ 
                     
                       
                         Ce 
                         2 
                       
                        
                       
                         O 
                         4 
                       
                     
                     ] 
                   
                 
                 + 
                 
                   [ 
                   
                     
                       Ce 
                       2 
                     
                      
                     
                       O 
                       3 
                     
                   
                   ] 
                 
               
             
           
         
       
     
     The rate of storage (r 2 ), R storage  and the rate of release (r 3 ), R release  of oxygen from ceria may be based on the following equations: 
     
       
         
           
             
               r 
               2 
             
             = 
             
               
                 a 
                 c 
               
                
               
                 A 
                 2 
               
                
               
                 exp 
                  
                 
                   ( 
                   
                     
                       - 
                       
                         E 
                         2 
                       
                     
                     RT 
                   
                   ) 
                 
               
                
               
                 
                   X 
                   
                     O 
                      
                     
                         
                     
                      
                     2 
                   
                 
                  
                 
                   ( 
                   
                     1 
                     - 
                     θ 
                   
                   ) 
                 
               
                
               
                 TOSC 
                 green 
               
             
           
         
       
       
         
           
             
               r 
               3 
             
             = 
             
               
                 a 
                 c 
               
                
               
                 A 
                 3 
               
                
               
                 exp 
                  
                 
                   ( 
                   
                     
                       - 
                       
                         E 
                         3 
                       
                     
                     RT 
                   
                   ) 
                 
               
                
               
                 
                   X 
                   A 
                 
                  
                 
                   ( 
                   θ 
                   ) 
                 
               
                
               
                 TOSC 
                 green 
               
             
           
         
       
     
     Where a c  is the catalyst activity, or the aging parameter of the catalyst. The aging parameter of the catalyst is indicative of the oxygen storage state of the catalyst. For example, as the catalyst ages, its capacity to store oxygen may diminish. In one example, an aging parameter of one indicates a fresh catalyst, with decreasing aging parameters indicating decreased capacity to store oxygen. The aging parameter may be based on bulk estimates of upstream air/fuel ratio, downstream air/fuel ratio, air mass, and temperature. In some embodiments, the aging parameter may be computed from the predetermined catalyst gain, described with respect to  FIG. 2 . In another embodiment, a HEGO model is used in series with the catalyst model to estimate the downstream HEGO voltage and then, using the measured HEGO voltage, an error is computed which is used to update catalyst activity. The terms A and E indicate the pre-exponential factor and activation energy, respectively. A and E are tunable parameters which may be optimized offline, using a genetic algorithm or other non-linear constrained optimization. 
     At  512 , the fractional oxidation state (FOS) and the total oxygen storage capacity (TOSC) are determined. The FOS may be determined using the equation for θ above, and further based on the equation (6): 
     
       
         
           
             λ 
             = 
             
               
                 1 
                 
                   ( 
                   
                     2 
                     + 
                     
                       y 
                       2 
                     
                   
                   ) 
                 
               
                
               
                 
                   ( 
                   
                     
                       [ 
                       CO 
                       ] 
                     
                     + 
                     
                       [ 
                       NO 
                       ] 
                     
                     + 
                     
                       
                         2 
                          
                         
                           [ 
                           
                             CO 
                             2 
                           
                           ] 
                         
                       
                        
                       
                         [ 
                         
                           
                             H 
                             2 
                           
                            
                           O 
                         
                         ] 
                       
                     
                     + 
                     
                       2 
                        
                       
                         [ 
                         
                           O 
                           2 
                         
                         ] 
                       
                     
                   
                   ) 
                 
                 
                   ( 
                   
                     
                       [ 
                       CO 
                       ] 
                     
                     + 
                     
                       [ 
                       
                         CO 
                         2 
                       
                       ] 
                     
                     + 
                     
                       [ 
                       
                         CH 
                         y 
                       
                       ] 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     As the overall balance of the elemental species (e.g., C, H, and O) does not change (unless there is storage or release within the catalyst), the amount of change in oxygen from the inlet concentration may be attributed to a change in the ceria FOS. Further, this equation may be used to validate the model by comparing the calculated species concentrations to the measured air-fuel ratio, both upstream and downstream of the catalyst. 
     The TOSC represents the total oxygen storage capacity and as each ceria (Ce 2 O 3 ) molecule stores half a mole of oxygen, the TOSC may be equivalent to half the total ceria capacity. 
     At  514 , tailpipe emissions may be calculated, using change in the concentration of the species at the outlet of the catalyst. In some embodiments, if the emissions of the regulated species, NOx, CO, and HC, are above a threshold, engine operation may be adjusted to reduce emissions, such as increasing EGR in order to lower NOx. Upon calculating tailpipe emissions, method  500  returns. 
     Thus, the methods  400  and  500  presented above with respect to  FIGS. 4 and 5  provide for a method for an engine including a catalyst. The method comprises determining catalyst activity based on an error between predicted exhaust gas sensor output and measured exhaust gas sensor output; applying the catalyst activity and a plurality of inlet exhaust species concentrations to a catalyst model including a set of axially-averaged mass balances and energy balances of a fluid phase and washcoat of the catalyst to determine a total oxygen storage capacity and fractional oxidation state of the catalyst; maintaining a desired air-fuel ratio based on the total oxygen storage capacity and fractional oxidation state of the catalyst; and indicating catalyst degradation if the catalyst activity or total oxygen storage capacity is less than a threshold. In this way, each exhaust gas species may be input into a catalyst model, which axially averages catalyst dynamics, such as temperature, composition, etc. Based on the catalyst model, air-fuel ratio may be controlled, and catalyst degradation may be indicated. 
     While the embodiment described with respect to  FIGS. 4 and 5  calculates the mass balance for seven separate exhaust gas species (CO, HC, NOx, H 2 , H 2 O, O 2 , and CO 2 ), thus allowing monitoring of each species, in some embodiments only one or a combination of the species may be monitored. For example, rather than calculate a mass balance for each of the separate species, the species may be grouped into oxidants (e.g., O 2 , and NOx) and reductants (e.g., HC, CO, and H 2 ). Additionally or alternatively, only the change in concentration of desired regulated emissions, such as CO, HC, and NOx, may be monitored. 
     It will be appreciated that the configurations and methods disclosed herein are exemplary in nature, and that these specific embodiments are not to be considered in a limiting sense, because numerous variations are possible. For example, the above technology can be applied to V-6, I-4, I-6, V-12, opposed 4, and other engine types. The subject matter of the present disclosure includes all novel and non-obvious combinations and sub-combinations of the various systems and configurations, and other features, functions, and/or properties disclosed herein. 
     The following claims particularly point out certain combinations and sub-combinations regarded as novel and non-obvious. These claims may refer to “an” element or “a first” element or the equivalent thereof. Such claims should be understood to include incorporation of one or more such elements, neither requiring nor excluding two or more such elements. Other combinations and sub-combinations of the disclosed features, functions, elements, and/or properties may be claimed through amendment of the present claims or through presentation of new claims in this or a related application. Such claims, whether broader, narrower, equal, or different in scope to the original claims, also are regarded as included within the subject matter of the present disclosure.