Method and Device for Controlling a Secondary Air Mass Flow in a Secondary Air Supply of an Internal Combustion Engine

Abstract

A method for controlling a secondary air mass flow in a secondary air supply of an internal combustion engine. The internal combustion engine comprises the secondary air supply, a device for determining a pressure in the secondary air supply, and an exhaust gas lambda sensor for determining a current exhaust gas lambda value. The method includes a) determining a first secondary air mass flow) related to the effective throttle area using a throttle equation and depending on a pressure in the secondary air supply, b) determining a second secondary air mass flow based on the measured exhaust gas lambda value and taking into account a primary air mass flow of the internal combustion engine and a supplied fuel mass flow, c) deriving an effective throttle area from the first secondary air mass flow determined in step a) and the second secondary air mass flow determined in step b), and d) controlling the secondary air mass flow in the secondary air supply using the effective throttle area determined in step c) or a quantity derived therefrom.

Claims

1. A method for controlling a secondary air mass flow in a secondary air supply of an internal combustion engine, wherein the internal combustion engine comprises the secondary air supply, a device for determining a pressure in the secondary air supply, and an exhaust gas lambda sensor for determining a current exhaust gas lambda value, the method comprising: a) determining a first secondary air mass flow related to the effective throttle area using a throttle equation and depending on a pressure in the secondary air supply; b) determining a second secondary air mass flow based on the measured exhaust gas lambda value and taking into account a primary air mass flow of the internal combustion engine and a supplied fuel mass flow; c) deriving an effective throttle area from the first secondary air mass flow determined in step a) and the second secondary air mass flow determined in step b); and d) controlling the secondary air mass flow in the secondary air supply using the effective throttle area determined in step c) or a variable derived therefrom.

2. The method according to claim 1, wherein the first secondary air mass flow in step a) is determined by way of the throttle equation, which relates the first secondary air mass flow to the pressure in the secondary air supply and the effective throttle area.

3. The method according to claim 1, wherein the effective throttle area in step c) is derived using a mass flow balance depending on an exhaust gas lambda value.

4. The method according to claim 1, wherein the effective throttle area in step c) is derived in consideration of the time dynamics of the exhaust gas lambda probe.

5. The method according to claim 1, wherein the effective throttle area in step c) is derived using the recursive Least Mean Square method or the recursive Normalized Least Mean Square method.

6. The method according to claim 5, wherein the difference between the secondary air mass flow derived from the pressure and the secondary air mass flow derived from the exhaust gas lambda value is used as the error term of the recursive Least Mean Square method or the recursive Normalized Least Mean Square method.

7. The method according to claim 1, wherein the secondary air mass flow is controlled as a function of an actual value of the secondary air mass flow determined using the effective throttle area.

8. The method according to claim 1, wherein the secondary air mass flow is controlled as a function of an actual value of the secondary air mass flow, which is derived from the first secondary air mass flow related to the effective throttle area and the effective throttle area determined in step c).

9. The method according to claim 1, wherein the secondary air mass flow is controlled as a function of the actual value of the secondary air mass flow and a target value of the secondary air mass flow predetermined as a function of the time.

10. The method according to claim 9, wherein the target value of the secondary air mass flow is specified in the form of a time-dependent target secondary air mass flow or in the form of a time-dependent target exhaust gas air lambda value or in the form of a time-dependent target pressure in the secondary air supply.

11. The method according to claim 9, wherein a control variable for an actuator is derived from the actual value and from the target value of the secondary air mass flow, and wherein a controllable secondary air pump or a controllable electrical additional compressor is used as the actuator.

12. A control device for controlling a secondary air mass flow in a secondary air supply of an internal combustion engine, the internal combustion engine comprising: the secondary air supply for supplying secondary air, a device for determining the pressure in the secondary air supply, and an exhaust gas lambda sensor for determining a current exhaust gas lambda value, wherein the control device is configured to: a) determine a first secondary air mass flow related to the effective throttle area using a throttle equation and depending on a pressure in the secondary air supply; b) determine a second secondary air mass flow based on the measured exhaust gas lambda value and taking into account the primary air mass flow of the internal combustion engine and the supplied fuel mass flow; c) derive an effective throttle area from the first secondary air mass flow determined in step a) and the second secondary air mass flow determined in step b); and d) control the secondary air mass flow in the secondary air supply using the effective throttle area determined in step c) or a quantity derived therefrom.

13. The control device according to claim 12, wherein the control device is further configured to control the secondary air mass flow by way of a controllable secondary air pump or by way of a controllable electrical additional compressor or a controllable secondary air valve.

14. An exhaust gas system of an internal combustion engine comprising the control device according to claim 12.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Preferred embodiments are described in more detail below with reference to the accompanying drawings. The figures show:

[0032] FIG. 1 shows an overview diagram of an exhaust gas system;

[0033] FIG. 2 shows a physical model of the secondary air supply;

[0034] FIG. 3 shows a schematic illustration of an estimation algorithm for determining the effective throttle area A.sub.eff.

DETAILED DESCRIPTION

[0035] One measure to increase the temperature in the exhaust aftertreatment system is secondary air injection. An external mass air flow is introduced into the exhaust manifold at the exhaust valves of the engine, which reacts exothermically with a rich combustion chamber lambda on the hot surfaces of the manifold and turbocharger.

[0036] The method and control device described below for controlling a secondary air mass flow in a secondary air supply of an internal combustion engine is based on a physical model that describes the secondary air supply. This model is illustrated below with reference to FIG. 1. In FIG. 1, an internal combustion engine 1 is shown along with an exhaust gas line 2 and a secondary air supply 3. In the combustion chambers of the internal combustion engine 1, the supplied mass fuel flow {dot over (m)}.sub.inj is partially converted with the supplied primary mass air flow {dot over (m)}.sub.air. A secondary mass flow of air {dot over (m)}.sub.secAir is supplied to the exhaust gas mass flow in the exhaust gas line 2 via the secondary air supply 3, wherein the unburned fuel in the exhaust gas line 2 reacts exothermically with the supplied secondary air. An exhaust gas lambda sensor 4 is arranged in the exhaust gas line 2 and is configured to determine the exhaust gas lambda value .sub.sens.

[0037] The secondary air supply 3 may be considered a throttle. A pressure sensor 5 is arranged in the secondary air supply 3 and is configured to determine the pressure p.sub.S in the secondary air supply 3. The secondary air mass flow {dot over (m)}.sub.secAir is determined based on the pressure differential between the modeled pressure p.sub.3 in the exhaust gas line 2 and the measured pressure p.sub.S in the secondary air supply 3 and may be described by an effective throttle area A.sub.eff, which is also depicted in FIG. 1.

[0038] The physical model mentioned above describes the relationship between the pressure p.sub.S in the secondary air supply 3 and the measured exhaust gas lambda value .sub.sens. This physical model is shown schematically in FIG. 2. Since the secondary air supply 3 can be represented as a throttle, the secondary air mass flow can be determined in step 6, based on the pressure p.sub.S in the secondary air supply 3, the temperature T.sub.S in the secondary air supply 3, the pressure p.sub.3 in the exhaust gas line 2 and the effective throttle area A.sub.eff by way of a choke equation {dot over (m)}.sub.secAir. Based on the secondary air mass flow {dot over (m)}.sub.secAir, the primary air mass flow {dot over (m)}.sub.air and the fuel mass flow {dot over (m)}.sub.inj determined in this way, a lambda calculation 7 can then be performed to obtain a computational lambda value .sub.sum. This calculated lambda value .sub.sum is then converted into the actual exhaust gas lambda value .sub.sens(t) measured by the exhaust gas lambda sensor 4 by applying a lambda dynamic 8, i.e. by taking into account the dynamics and the dead time of the exhaust gas lambda sensor 4 represented by the parameters .sub.sens and .sub.sens.

[0039] Alternatively, the mass flow determined via the throttle equation may be delayed {dot over (m)}.sub.secAir to the position of the lambda sensor (PT1+dead time). The area is adapted by comparing the delayed {dot over (m)}.sub.secAir (from throttle) with the lambda-based mass flow (via lambda sensor). To determine the lambda-based mass flow, the difference between the measured exhaust gas lambda (at the sensor) and target combustion chamber lambda (in the combustion chamber) is taken into account (here too, it is advantageous to bring the two lambdas into phase, i.e., to delay the combustion chamber lambda with respect to the location of the sensor (PT1+dead time)).

[0040] The calculations underlying the physical model shown in FIG. 2 are shown below. The secondary air mass flow {dot over (m)}.sub.secAir through the secondary air supply 3, which can be described as a throttle, can be represented by the throttle equation

[00001]$\begin{array}{cc}{\stackrel{.}{m}}_{\mathrm{secAir}}=A_{\mathrm{eff}}.Math.p_S.Math.\sqrt{\frac{2}{R.Math.T_S}}.Math.\left({}_S\right)& \left(1\right)\end{array}$ [0041] wherein

[00002]${}_S=\frac{p_3}{p_S}$

denotes the pressure ratio between the pressure p.sub.3 in the exhaust gas line 2 and the pressure p.sub.S in the secondary air supply 3. A.sub.eff is the effective throttle area, p.sub.S is the pressure in the secondary air supply 3 and T.sub.S is the temperature of the secondary air. (.sub.S) denotes a flow function dependent on the pressure ratio .sub.S.

[00003]$\begin{array}{cc}\left(\right)=\{\begin{array}{cc}\text{?}.Math.\sqrt{2.Math.\frac{1-}{1-{}_{\mathrm{crit}}}-{\left(\frac{1-}{1-{}_{\mathrm{crit}}}\right)}^2},& \mathrm{if}{}_{\mathrm{crit}}1\\ {}_{\mathrm{crit}},& \mathrm{if}0{}_{\mathrm{crit}}\end{array}& \left(2\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

wherein .sub.=0.528 and .sub.=0.484.

[0042] In the subsequent lambda calculation 7 of the model shown in FIG. 2, the calculated lambda .sub.sum is calculated based on the total supplied air ({dot over (m)}.sub.secAir+{dot over (m)}.sub.air) to

[00004]$\begin{array}{cc}\text{?}=\frac{{\stackrel{.}{m}}_{\mathrm{air}}+{\stackrel{.}{m}}_{\mathrm{secAir}}}{{}_{\mathrm{sto}}.Math.{\stackrel{.}{m}}_{\mathrm{inj}}}& \left(3\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

wherein .sub.sto=14.7 denotes the stoichiometric ratio in the conversion of fuel with air. If the equation is transformed using the combustion chamber lambda,

[00005]$\begin{array}{cc}{}_{\mathrm{engine}}=\frac{{\stackrel{.}{m}}_{\mathrm{air}}}{{}_{\mathrm{sto}}.Math.{\stackrel{.}{m}}_{\mathrm{inj}}}& \left(4\right)\end{array}$

the result is

[00006]$\begin{array}{cc}\text{?}={}_{\mathrm{engine}}.Math.\left(1+\frac{{\stackrel{.}{m}}_{\mathrm{secAir}}}{{\stackrel{.}{m}}_{\mathrm{air}}}\right)& \left(5\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

solved for {dot over (m)}.sub.secAir:

[00007]$\begin{array}{cc}\stackrel{.}{m}\text{?}={\stackrel{.}{m}}_{\mathrm{air}}.Math.\left(\frac{{}_{\mathrm{exhaust},\mathrm{sensor}}}{{}_{\mathrm{engine}}}-1\right)& \left(6\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

[0043] For example, the dynamic response of the lambda path may be modeled in the subsequent lambda dynamics 8 taking into account the dynamics and dead time of the exhaust gas lambda sensor 4. For example, the following differential equation may be used:

[00008]$\begin{array}{cc}\text{?}\frac{d{}_{\mathrm{sens}}\left(t\right)}{\mathrm{dt}}+{}_{\mathrm{sens}}\left(t\right)={}_{\mathrm{sum}}\left(t-{}_{\mathrm{sens}}\right)& \left(6\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

[0044] wherein .sub.sum denotes the calculated determined lambda, .sub.sens and .sub.sens denotes the parameters used to model the lambda dynamics, and .sub.sens(t) denotes the exhaust gas lambda value sensed by the exhaust gas lambda sensor 4.

[0045] Based on the model equations for the physical model shown in FIG. 2, an estimation algorithm for the effective throttle area A.sub.eff is now built. The procedure for determining the effective throttle area A.sub.eff is illustrated schematically in FIG. 3. The aim is to determine the effective throttle area A.sub.eff. The effective throttle area A.sub.eff determined in this way can then be used for a control of the secondary air mass flow.

[0046] First, the lower path of the diagram shown in FIG. 3 for estimating the effective throttle area A.sub.eff will be described. In the lower path, the corresponding secondary air mass flow {dot over (m)}.sub.secAirLambda is calculated based on the exhaust lambda value .sub.sens(t) measured by the exhaust gas lambda sensor 4 using an inverted lambda calculation 9 and the primary air mass flow {dot over (m)}.sub.air and the fuel mass flow {dot over (m)}.sub.inj. To determine {dot over (m)}.sub.secAirLambda, the formulas

[00009]$\begin{array}{cc}\stackrel{.}{m}\text{?}={}_{\mathrm{sto}}.Math.{\stackrel{.}{m}}_{\mathrm{inj}}.Math.{}_{\mathrm{sens}}-{\stackrel{.}{m}}_{\mathrm{air}}& \left(7\right)\end{array}$$\mathrm{and}$$\begin{array}{cc}\stackrel{.}{m}\text{?}={\stackrel{.}{m}}_{\mathrm{air}}.Math.\left(\frac{{}_{\mathrm{sens}}}{\text{?}}-1\right)& \left(8\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

are used, which arise from the definition of lambda, wherein .sub.sto denotes the stoichiometric ratio of air to fuel, {dot over (m)}.sub. denotes the supplied flow of fuel mass, .sub.sens denotes the measured exhaust gas lambda value, .sub.engine denotes the combustion chamber lambda, and {dot over (m)}.sub. denotes the primary air mass flow.

[0047] In the upper path of the diagram shown in FIG. 3, in the first step 10, the secondary air mass flow {dot over (m)}.sub. related to the effective throttle area A.sub.eff (without effective flow area) is determined by way of the flow equation

[00010]$\begin{array}{cc}\stackrel{.}{m}\text{?}=p_S.Math.\sqrt{\frac{2}{R.Math.T_S}}.Math.\left({}_S\right)& \left(9\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

[0048] Here .sub. denotes the pressure ratio

[00011]$\text{?}=\frac{p_3}{p_S},$$\text{?}\text{indicates text missing or illegible when filed}$

p.sub.S denotes the pressure in the secondary air supply 3, T.sub.S denotes the temperature in the secondary air supply 3, and p.sub.3 denotes the pressure in the exhaust gas line 2. () is defined in Formula (2).

[0049] Using the subsequent lambda dynamics 11, the dynamics and dead time of the exhaust gas lambda sensor 4, which are described by the parameters .sub.sens and .sub.sens, are applied to the secondary air mass flow {dot over (m)}.sub.secAirNoAeff determined in this way and related to the effective throttle area. In this way, a time-dependent secondary air mass flow related to the effective throttle area is obtained {dot over (m)}.sub.secAirNoAeffDly(t), which is brought into phase with the secondary air mass flow {dot over (m)}.sub.secAirLambda determined based on the measured exhaust gas lambda value .sub.sens(t). To apply the lambda dynamics to the time dependent secondary air mass flow {dot over (m)}.sub.secAirNoAeffDly(t) related to the effective throttle area, the following differential equation is used

[00012]$\begin{array}{cc}\text{?}\frac{d{\stackrel{.}{m}}_{\mathrm{secAirNoAeffDly}}\left(t\right)}{\mathrm{dt}}+{\stackrel{.}{m}}_{\mathrm{secAirNoAeffDly}}\left(t\right)={\stackrel{.}{m}}_{\mathrm{secAirNoAeff}}\left(t-{}_{\mathrm{sens}}\right)& \left(10\right)\end{array}$$\text{?}\text{indicates text missing or illegible when filed}$

Next, the effective flow area A.sub.eff is determined based on the secondary air mass flow {dot over (m)}.sub. determined in the lower path and the secondary air mass flow {dot over (m)}.sub. determined in the upper path related to the effective flow area. This step is drawn as step 12 in FIG. 3.

[0050] To determine the effective throttle area A.sub.eff, one could divide the size {dot over (m)}.sub.secAirLambda(t) by the size {dot over (m)}.sub.secAirNoAeffDly(t). However, it has been shown that such quotient formation can lead to inaccurate results. In particular, when the secondary air mass flow becomes very small or even approaches zero, inaccurate values for the effective throttle area A.sub.eff may be obtained.

[0051] For this reason, it is advantageous to use a Recursive-Least-Square algorithm, in the present case, for example, a LMS (least mean square) algorithm or further preferably an NLMS (normalized least mean square) algorithm, to recursively determine the effective throttle area A.sub.eff. The starting point for the recursive determination of the effective throttle area is the equation

[00013]$\begin{array}{cc}{\stackrel{.}{m}}_{\mathrm{secAirMod},k}={\stackrel{.}{m}}_{\mathrm{secAirNoAeffDly},k}.Math.A_{\mathrm{effEst},k-1}& \left(11\right)\end{array}$ [0052] which establishes a connection between the secondary air mass flow determined by way of the throttle equation per effective throttle area {dot over (m)}.sub.secAirNoAeffDly,k and the secondary air mass flow determined by way of the exhaust gas lambda sensor 4 {dot over (m)}.sub.secAirMod,k.

[0053] If an LMS (Least Mean Square) algorithm is used as the adaptation algorithm, the derivation results in the adaption factor

[00014]$\frac{K.Math.w}{w^T.Math.w}.Math.\mathrm{error}$

and the scalar case

[00015]$\frac{K}{w}.Math.\mathrm{error},$

where w denotes the regressor, K denotes a proportionality constant, and error denotes the error. In the present case, the regressor w is the secondary air mass flow obtained from the throttle equation without an effective throttle area {dot over (m)}.sub.secAirNoAeffDly,k. However, if the secondary air mass flow approaches zero, the LMS algorithm may, for example, encounter the problem that the adaptation factor becomes infinity.

[0054] To avoid this problem, it is advantageous to use a normalized Least Mean Square (NLMS) algorithm in the implementation. The adaptation factor for the NLSM algorithm is

[00016]$\frac{K.Math.w}{1+w^T.Math.w}.Math.\mathrm{error},$

which in the scalar case results in

[00017]$\frac{K.Math.w}{1+w^2}.Math.\mathrm{error}.$

For the recursive estimation of the effective throttle area, using a modified NLMS algorithm results in

[00018]$\begin{array}{cc}{A}_{\mathrm{effEst},k}=A_{\mathrm{effEst},k-1}+\frac{K_{\mathrm{id}}.Math.{\stackrel{.}{m}}_{\mathrm{secAirNoAeffDly},k}}{1+K_{\mathrm{id}}.Math.{\stackrel{.}{m}}_{\mathrm{secAirNoAeffDly},k}^2}.Math.\left({\stackrel{.}{m}}_{\mathrm{secAirLamda},k}-{\stackrel{.}{m}}_{\mathrm{secAirMod},k}\right)& \left(12\right)\end{array}$

[0055] K.sub.id is a setting parameter for the estimation speed. This difference equation is repeated at each time step.

[0056] The effective throttle area A.sub.eff may thus be determined from the lambda information provided by the lambda sensor. In addition, no further calibration is necessary. The throttle equation may then be utilized for the forward path ({dot over (m)}.sub.secAirtarget.fwdarw.P.sub.S,target) and the reverse path (p.sub.s,1st.fwdarw.{dot over (m)}.sub.secAir) to consistently calculate both paths. In particular, by way of the NLMS algorithm described above, the lambda information can be transferred to an effective area at run time and be coupled into the pressure modeling. The control can then be carried out, for example via a conventional PID regulator with ({dot over (m)}.sub.secAirtarget(t){dot over (m)}.sub.secAir(t)) as control variable.

[0057] The disclosure can also be extended to other exhaust gas topologies. For example, if there are multiple valves in the secondary air supply, it would be possible to use a different reference quantity for modeling the throttle instead of p.sub.S, such as exhaust gas back pressure or boost pressure. Instead of pressure sensors, modeled variables may also be used.

[0058] The following describes how such an effective throttle area, A.sub.eff, can be used for controlling the secondary air mass flow. For this purpose, the actual value of the secondary air mass flow is first determined.

Determining the Actual Value of the Secondary Air Mass Flow

[0059] The determination of the effective throttle area A.sub.eff described above, in particular by way of the LMS or NLMS algorithm, allows an accurate determination of the actual value of the secondary air mass flow in the secondary air supply. The starting point for this is again the throttle equation. Based on the determined estimate of the effective throttle area A.sub.eff, the actual value of the secondary air mass flow is obtained according to equation (1) as

[00019]$\stackrel{.}{m}\text{?}\left(t\right)=A_{\mathrm{eff}}.Math.p_S.Math.\sqrt{\frac{2}{R.Math.T_S}}.Math.\left({}_S\right)$$\text{?}\text{indicates text missing or illegible when filed}$

wherein

[00020]${}_S=\frac{p_3}{p_S}$

denotes the pressure ratio between the pressure p.sub.3 in the exhaust gas line 2 and the pressure p.sub.S in the secondary air supply 3. Here, p.sub.3 is the pressure in the exhaust gas line 2, p.sub.S is the pressure in the secondary air supply 3, and T.sub.S is the temperature of the secondary air. In the time-discrete notation used to determine A.sub.effEst,k, in which the index k numbers the discrete time steps, the secondary air mass flow is given by {dot over (m)}.sub.secAir,k

[00021]$\begin{array}{cc}{\stackrel{.}{m}}_{\mathrm{secAir},k}={\stackrel{.}{m}}_{\mathrm{secAirNoAeffDly},k}.Math.A_{\mathrm{effEst},k}& \left(13\right)\end{array}$ [0060] wherein {dot over (m)}.sub.secAirNoAeffDly,k is the secondary air mass flow related to the effective throttle area, which exhibits the lambda dynamics described by equation (6), and A.sub.effEst,k is the estimated value of the effective throttle area for the time step k.

Provide a Time-Dependent Target Value of the Secondary Air Mass Flow

[0061] To control the secondary air mass flow in the secondary air supply as a function of time, a target value of the secondary air mass flow is also required, which specifies the time curve of the secondary air mass flow, in particular during the start phase of the internal combustion engine. To provide this target value, there are three options:

[0062] According to a first option, the target value {dot over (m)}.sub.secAirtargetl(t) may be directly predetermined as a function of the time.

[0063] According to a second option, the target value .sub.exhaust,target(t) of the exhaust gas lambdas as well as the target value .sub.engine,target(t) of the combustion chamber lambdas are predetermined as a function of time. By way of the equation

[00022]$\begin{array}{cc}{\stackrel{.}{m}}_{\mathrm{secAirTarget}}\left(t\right)={\stackrel{.}{m}}_{\mathrm{air}}.Math.\left(\frac{{}_{\mathrm{exhaust},\mathrm{target}}\left(t\right)}{{}_{\mathrm{engine},\mathrm{target}}\left(t\right)}-1\right)& \left(14\right)\end{array}$$\mathrm{with}{}_{\mathrm{engine},\mathrm{target}}\left(t\right)=\frac{{\stackrel{.}{m}}_{\mathrm{air}}}{{}_{\mathrm{sto}}.Math.{\stackrel{.}{m}}_{\mathrm{inj}}}\mathrm{and}{}_{\mathrm{exhaust},\mathrm{target}}\left(t\right)=\frac{{\stackrel{.}{m}}_{\mathrm{air}}+{\stackrel{.}{m}}_{\mathrm{secAirtarget}}\left(t\right)}{{}_{\mathrm{sto}}.Math.{\stackrel{.}{m}}_{\mathrm{inj}}}$ [0064] the target values .sub.exhaust,target(t) of the exhaust gas lambdas and .sub.engine,target(t) the combustion chamber lambdas may be converted to the secondary air mass flow target value {dot over (m)}.sub.secAirtarget(t). Here .sub.sto=14.7 denotes the stoichiometric ratio in the conversion of fuel with air, {dot over (m)}.sub.air denotes the primary air mass flow and {dot over (m)}.sub.inj denotes the fuel mass flow.

[0065] A third option is to specify the target pressure p.sub.s,target(t) in the secondary air supply 3 as a function of time. From this target pressure p.sub.s,target(t), the target value {dot over (m)}.sub.secAirtarget(t) of the secondary air mass flow can then be derived by way of the throttle equation:

[00023]$\begin{array}{cc}{\stackrel{.}{m}}_{\mathrm{secAirtarget}}\left(t\right)=A_{\mathrm{eff}}.Math.p_{S,\mathrm{Soll}}\left(t\right).Math.\sqrt{\frac{2}{R.Math.T_S}}.Math.\left({}_{S,\mathrm{target}}\left(t\right)\right)& \left(15\right)\end{array}$

We are using the inverted throttle equation

[00024]$\begin{array}{cc}{p}_{S,\mathrm{target}}\left(t\right)=\frac{{\stackrel{.}{m}}_{\mathrm{secAirtarget}}\left(t\right)}{(A_{\mathrm{eff}}.Math.\sqrt{\frac{2}{R.Math.T_S}.Math.\left({}_{S,\mathrm{target}}\left(t\right)\right))}}& \left(16\right)\end{array}$ [0066] wherein

[00025]${}_{S,\mathrm{target}}\left(t\right)=\frac{p_3}{p_{S,\mathrm{target}}\left(t\right)}$

denotes the pressure ratio between the pressure p.sub.3 in the exhaust gas line 2 and the pressure p.sub.s,target(t) in the secondary air supply 3. p.sub.3 is the pressure in the exhaust gas line 2, p.sub.s,target(t) is the target pressure in the secondary air supply 3, and T.sub.S is the temperature of the secondary air.

Determining the Control Deviation and a Control Variable for an Actuator

[0067] The control deviation results from the difference between the target value {dot over (m)}.sub.secAirtarget(t) of the secondary air mass flow and the actual value {dot over (m)}.sub.secAir(t) of the secondary air mass flow

[00026]$\mathrm{Control}\mathrm{deviation}={\stackrel{.}{m}}_{\mathrm{secAirtarget}}\left(t\right)-{\stackrel{.}{m}}_{\mathrm{secAir}}\left(t\right)$

[0068] The variable for the actuator for controlling the secondary air mass flow can then be determined from the control deviation. To control the secondary air mass flow, there are then different options depending on the configuration.

[0069] According to a first option, the secondary air mass flow may be controlled by way of a controlled secondary air pump wherein the air is withdrawn from the air filter.

[0070] According to a second option for controlling the secondary air mass flow, in particular in turbocharger engines, an electrical additional compressor can be provided in the secondary air section, wherein the air is preferably drawn downstream of the turbocharger.

[0071] According to a third option for controlling the secondary air mass flow, in turbocharger engines, the air can be drawn downstream of the turbocharger and the existing charging system used to adjust the target pressure required for the secondary air mass flow. Here, too, it may be advantageous to provide an additional electrical compressor, if necessary.

[0072] According to another option, a controlled valve can also be provided in the secondary air supply, for example in the form of an adjustable throttle valve, for controlling the secondary air mass flow.

[0073] The features disclosed in the foregoing description, claims and drawings may be of importance, both individually and in any combination, for the realization of the disclosure in its various embodiments.