Patent Publication Number: US-10329986-B2

Title: Model-based monitoring for selective catalytic reduction device in aftertreatment assembly

Description:
INTRODUCTION 
     The disclosure relates generally to control of operation of an aftertreatment assembly, and more particularly, to model-based monitoring of a selective catalytic reduction (SCR) device in an aftertreatment assembly. Oxides of nitrogen, referred to herein as NOx, are a by-product of the combustion process. The oxides of nitrogen are created by the disassociation of nitrogen and oxygen molecules in the high temperatures of a combustion chamber. Many devices employ aftertreatment devices, such as selective catalytic reduction (SCR) devices, to convert the oxides of nitrogen to other constituents, in the presence of a catalyst. The efficiency of the aftertreatment device may decline after a period of use. 
     SUMMARY 
     An aftertreatment assembly includes a selective catalytic reduction (SCR) device having a catalyst and configured to receive an exhaust gas. A controller is operatively connected to the SCR device. The controller having a processor and a tangible, non-transitory memory on which is recorded instructions for executing a method of model-based monitoring of the SCR device. The method relies on a physics-based model that may be implemented in a variety of forms. Execution of the instructions by the processor causes the controller: to obtain at least one estimated parameter, including obtaining an estimated nitrogen oxide (NOx) concentration in the exhaust gas exiting the SCR device. 
     The controller is configured to obtain at least one threshold parameter based at least partially on a catalyst degradation model, including obtaining a threshold NOx concentration (y T ) in the exhaust gas exiting the SCR device. The catalyst degradation model is based at least partially on a predetermined threshold storage capacity (Θ T ). A catalyst status is determined based on a comparison of the estimated parameter and the threshold parameter, via the controller. The operation of the assembly is controlled based at least partially on the catalyst status. 
     The catalyst degradation model is based at least partially on an inlet NOx concentration (u 1 ) in the exhaust gas entering the SCR device, an inlet dose (u 2 ) injected by the reductant injector, a plurality of predetermined parameters (r R , r O , r D , r A , M WNH3  and M WNOx ), a flow rate (F) of the exhaust gas received at the SCR device and a volume (V) and temperature (T) of the SCR device. The catalyst degradation model is based at least partially on respective rates of change over time (t) of an outlet NOx concentration (y 1 ) of the exhaust gas exiting the SCR device, an ammonia coverage ratio (θ), and an outlet ammonia concentration (y 2 ), respectively designated as dy 1 /dt, dθ/dt and dy 2 /dt, and calculated as: 
                 dy   1     dt     =       1   V     ⁢     (       Fu   1     -     Fy   1     -       r   R     ⁢     y   1     ⁢     Θ   T     ⁢   θ       )                       d   ⁢           ⁢   θ     dt     =             r   A     ⁢     y   2         F   ⁢           ⁢     M     WNH   3           ⁢     (     1   -   θ     )       -       r   D     ⁢   θ     -           r   R     ⁢     y   1         F   ⁢           ⁢     M     WNO   x           ⁢   θ     -       r   O     ⁢   θ                       dy   2     dt     =       1   V     ⁢     (       Fu   2     -     Fy   2     -       r   A     ⁢     Θ   T     ⁢       y   2     ⁡     (     1   -   θ     )         +       r   D     ⁢   F   ⁢           ⁢     M     WNH   3       ⁢     Θ   T     ⁢   θ       )             
In one embodiment, the catalyst degradation model is represented as:
 
     
       
         
           
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     Obtaining the estimated parameter includes: obtaining an estimated storage capacity (Θ) of the catalyst based at least partially on an extended Kalman filter and a capacity aging model. The capacity aging model is based at least partially on a sample time (k), an outlet NOx concentration (y 1 ) of the exhaust gas exiting the SCR device, an outlet ammonia concentration (y 2 ), an inlet NOx concentration (u 1 ) of the exhaust gas entering the SCR device, an inlet dose (u 2 ) injected by the reductant injector, a plurality of predetermined parameters (r R , r O , r D , r A ), and a predetermined look-up factor (K(T, F/V)) of a flow rate (F) of the exhaust gas entering the SCR device, a volume (V) and a temperature (T) of the SCR device. 
     The assembly may include an outlet NOx sensor in communication with the exhaust gas downstream of the SCR device. Obtaining the estimated parameter may include obtaining a measurement (y S ) of the outlet NOx concentration in the exhaust gas, via the outlet NOx sensor. An ammonia coverage ratio (θ) and storage capacity (Θ) is obtained based at least partially on the extended Kalman filter applied to the capacity aging model. The capacity aging model is based at least partially on the measurement (y S ) of the output NOx sensor, a catalyst NOx conversion efficiency ({circumflex over (η)}), a catalyst ammonia conversion efficiency ( ), with the capacity aging model being characterized as: 
     
       
         
           
             
               
                 
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     Obtaining the estimated parameter may include obtaining respective updated values of the catalyst NOx conversion efficiency ( ), the catalyst ammonia conversion efficiency ({circumflex over (ξ)}), the outlet NOx concentration (y 1 ) and the outlet ammonia concentration (y 2 ). The respective updated values are applied to the extended Kalman filter and the capacity aging model. The respective updated values are obtained as follows: 
     
       
         
           
             
               
                 
                   
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     The controller may be configured to send an output of the extended Kalman filter and the capacity aging model to a model predictive control (MPC) module. The MPC module is employed to obtain an optimized value of inlet dose (u 2 ). The reductant injector is commanded to inject the optimized value of the inlet dose (u 2 ). Determining the catalyst status may include comparing a first integration of the outlet NOx concentration over time (∫y 1 dt) and a second integration of the threshold NOx concentration over time (∫y T  dt). If the first integration exceeds the second integration, then a diagnostic signal may be generated by the controller (∫y 1 dt&gt;∫y T  dt). 
     Determining the catalyst status may include comparing the estimated storage capacity (Θ) and the threshold storage capacity (Θ T ). If the estimated storage capacity (Θ) falls below the threshold storage capacity (Θ T ), then a diagnostic signal may be generated by the controller (Θ&lt;Θ T ). The controller may be configured to set a first flag as true if the first integration exceeds the second integration, and set a second flag as true if the estimated storage capacity (Θ) falls below the threshold storage capacity (Θ T ). If at least one of the first flag and the second flag is true, then the controller may be configured to generate a diagnostic report. If both of the first and the second flags are true, then the controller may be configured to command the engine to reduce production of the exhaust gas. 
     The above features and advantages and other features and advantages of the present disclosure are readily apparent from the following detailed description of the best modes for carrying out the disclosure when taken in connection with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic fragmentary view of an aftertreatment assembly having a controller; 
         FIG. 2  is a flowchart for a method executable by the controller of  FIG. 1 ; 
         FIG. 3  is a schematic diagram of a control structure embodying the method of  FIG. 2 , in accordance with a first embodiment; 
         FIG. 4  is a schematic diagram of a control structure embodying the method of  FIG. 2 , in accordance with a second embodiment; and 
         FIG. 5  is a schematic diagram of a control structure embodying the method of  FIG. 2 , in accordance with a third embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Referring to the drawings, wherein like reference numbers refer to like components,  FIG. 1  schematically illustrates an aftertreatment assembly  12 , which may be part of a device  10 . The device  10  may be a mobile platform, such as, but not limited to a, standard passenger car, sport utility vehicle, light truck, heavy duty vehicle, ATV, minivan, bus, transit vehicle, bicycle, robot, farm implement, sports-related equipment, boat, plane, train or other transportation device. The device  10  may take many different forms and include multiple and/or alternate components and facilities. 
     The device  10  includes an internal combustion engine  14 , referred to herein as engine  14 . The engine  14  is configured to combust an air-fuel mixture in order to generate output torque and may include a spark-ignition engine, a compression-ignition engine, a piston-driven engine or other type of engine available to those skilled in the art. The combustion of the air-fuel mixture produces an exhaust gas  16 , which is expelled from the engine  14  to the aftertreatment assembly  12 . The assembly  12  may include an oxidation catalyst  20 , which is configured to convert nitrogen monoxide, a NOx form not easily treated in a selective catalytic reduction (SCR) device, into nitrogen dioxide, a NOx form easily treated in a selective catalytic reduction (SCR) device. 
     Referring to  FIG. 1 , a reductant injector  22  injects a reductant, such as urea, into the stream of exhaust gas  16 . The reductant may be directly sprayed into the stream of the exhaust gas  16 . A mixer device  24  may be employed for providing a substantially even distribution. The assembly  12  includes a selective catalytic reduction (SCR) device  26  having a catalyst and configured to receive the exhaust gas  16 . The SCR device  26  is configured to utilize constituents of the injected reductant to convert NOx to other constituents, as understood by those skilled in the art. 
     Referring to  FIG. 1 , an inlet NOx sensor  30  and an outlet NOx sensor  34  are configured to detect and quantify the NOx concentration in the flow of exhaust gas  16  entering and exiting SCR device  26 , respectively. An outlet ammonia sensor  36  is configured to detect and quantify the ammonia concentration in the exhaust gas flow exiting SCR device  26 . A first temperature sensor  38  and a second temperature sensor  40  are configured to detect temperature of the exhaust gas  16  entering and exiting the SCR device  26 , respectively. It should be noted that the NOx concentration and the ammonia concentration in the exhaust gas  16  and other parameters described below may be quantified in other ways, including via “virtual sensing” and modeling based on other measurements and using sensors at other locations. For example, a virtual NOx sensor modeling engine output and conditions within the exhaust gas flow may be employed to estimate the NOx concentration entering the SCR device  26 . The gas or substrate temperature inside the SCT catalyst can be estimated based on a measurement of SCR inlet and outlet temperatures  38  and  40  and ambient temperature. 
     Referring to  FIG. 1 , the assembly  12  includes a controller C operatively connected to or in electronic communication with the engine  14 . Referring to  FIG. 1 , the controller C includes at least one processor P and at least one memory M (or any non-transitory, tangible computer readable storage medium) on which are recorded instructions for executing method  100  for model-based monitoring of the SCR device  26 , shown in  FIG. 2  and described below. The method  100  relies on a number of models, including a catalyst degradation model, which is developed and calibrated to a real SCR device  26 . The memory M can store controller-executable instruction sets, and the processor P can execute the controller-executable instruction sets stored in the memory M. 
     Referring now to  FIG. 2 , a flowchart of the method  100  stored on and executable by the controller C of  FIG. 1  is shown. The method  100  is also illustrated with respect to four embodiments. The controller C of  FIG. 1  is specifically programmed to execute the steps of the method  100 . The method  100  need not be applied in the specific order recited herein. Furthermore, it is to be understood that some steps may be eliminated. 
     Referring to  FIG. 2 , method  100  may begin with block  102 , where the controller C is programmed or configured to obtain at least one estimated parameter, including obtaining an estimated nitrogen oxide (NOx) concentration in the exhaust gas  16  exiting the SCR device  26 . In block  104  of  FIG. 2 , the controller C is programmed to obtain at least one threshold parameter based at least partially on a catalyst degradation model, including obtaining a threshold NOx concentration (y T ) in the exhaust gas exiting the SCR device. 
     The catalyst degradation model is based at least partially on a predetermined threshold storage capacity (Θ T ). The threshold storage capacity (Θ T ) may be based on various statutory requirements. To determine the threshold storage capacity (Θ T ), the maximum ammonia storage capability parameter of the SCR device  26  may be reduced until its simulated NOx output (under standard urea injection control) exceeds about 1.5 times (or another factor based) of the nominal NOx output from a nominal SCR plant model or test: 
                     ∫     NOx_outlet   ⁢     (     t   ,     Θ   T       )     ⁢   dt         ∫     NOx_outlet   ⁢     (     t   ,     Θ   nominal       )     ⁢   dt              FTP     =   1.5         
The calibration may be derived from simulation during a United States Federal Test Procedure (FTP). The calibrated threshold storage capacity (Θ T ) may be based on OBD (no board diagnostics), stored in the controller C and run in parallel with the physical SCR device  26 . As a non-limiting example, the following values may be employed in one embodiment: Θ nominal =0.4298, Θ T =0.137.
 
     The catalyst degradation model is based at least partially on NOx mass flow balance and ammonia (NH3) mass flow balance through the SCR catalyst. The catalyst degradation model is based at least partially on respective rates of change over time (t) of an outlet NOx concentration (y 1 ) of the exhaust gas exiting the SCR device, an ammonia coverage ratio (θ), and an outlet ammonia concentration (y 2 ), respectively designated as dy 1 /dt, dθ/dt and dy 2 /dt, and calculated as: 
                 dy   1     dt     =       1   V     ⁢     (       Fu   1     -     Fy   1     -       r   R     ⁢     y   1     ⁢     Θ   T     ⁢   θ       )                       d   ⁢           ⁢   θ     dt     =             r   A     ⁢     y   2         FM     WNH   3         ⁢     (     1   -   θ     )       -       r   D     ⁢   θ     -           r   R     ⁢     y   1         FM     WNO   x         ⁢   θ     -       r   O     ⁢   θ                       dy   2     dt     =       1   V     ⁢     (       Fu   2     -     Fy   2     -       r   A     ⁢     Θ   T     ⁢       y   2     ⁡     (     1   -   θ     )         +       r   D     ⁢     FM     WNH   3       ⁢     Θ   T     ⁢   θ       )             
In one embodiment, the catalyst degradation model is represented as:
 
             0   =       1   V     ⁢     (       Fu   1     -     Fy   1     -       r   R     ⁢     y   1     ⁢     Θ   T     ⁢   θ       )                       d   ⁢           ⁢   θ     dt     =             r   A     ⁢     y   2         FM     WNH   3         ⁢     (     1   -   θ     )       -       r   D     ⁢   θ     -           r   R     ⁢     y   1         FM     WNO   x         ⁢   θ     -       r   O     ⁢   θ                   0   =       1   V     ⁢     (       Fu   2     -     Fy   2     -       r   A     ⁢     Θ   T     ⁢       y   2     ⁡     (     1   -   θ     )         +       r   D     ⁢     FM     WNH   3       ⁢     Θ   T     ⁢   θ       )             
As shown above, the catalyst degradation model is based at least partially on an inlet NOx concentration (u 1 ) in the exhaust gas  16  entering the SCR device  26 , an inlet dose (u 2 ) injected by the reductant injector  22 , an outlet NOx concentration (y 1 ) of the exhaust gas exiting the SCR device, and an outlet ammonia concentration (y 2 ),—a plurality of predetermined chemical reaction parameters as a function of catalyst temperature (r R , r O , r D , r A , M WNH3  and M WNOx ), a flow rate (F) of the exhaust gas  16  received at the SCR device  26  and a volume (V) of the SCR device  26 . Here r R  is a NOx reduction rate, r O  is an ammonia oxidation rate, r A  is an adsorption rate, r D  is a desorption rate, each obtained by calibration in a test cell or laboratory conditions and stored in a look-up table. The predetermined parameters (r R , r O , r D , r A ) are proportional to an exponent of (−E/RT), where T is a catalyst temperature, R is a gas constant and E is an activation energy of reduction, oxidation, desorption and adsorption, respectively. Additionally, M WNH3  and M WNOx  are molecular weights of the NOx and the ammonia in the exhaust gas  16  received at the SCR device  26 .
 
     In block  106  of  FIG. 2 , the controller C is programmed to determine a catalyst status based on a comparison of the estimated parameter(s) determined in block  102  and the threshold parameter(s) determined in block  104 . In block  108  of  FIG. 2 , the controller C is programmed to control operation of the assembly  12  based at least partially on the catalyst status. 
     The method  100  is now illustrated with three embodiments described below.  FIG. 3  is a schematic diagram of a first control structure  200  having various modules or units embodying the method  100 , in accordance with a first embodiment.  FIGS. 4 and 5  illustrate a second and third control structure  300 ,  400 , respectively, embodying the method  100 . The various modules or units of the first, second and third control structures  200 ,  300  and  400  may be embedded as part of the controller C. 
     The embodiment in  FIG. 3  includes a rear oxidation catalytic device  28  positioned downstream of the SCR device  26  and configured to eliminate ammonia slip from SCR catalyst, thus minimizing cross-contamination of ammonia in the measurements of the outlet NOx sensor  34  as well. The inlet parameters  202  (including the inlet NOx concentration (u 1 ) in the exhaust gas  16  entering the SCR device  26  and the inlet dose (u 2 ) injected by the reductant injector  22 ) are input into the Catalyst Degradation Unit  220  containing the degradation catalyst model with threshold storage capacity (Θ T ), to obtain the rates of change over time (t) of an outlet NOx concentration (y 1 ) of the exhaust gas exiting the SCR device, an ammonia coverage ratio (θ), and an outlet ammonia concentration (y 2 ), respectively designated as y 1 , dθ/dt and y 2  y 2 , and described previously. 
     Referring to  FIG. 3 , the first control structure  200  includes a Measured Integration Unit  210  configured to integrate the current measured outlet NOx concentration over time (∫y 1 dt). A Threshold Integration Unit  230  is configured to obtain a second integration of the threshold NOx concentration predicted from the Catalyst Degradation Unit  220  over time (∫y T  dt). The Comparison Module  240  is configured to compare the outputs of the Measured Integration Unit  210  and the Threshold Integration Unit  230 , per block  106  of  FIG. 2 . 
       FIG. 4  is a schematic diagram of a second control structure  300  embodying the method  100 . The inlet parameters  302  (including the inlet NOx concentration (u 1 ) in the exhaust gas  16  entering the SCR device  26  and the inlet dose (u 2 ) injected by the reductant injector  22 ) are input into the Catalyst Degradation Unit  320 , to obtain the rates of change over time (t) of an predicted outlet NOx concentration (y T1 ) from the degradation catalyst model, an ammonia coverage ratio (θ), and a predicted outlet ammonia concentration (y T2 ). 
     The embodiment in  FIG. 4  does not include a rear oxidation catalytic device positioned downstream of the SCR device  26 . Cross-contamination of ammonia in the measurements of the outlet NOx sensor  34  is accommodated by feeding the model outputs and parameters  324  obtained from the Catalyst Degradation Unit  320  into a NOx Sensor Model  322 . The model parameters  324  include a predetermined look-up factor (K) which is stored as a function of a flow rate (F) of the exhaust gas  16  entering the SCR device  26 , a volume (V) and a temperature (T) of the SCR device  26 . The Sensor Model  322  predicts the measurement (y S ) of the output NOx sensor  34  with the modeled outlet NOx concentration (y T1 ), outlet ammonia concentration (y T2 ) and the model parameters  324 . The Sensor Model  322  may be characterized as:
 
 y   S   =y   T1   +K ( T,F/V ) y   T2  
 
     Referring to  FIG. 4 , the second control structure  300  includes a Measured Integration Unit  310  configured to obtain a first integration of the measured outlet NOx concentration by the NOx sensor affected by cross-sensitivity of NH3 presence over time (∫y 1 dt). A Threshold Integration Unit  330  is configured to obtain a second integration of the threshold NOx concentration from the NOx Sensor Model  322  over time (∫y s  dt). The Comparison Module  340  is configured to compare the outputs of the Measured Integration Unit  310  and the Threshold Integration Unit  330 , per block  106  of  FIG. 2 . 
       FIG. 5  is a schematic diagram of a third control structure  400  embodying the method  100 . The inlet parameters  402  may be input into a Catalyst Degradation Unit  420 , as described previously, with the results subsequently fed into the Threshold Integration Unit  430 , as in the first and second embodiments. 
     In the embodiment shown in  FIG. 5 , obtaining the estimated parameter in block  102  includes obtaining an estimated storage capacity (Θ) of the catalyst based at least partially on a Kalman Filter Module  404  and a Capacity Aging Model Unit  406 . The inlet parameters  402  (including the inlet NOx concentration (u 1 ) in the exhaust gas  16  entering the SCR device  26  and the inlet dose (u 2 ) injected by the reductant injector  22 ) are input into a Kalman Filter Module  404  and a Capacity Aging Model Unit  406 . The Kalman Filter Module  404  is constructed to estimate the current storage capacity (Θ) of the SCR catalyst as an indication to estimate or monitor catalyst aging. When Θ is aged and its estimation reduced in size to OBD threshold Θ T  (i.e. Θ&lt;Θ T ), a malfunction of the catalyst is detected. The capacity aging model stored in the Capacity Aging Model Unit  406  is characterized as follows: 
     
       
         
           
             
               
                 
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                     y 
                     ^ 
                   
                   2 
                 
               
             
           
         
       
     
     The first equation above describes the ammonia coverage ratio (θ), which is based at least partially on a sampling or sample time (k), an estimated catalyst NOx conversion efficiency ({circumflex over (η)}) updated at the sample time k, an estimated catalyst ammonia conversion efficiency (  updated at sample time k, the estimated outlet NOx concentration (y 1 ), an outlet ammonia concentration (y 2 ), the inlet NOx concentration (u 1 ) of the exhaust gas  16  entering the SCR device  26 , the inlet dose (u 2 ) injected by the reductant injector  22  and a plurality of predetermined parameters (r R , r O , r D , r A ). Here r R  is a NOx reduction rate, r O  is a NOx oxidation rate, r A  is an adsorption rate, r D  is a desorption rate, each obtained by calibration in a test cell or laboratory conditions and stored in a look-up table. 
     The second equation above describes the change in maximum ammonia storage capacity (Θ), assuming the storage capacity changes very slowly in a catalyst lifespan, such that Θ is considered as a constant between the sample time at (k+1) and k. The third equation above relates the measurement (y S ) of the output NOx sensor  34  with the modeled outlet NOx concentration (y 1 ), outlet ammonia concentration (y 2 ), and a predetermined look-up factor (K(T, F/V)) of a flow rate (F) of the exhaust gas  16  entering the SCR device  26 , a volume (V) and a temperature (T) of the SCR device  26 . In the embodiment shown in  FIG. 5 , a measurement (y S ) of the outlet NOx concentration in the exhaust gas  16  is obtained via the outlet NOx sensor  34 , positioned downstream of the SCR device  26 . 
     Based on these three equations, the Kalman Filter Module  404  (storing a designed linear time-varying extended Kalman filter available to those skilled in the art) is applied to estimate the ammonia coverage ratio (θ) and the estimated storage capacity (Θ) at the sample time (k+1). Furthermore, respective updated values of the catalyst NOx conversion efficiency  ), the catalyst ammonia conversion efficiency ({circumflex over (ξ)}), the outlet NOx concentration (ŷ 1 ) and the outlet ammonia concentration (ŷ 2 ) may be obtained as follows from the previously estimated ammonia coverage ratio (θ(k)) and the estimated storage capacity (Θ(k)) at the sample time (k): 
     
       
         
           
             
               
                 
                   
                     y 
                     ^ 
                   
                   1 
                 
                 ⁡ 
                 
                   ( 
                   k 
                   ) 
                 
               
               = 
               
                 
                   ( 
                   
                     1 
                     - 
                     
                       
                         η 
                         ^ 
                       
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   u 
                   1 
                 
               
             
             , 
             
               
                 
                   η 
                   ^ 
                 
                 ⁡ 
                 
                   ( 
                   k 
                   ) 
                 
               
               = 
               
                 
                   
                     r 
                     R 
                   
                   ⁢ 
                   
                     Θ 
                     ^ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     θ 
                     ^ 
                   
                 
                 
                   
                     ( 
                     
                       F 
                       / 
                       V 
                     
                     ) 
                   
                   + 
                   
                     
                       r 
                       R 
                     
                     ⁢ 
                     
                       Θ 
                       ^ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       θ 
                       ^ 
                     
                   
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   ξ 
                   ^ 
                 
                 ⁡ 
                 
                   ( 
                   k 
                   ) 
                 
               
               = 
               
                 1 
                 - 
                 
                   
                     
                       y 
                       ^ 
                     
                     2 
                   
                   
                     u 
                     2 
                   
                 
               
             
             , 
             
               
                 
                   y 
                   ^ 
                 
                 2 
               
               = 
               
                 
                   
                     
                       r 
                       D 
                     
                     ⁢ 
                     
                       Θ 
                       ^ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       θ 
                       ^ 
                     
                   
                   + 
                   
                     
                       ( 
                       
                         F 
                         / 
                         V 
                       
                       ) 
                     
                     ⁢ 
                     
                       u 
                       2 
                     
                   
                 
                 
                   
                     ( 
                     
                       F 
                       / 
                       V 
                     
                     ) 
                   
                   + 
                   
                     
                       r 
                       A 
                     
                     ⁢ 
                     
                       
                         Θ 
                         ^ 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             θ 
                             ^ 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     The respective updated values may be used to update the Capacity Aging Model Unit  406 . The Kalman Filter Module  404  is executed again to estimate θ and Θ. The controller C may be configured to send the output parameters  408  of the Kalman Filter Module  404  and the Capacity Aging Model  406  to a model predictive control (MPC) module  450 . The MPC module  450  is employed to obtain an optimized value  452  of inlet dose (u 2 ) adjusted based on the current estimated ammonia coverage ratio θ and storage capacity Θ. The reductant injector  22  of  FIG. 1  is commanded to inject the optimized value  452  (obtained from the MPC Module  450 ) of the inlet dose (u 2 ). 
     The capacity aging model stored in the Capacity Aging Model Unit  406  may be extended to a general format as follows: 
     
       
         
           
             
               
                 
                   θ 
                   ⁡ 
                   
                     ( 
                     
                       k 
                       + 
                       1 
                     
                     ) 
                   
                 
                 - 
                 
                   θ 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
               
               dt 
             
             = 
             
               f 
               ⁡ 
               
                 ( 
                 
                   θ 
                   , 
                   
                     u 
                     1 
                   
                   , 
                   
                     u 
                     2 
                   
                   , 
                   T 
                   , 
                   F 
                   , 
                   Θ 
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 
                   Θ 
                   ⁡ 
                   
                     ( 
                     
                       k 
                       + 
                       1 
                     
                     ) 
                   
                 
                 - 
                 
                   Θ 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
               
               dt 
             
             = 
             0 
           
         
       
       
         
           
             
               y 
               s 
             
             = 
             
               g 
               ⁡ 
               
                 ( 
                 
                   θ 
                   , 
                   
                     u 
                     1 
                   
                   , 
                   
                     u 
                     2 
                   
                   , 
                   T 
                   , 
                   F 
                   , 
                   Θ 
                 
                 ) 
               
             
           
         
       
     
     As noted above per block  106  of  FIG. 2 , the controller C is programmed to determine a catalyst status based on a comparison of the estimated and the threshold parameter(s). In block  108  of  FIG. 2 , the controller C is programmed to control operation of the assembly  12  based at least partially on the catalyst status. Determining the catalyst status may include comparing the first integration of the outlet NOx concentration over time (from the Measured Integration Units  210 ,  310 ,  410  of  FIGS. 3-5 ) and the second integration of the threshold NOx concentration over time (from the Threshold Integration Units  230 ,  330 ,  430  of  FIGS. 3-5 ). If the first integration exceeds the second integration, then a diagnostic signal may be generated by the controller C (∫y 1 dt&gt;∫y T  dt). 
     In the embodiment of  FIG. 5 , determining the catalyst status per block  106  includes comparing the estimated storage capacity (Θ) and the threshold storage capacity (Θ T ). If the estimated storage capacity (Θ) falls below the threshold storage capacity (Θ T ), then a diagnostic signal may be generated by the controller C (Θ&lt;Θ T ). The diagnostic signal may be in a number of formats. For example, the diagnostic signal may include indicating a message on a display  42  of the device  10 . For example, the diagnostic signal may include illuminating a “check engine” light on the display  42 . The diagnostic signal may include a diagnostic report sent to a fleet manager of the device  10 . 
     The controller C may be configured to set a first flag as true if the first integration exceeds the second integration, and set a second flag as true if the estimated storage capacity (Θ) falls below the threshold storage capacity (Θ T ). If at least one of the first flag and the second flag is true, controller C may be configured to generate a diagnostic report. If both of the first and the second flags are true, controller C may be configured to command the engine  14  to reduce production of the exhaust gas  16 , for example, by shifting to a predefined operating mode with reduced speed. 
     In summary, the method  100  provides an efficient way to monitor and control the assembly  12 . In one embodiment, the threshold storage capacity (Θ T ) is set to the point where the emission from the SCR device  26  increases to 1.5 times the nominal rate. The calibrated maximum storage capability parameter may be stored in the controller C and run in parallel with the physical SCR device  26 . The estimated NOx and ammonia concentration exiting the SCR device  26  are compared with the threshold NOx and NH3 ammonia concentration simulated with the catalyst degradation model. 
     The controller C of  FIG. 1  may be an integral portion of, or a separate module operatively connected to, other controllers of the device  10 , such as the engine controller. The controller C includes a computer-readable medium (also referred to as a processor-readable medium), including any non-transitory (e.g., tangible) medium that participates in providing data (e.g., instructions) that may be read by a computer (e.g., by a processor of a computer). Such a medium may take many forms, including, but not limited to, non-volatile media and volatile media. Non-volatile media may include, for example, optical or magnetic disks and other persistent memory. Volatile media may include, for example, dynamic random access memory (DRAM), which may constitute a main memory. Such instructions may be transmitted by one or more transmission media, including coaxial cables, copper wire and fiber optics, including the wires that comprise a system bus coupled to a processor of a computer. Some forms of computer-readable media include, for example, a floppy disk, a flexible disk, hard disk, magnetic tape, any other magnetic medium, a CD-ROM, DVD, any other optical medium, punch cards, paper tape, any other physical medium with patterns of holes, a RAM, a PROM, an EPROM, a FLASH-EEPROM, any other memory chip or cartridge, or any other medium from which a computer can read. 
     Look-up tables, databases, data repositories or other data stores described herein may include various kinds of mechanisms for storing, accessing, and retrieving various kinds of data, including a hierarchical database, a set of files in a file system, an application database in a proprietary format, a relational database management system (RDBMS), etc. Each such data store may be included within a computing device employing a computer operating system such as one of those mentioned above, and may be accessed via a network in any one or more of a variety of manners. A file system may be accessible from a computer operating system, and may include files stored in various formats. An RDBMS may employ the Structured Query Language (SQL) in addition to a language for creating, storing, editing, and executing stored procedures, such as the PL/SQL language mentioned above. 
     The detailed description and the drawings or figures are supportive and descriptive of the disclosure, but the scope of the disclosure is defined solely by the claims. While some of the best modes and other embodiments for carrying out the claimed disclosure have been described in detail, various alternative designs and embodiments exist for practicing the disclosure defined in the appended claims. Furthermore, the embodiments shown in the drawings or the characteristics of various embodiments mentioned in the present description are not necessarily to be understood as embodiments independent of each other. Rather, it is possible that each of the characteristics described in one of the examples of an embodiment can be combined with one or a plurality of other desired characteristics from other embodiments, resulting in other embodiments not described in words or by reference to the drawings. Accordingly, such other embodiments fall within the framework of the scope of the appended claims.