Patent Publication Number: US-8109079-B2

Title: Apparatus, system, and method for controlling ammonia slip from an SCR catalyst

Description:
FIELD 
     This disclosure relates to controlling nitrogen oxides (NO x ) emissions for internal combustion engines, and more particularly to apparatus, systems and methods for controlling ammonia slip from a selective catalytic reduction (SCR) catalyst. 
     BACKGROUND 
     Emissions regulations for internal combustion engines have become more stringent over recent years. The regulated emissions of NO x  and particulates from internal combustion engines are low enough that in many cases the emissions levels cannot be met with improved combustion technologies. Therefore, the use of aftertreatment systems on engines to reduce emissions is increasing. For reducing NO x  emissions, NO x  reduction catalysts, including selective catalytic reduction (SCR) systems, are utilized to convert NO x  (NO and NO 2  in some fraction) to N 2  and other compounds. SCR systems utilize a reductant, typically ammonia, to reduce the NO x . Currently available SCR systems can produce high NO x  conversion rates allowing the combustion technologies to focus on power and efficiency. However, currently available SCR systems also suffer from a few drawbacks. 
     SCR systems generate ammonia to reduce the NO x . When just the proper amount of ammonia is available at the SCR catalyst under the proper conditions, the ammonia is utilized to reduce NO x . However, if the reduction reaction rate is too slow, or if there is excess ammonia in the exhaust, ammonia can slip out the exhaust pipe. Ammonia is an extreme irritant and an undesirable emission. Accordingly, slips of even a few tens of ppm are problematic. Additionally, due to the undesirability of handling pure ammonia, many systems utilize an alternate compound such as urea, that vaporizes and decomposes to ammonia in the exhaust stream. Presently available SCR systems treat injected urea as injected ammonia, and do not account for the vaporization and hydrolysis of urea to component compounds such as ammonia and isocyanic acid. As a result, the urea can decompose to ammonia downstream of the SCR causing ammonia slip, and less ammonia may be available for NO x  reduction than the control mechanism estimates causing higher NO x  emissions at the tailpipe. 
     SCR systems that utilize urea dosing to generate ammonia depend upon the real-time delivery of urea to the SCR catalyst as engine NO x  emissions emerge. Urea dosers have relatively slow physical dynamics compared to other chemical injectors such as hydrocarbon injectors. Therefore, urea doser dynamics can substantially affect an SCR controls system. 
     Some currently available SCR systems account for the dynamics of the urea dosing and the generally fast transient nature of the internal combustion engine by utilizing the inherent ammonia storage capacity of many SCR catalyst formulations. 
     One currently available method introduces a time delay at the beginning of an engine NO x  spike before urea dosing begins (or ramps up), and a time delay after the NO x  spike before urea dosing ends (or ramps down). Ordinarily, an engine NO x  spike will cause a temperature increase in the exhaust gas and SCR catalyst, which may result in the release of stored ammonia on the catalyst. This is especially true when engine power output is used as a substitute for directly estimating engine NO x  emissions. The ammonia release provides ammonia for reducing engine out NO x  while delaying urea injection prevents excess ammonia from slipping out the exhaust. On the NO x  decrease, normally the temperature of the engine exhaust and SCR catalyst decrease, and therefore continued urea injection (the delay before ramping down urea injection) provides ammonia to store on the SCR catalyst and recharge the catalyst. 
     In many ordinary circumstances, the time delay method causes desirable results in the SCR catalyst. However, in some cases the time delay method can produce undesirable results and even responses that are opposite from an optimal response. For example, a decrease in EGR fraction for any reason causes an engine out NO x  spike with a decrease in exhaust temperature. In a time delay system utilizing engine-out power as a substitute for NO x  emissions, the change will likely be ignored and a standard amount of injected urea will cause an increase in NO x  emissions. In a time delay system that recognizes the engine out NO x  spike, the system delays injecting ammonia-creating urea. Because the temperature on the SCR catalyst is relatively lower, the amount of NO x -reducing ammonia released from the catalyst is reduced, which results in a NO x  emissions increase. At the end of the NO x  spike event, the exhaust temperature increases (from restoration of the designed EGR fraction) while the NO x  emissions decreases. The SCR catalyst ejects ammonia from the reduced storage capacity while the urea injector continues to add ammonia to the system without NO x  available for reduction. Therefore, the system can slip significant amounts of ammonia on the down cycle. 
     Other currently available systems determine whether the SCR catalyst is at an ammonia storing (adsorption) or ammonia ejecting (desorption) temperature. When the SCR catalyst is storing ammonia, the system injects urea until the catalyst is full. When the SCR catalyst is ejecting ammonia, the system halts injection and allows stored ammonia to release and reduce NO x . 
     Presently available systems tracking the SCR catalyst temperature suffer from a few drawbacks. For example, the amount of ammonia stored on the SCR catalyst varies with temperature. However, presently available systems assume a storage amount below a specified temperature, and zero storage above the specified temperature. Therefore, the controls may toggle significantly around the specified temperature, significantly overestimate ammonia storage capacity just below the specified temperature, and significantly underestimate ammonia storage capacity just above the specified temperature. Such systems utilize the “normalized stoichiometric ratio” (NSR) to determine baseline urea injection, but do not account for variances in the NO x  composition and NH 3  to isocyanic acid ratio of the urea when determining the NSR. Further, such systems do not account for the incomplete vaporization and hydrolysis of urea that occurs in many systems and may therefore not inject sufficient urea to reduce NO x  and/or provide the desired ammonia for storage. 
     Also, many known SCR systems do not utilize an ammonia oxidation (AMOX) catalyst downstream of the SCR catalyst to convert at least some ammonia slipping from the SCR catalyst to N 2  and other less harmful compounds. For those conventional SCR systems that do employ an AMOX catalyst, the operating conditions and conversion capability of the AMOX catalyst are not factored into the reductant dosing rate, ammonia storage control, ammonia slippage control, and NO x  conversion efficiency feedback of such systems. 
     SUMMARY 
     The subject matter of the present application has been developed in response to the present state of the art, and in particular, in response to the problems and needs in the art that have not yet been fully solved by currently available exhaust aftertreatment systems. Accordingly, the subject matter of the present application has been developed to provide apparatus, systems, and methods for controlling ammonia slip from an SCR catalyst that overcomes at least some shortcomings of the prior art aftertreatment systems. 
     For example, according to one representative embodiment, an apparatus for controlling ammonia slip from an SCR catalyst can include various modules. The SCR catalyst is in exhaust receiving communication with an exhaust gas stream produced by an engine system. The exhaust gas stream flows from an engine of the engine system to a tailpipe of the engine system. 
     The apparatus includes a NO x  reduction target module that is configured to determine a NO x  reduction requirement. The NO x  reduction requirement includes an amount of NO x  in the exhaust gas stream that is to be reduced on the SCR catalyst. The apparatus also includes an ammonia target module that is configured to determine an ammonia addition requirement. The ammonia addition requirement includes an amount of ammonia added to the exhaust gas stream to achieve the NO x  reduction requirement. The apparatus further includes a reductant target module that is configured to determine a reductant injection requirement. The reductant injection requirement includes an amount of reductant added to the exhaust gas stream to achieve the ammonia addition requirement. 
     The apparatus additionally includes a reductant modifier module that is configured to determine a reductant modifier requirement. The reductant target module is also configured to adjust the amount of reductant added to the exhaust gas stream in response to the reductant modifier requirement. The apparatus also includes an ammonia oxidation (AMOX) conversion module that is configured to determine a conversion capability of an AMOX catalyst in exhaust receiving communication with the SCR catalyst. The reductant modifier module is communicable in data receiving communication with the AMOX conversion module to receive the conversion capability of the AMOX catalyst. Further, the reductant modifier requirement is based at least partially on the conversion capability of the AMOX catalyst. 
     According to some implementations of the apparatus, the AMOX conversion module is configured to determine an amount of ammonia slipping from the tailpipe. 
     In yet some implementations, the AMOX conversion module includes an AMOX catalyst bed temperature module, an AMOX catalyst degradation module, and/or a tailpipe ammonia slip target module. The AMOX catalyst bed temperature module can be configured to estimate a temperature of a catalyst bed of the AMOX catalyst. The AMOX catalyst degradation module can be configured to determine an AMOX catalyst degradation factor. The tailpipe ammonia slip target module can be configured to determine a tailpipe ammonia slip target. In certain implementations, the conversion capability is a function of the estimated temperature of the AMOX catalyst bed, the AMOX catalyst degradation factor, and/or the tailpipe ammonia slip target. In some instances, the tailpipe ammonia slip target includes a value selected from the group consisting of an ammonia slip target average value and an ammonia slip target maximum value. 
     According to some implementations, the conversion capability is a function of the flow rate of the exhaust gas stream, the amount of ammonia entering the AMOX catalyst, a desired amount of ammonia exiting the AMOX catalyst, the temperature of AMOX catalyst bed, the amount of NO x  entering the AMOX catalyst, a ratio of NO to NO 2  in the exhaust gas entering the AMOX catalyst, and a catalyst degradation factor. 
     In some implementations, the AMOX conversion module is configured to determine a thermal mass of the AMOX catalyst. The apparatus can further include an ammonia storage module that is configured to determine an ammonia storage condition of the SCR catalyst. The ammonia addition requirement can be a function of the ammonia storage condition of the SCR catalyst. In certain instances, the ammonia storage condition is a function of the thermal mass of the AMOX catalyst. 
     According to some embodiments, the apparatus can also include a corrected tailpipe NO x  module that is configured to determine a corrected tailpipe NO x  sensor signal based at least partially on input from a tailpipe NO x  sensor and the condition of the tailpipe NO x  sensor. The measured amount of NO x  exiting the tailpipe can be interpreted from the corrected tailpipe NO x  sensor signal. In certain instances, the corrected tailpipe NO x  sensor signal is a function of the conversion capability. 
     According to another embodiment, a computer program product that includes a computer readable medium with computer usable program code executable to perform operations for controlling ammonia conversion on an AMOX catalyst of an SCR system. The SCR system is coupled to an internal combustion engine and operable to inject urea into an exhaust gas stream. The urea reduces to ammonia in the exhaust gas stream and the ammonia facilitates a reduction of NOx emissions in the exhaust gas stream on an SCR catalyst positioned upstream of the AMOX catalyst. The operations of the computer program product can include determining an ammonia conversion capability of the AMOX catalyst and adjusting the amount of urea injected into the exhaust gas stream in response to the determined ammonia conversion capability. 
     In some implementations of the computer program product, determining the ammonia conversion capability of the AMOX catalyst can include determining the temperature of a catalyst bed of the AMOX catalyst, determining a desired amount of ammonia slip from the AMOX catalyst, and determining an AMOX catalyst degradation factor. The ammonia conversion capability can be a function of the determined temperature of the AMOX catalyst bed, the desired amount of ammonia slip from the AMOX catalyst, and/or the AMOX catalyst degradation factor. 
     According to some implementations, the computer program product can further include determining an amount of ammonia slipping from the AMOX catalyst and adjusting the amount of urea injected into the exhaust gas stream in response to the determined amount of ammonia slipping from the AMOX catalyst. Similarly, in some implementations, the computer program can also include determining a thermal mass of the AMOX catalyst and adjusting the amount of urea injected into the exhaust gas stream in response to the determined thermal mass of the AMOX catalyst. 
     According to yet another embodiment, a method for controlling ammonia slip from a selective catalytic reduction (SCR) catalyst in exhaust receiving communication with an exhaust gas stream can include determining a NO x  reduction requirement, an ammonia addition requirement, a reductant injection requirement, an AMOX catalyst conversion capability, and a reductant limiting requirement. The NO x  reduction requirement can include an amount of NO x  in the exhaust gas stream to be reduced on a selected catalytic reduction (SCR) catalyst. The ammonia addition requirement can include an amount of ammonia added to the exhaust gas stream to achieve the NO x  reduction requirement. The reductant injection requirement can include an amount of reductant added to the exhaust gas stream to achieve the ammonia addition requirement. The reductant limiting requirement can be a function of the AMOX catalyst conversion capability. The method can also include limiting the amount of reductant added to the exhaust gas stream in response to the reductant limiting requirement. 
     In certain implementations, determining the AMOX catalyst conversion capability can include determining an estimated temperature of a catalyst bed of the AMOX catalyst, determining a tailpipe ammonia slip target, and determining an AMOX catalyst degradation factor. 
     The method can, in some implementations, include determining an AMOX catalyst thermal mass, wherein the ammonia addition requirement is a function of the AMOX catalyst thermal mass. 
     According to another embodiment, a system for controlling ammonia slip in an engine system that has a SCR system can include an internal combustion engine that produces an exhaust gas stream. The system also includes a SCR catalyst operable to reduce NO x  emissions in the exhaust gas stream in the presence of a reductant and a reductant injector that injects reductant into the exhaust gas stream upstream of the SCR catalyst. The system includes an AMOX catalyst downstream of the SCR catalyst. The AMOX catalyst is operable to reduce ammonia emissions in the exhaust gas stream. Additionally, the system includes a controller with an AMOX conversion module that is operable to determine an ammonia conversion capability of the AMOX catalyst. The controller is also communicable in electronic communication with the reductant injector to control the amount of reductant injected into the exhaust gas stream. The amount of reductant injected into the exhaust gas stream is a function of the ammonia conversion capability of the AMOX catalyst. 
     According to some implementations, the ammonia conversion capability is a function of a temperature of a bed of the AMOX catalyst, a degradation factor of the AMOX catalyst, and a desired tailpipe ammonia slip amount. In some instances, the ammonia conversion capability can also be a function of a flow rate of the exhaust gas stream, an amount of ammonia entering the AMOX catalyst, an amount of NO x  entering the AMOX catalyst, and a ratio of NO to NO 2  in the exhaust gas entering the AMOX catalyst. 
     In certain implementations, the AMOX conversion module of the system is operable to determine a thermal mass of the AMOX catalyst. The amount of reductant injected into the exhaust gas stream can be a function of the thermal mass of the AMOX catalyst. 
     Reference throughout this specification to features, advantages, or similar language does not imply that all of the features and advantages that may be realized with the subject matter of the present disclosure should be or are in any single embodiment. Rather, language referring to the features and advantages is understood to mean that a specific feature, advantage, or characteristic described in connection with an embodiment is included in at least one embodiment of the present disclosure. Thus, discussion of the features and advantages, and similar language, throughout this specification may, but do not necessarily, refer to the same embodiment. 
     Furthermore, the described features, advantages, and characteristics of the subject matter of the present disclosure may be combined in any suitable manner in one or more embodiments. One skilled in the relevant art will recognize that the subject matter may be practiced without one or more of the specific features or advantages of a particular embodiment. In other instances, additional features and advantages may be recognized in certain embodiments that may not be present in all embodiments. These features and advantages will become more fully apparent from the following description and appended claims, or may be learned by the practice of the subject matter as set forth hereinafter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In order that the advantages of the subject matter may be more readily understood, a more particular description of the subject matter briefly described above will be rendered by reference to specific embodiments that are illustrated in the appended drawings. Understanding that these drawings depict only typical embodiments of the subject matter and are not therefore to be considered to be limiting of its scope, the subject matter will be described and explained with additional specificity and detail through the use of the drawings, in which: 
         FIG. 1  is a schematic block diagram of an internal combustion engine having an exhaust after-treatment system according to one representative embodiment; 
         FIG. 2  is a schematic block diagram of the exhaust after-treatment system of  FIG. 1  according to one representative embodiment; 
         FIG. 3  is a schematic block diagram of a controller of the exhaust after-treatment system of  FIG. 2  according to one representative embodiment; 
         FIG. 4  is a schematic block diagram of a NO x  reduction target module of the controller of  FIG. 3  according to one representative embodiment; 
         FIG. 5A  is a schematic block diagram of a feedforward ammonia target module of the controller of  FIG. 3  according to one representative embodiment; 
         FIG. 5B  is a schematic block diagram of a feedback ammonia target module of the controller of  FIG. 3  according to one representative embodiment; 
         FIG. 6  is a schematic block diagram of a reductant target module of the controller of  FIG. 3  according to one representative embodiment; 
         FIG. 7  is a schematic block diagram of a reductant hydrolysis module of the reductant target module of  FIG. 6  according to one representative embodiment; 
         FIG. 8  is a schematic block diagram of an inverse reductant hydrolysis module of the reductant target module of  FIG. 6  according to one representative embodiment; 
         FIG. 9  is a schematic flow chart diagram of a control system operable to determine ammonia and isocyanic acid flow into an SCR catalyst according to one embodiment; 
         FIG. 10  is a schematic block diagram of an ammonia storage module of the controller of  FIG. 3  according to one representative embodiment; 
         FIG. 11  is a schematic block diagram of a current ammonia storage level module of the ammonia storage module of  FIG. 10  according to one representative embodiment; 
         FIG. 12  is a schematic flow chart diagram of a control system operable to determine the storage level of ammonia on an SCR catalyst; 
         FIG. 13  is a schematic flow chart diagram of a control system operable to determine the amount of ammonia slip from an SCR catalyst; 
         FIG. 14  is a schematic block diagram of an AMOX catalyst ammonia conversion module of the controller of  FIG. 3  according to one representative embodiment; 
         FIG. 15  is a schematic block diagram of a reductant modifier module of the controller of  FIG. 3  according to one representative embodiment; 
         FIG. 16  is a schematic block diagram of a corrected tailpipe NO x  module of the reductant modifier module of  FIG. 15  according to one representative embodiment; and 
         FIG. 17  is a method of reducing NO x  emissions using ammonia storage on an SCR catalyst. 
     
    
    
     DETAILED DESCRIPTION 
     Many of the functional units described in this specification have been labeled as modules, in order to more particularly emphasize their implementation independence. For example, a module may be implemented as a hardware circuit comprising custom VLSI circuits or gate arrays, off-the-shelf semiconductors such as logic chips, transistors, or other discrete components. A module may also be implemented in programmable hardware devices such as field programmable gate arrays, programmable array logic, programmable logic devices or the like. 
     Modules may also be implemented in software for execution by various types of processors. An identified module of executable code may, for instance, comprise one or more physical or logical blocks of computer instructions, which may, for instance, be organized as an object, procedure, or function. Nevertheless, the executables of an identified module need not be physically located together, but may comprise disparate instructions stored in different locations which, when joined logically together, comprise the module and achieve the stated purpose for the module. 
     Indeed, a module of executable code may be a single instruction, or many instructions, and may even be distributed over several different code segments, among different programs, and across several memory devices. Similarly, operational data may be identified and illustrated herein within modules, and may be embodied in any suitable form and organized within any suitable type of data structure. The operational data may be collected as a single data set, or may be distributed over different locations including over different storage devices, and may exist, at least partially, merely as electronic signals on a system or network. 
     Reference throughout this specification to “one embodiment,” “an embodiment,” or similar language means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the present invention. Thus, appearances of the phrases “in one embodiment,” “in an embodiment,” and similar language throughout this specification may, but do not necessarily, all refer to the same embodiment. 
     Furthermore, the described features, structures, or characteristics of the subject matter described herein may be combined in any suitable manner in one or more embodiments. In the following description, numerous specific details are provided, such as examples of controls, structures, algorithms, programming, software modules, user selections, network transactions, database queries, database structures, hardware modules, hardware circuits, hardware chips, etc., to provide a thorough understanding of embodiments of the subject matter. One skilled in the relevant art will recognize, however, that the subject matter may be practiced without one or more of the specific details, or with other methods, components, materials, and so forth. In other instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of the disclosed subject matter. 
     Internal Combustion Engine System 
       FIG. 1  depicts one embodiment of an internal combustion engine system  10 . The main components of the engine system  10  include an internal combustion engine  11  and an exhaust gas after-treatment system  100  coupled to the engine. The internal combustion engine  11  can be a compression ignited internal combustion engine, such as a diesel fueled engine, or a spark-ignited internal combustion engine, such as a gasoline fueled engine operated lean. The engine system  10  further includes an air inlet  12 , intake manifold  14 , exhaust manifold  16 , turbocharger turbine  18 , turbocharger compressor  20 , temperature sensors (e.g., temperature sensor  24 ), pressure sensors (e.g., pressure sensor  26 ), and air-flow sensor  56 . The air inlet  12  is vented to the atmosphere and connected to an inlet of the intake manifold  14  to enable air to enter the intake manifold. The intake manifold  14  includes an outlet operatively coupled to compression chambers of the internal combustion engine  11  for introducing air into the compression chambers. 
     Within the internal combustion engine  11 , the air from the atmosphere is combined with fuel to power the engine. Combustion of the fuel and air produces exhaust gas that is operatively vented to the exhaust manifold  16 . From the exhaust manifold  16 , a portion of the exhaust gas may be used to power the turbocharger turbine  18 . The turbine  18  drives the turbocharger compressor  20 , which may compress at least some of the air entering the air inlet  12  before directing it to the intake manifold  14  and into the compression chambers of the engine  11 . 
     The exhaust gas after-treatment system  100  is coupled to the exhaust manifold  16  of the engine  11 . At least a portion of the exhaust gas exiting the exhaust manifold  16  can pass through the exhaust after-treatment system  100 . In certain implementations, the engine system  10  includes an exhaust gas recirculation (EGR) valve (not shown) configured to open to allow a portion of the exhaust gas to recirculate back into the compression chambers for altering the combustion properties of the engine  11 . 
     Generally, the exhaust gas after-treatment system  100  is configured to remove various chemical compound and particulate emissions present in the exhaust gas received from the exhaust manifold  16  and not recirculated back into the engine  11 . As illustrated in  FIG. 2 , the exhaust gas after-treatment system  100  includes controller  130 , oxidation catalyst  140 , particulate matter (PM) filter  142 , SCR system  150  having an SCR catalyst  152 , and ammonia oxidation (AMOX) catalyst  160 . In an exhaust flow direction, indicated by directional arrow  144 , exhaust may flow from the exhaust manifold  16 , through the oxidation catalyst  140 , through the particulate filter  142 , through the SCR catalyst  152 , through the AMOX catalyst  160 , and then be expelled into the atmosphere. In other words, the particulate filter  142  is positioned downstream of the oxidation catalyst  140 , the SCR catalyst  152  is positioned downstream of the particulate filter  142 , and the AMOX catalyst  160  is positioned downstream of the SCR catalyst  152 . Generally, exhaust gas treated in the exhaust gas after-treatment system  100  and released into the atmosphere consequently contains significantly fewer pollutants, such as diesel particulate matter, NO x , hydrocarbons, such as carbon monoxide and carbon dioxide, than untreated exhaust gas. 
     The oxidation catalyst  140  can be any of various flow-through, diesel oxidation catalysts (DOC) known in the art. Generally, the oxidation catalyst  140  is configured to oxidize at least some particulate matter, e.g., the soluble organic fraction of soot, in the exhaust and reduce unburned hydrocarbons and CO in the exhaust to less environmentally harmful compounds. For example, the oxidation catalyst  140  may sufficiently reduce the hydrocarbon and CO concentrations in the exhaust to meet the requisite emissions standards. 
     The particulate filter  142  can be any of various particulate filters known in the art configured to reduce particulate matter concentrations, e.g., soot and ash, in the exhaust gas to meet requisite emission standards. The particulate filter  142  can be electrically coupled to a controller, such as controller  130 , that controls various characteristics of the particulate filter, such as, for example, the timing and duration of filter regeneration events. In some implementations, the particulate filter  142  and associated control system is similar to, or the same as, the respective particulate filters and control systems described in U.S. patent application Ser. Nos. 11/227,320; 11/227,403; 11/227,857; and 11/301,998, which are incorporated herein by reference. 
     The SCR system  150  includes a reductant delivery system  151  that includes a reductant source  170 , pump  180  and delivery mechanism  190 . The reductant source  170  can be a container or tank capable of retaining a reductant, such as, for example, ammonia (NH 3 ), urea, diesel fuel, or diesel oil. The reductant source  170  is in reductant supplying communication with the pump  180 , which is configured to pump reductant from the reductant source to the delivery mechanism  190 . The delivery mechanism  190  can include a reductant injector schematically shown at  192  positioned upstream of the SCR catalyst  152 . The injector is selectively controllable to inject reductant directly into the exhaust gas stream prior to entering the SCR catalyst  152 . In some embodiments, the reductant can either be ammonia or urea, which decomposes to produce ammonia. As will be described in more detail below, in these embodiments, the ammonia reacts with NO x  in the presence of the SCR catalyst  152  to reduce the NO x  to less harmful emissions, such as N 2  and H 2 O. The SCR catalyst  152  can be any of various catalysts known in the art. For example, in some implementations, the SCR catalyst  152  is a vanadium-based catalyst, and in other implementations, the SCR catalyst is a zeolite-based catalyst, such as a Cu-Zeolite or a Fe-Zeolite catalyst. In one representative embodiment, the reductant is aqueous urea and the SCR catalyst  152  is a zeolite-based catalyst. 
     The AMOX catalyst  160  can be any of various flow-through catalysts configured to react with ammonia to produce mainly nitrogen. Generally, the AMOX catalyst  160  is utilized to remove ammonia that has slipped through or exited the SCR catalyst  152  without reacting with NO x  in the exhaust. In certain instances, the system  10  can be operable with or without an AMOX catalyst. Further, although the AMOX catalyst  160  is shown as a separate unit from the SCR catalyst  152 , in some implementations, the AMOX catalyst can be integrated with the SCR catalyst, e.g., the AMOX catalyst and the SCR catalyst can be located within the same housing. 
     The exhaust after-treatment system  100  includes various sensors, such as temperature sensors  124 A-F, pressure sensor  126 , oxygen sensor  162 , NO x  sensors  164 A-D, NH 3  sensors  166 A-C, dual ammonia/NO x  sensors (not shown) and the like, that are disposed throughout the exhaust gas after-treatment system. The various sensors may be in electrical communication with the controller  130  to monitor operating conditions and control the engine system  10 , including the exhaust after-treatment system  100 . In the illustrated embodiment, the exhaust gas after-treatment system  100  includes NO x  sensor  164 A upstream of the oxidation catalyst  140 , NO x  sensor  164 B embedded within the SCR catalyst  152 , NO x  sensor  164 C intermediate the SCR catalyst and AMOX catalyst  160 , and NO x  sensor  164 D downstream of the AMOX catalyst. Further, the illustrated exhaust gas after-treatment system  100  includes NH 3  sensor  166 A upstream of the SCR catalyst  125 , NH 3  sensor  166 B embedded within the SCR catalyst  152 , and NH 3  sensor  166 C downstream of the AMOX catalyst  160 . 
     Although the exhaust after-treatment system  100  shown includes one of an oxidation catalyst  140 , particulate filter  142 , SCR catalyst  152 , and AMOX catalyst  160  positioned in specific locations relative to each other along the exhaust flow path, in other embodiments, the exhaust after-treatment system may include more than one of any of the various catalysts positioned in any of various positions relative to each other along the exhaust flow path as desired. Further, although the oxidation catalyst  140  and AMOX catalyst  160  are non-selective catalysts, in some embodiments, the oxidation and AMOX catalysts can be selective catalysts. 
     The controller  130  controls the operation of the engine system  10  and associated sub-systems, such as the engine  11  and exhaust gas after-treatment system  100 . The controller  130  is depicted in  FIG. 2  as a single physical unit, but can include two or more physically separated units or components in some embodiments if desired. Generally, the controller  130  receives multiple inputs, processes the inputs, and transmits multiple outputs. The multiple inputs may include sensed measurements from the sensors and various user inputs. The inputs are processed by the controller  130  using various algorithms, stored data, and other inputs to update the stored data and/or generate output values. The generated output values and/or commands are transmitted to other components of the controller and/or to one or more elements of the engine system  10  to control the system to achieve desired results, and more specifically, achieve desired exhaust gas emissions. 
     The controller  130  includes various modules for controlling the operation of the engine system  10 . For example, the controller  130  includes one or more modules for controlling the operation of the particulate filter  142  as described above. The controller  130  also includes one or more modules for controlling the operation of the SCR system  150 . The controller  130  further includes one or more modules for controlling the operation of the engine  11 . Additionally, in the event the oxidation catalyst  140  and AMOX catalyst  160  are selectively controllable, the controller  130  can include one or more modules for controlling the operation of the respective oxidation and AMOX catalysts. 
     Referring to  FIG. 3 , and according to one embodiment, the controller  130  includes several modules for controlling operation of the SCR system  150  to provide efficient reduction of NO x  during transient and steady state operations, while reducing ammonia slip from the tailpipe. More specifically, the controller  130  includes a NO x  reduction target module  300 , at least one ammonia target module (e.g., feedforward ammonia target module  310  and feedback ammonia target module  344 ) a reductant target module  330 , an NH 3  storage module  350 , an AMOX NH 3  conversion module  380 , a reductant limiting module  390 , and a corrected tailpipe NO x  module  397 . Generally, the modules are independently and/or cooperatively operated to achieve optimal NO x  conversion efficiency on the SCR catalyst  152  while minimizing ammonia slip and urea consumption. The controller  130  is communicable in data receiving and/or transmitting communication with several sub-systems of the engine system  10 , such as engine controls  167 , PM filter system controls  168 , and SCR system controls  169 . 
     NO x  Reduction Target Module 
     Referring to  FIG. 4 , the NO x  reduction target module  300  is operable to determine a NO x  reduction requirement  304 . The NO x  reduction requirement represents the amount of NO x  that should be reduced from the exhaust gas stream on the SCR catalyst  152  to achieve a predetermined exhaust gas emissions limit. In other words, the NO x  reduction target module  300  determines the NO x  reduction requirement  304  necessary to achieve the desired tailpipe NO x  level  306 . The desired amount of NO x  at the tailpipe, e.g., desired tailpipe NO x  level  306  (see  FIGS. 4 and 16 ), is representative of the amount of NO x  allowed to exit the tailpipe pursuant to regulated emissions standards. 
     Generally, the NO x  reduction requirement  304  is expressed as the fraction of the NO x  in the exhaust gas stream to be reduced. The NO x  reduction requirement can also be expressed in terms of a NO x  reduction rate or the rate at which NO x  should be reduced to achieve the predetermined exhaust gas emissions limit. In certain implementations, the NO x  reduction target module  300  is communicable in data receiving communication with the NO x  sensor  164 A to determine the amount of NO x  present in the exhaust gas stream prior to entering the SCR catalyst  152 . Alternatively, or additionally, in some implementations, the amount of NO x  present in the exhaust gas stream can be estimated via operation of an engine operating conditions module  302 . The engine operating conditions module  302  compares the operating conditions of the engine  11  against a stored operating map containing predetermined exhaust NO x  levels for various operating conditions of the engine to determine an estimated amount of NO x  in the exhaust gas stream. The NO x  reduction target module  300  compares the actual or estimated amount of NO x  in the exhaust gas stream at the engine outlet to a desired level of NO x    306  in the exhaust gas emitted from the tailpipe to determine the NO x  reduction requirement  304 . 
     Ammonia Target Modules 
     The controller  130  includes an ammonia target module operable to determine an ammonia addition requirement. As defined herein, the ammonia addition requirement is the amount of ammonia that should be added to the exhaust gas stream to reduce the NO x  in the exhaust gas stream to the desired level for meeting the emissions standards. In certain embodiments, the controller  130  includes the feedforward ammonia target module  310  for determining an ammonia addition requirement  326  using a feedforward methodology (see  FIG. 5A ). In other embodiments, the controller  130  includes the feedback ammonia target module  344  for determining an ammonia addition requirement  348  using a feedback methodology (see  FIG. 5B ). In yet some embodiments, the controller  130  includes both the feedforward ammonia target module  310  and the feedback ammonia target module  344 . 
     Referring first to  FIG. 5A , the feedforward ammonia target module  310  receives as input the NO x  reduction requirement  304  from the NO x  reduction target module  311  (see  FIG. 4 ), an NH 3  storage modifier  352  from the NH 3  storage module  350  (see  FIG. 10 ), and a current SCR catalyst inlet NH 3  flow rate  335  from the reductant hydrolysis module  333  (see  FIG. 7 ) and utilized by the module  310  to determine the ammonia addition requirement  326 . In the representative illustrated embodiment, the feedforward ammonia target module  310  includes a NO x  reduction efficiency module  312 , an SCR catalyst inlet NO 2 /NO x  ratio module  314 , an SCR catalyst inlet exhaust properties module  316 , an SCR catalyst bed temperature module  318 , an SCR catalyst inlet NO x  module  320 , an SCR catalyst space velocity module  322 , and a NO x  reduction reaction rate module  324 . 
     The NO x  reduction efficiency module  312  is operable to determine the maximum efficiency of NO x  reduction on the SCR catalyst  152 . Generally, the NO x  reduction efficiency module  312  considers a desired NO x  conversion efficiency and the condition of the SCR catalyst. 
     The desired NO x  conversion efficiency can be any of various efficiencies and be dependent on the difference between the amount of NO x  in the exhaust gas stream at the engine outlet with the desired amount of NO x  in the exhaust gas stream at the tailpipe outlet. For example, in some implementations, the desired NO x  conversion efficiency of the SCR catalyst  152  can be the efficiency necessary for achieving the desired tailpipe NO x  level  306  at the SCR catalyst outlet. However, in embodiments having an AMOX catalyst, the desired NO x  conversion efficiency of the SCR catalyst  152  can be lower than if no AMOX catalyst is being used because the AMOX catalyst can reduce ammonia slipping from the SCR catalyst. 
     The condition of the SCR catalyst  152  affects the efficiency of the SCR catalyst. The more degraded the condition of the SCR catalyst, the lower the maximum efficiency of NO x  reduction on the SCR catalyst  152 . Accordingly, the NO x  reduction efficiency module  312  is operable to compare the desired NO x  conversion efficiency with the maximum NO x  conversion efficiency of the SCR catalyst  152  and output the smaller of the two efficiencies to the feedforward ammonia target module  310 . The feedforward ammonia target module  310  then utilizes the smaller of the desired and maximum NO x  conversion efficiencies determined by the NO x  reduction efficiency module  312  to determine the ammonia addition requirement  326 . Generally, the lower the smaller NOx conversion efficiency, the lower the ammonia addition requirement  326 . The NO x  reduction efficiency module  312  can determine the maximum NO x  conversion efficiency of the SCR catalyst  152  in various ways, such as described in pending U.S. patent application entitled “APPARATUS, SYSTEM, AND METHOD FOR ESTIMATING A MAXIMUM NH 3  CONVERSION CAPABILITY OF A SELECTIVE CATALYTIC REDUCTION CATALYST,” attorney docket number 8-02-12952, which is incorporated herein by reference. Moreover, the condition of the SCR catalyst  152  can be indicated by an SCR catalyst degradation factor. The SCR catalyst degradation factor can be determined by an SCR catalyst degradation factor module, such as module  368  described below in relation to  FIG. 11 , according to any of various ways. For example, the SCR catalyst degradation factor module can determine the SCR catalyst degradation factor in a manner similar to that described in pending U.S. patent application entitled “APPARATUS, SYSTEM, AND METHOD FOR DETERMINING DEGRADATION OF A SELECTIVE CATALYTIC REDUCTION CATALYST,” attorney docket number 8-02-12959, which is incorporated herein by reference. 
     The SCR catalyst inlet NO 2 /NO x  ratio module  314  is operable to predict the NO 2 /NO x  ratio of the exhaust gas in the exhaust gas stream at the inlet of the SCR catalyst  152 . In some implementations, the NO 2 /NO x  ratio is expressed as the following ratio: 
                     NO   2       NO   +     NO   2               (   1   )               
where NO is the mass concentration of nitrogen monoxide in a predetermined volume of exhaust gas and NO 2  is the mass concentration of nitrogen dioxide in the predetermined volume of exhaust gas.
 
     The SCR catalyst inlet exhaust properties module  316  is operable to determine various properties of the exhaust gas at the inlet of the SCR catalyst  152 . The properties can include, for example, the mass flow rate of the exhaust and the temperature of the exhaust. In some implementations, the exhaust gas properties are predicted based on predetermined exhaust property values for predetermined operating conditions of the engine system  10 . For example, the SCR catalyst inlet exhaust properties module  316  can include an exhaust properties map, table or vector comparing predetermined exhaust property values with engine system operating conditions, such as the operating load and/or speed of the engine  11 . In certain implementations, the SCR catalyst inlet exhaust properties module  316  determines the exhaust gas properties by processing input from any of various sensors known in the art, such as mass flow and temperatures sensors. 
     The SCR catalyst bed temperature module  318  is operable to determine the bed temperature of the SCR catalyst  152 . The bed temperature of the SCR catalyst  152  can be determined based on one or more temperature sensors embedded in the SCR catalyst, such as temperature sensor  124 D, or predicted by a module (see, e.g., AMOX catalyst bed temperature module  386  of  FIG. 13 ) that uses various operating parameters of the system, such as the exhaust gas mass flow rate and temperature before and after the SCR catalyst  152 . Accordingly, although the illustrated embodiments use an SCR catalyst bed temperature sensor  124 D for determining the temperature of the SCR catalyst bed, in other embodiments, the sensor is replaced or supplemented with an SCR catalyst bed temperature module operable to predict or estimate the temperature of the SCR catalyst bed. 
     The SCR catalyst inlet NO x  module  320  is operable to determine the concentration of NO x  in the exhaust gas at the inlet of the SCR catalyst  152 . The NO x  concentration can be predicted based on predetermined exhaust conditions corresponding to predetermined operating conditions of the engine system  10 . For example, the module  320  can access an exhaust properties map, table or vector such as described above to estimate the NO x  concentration in the exhaust. Alternatively, or additionally, the concentration of NO x  in the exhaust gas upon entering the SCR catalyst  152  can be measured using the first NO x  sensor  164 A positioned upstream of the SCR catalyst. 
     The SCR catalyst space velocity module  322  is operable to determine the space velocity of the SCR catalyst  152 . Generally, the space velocity of the SCR catalyst  152  represents the amount of NO x  in the exhaust gas stream that is reactable within the SCR catalyst over a given time. Accordingly, the space velocity of the SCR catalyst  152  typically is represented in terms of per unit time, e.g., 1/hour, 1,000/hour, etc. The space velocity of the SCR catalyst  152  is based on various exhaust gas and catalyst conditions. For example, the space velocity can be based at least partially on the volume and/or reaction, or bed, surface area of the SCR catalyst, and the density, viscosity and/or flow rate of the exhaust gas. In some implementations, the SCR catalyst space velocity module  322  determines the space velocity of the SCR catalyst  152  by receiving inputs concerning operating conditions of the engine system  10 , and, based on the operation conditions, obtaining the space velocity of the SCR for the given conditions by accessing a table or map stored on the module. The table can include various predetermined space velocities obtained via experimental testing and calibration for a given SCR catalyst operating under the various operating conditions achievable by the engine system  10 . 
     The NO x  reduction reaction rate module  324  is operable to predict the rate at which ammonia reacts with and reduces NO x  on the SCR catalyst  152 . The predicted NO x  reaction rate is at least partially dependent on the NO x  composition or concentration of the exhaust gas and the frequency of the various types of NO x  reduction reactions occurring on the SCR catalyst  152 . Generally, NO x  is reduced by ammonia in one of the following three most active stoichiometric chemical reactions: 
     
       
         
           
             
               
                 
                   
                     
                       NH 
                       3 
                     
                     + 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       NO 
                     
                     + 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         NO 
                         2 
                       
                     
                   
                   → 
                   
                     
                       N 
                       2 
                     
                     + 
                     
                       
                         3 
                         2 
                       
                       ⁢ 
                       
                         H 
                         2 
                       
                       ⁢ 
                       O 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       NH 
                       3 
                     
                     + 
                     NO 
                     + 
                     
                       
                         1 
                         4 
                       
                       ⁢ 
                       
                         O 
                         2 
                       
                     
                   
                   → 
                   
                     
                       N 
                       2 
                     
                     + 
                     
                       
                         3 
                         2 
                       
                       ⁢ 
                       
                         H 
                         2 
                       
                       ⁢ 
                       O 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       NH 
                       3 
                     
                     + 
                     
                       
                         3 
                         4 
                       
                       ⁢ 
                       
                         NO 
                         2 
                       
                     
                   
                   → 
                   
                     
                       7 
                       8 
                     
                     ⁢ 
                     
                       N 
                       2 
                     
                     ⁢ 
                     
                       3 
                       2 
                     
                     ⁢ 
                     
                       H 
                       2 
                     
                     ⁢ 
                     O 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     The predicted NO x  reaction rate is also at least partially dependent on the ammonia concentration rate, the bed temperature of the SCR catalyst  152 , and the space velocity of the SCR catalyst. Further, in some implementations, the predicted NO x  reaction rate is also at least partially dependent on the degradation factor or condition of the SCR catalyst  152 . The predicted NO x  reaction rate can be expressed as the sum of a predicted NO x  reaction rate for reducing NO according to Equations 2 and 3 above and a predicted NO x  reaction rate for reducing NO 2  according to Equations 3 and 4 above. 
     Based at least partially on the desired NO x  conversion efficiency, the NO 2 /NO x  ratio of the exhaust gas, the exhaust flow rate, the temperature and condition of the SCR catalyst  152  bed, the amount of NO x  and NH 3  at the inlet of the SCR catalyst, and NO x  reduction reaction rate, the ammonia target module determines the ammonia addition requirement  326 . In some embodiments, the ammonia addition requirement  326  is also at least partially based on an NH 3  storage modifier  352  determined by an NH 3  storage module  350  as will be described in more detail below (see  FIG. 7 ). 
     According to another embodiment shown in  FIG. 5B , the ammonia addition requirement, e.g., ammonia addition requirement  348 , can be determined by the feedback ammonia target module  344 . The feedback ammonia target module  344  receives as input the desired tailpipe NO x  level  306 , the amount of NH 3  exiting the tailpipe as sensed by the tailpipe NH 3  sensor  166 C, the NH 3  storage modifier  352 , and a corrected tailpipe NO x  value  399  (see  FIG. 16 ). Further, the feedback ammonia target module  344  includes an exhaust flow properties module  345  and a tailpipe NO x  feedback module  347 . In contrast to the feedforward ammonia target module  310 , the feedback ammonia target module  344  relies mainly on the properties of the exhaust gas stream after passing through the SCR catalyst  152  and adjusts the reductant dosing rate to compensate for errors and inconsistencies in the SCR system  150 . 
     The exhaust flow properties module  345  is operable to determine various conditions of the exhaust gas stream, e.g., temperature, flow rate, etc., in a manner similar to that described above in relation to SCR catalyst inlet exhaust properties module  316 . 
     The tailpipe NO x  feedback module  347  is operable to determine a tailpipe NO x  feedback value that can be utilized by the feedback ammonia target module  344  for determining the ammonia addition requirement  348 . The tailpipe NO x  feedback value accounts for inconsistencies in the SCR system  150 , such as modeling errors, catalyst aging, sensor aging, reductant concentration variations, reductant injector delays, which can reduce the efficiency of the system. Therefore, the tailpipe NO x  feedback module  396  is operable to modulate the tailpipe NO x  feedback value to increase the efficiency of the SCR system  150  and achieve the desired NO x  conversion efficiency despite inconsistencies that may be present in the system. 
     The tailpipe NO x  feedback module  347  generates the tailpipe NO x  feedback value by comparing the sensed amount of NO x  as detected by the tailpipe NO x  sensor  164 D with the desired or targeted tailpipe NO x  amount  306 . Accordingly, the tailpipe NO x  feedback value is at least partially dependent on the difference between the sensed tailpipe NO x  and the targeted or desired tailpipe NO x    306 . Generally, the greater the difference between the sensed tailpipe NO x  and the targeted tailpipe NO x    306 , the higher the ammonia addition requirement  348 . For example, if the sensed amount of tailpipe NO x  is relatively high compared to the targeted tailpipe NO x    306 , then the feedback ammonia target module  344  can increase the ammonia addition requirement  348 . As will be explained in more detail below, an increase in the ammonia addition requirement  348  can result in more reductant being added to the exhaust gas stream for increased NO x  conversion on the SCR catalyst  152 . Conversely, if the sensed amount of tailpipe NO x  is relatively low compared to the targeted tailpipe NO x    306 , then the feedback ammonia target module  344  can decrease the ammonia addition requirement, which may consequently result in less reductant being added to the exhaust gas stream to conserve reductant, and thus increase the efficiency of the SCR system  150 . 
     In certain embodiments, because of the cross-sensitivity of some NO x  sensors to ammonia, the feedback ammonia target module  344  is utilized by the SCR system  150  to generate the ammonia addition requirement only when ammonia is not slipping from the SCR system  150 , e.g., slipping out of the tailpipe. Whether ammonia is slipping from the tailpipe can be sensed by the tailpipe NH 3  sensor  166 C and/or predicted by the AMOX NH 3  conversion module  380 , as will be described in more detail below. 
     In certain embodiments, the controller  130  includes a control logic selection algorithm (not shown) configured to select one of the ammonia addition requirements  326 ,  348  to act as the ammonia addition requirement for the SCR system  150  based at least partially on whether NH 3  is slipping from the tailpipe. In other words, the module used for determining the ammonia addition requirement for the SCR system  150  is switchable based on whether the SCR system is operating in a tailpipe NH 3  slip U mode or a tailpipe NH 3  non-slip mode. More specifically, when NH 3  is slipping from the tailpipe, the ammonia addition requirement  326  determined by the feedforward ammonia target module  310  is communicated to the reductant target module  330  and used in the determination of the reductant injection requirement  332  (see  FIG. 8 ). Conversely, when NH 3  is not slipping from the tailpipe, the ammonia addition requirement  348  determined by the feedback ammonia target module  344  is communicated to the reductant target module  330  and used in the determination of the reductant injection requirement  332 . In some implementations, the control logic selection algorithm of the controller  130  determines the ammonia addition requirement based on a combination, e.g., an average, of the ammonia addition requirements  326 ,  348  regardless of whether ammonia is slipping from the tailpipe. In certain implementations, the ammonia addition requirement  326  can be adjusted according to the ammonia addition requirement  348 . 
     In some embodiments, the feedback ammonia target module  344  includes a signal correction algorithm (not shown) configured to filter the signal from the tailpipe NO x  sensor  164 D such that the signal is suitable for yielding a more accurate NO x  concentration at the tailpipe when ammonia is slipping from the tailpipe. Accordingly, in some implementations, the ammonia addition requirement  348  generated by the feedback ammonia target module  344  can be communicated to the reductant target module  330  during operation in the tailpipe NH 3  slip or non-slip mode. 
     As described above, the controller  130  can utilize the feedforward ammonia target module  310 , the feedback ammonia target module  344 , or both to determine an ammonia addition requirement for the SCR system  150 . Once determined, the ammonia addition requirement, e.g., ammonia addition requirement  326 , ammonia addition requirement  348 , or combination of both, is communicated to the reductant target module  330 , or more specifically, the inverse reductant hydrolysis module  334  of the reductant target module. As used hereafter, the ammonia addition requirement communicated to the reductant target module  330  will be referenced as the ammonia addition requirement  326 . Nevertheless, it is recognized that any reference to the ammonia addition requirement  326  can be substituted with the ammonia addition requirement  348  or a combination of the ammonia addition requirements  326 ,  348 . 
     Reductant Target Module 
     Referring to  FIG. 6 , the reductant target module  330  includes a reductant hydrolysis module  333  and an inverse reductant hydrolysis module  334 . As will be described in more detail below, the reductant hydrolysis module  333  is operable to determine a current SCR catalyst inlet NH 3  flow rate  335  and a current SCR catalyst inlet HNCO flow rate  336  based on the current reductant dosing rate (see  FIG. 7 ). The current SCR catalyst inlet NH 3  flow rate  335  and current SCR catalyst inlet HNCO flow rate  336  are then communicated to other various modules of the control system  150 . In contrast to the reductant hydrolysis module  333 , the inverse reductant hydrolysis module  334  is operable to receive the ammonia addition requirement  326  from the ammonia target module  310  and determine a reductant injection requirement or dosing rate  332 , i.e., the amount of reductant necessary to achieve the ammonia addition requirement  326  (see  FIG. 8 ). Based on the reductant injection requirement  332 , the controller  130  commands the SCR system controls to inject an amount of reductant corresponding to the reductant injection requirement  332 . 
     The reductant can be any of various reductants known in the art. For example, in one implementation, the reductant is ammonia. In other implementations, the reductant is urea, which breaks down into ammonia and other components as will be described in more detail below. 
     Reductant Hydrolysis Module 
     Referring back to  FIG. 7 , the reductant hydrolysis module  333  includes an NH 3  conversion efficiency table  337 , an isocyanic acid (HNCO) conversion efficiency table  338 , and an SCR catalyst inlet exhaust properties module  339 . The SCR catalyst inlet exhaust properties module  339  is operable to determine the mass flow rate of the exhaust gas stream in a manner similar to that described above in relation to SCR catalyst inlet exhaust properties module  316  of  FIG. 5 . The reductant hydrolysis module  333  is communicable in data receiving communication with the reductant delivery mechanism  190  for receiving a current reductant dosing rate  383  and the exhaust temperature sensor  124 B for receiving the temperature of the exhaust. 
     As described above, in implementations where the reductant is urea, the reductant hydrolysis module  333  is operable to determine the amount of ammonia and isocyanic acid entering the SCR catalyst  152 . According to one embodiment, the reductant hydrolysis module  333  is operable to follow the schematic flow chart  400  of  FIG. 9  to determine the current SCR catalyst inlet NH 3  and HNCO flow rates  335 ,  336 , respectively. The exhaust temperature is sensed, such as by the temperature sensor  124 B, or estimated, at  410  and the exhaust mass flow rate is estimated by the SCR catalyst inlet exhaust properties module  339  at  420 . Based at least partially on the exhaust temperature determined at  410  and the exhaust mass flow rate determined at  420 , the conversion efficiency of urea to NH 3  is determined at  430  and the conversion efficiency of urea to isocyanic acid (HNCO) is determined at  440 . Accordingly, the conversion efficiencies of urea to NH 3  and isocyanic acid are a function of the exhaust gas temperature and mass flow rate. The NH 3  and HNCO conversion efficiencies are determined by comparing the exhaust gas temperature and mass flow rate to one or more predetermined efficiency values listed on NH 3  and HNCO conversion efficiency look-up tables  337 ,  338 , respectively. 
     According to the reductant injection requirement  332  received by the SCR inlet ammonia and isocyanic acid module  360  from the reductant target module  330 , urea is injected into the exhaust gas stream by a urea injector at  450 . The urea is mixed with the exhaust gas stream flowing through an exhaust pipe between the urea injector and the surface of SCR catalyst  152 . As the urea flows along the exhaust pipe, it reacts with the exhaust gas to form NH 3  at  460  and HNCO at  470 . The NH 3  and HNCO in the exhaust gas stream then enter the SCR catalyst  152  as the current SCR catalyst inlet NH 3  flow rate  335  and current SCR catalyst inlet HNCO flow rate  336 , respectively. After the HNCO enters the SCR catalyst  152 , the catalyst bed promotes a reaction between at least a portion of the HNCO and water (H 2 O) in the exhaust gas stream to form additional NH 3  at  480 . The current SCR catalyst inlet NH 3  flow rate  335  and the current HNCO to NH 3  flow rate  341 , i.e., the NH 3  from the conversion of HNCO to NH 3  occurring within the SCR catalyst  152  at  480 , are combined to provide an estimation of the total amount of ammonia within the SCR catalyst, e.g., the current SCR catalyst NH 3  flow rate  343 . The estimated amount of HNCO that is not converted to NH 3  at  480  flows through and out of the SCR catalyst  152  at an SCR catalyst outlet HNCO flow rate  349 . 
     As discussed above, the amount of urea converted to NH 3  is at least partially dependent on the NH 3  conversion efficiency. In an ideal situation, the NH 3  conversion efficiency is 100% such that the all the urea converts to 2-parts ammonia and 1-part carbon dioxide without any intermediate conversion to HNCO according to the following equation:
 
NH 2 —CO—NH 2 ( aq )+H 2 O→2NH 3 ( g )+CO 2   (5)
 
     In actuality, the NH 3  conversion efficiency is typically less than 100% such that the urea converts to ammonia and isocyanic acid according to the following equation:
 
NH 2 —CO—NH 2 ( s )→NH 3 ( g )+HNCO( g )  (6)
 
     The remaining isocyanic acid converts to ammonia and carbon dioxide 
     CO 2  according to the HNCO conversion efficiency. In ideal situations, the HNCO conversion efficiency is 100% such that all the isocyanic acid converts to 1-part ammonia and 1-part carbon dioxide within the SCR catalyst  152  according to the following equation:
 
HNCO( g )+H 2 O( g )→NH 3 ( g )+CO 2 ( g )  (7)
 
     Typically, however, the HNCO conversion efficiency is less than 100% such that some of the HNCO is converted to ammonia and carbon dioxide and the remaining portion of HNCO is unconverted within the SCR catalyst  152 . 
     The flow rate of NH 3  into the SCR catalyst  152  ({dot over (n)} NH     3   (s)) per flow rate of injected urea ({dot over (n)} urea  (s)) is estimated according to the following equation: 
                           n   .       NH   3       ⁡     (   s   )             n   .     urea     ⁡     (   s   )         =       1       τ   ⁢           ⁢   s     +   1       ⁢     (     1   -     ⅇ       -   x     /   L         )     ⁢       η     NH   3       ⁡     (       m   .     ,   T     )                 (   8   )               
where τ is the mixing time constant, s is a complex variable used for Laplace transforms, L is the characteristic mixing length, x is the distance from the urea injector to the SCR catalyst inlet or face, and η NH     3    is the NH 3  conversion efficiency from urea, which is based on the mass flow rate ({dot over (m)}) and temperature (T) of the exhaust gas. The complex variable s can be expressed as a σ+jω, where σ represents the amplitude and ω represents the frequency of a sinusoidal wave associated with a given urea dosing rate input. The mixing time constant is predetermined based at least partially on the Federal Test Procedure (FTP) heavy-duty transient cycle for emission testing of heavy-duty on-road engines. Assuming 100% conversion efficiency, the mixing time constant is tuned with the FTP data to eliminate transient mismatches. The characteristic length L is defined as the major linear dimension of the exhaust pipe that is substantially perpendicular to the exhaust gas flow. For example, for a cylindrical exhaust pipe, the major linear dimension is the diameter of the pipe. In some embodiments, the distance from the urea injector to the SCR catalyst face x is between about 5 and 15 times the characteristic length. In specific implementations, the distance x is about 10 times the characteristic length.
 
     Similarly, the flow rate of isocyanic acid (HNCO) into the SCR catalyst  152  ({dot over (n)} HNCO (s)) per flow rate of injected urea ({dot over (n)} urea (s)) is estimated according to the following equation: 
                           n   .     HNCO     ⁡     (   s   )             n   .     urea     ⁡     (   s   )         =       1       τ   ⁢           ⁢   s     +   1       ⁢     (     1   -     ⅇ       -   x     /   L         )     ⁢       η   HNCO     ⁡     (       m   .     ,   T     )                 (   9   )               
where η HNCO  is the HNCO conversion efficiency from urea. The conversion efficiencies of urea to ammonia (η NH     3   ) and urea to isocyanic acid (η HNCO ) is predetermined based on operating parameters of the engine system  10 . In some implementations, the conversion efficiencies are tuned by comparing a measurement of the NH 3  and HNCO at the inlet of the SCR catalyst  152  with the expected amount of NH 3  and HNCO based on the stoichiometric reaction of Equation 6 while dosing urea into exhaust at specific mass flow rates and temperatures.
 
Inverse Reductant Hydrolysis Module
 
     Referring now to  FIG. 8 , based at least partially on the ammonia addition requirement  326  received from the ammonia target module  310 , the inverse reductant hydrolysis module  334  of the reductant target module  330  is operable to determine the reductant injection requirement  332  to achieve the ammonia addition requirement  326  generated by the ammonia target module  310 . In some implementations, the process used by the inverse reductant hydrolysis module  334  to determine the reductant injection requirement  332  is similar to the process illustrated in flow chart  400 , but inverted. In other words, the same techniques used in flow chart  400  to determine the current SCR catalyst inlet NH 3  flow rate  335  can be used to determine the reductant injection requirement  332 , but in a different order. 
     For example, in the flow chart  400 , the actual urea dosing rate is known and used to determine the flow of NH 3  into the SCR catalyst  152 . In contrast, in the process used by the inverse reductant hydrolysis module  334 , the ammonia addition requirement  326 , e.g., the desired or estimated flow of NH 3  into the SCR catalyst  152 , is known and used to determine the corresponding reductant injection requirement, e.g., dosing rate, necessary to achieve the desired NH 3  flow rate. The reductant injection requirement  332  is determined by predicting the hydrolysis rates and conversion efficiencies of urea to NH 3  and HNCO based on the temperature and mass flow rate of the exhaust gas stream. For example, the inverse reductant hydrolysis module  334  can include an NH 3  conversion efficiency table, HNCO conversion efficiency table, and an SCR catalyst inlet exhaust properties module similar to the reductant hydrolysis module  333 . Alternatively, the inverse reductant hydrolysis module  334  can access the NH 3  conversion efficiency table  337 , HNCO conversion efficiency table  338 , and output of the SCR catalyst inlet exhaust properties module  339  of the reductant hydrolysis module  333 . 
     In some implementations, with the desired flow rate of NH 3  into the SCR catalyst  152  ({dot over (n)} NH     3   (s)), e.g., the ammonia addition requirement, known, the reductant injection requirement  332  is determined from Equation 8 above by solving for the flow rate of injected urea {dot over (n)} urea (s). In one specific implementation, the reduction injection requirement  332  expressed in terms of mL/hr of urea is approximately equal to: 
                       mL   hr     ⁢   Urea     ≈     1.85   *     f   ⁡     (   a   )       *     m   .     ⁢     NO   x               (   10   )               
where {dot over (m)}NO x  is equal to the mass flow rate of the total amount of NO x  in the exhaust gas stream expressed in terms of grams/hour and ƒ(α) is a non-dimensional piecewise function where α is equal to the NO 2 /NO x  ratio expressed above in Equation 1. When NO is greater than or equal to NO 2 , i.e., NO 2 /NO x ≦0.5,ƒ(α) is equal to about one, and when NO is less than or equal to NO 2 , i.e., NO 2 /NO x ≧0.5ƒ(α) is equal to:
 
     
       
         
           
             
               
                 
                   
                     2 
                     ⁢ 
                     
                       ( 
                       
                         a 
                         + 
                         1 
                       
                       ) 
                     
                   
                   3 
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     In another specific embodiment, the reduction injection requirement  332  is determined based on the ideal stoichiometric conversion of urea to ammonia and the ideal stoichiometric reduction of NO x  on the SCR catalyst  152 . When the level of NO in the exhaust gas stream is greater than or equal to the level of NO 2  in the exhaust gas, the amount of urea for reducing one gram of NO x  is represented by Equation 12 below. When the level of NO in the exhaust gas is less than or equal to the level of NO 2  in the exhaust gas, the amount of urea for reducing one gram of NO x  is represented by Equation 13 below, where α is equal to the NO 2 /NO x  ratio expressed above in Equation 1. As used in Equations 12 and 13, MW Urea  is the molecular weight of the urea to be injected and MW NOx  is the molecular weight of NO x  in the exhaust gas stream. 
     
       
         
           
             
               
                 
                   0.5 
                   * 
                   
                     ( 
                     
                       
                         MW 
                         Urea 
                       
                       
                         MW 
                         NOx 
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   0.5 
                   * 
                   
                     ( 
                     
                       
                         MW 
                         Urea 
                       
                       
                         MW 
                         NOx 
                       
                     
                     ) 
                   
                   * 
                   
                     
                       2 
                       ⁢ 
                       
                         ( 
                         
                           a 
                           + 
                           1 
                         
                         ) 
                       
                     
                     3 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Based on Equations 12 and 13, the flow rate of urea in terms of grams per second can be expressed in terms of the mass flow rate of NO x  ({dot over (m)} NOx ) in the exhaust gas stream. For example, when the amount of NO in the exhaust gas stream is more than or equal to the amount of NO 2  in the exhaust gas stream, the flow rate of urea can be expressed according to the following equation: 
                       m   .     NOx       0.5   *     (       MW   Urea       MW   NOx       )               (   14   )               
where MW Urea  is the molecular weight of urea, and MW NOx  is the molecular weight of NO x  in the exhaust gas stream. When the amount of NO in the exhaust gas stream is less than or equal to the amount of NO 2  in the exhaust gas stream, the flow rate of urea can be expressed according to the following equation:
 
     
       
         
           
             
               
                 
                   
                     
                       m 
                       . 
                     
                     NOx 
                   
                   
                     0.5 
                     * 
                     
                       ( 
                       
                         
                           MW 
                           Urea 
                         
                         
                           MW 
                           NOx 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         2 
                         ⁢ 
                         
                           ( 
                           
                             a 
                             + 
                             1 
                           
                           ) 
                         
                       
                       3 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     In some implementations, the inverse reductant hydrolysis module  334  is communicable in data receiving communication with the reductant modifier module  390  to receive a reductant modifier requirement  342  (see  FIG. 15 ). As will be described in more detail below, the reductant modifier requirement  342  includes instructions for increasing or decreasing the reductant injection requirement  332  based on whether one or more reductant limiting conditions are present. Accordingly, the inverse reductant hydrolysis module  334  is operable to modify the reductant injection requirement  332  according to the reductant modifier requirement  342 . 
     Ammonia Storage Module 
     Referring to  FIG. 10 , the NH 3  storage module  350  is operable to determine an ammonia storage modifier or storage compensation command  352 . Generally, the ammonia storage modifier  352  includes information regarding the state of ammonia storage on the SCR catalyst  152 . More specifically, the ammonia storage modifier  352  includes instructions on whether ammonia entering the SCR catalyst  152  should be increased or decreased, e.g., whether the ammonia addition requirement should be increased or decreased. The ammonia target module  310  is communicable in data receiving communication with the NH 3  storage module  350  to receive the ammonia storage modifier  352  as an input value. Based on the ammonia storage modifier  352 , the ammonia target module  310  is operable to adjust, e.g., increase or decrease, the ammonia addition requirement  326  to compensate for modulations in the ammonia storage level on the SCR catalyst  152  and maintain a sufficient amount of stored NH 3  on the SCR catalyst for transient operations of the engine  11 . 
     As discussed above, the performance of the SCR system  150  is defined by the conversion efficiency of NO x  in the exhaust gas stream and the amount of ammonia that has slipped out of the tail-pipe over both steady-state and transient duty cycles. During transient duty cycles, the response of conventional control systems that monitor only the NO x  level at the tailpipe outlet typically are limited by the dynamics of the reductant dosing system, the cross-sensitivity of the NO x  sensor to NH 3 , and other factors. Accordingly, conventional control systems may have unstable feedback controls during transient duty cycles. To improve the response and feedback controls during transient duty cycles, the SCR system  150  utilizes NH 3  stored on the SCR catalyst to manage transient NO x  spikes that may occur during transient operation or cycles of the engine  11 . Further, NH 3  stored on the SCR catalyst  152  can be used to reduce NO x  when engine system operating conditions, such as low SCR catalyst bed temperatures, require a reduction or elimination of reductant dosing. The NH 3  storage module  350  is configured to monitor and regulate the amount of ammonia stored on the SCR catalyst  152  such that a sufficient amount of stored NH 3  is maintained on the SCR catalyst to accommodate transient NO x  variations and low catalyst bed temperatures as well as reduce NH 3  slip. 
     The NH 3  storage module  350  includes a current NH 3  storage level module  354  and a target NH 3  storage level module  356 . The modules  354 ,  356  process one or more inputs received by the NH 3  storage module  350  as will be explained in more detail below. 
     Current Ammonia Storage Level Module 
     Referring to  FIG. 11 , the current NH 3  storage level module  354  is communicable in data receiving communication with several sensors for receiving data sensed by the sensors. In the illustrated embodiment, the several sensors include at least the SCR catalyst bed temperature sensor  124 C, NH 3  sensors  166 A-C, and NO x  sensors  164 A-D. The current NH 3  storage level module  354  also is capable of receiving an AMOX NH 3  conversion capability  382  value and a corrected tailpipe NO x  value  399  as will be described in further detail below. 
     The current NH 3  storage level module  354  also includes an SCR catalyst inlet exhaust properties module  358 , an NH 3  flux module  364 , an SCR catalyst inlet NO 2 /NO x  ratio module  366 , an SCR catalyst degradation factor module  368 , an SCR catalyst NH 3  slip module  369 , and an NH 3  desorption module  375 . Based on input received from the sensors  124 C,  166 A,-C,  164 A-D, the AMOX NH 3  conversion capability  382  (if an AMOX catalyst is used), the tailpipe NO x  feedback value  399 , and operation of the modules  358 ,  364 ,  366 ,  368 ,  369 ,  375 , the current NH 3  storage level module  354  is operable to determine the current NH 3  storage level  370  (e.g., an estimate of the current amount of NH 3  stored on the SCR catalyst  152  based at least partially on the SCR catalyst bed temperature), the current NH 3  slip  372  (e.g., an estimate of the current amount of NH 3  exiting the SCR catalyst), and the NH 3  maximum storage capacity  374  (e.g., an estimate of the maximum amount of NH 3  capable of being stored on the SCR catalyst based under current conditions). The fraction of the available storage on the SCR catalyst that is filled can be determined by dividing the current NH 3  storage level  370  by the NH 3  maximum storage capacity  374 . 
     The NO x  sensor  164 B being embedded within the SCR catalyst  152  provides several advantages over prior art systems. For example, placing the NO x  sensor  164 B inside the SCR catalyst  152  improves the monitoring of stored ammonia on the catalyst by reducing the signal-to-noise ratio of the NO x  sensor. The NO x  sensor  164 B can be used with other NO x  sensors in the exhaust aftertreatment system  100  to quantify the spatial distribution of stored ammonia. 
     The SCR catalyst inlet exhaust properties module  358  is similar to SCR catalyst inlet exhaust properties module  316  of the ammonia target module  310 . For example, the exhaust properties module  358  is operable to determine various properties of the exhaust, such as the temperature and flow rate of the exhaust. 
     The NH 3  flux module  364  is operable to determine the rate at which NH 3  flows into the SCR catalyst  152 . The NH 3  flux module  364  can also process data concerning the amount of NH 3  present at the tailpipe outlet as sensed by the NH 3  sensor  166 C. The NH 3  sensor  166 C at the tailpipe outlet assists in the measurement and control of the tailpipe NH 3  slip by providing information regarding the tailpipe NH 3  slip to various modules of the controller  130 . In some instances, the modules, e.g., the target NH 3  storage level module  356  and the reductant modifier module  390 , adjust the urea dosing rate and the ammonia storage targets based at least partially on the tailpipe NH 3  slip information received from the NH 3  sensor. 
     The SCR catalyst inlet NO 2 /NO x  ratio module  366  is similar to the SCR catalyst inlet NO 2 /NO x  ratio module  314  of the ammonia target module  310 . For example, the SCR catalyst inlet NO 2 /NO x  ratio module  366  is operable to predict the NO 2 /NO x  ratio of the exhaust gas in the exhaust gas stream according to Equation 1. 
     The SCR catalyst degradation factor module  368  is operable to determine a degradation factor or condition of the SCR catalyst  152  in a manner the same as or similar to the NO x  reduction efficiency module  312  of the ammonia target module  310  described above. 
     According to one embodiment, the current NH 3  storage level module  354  determines the estimated current NH 3  storage level  370  by utilizing, at least in part, the current condition of the SCR catalyst bed, the size and properties of the SCR catalyst bed, and the ammonia flux entering the SCR catalyst. Referring to  FIG. 12 , and according to one exemplary embodiment, the NH 3  storage level module  354  utilizes the schematic flow chart  500  to determine the current NH 3  storage level  370  on the SCR catalyst  152 . The reducant target module  330  is operable to determine the reductant injection requirement  332 , e.g., urea dosing rate, at  510 . Alternatively, the current NH 3  storage level module  354  is communicable in data receiving communication with the reductant delivery mechanism  190  for receiving the current reductant dosing rate  383 . The SCR catalyst bed temperature sensor  124 C senses, or a bed temperature module estimates, the temperature of the SCR catalyst bed temperature at  520 . 
     Based at least partially on the temperature of the SCR catalyst bed as determined at  520 , the NH 3  maximum storage capacity  374  is generated by the current NH 3  storage level module  354  at  530 . The NH 3  maximum storage capacity  374  is dependent on the temperature of the SCR catalyst bed and can be determined by comparing the SCR catalyst bed temperature against a pre-calibrated look-up table. The urea dosing rate, which corresponds to the ammonia flux entering the SCR catalyst  152 , and SCR catalyst bed temperature are used to determine an NH 3  fill-up or adsorption time constant and the SCR catalyst bed temperature and NO x  flux are used to determine an NH 3  removal or desorption time constant. The time constants can be retrieved from respective look-up tables  540 ,  550  stored on, for example, the current NH 3  storage level module  354 . 
     A determination of the SCR catalyst mode is made at  560 . Based on whether the SCR catalyst  152  is in an NH 3  fill-up mode or an NH 3  removal mode, the corresponding time constant (τ) is used to calculate the current NH 3  storage level (NH 3 Storage) at  570  according to the following first order dynamics equation: 
                       NH   3     ⁢   Storage     =       NH   3     ⁢       Storage   Max     ⁡     (     1       τ   ⁢           ⁢   s     +   1       )                 (   16   )               
where NH 3 Storage MAX  is the NH 3  maximum storage capacity  374  of the SCR catalyst  152  and s is the complex variable used for Laplace transforms. In other words, if it is determined at  560  that more ammonia should be stored on the SCR catalyst  152 , the NH 3  adsorption time constant determined at  540  is used in Equation 16 to determine the current NH 3  storage level  370 . Alternatively, if it is determined at  560  that ammonia should be removed from the SCR catalyst  152 , the NH 3  desorption time constant determined at  550  is used in Equation 16 to determine the current NH 3  storage level  370 . Accordingly, the current NH 3  storage level  370  is at least partially based on the ammonia flux, temperature of the catalyst and degradation of the catalyst.
 
     In at least one embodiment, the storage mode, e.g., fill-up or removal mode, of the SCR catalyst  152  is determined by the NH 3  storage module  350  by comparing the NH 3  maximum storage capacity  374  with the current NH 3  storage level  370 . If the NH 3  maximum storage capacity  374  is less than the current NH 3  storage level  370  then the SCR catalyst  152  is in the desorption mode. Similarly, if the NH 3  maximum storage capacity  374  is more than the current NH 3  storage level  370  then the SCR catalyst  152  is in the adsorption mode. 
     The look-up tables utilized at  540 ,  550  include a listing of the adsorption and desorption time constants, respectively, corresponding to various possible urea dosing rates and SCR catalyst bed temperatures. In certain implementations, the adsorption time constants can be calibrated using steady-state testing. For example, the engine  11  can be run at specific steady state modes such that the temperature of SCR catalyst bed reaches and is held at a specific temperature corresponding to each mode. Prior to reaching each mode, the SCR catalyst  152  is clean such that the catalyst bed does not contain stored ammonia, i.e., the amount of NO x  coming out of the engine is the same as the amount of NO x  coming out of the SCR catalyst. For each respective mode, the reductant target module  330  is operable to communicate to the reductant delivery mechanism  190  to inject an amount of reductant necessary to achieve 100% conversion of NO x . The amount of reductant can vary for different stoichiometric reactions rates ranging, for example, between about 0.5 to about 2.0. The amount of time between the initial reductant dosing and ammonia slippage from the SCR catalyst  152  is determined for each mode at each stoichiometric reaction dosing rate and used to calibrate the adsorption time constants in the NH 3  fill-up time constant table. 
     The desorption time constants in the NH 3  removal time constant table can be calibrated during the same test used for calibrating the adsorption time constants. For example, after NH 3  begins to slip from the SCR catalyst  152  as described above, the NH 3  slip and NO x  leaving the SCR catalyst are monitored until they stabilize or become constant. Once the NH 3  slip and SCR catalyst outlet NO x  are stable, the urea dosing is discontinued and the amount of time between discontinuation of urea dosing and the SCR catalyst outlet NO x  to equal the engine outlet NO x  is determined for each mode at each stoichiometric reaction dosing rate. 
     If desired, the adsorption and desorption time constants can be further calibrated to compensate for transient operation of the engine  11 . For example, the Fourier Transform Infrared (FTIR) measurements of ammonia slip values and the time between the beginning of a transient FTP cycle and slippage from the SCR catalyst can be used to fine-tune the adsorption and desorption time constants. More specifically, the time constants can be adjusted based on a least squares approach that can provide the best first order model fit to the transient data. 
     The target NH 3  storage level module  356  is operable to determine a target NH 3  storage level based at least in part on the NH 3  maximum storage capacity  374  determined by the current NH 3  storage level module  354 . Generally, the target NH 3  storage level module  356  determines the target NH 3  storage level by multiplying the NH 3  maximum storage capacity  374  by an ammonia storage level fraction. The ammonia storage level fraction can be any of various fractions, such as fifty percent, seventy-five percent, ninety percent, and one-hundred percent. The ammonia storage level fraction is determined based at least partially on the SCR catalyst degradation factor and user defined maximum allowable ammonia slip. 
     Once the current NH 3  storage level  370  and the target NH 3  storage level are determined, the NH 3  storage module  350  utilizes the current NH 3  storage level  370  as feedback and compares the current NH 3  storage level and the target NH 3  storage level. If the current NH 3  storage level is less than the target NH 3  storage level, the ammonia storage modifier  352  is set to a positive value. If the current NH 3  storage level  370  is more than the target NH 3  storage level, the ammonia storage modifier  352  is set to a negative value. The positive and negative values can vary depending on how much less or more the current NH 3  storage level  370  is compared to the target NH 3  storage level. The ammonia storage modifier  352  is communicated to the ammonia target module  310  (see  FIG. 5 ). An ammonia storage modifier  352  with a positive value indicates to the ammonia target module  310  that the ammonia addition requirement  326  should be correspondingly increased. In contrast, an ammonia storage modifier  352  with a negative value indicates to the ammonia target module  310  that the ammonia addition requirement  326  should be correspondingly decreased. 
     The amount of NH 3  storage on the catalyst  152  can be controlled by controlling any of various inputs into the SCR system  150 . For example, referring to  FIG. 12 , the amount of ammonia storage on the SCR catalyst  152  is dependent on the following separately controllable factors: the urea dosing rate, the SCR catalyst bed temperature, and the SCR catalyst maximum capacity. Accordingly, the controller  130  can be operable to selectively or cooperatively control the current NH 3  storage level on the SCR catalyst  152 . 
     The ammonia storage modifier  352  also can be adjusted according to the current NH 3  storage slip  372 , the presence or absence of an AMOX catalyst, such as AMOX catalyst  160 , and if an AMOX catalyst is used, the conversion capability  382  of the AMOX catalyst. 
     According to one embodiment, the SCR catalyst ammonia slip module  369  determines the estimated current NH 3  slip  372  from the SCR catalyst  152  by utilizing, at least in part, the ammonia and NO x  flux entering the catalyst, the size and properties of the SCR catalyst bed, and the ratio of NO to NO 2 . Referring to  FIG. 13 , and according to one exemplary embodiment, the ammonia slip module  369  utilizes the schematic flow chart  600  to determine the current NH 3  slip  372  from the SCR catalyst  152 . The amount of NO x  at the inlet of the SCR catalyst  152  is determined at  610  and the amount of NO x  at the outlet of the SCR catalyst is determined at  614 . The NO x  inlet amount can be sensed by the NO x  sensor  164 A and the NO x  outlet amount can be sensed by the NO x  sensor  164 C or NO x  sensor  164 D. To account for any degradation of the sensor  164 D, the output of the NO x  sensor  164 D can be corrected as described above in relation to corrected tailpipe NO x  module  362 . The ratio of NO to NO 2  in the exhaust gas stream at the inlet of the SCR catalyst  152  is determined at  612  and the ratio of NO to NO 2  in the exhaust gas stream at the outlet of the SCR catalyst is determined at  616 . In some implementations, the SCR catalyst NO 2 /NO x  ratio module  366  is operable to determine the NO to NO 2  ratios at the inlet and outlet of the SCR catalyst  152 , respectively. 
     At  620 , the amount of ammonia consumed within the SCR catalyst  152  is calculated based on the net loss, e.g., conversion, of NO and NO 2  from the exhaust gas stream. In some implementations, the calculation is performed by the current NH 3  storage level module  354 . Based at least partially on the flow of NH 3  into the SCR catalyst  152  determined at  630  and the amount of ammonia consumed within the SCR catalyst  152 , the excess amount of NH 3  within the SCR catalyst is estimated at  640 . As described above, the amount of NH 3  flowing into the SCR catalyst  152  can be determined by utilizing flow chart  400  of  FIG. 10 . 
     Further, based at least partially on the current NH 3  storage level  370  determined at  650 , the flow rate of the exhaust gas stream into and through the SCR catalyst  152  determined at  652 , and the temperature of the SCR catalyst bed determined at  653 , the amount of ammonia desorbed from the bed of the SCR catalyst  152  is estimated at  660 . Generally, desorption of ammonia occurs when there is a specific increase in the temperature of the SCR catalyst bed. The amount of temperature increase necessary to effect desorption of ammonia is at least partially dependent on the condition and type of SCR catalyst being used. As shown in  FIG. 11 , the current NH 3  storage level module  354  can include the desorbed NH 3  module  375 , which is operable to estimate the amount of ammonia desorbed from the bed of the SCR catalyst  152 . In certain implementations, the NH 3  storage level module  354  estimates the amount of ammonia desorbed from the SCR catalyst bed based on the excess NO x  flux available for reduction reaction on the SCR catalyst surface. 
     Based at least partially on the excess amount of NH 3  within the SCR catalyst  152 , the amount of NH 3  desorbed from the SCR catalyst bed, and the amount of NH 3  stored on the SCR catalyst relative to the NH 3  maximum storage capacity  374  of the catalyst, i.e., the fraction of the SCR catalyst occupied by stored ammonia, the amount of NH 3  slipping from the SCR catalyst is estimated at  680 . The amount of NH 3  slipping from the SCR catalyst  152  is equal to the sum of the excess amount of NH 3  determined at  640  and the desorbed amount of NH 3  determined at  660 . The fraction of the SCR catalyst occupied by stored ammonia is determined at  670  by dividing the NH 3  stored on the catalyst as determined at  650  by the NH 3  maximum storage capacity determined, for example, at  530  of flow diagram  500 . Generally, if the total amount of NH 3  stored on the SCR catalyst  152  is greater than the NH 3  maximum storage capacity  374 , i.e., the ammonia stored fraction determined at  670  is greater than one, then ammonia slip from the catalyst is occurring and the amount of slip is determined at  680 . If the total amount of NH 3  within the SCR catalyst is less than the NH 3  maximum storage capacity  374 , i.e., the ammonia stored fraction is less than one, then ammonia slip is not occurring and the amount of ammonia slip is not calculated at  680 . In other words, the model used to compute the ammonia slip at  680  does not become active until the SCR catalyst  152  is full with ammonia, or the SCR catalyst bed temperature and rate of increase of the SCR catalyst bed temperatures are above predetermined thresholds. 
     The amount of NH 3  slip from the catalyst  152  can be controlled by controlling any of various inputs into the SCR system  150 . For example, referring to  FIG. 13 , the amount of ammonia slip from the SCR catalyst  152  is dependent on the following separately controllable factors: the amount of NH 3  flowing into the SCR catalyst as determined at  630 ; the exhaust flow rate as determined at  652 ; and the current in NH 3  storage level as determined using flow chart  500 . Accordingly, the controller  130  can be operable to selectively or cooperatively control the NH 3  slip from the SCR catalyst. 
     If the current NH 3  storage slip  372  is relatively high, such as when the temperature of the SCR catalyst bed exceeds a predetermined level, then the NH 3  storage module is operable to decrease the ammonia storage modifier  352 . In contrast, if the current NH 3  storage slip  372  is relatively low, then the NH 3  storage module is operable to increase or hold steady the ammonia storage modifier  352 . 
     AMOX Ammonia Conversion Module 
     According to one embodiment shown in  FIG. 14 , the AMOX NH 3  conversion module  380  determines an AMOX NH 3  conversion capability or efficiency  382 , a tailpipe NH 3  slip  384  and an AMOX catalyst thermal mass  385 . Generally, the NH 3  conversion capability  382  represents an estimate of the ability of the AMOX catalyst  160  to convert NH 3  to N 2  and other less harmful or less noxious components. The tailpipe NH 3  slip  384  represents an estimate of the amount of NH 3  exiting the AMOX catalyst  160 . As will be described in more detail below, the AMOX thermal mass  385  is a measure of the AMOX catalyst&#39;s ability to conduct and store heat. 
     The AMOX NH 3  conversion module  380  receives input regarding the exhaust gas flow rate  700  entering the AMOX catalyst  160  and the amount of NH 3  entering the AMOX catalyst. In some implementations, the exhaust gas flow rate  700  is determined by the SCR catalyst inlet exhaust properties module  358  of current NH 3  storage level module  354  (see  FIG. 11 ) or other similar module. The amount of NH 3  entering the AMOX catalyst  160  can be represented by an NH 3  input  712  and/or the current NH 3  slip  372 . More specifically, in some implementations, the AMOX NH 3  conversion module  380  is communicable in data receiving communication with the current NH 3  storage level module  354  to receive the current NH 3  slip  372 . In these implementations, the amount of NH 3  entering the AMOX catalyst  160  can be set to the current NH 3  slip  372 . In some implementations, the control system  150  can include an NH 3  sensor between the SCR catalyst  152  and the AMOX catalyst  160 . In these implementations, the amount of NH 3  entering the AMOX catalyst  160  can be set to the output of the NH 3  sensor. Alternatively, in certain instances, the amount of NH 3  entering the AMOX catalyst  160  can be set to a combination of the current NH 3  slip  372  and the output of the NH 3  sensor, such as an average of the current NH 3  slip  372  and the output of the NH 3  sensor. The AMOX NH 3  conversion module  380  can also be communicable in data receiving communication with various other sensors, such as temperature sensors  124 D,  124 E and NO x  sensor  164 C. 
     The AMOX NH 3  conversion module  380  includes several modules including, but not limited to, an AMOX catalyst bed temperature module  386 , an NO 2 /NO x  ratio module  387 , an AMOX catalyst degradation module  388 , and a tailpipe NH 3  slip target module  389 . 
     The AMOX catalyst bed temperature module  386  is operable to estimate the temperature of the AMOX catalyst bed. In one implementation, the AMOX catalyst bed temperature module  386  utilizes the input from the temperature sensors  124 D,  124 E to determine the difference between the temperature of the exhaust at the inlet of the AMOX catalyst  160  and the temperature of the exhaust at the outlet of the AMOX catalyst. Based at least partially on the temperature differential and mass flow rate properties of the exhaust gas stream, the AMOX catalyst bed temperature module  386  calculates the temperature of the AMOX catalyst bed. Alternatively, or in addition to estimating the AMOX catalyst bed temperature as described above, the SCR system  150  can include a temperature sensor (not shown) coupled to the AMOX catalyst  160 . The AMOX catalyst bed temperature module  386  can utilize the output of the AMOX catalyst temperature sensor to determine the temperature of the AMOX catalyst bed. 
     Similar to the SCR catalyst NO 2 /NO x  ratio module  366  of the current NH 3  storage level module  354 , the NO 2 /NO x  ratio module  387  of the AMOX NH 3  conversion module  380  is operable to determine the ratio of NO 2  to NO x  according to Equation 1 above, where NO 2  is the amount of nitrogen dioxide at the inlet of the AMOX catalyst  160  and NO is the amount of nitrogen oxide at the inlet of the AMOX catalyst as sensed by the NO x  sensor  164 C. 
     Similar to the SCR catalyst degradation factor module  368  of current NH 3  storage level module  354 , the AMOX catalyst degradation module  388  is operable to determine an AMOX catalyst degradation factor indicating the condition of the AMOX catalyst. In certain implementations, the catalyst degradation factor is determined by an algorithm that compares the conversion efficiency of the “aged” AMOX catalyst at predetermined engine operating conditions and urea dosing rates with the conversion efficiency of a “fresh” AMOX catalyst under the same predetermined conditions and dosing rates. 
     The tailpipe NH 3  slip target module  389  is operable to determine a tailpipe NH 3  slip target, i.e., the desired amount of NH 3  allowed to exit the AMOX catalyst  160 . The tailpipe NH 3  slip target is based at least partially on a desired average amount of NH 3  slip from the AMOX catalyst and/or a desired maximum amount of NH 3  slip from the AMOX catalyst. In some instances, both the desired average amount of NH 3  slip from the AMOX catalyst and desired maximum amount of NH 3  slip from the AMOX catalyst are used to ensure that actual tailpipe slip levels remain below a human detectable threshold. Further, the tailpipe NH 3  slip target can be based on other factors, such as current emissions standards and customer-based specifications. 
     Based at least partially on at least one of the flow rate of exhaust, NO x , and ammonia entering the AMOX catalyst  160 , the temperature of the AMOX catalyst bed, the ratio of NO 2 /NO x  at the inlet of the AMOX catalyst, the catalyst degradation factor, and the tailpipe NH 3  slip target, the AMOX NH 3  conversion module  380  estimates the AMOX NH 3  conversion capability  382 , the tailpipe NH 3  slip  384 , and the AMOX catalyst thermal mass  385 . For example, in some implementations, the AMOX NH 3  conversion capability  382  and the tailpipe NH 3  slip  384  are dependent on the amount of NO x  entering the AMOX catalyst, the temperature of the AMOX catalyst, and a space velocity of the AMOX catalyst. Further, in some instances, the AMOX catalyst thermal mass  385  is based at least partially on the geometric dimensions of the AMOX catalyst, and the material properties of the AMOX catalyst, such as the thermal conductivity and volumetric heat capacity of the AMOX catalyst. In some instances, the AMOX NH 3  conversion capability  382 , the tailpipe NH 3  slip  384 , and the AMOX catalyst thermal mass  385  can be estimated by accessing a multi-dimensional, pre-calibrated look-up table stored on the controller  130 . 
     Generally, the higher the AMOX catalyst conversion capability  382 , the more tolerance the SCR system  150  has to NH 3  slipping from the SCR catalyst  152 . Accordingly, if the AMOX catalyst conversion capability  382  is relatively high, more NH 3  can be allowed to slip from the SCR catalyst  152 . However, with more NH 3  slipping from the SCR catalyst  152 , more NH 3  storage sites on the surface of the SCR catalyst  152  may be vacant, thus requiring an increase in the ammonia addition requirement  326 . In such an instance, the NH 3  storage module  350  can increase the ammonia storage modifier  352 , which in turn can increase the ammonia addition requirement  326 . In contrast, when the AMOX catalyst conversion capability  382  is relatively low, less NH 3  slippage from the SCR catalyst  152  is tolerated, resulting in less NH 3  removed from storage on the SCR catalyst. If more NH 3  slips from the SCR catalyst  152  and the AMOX catalyst conversion capability  382  is relatively low, the tailpipe NH 3  slip may correspondingly increase. Therefore, in these instances, the NH 3  storage module  350  can decrease or hold steady the ammonia storage modifier  352  to decrease or hold-steady the ammonia addition requirement  326 , and/or the AMOX NH 3  conversion module  380  can modulate the effectiveness of the AMOX catalyst  160 , such that tailpipe NH 3  slip is controlled. 
     In some implementations, the AMOX catalyst thermal mass value  385  is dependent on the material properties of the AMOX catalyst bed, such as thermal conductivity and volumetric heat capacity. Generally, the thermal mass  385  is a measure of the AMOX catalyst&#39;s ability to conduct and store heat. The AMOX NH 3  conversion module  380  can communicate the AMOX catalyst thermal mass value  385  to the NH 3  storage module  350 , which can use the thermal mass value in its determination of the ammonia storage modifier  352 . 
     As described above, the AMOX NH 3  conversion capability and AMOX catalyst thermal mass  385  is communicated to and processed by various other modules of the controller  130 . For example, the AMOX NH 3  conversion capability  382  and AMOX catalyst thermal mass  385  is received by the NH 3  storage module  350  and used to determine the ammonia storage modifier  352  (see  FIG. 10 ). Further, the AMOX NH 3  conversion capability  382  is used by the corrected tailpipe NO x  module  399  to determine the tailpipe NO x  feedback value  399  (see  FIG. 16 ). 
     The tailpipe NH 3  slip  384  determined by the AMOX embedded model NH 3  conversion module  380  can be communicated to other modules of the controller  130 . For example, the determined tailpipe NH 3  slip  384  can be communicated to the reductant modifier module  390  (see  FIG. 15 ) and corrected tailpipe NO x  module  397  (see  FIG. 16 ) to replace or supplement the tailpipe NH 3  slip measurement input communicated from the NH 3  sensor  166 C. For example, in certain instances, the input value for the tailpipe NH 3  into the modules  390 ,  397  can be an average of the determined tailpipe NH 3  slip  384  and the tailpipe NH 3  slip measurement from the sensor  166 C to provide a more accurate indication of the actual amount of NH 3  slipping from the tailpipe. 
     Reductant Modifier Module 
     Referring to  FIG. 15 , the reductant modifier module  390  is operable to determine a reductant modifier requirement  342  based at least in part on whether any of various reductant limiting conditions have been met. The reductant modifier module  390  includes a reducant modifier conditions module  394  and an SCR catalyst inlet exhaust properties module  395 . Generally, the reductant modifier module  390  is operable to either reduce reductant dosing, prevent reductant dosing or leave reductant dosing unchanged when certain predetermined conditions of the exhaust aftertreatment system  100  are met. 
     The reductant modifier conditions module  394  is operable to monitor the operating conditions of the engine system  10  and determine if one or more reductant limiting conditions are met. In some embodiments, the reductant limiting conditions include, but are not limited to, an exhaust gas temperature limit, an ammonia slip reductant rate limit, and an SCR catalyst bed temperature limit. 
     Reductant dosing at high exhaust gas temperatures can cause cyanuric acid and polymers (e.g., melamine) to form on the injector and exhaust pipe walls, which can lead to performance degradation of and damage to the system. For example, the formation of melamine can clog the nozzle. To prevent cyanuric acid from forming, the reductant modifier module  390 , including the reductant modifier conditions module  394 , monitors the exhaust gas temperature and prevents reductant dosing, e.g., via instructions in the reductant modifier requirement  342 , if the exhaust gas temperature exceeds a predetermined exhaust gas temperature limit. The current exhaust gas temperature can be sensed by at least one of the temperature sensors, e.g., exhaust temperature sensor  124 C and/or predicted by an SCR catalyst inlet exhaust properties module  395  similar to module  358 . 
     Reductant dosing at high SCR catalyst storage levels and SCR catalyst bed temperature ramps can cause ammonia to slip from the SCR catalyst  152 . To reduce ammonia slip in these situations, the reductant modifier module  390  monitors the current NH 3  storage level  370  and the modulations of the SCR catalyst bed temperature as sensed by the temperature sensor  124 D (or predicted by an SCR catalyst bed temperature module as described above). If the current NH 3  storage level  370  exceeds a predetermined NH 3  storage level associated with NH 3  slip, or if the modulation in SCR catalyst bed temperature exceeds a predetermined SCR catalyst bed temperature change, then the reductant modifier module reduces the reductant dosing rate, e.g., via instructions in the reductant modifier requirement, such that NH 3  slip from the SCR catalyst  152  is controlled. 
     The reductant modifier module  390  is also operable to prevent reductant dosing in the event a specific component or components of the SCR system  150  has malfunctioned or is otherwise not ready for operation. 
     Corrected Tailpipe NO x  Module 
     Referring to  FIG. 16 , the corrected tailpipe NO x  module  397  of the controller  130  is operable to determine the corrected tailpipe NO x  value  399 . The corrected tailpipe NO x  module  397  is communicable in data receiving communication with the tailpipe NO x  sensor  164 D and tailpipe NH 3  sensor  166 C. The corrected tailpipe NO x  module  397  is also communicable in data receiving communication with the current NH 3  storage level module  354  to receive the estimated current NH 3  slip  372  or the estimated amount of NH 3  exiting the SCR catalyst  152 . Further, the corrected tailpipe NO x  module  397  is communicable in data receiving communication with the AMOX NH 3  conversion module  380  to receive the AMOX NH 3  conversion capability  382 . The corrected tailpipe NO x  module  397  also includes a sensor degradation module  398  that is operable to determine a tailpipe NO x  sensor degradation factor based at least partially on the type of sensor, age of sensor, and operating conditions of the engine system  10 . In some instances, the tailpipe NO x  sensor degradation factor is determined by an algorithm that compares the NO x  sensor measurements at pre-determined operating conditions having known NO x  values. The degradation factor indicates an amount, e.g., a percentage, the measured NO x  sensor value should be adjusted to account for degradation of the NO x  sensor and inaccuracies associated with the degraded NO x  sensor measurements. In some implementations, the corrected tailpipe NO x  value is about is about 10% higher than the measured tailpipe NO x  value. 
     The corrected tailpipe NO x  module  397  processes the sensed tailpipe NO x  amount, the sensed tailpipe NH 3  amount, the estimated NH 3  slip  372 , the NO x  sensor degradation factor, and the AMOX conversion capability  382  to determine the corrected tailpipe NO x  value  399 . The corrected tailpipe NO x  value  399  can replace the sensed amount of NO x  detected by the tailpipe NO x  sensor  164 D in the reductant modifier requirement  342  calculation by the reductant modifier module  390  for a more accurate indication of the amount of NO x  leaving the tailpipe and a more accurate reductant modifier requirement. Additionally, the corrected tailpipe NO x  value  399  can be communicated to and processed by the current NH 3  storage level module  354 . 
     Exemplary Method for Reducing NO x  Emissions 
     Referring to  FIG. 17 , and according to one representative embodiment, a method  800  for reducing NO x  emissions using ammonia storage on an SCR catalyst is shown. The method  800  starts at  802  and includes determining  804  a NO x  reduction requirement. In some implementations, determining  804  a NO x  reduction requirement includes operating the NO x  reduction target module  300  to estimate the NO x  reduction requirement  304 . The method  800  also includes determining  806  an ammonia addition requirement. In some implementations, determining  806  an ammonia addition requirement includes operating the ammonia target module  310  to estimate the ammonia addition requirement  326 . The method  800  further includes determining  808  an ammonia storage modifier. In some implementations, determining  808  an ammonia storage modifier includes operating the NH 3  storage module  350  to estimate the ammonia storage modifier  352 . 
     After an ammonia storage modifier is determined, the method  800  includes comparing  810  the ammonia storage modifier to a predetermined value, such as zero. If the ammonia storage modifier is greater than or less than the predetermined value, then the method  800  includes adjusting  812 , such as by adding, the ammonia addition requirement determined at  808  by an amount corresponding to the ammonia storage modifier amount. If the ammonia storage modifier is approximately equal to the predetermined value, then the ammonia addition requirement determined at  808  is not adjusted. The method  800  includes determining  814  a reductant injection requirement  814  based on either the ammonia addition requirement determined at  808  or the adjusted addition requirement determined at  812 . In some implementations, determining  814  a reductant injection requirement includes operating the reductant target module  330  to calculate the reduction injection requirement  332 . The method  800  can also include determining  815  an AMOX catalyst NH 3  conversion capability  382  by operation of the AMOX NH 3  conversion module  380 . 
     The method  800  further includes determining  816  a reductant modifier. In some implementations, determining  816  a reductant modifier includes operating the reductant modifier module  390  to calculate the reductant modifier requirement  342 . After a reductant modifier is determined, the method  800  includes comparing  820  the reductant modifier to a predetermined value, such as zero. If the reductant modifier is greater than or less than the predetermined value, then the method  800  includes adjusting  822  the reductant injection requirement determined at  816  by an amount corresponding to the reductant modifier amount. If the reductant modifier is approximately equal to the predetermined value, then the reductant injection requirement determined at  808  is not adjusted. The method includes injecting  824  an amount of reductant corresponding to the reductant injection requirement determined at either  816  or  822  into the exhaust gas stream. 
     The schematic flow chart diagrams and method schematic diagrams described above are generally set forth as logical flow chart diagrams. As such, the depicted order and labeled steps are indicative of representative embodiments. Other steps and methods may be conceived that are equivalent in function, logic, or effect to one or more steps, or portions thereof, of the methods illustrated in the schematic diagrams. Additionally, the format and symbols employed are provided to explain the logical steps of the schematic diagrams and are understood not to limit the scope of the methods illustrated by the diagrams. Although various arrow types and line types may be employed in the schematic diagrams, they are understood not to limit the scope of the corresponding methods. Indeed, some arrows or other connectors may be used to indicate only the logical flow of a method. For instance, an arrow may indicate a waiting or monitoring period of unspecified duration between enumerated steps of a depicted method. Additionally, the order in which a particular method occurs may or may not strictly adhere to the order of the corresponding steps shown. 
     The present invention may be embodied in other specific forms without departing from its spirit or essential characteristics. The described embodiments are to be considered in all respects only as illustrative and not restrictive. The scope of the invention is, therefore, indicated by the appended claims rather than by the foregoing description. All changes which come within the meaning and range of equivalency of the claims are to be embraced within their scope.