DE102011014767B4 - Method for operating an internal combustion engine - Google Patents

Abstract

Method for operating an internal combustion engine, wherein a model of the internal combustion engine used to control and regulate the internal combustion engine is calculated using the explicit Euler method with a fixed increment, characterized in that the explicit Euler method uses a supplementary, operating point-dependent term for the required fixed step size is stabilized by shifting the eigenvalues of the system matrix.

Description

The present invention relates to a method for operating an internal combustion engine having the features of patent claim 1.

It is generally known that the control and regulation of an internal combustion engine is based on physical model approaches that are calculated using numerical solution methods. For example, the so-called filling and emptying method can be used to describe the intake and exhaust system of an internal combustion engine. Such physical approaches are generally non-linear time-continuous models. These models have different dynamics and bandwidths from working point to working point. If the internal combustion engine on which it is based has, for example, a throttle valve located in the intake system for setting the fresh air mass to be supplied to the cylinders and if the throttle valve is almost closed, the pressure ratio across the throttle valve changes only very slowly, since only a small mass flow develops across the throttle valve. If, on the other hand, the throttle valve is wide open, a large mass flow can develop over the throttle valve, causing the pressure ratio to change very quickly. In other words, with the throttle fully open, the pressure ratio across the throttle is practically unity and the reservoirs before and after the throttle act as a single reservoir. Such effects can also occur in an internal combustion engine that has an exhaust gas turbocharger, with large mass flows also forming over the compressor or the turbine and the containers in front of and behind the compressor or the turbine act like a single container. Large integration step sizes are sufficient for calculating slow transient processes, whereas fast state changes require small step sizes. In other words, there are very rapid pressure changes between the two containers, for which a certain fixed increment may not be sufficient for their calculation. Since the two aforementioned throttle valve scenarios or mass flow changes over the compressor and the turbine occur constantly during operation of the internal combustion engine, a numerical solution method with variable increments can be used to calculate the respective model. This is able to adapt the integration step size to the model to be calculated during the calculation. As an alternative, a method with a fixed step size can be used, whereby this is to be selected so small that the fast transient processes can be calculated in a stable manner. However, this means that the calculation of the model consumes a comparatively large amount of computing time, depending on the bandwidth of the model. While the choice of the calculation method is still relatively free in a development environment, it is not so in an implementation in the control unit of an internal combustion engine. Here it is not possible to wait until the method has calculated a solution in a very stiff dynamic range. Rather, the solution must be available at defined times. For this reason, the integration step size is fixed in the control unit. For example, an increment of 10 ms is provided for the calculation of the intake system. Among the methods with a fixed step size, there are numerous methods that differ significantly in terms of their convergence and stability behavior. With the same step size, complex methods often have better stability than simple approaches, but they also require more computing steps and therefore more computing time. Since this is very limited in the control unit of an internal combustion engine, it is necessary to use a simple numerical method for the approximate solution, for example the so-called explicit Euler method for the numerical solution of differential equation systems. However, practical tests have shown that when the solution method is changed from a method with a variable step size to the explicit Euler method with a fixed integration step size of 10 ms for a model of an internal combustion engine implemented in a development environment, the simulation is unstable in most of the operating points is.

For the stable calculation of a model of an internal combustion engine used to operate an internal combustion engine by means of a numerical solution method with a fixed increment, it is necessary from DE 10 2009 007 808 A1 previously known to shift all eigenvalues of the linearized model in operating points of the internal combustion engine, which are critical with regard to the stability of the calculation, into a stable area of the numerical solution method. This approach builds on the knowledge that the eigenvalues of a linearized model or a linearized system, consisting of two containers coupled by a separation point, are determined in particular by the eigenvalue contribution generated by the separation point. A separation point between two containers is preferably a throttle valve or a compressor in the intake line or a turbine in the exhaust line of an internal combustion engine. A decision as to whether the numerical solution method is stable is preferably made on the basis of an examination of the stability criterion for time-discrete systems in the state representation, as is described in more detail there in the exemplary embodiment. In terms of this approach, the particular whose eigenvalue contribution generated by the mass flow via a separation point between two containers corresponds to at least one eigenvalue contribution of several possible eigenvalue contributions of the system matrix of a linear differential equation system, with the numerical solution method then working stably if all the eigenvalues of the system matrix fulfill a condition dependent on the increment of the numerical solution method. The differential equation system describes at least one partial model of an internal combustion engine and the system matrix is formed by linearization of the underlying differential equation system, as is described in more detail there in the exemplary embodiment. According to this approach, the shifting of at least one eigenvalue, which is determined in particular by the eigenvalue contribution of the mass flow across a separation point between two containers, into a stable range takes place by means of a scaling factor. In particular, the scaling factor describes a correction mass flow. In one embodiment of this approach, the scaling factor is selected as a function of the eigenvalue contribution of the mass flow via the respective separation point, with the criterion in a particularly advantageous embodiment of this approach being that the eigenvalue contribution is equal to a limit value, and only then does the underlying model intervene if the eigenvalue contribution of the mass flow over the respective separation point is smaller than this limit value. This approach therefore has the advantage that models for describing the operating behavior of an internal combustion engine, which are used in particular for the control and regulation of an internal combustion engine, are calculated with numerical solution methods such as the explicit Euler method and a fixed step size of 10 ms, for example, and in can be implemented with real-time capability in the control unit of an internal combustion engine. Since critical components, such as the throttle valve or the compressor in the intake line or the turbine in the exhaust line, provide too large eigenvalue contributions in the surrounding states at some operating points, a real-time capable calculation of the associated models, for example with the explicit Euler method, is not consistently possible . An approach for influencing the dynamics in the relevant volumes and working points is advantageously used. This approach produces no steady-state error and very little dynamic error, particularly for the throttle valve. Overall, these measures create a real-time model that does not generate any additional stationary errors.

From the WO 1996/032579 A1 a method for model-based determination of the air mass flowing into the cylinders of an internal combustion engine is known. The air mass actually flowing into the cylinder is calculated with the help of an intake manifold charge model, which supplies a load variable from the input variables throttle valve opening angle, ambient pressure and parameters that represent the valve control, on the basis of which the injection time is determined. In addition, this load variable is used for the prediction in order to estimate the load variable at a point in time that is at least one sampling step later than the current calculation of the injection time.

the DE 697 23 754 T2 describes an air-fuel ratio control device for internal combustion engines. The ratio control device includes a catalyst arranged in an exhaust system of the internal combustion engine to purify an exhaust gas discharged from the internal combustion engine, a first exhaust gas sensor arranged in the exhaust system to detect the air-fuel ratio of the exhaust gas upstream of the catalyst , and a second exhaust gas sensor disposed in the exhaust system for detecting the concentration of a component of the exhaust gas passed through the catalyst downstream of the catalyst. The ratio control device further includes a correction quantity calculator for determining a correction quantity to correct an air-fuel ratio of the internal combustion engine based on an output from the second exhaust gas sensor in order to match the concentration of the component of the exhaust gas downstream of the catalyst with a predetermined appropriate value .

Further provided is a control means for controlling the air-fuel ratio of the internal combustion engine based on the correction amount determined by the correction amount calculation means and an output of the first exhaust gas sensor to converge the concentration of the component of the exhaust gas downstream of the catalyst toward the predetermined appropriate value . The apparatus includes a state predictor for estimating the concentration of the component of the exhaust gas downstream of the catalyst detected by the second exhaust gas sensor after a dead time in an exhaust system including the catalyst between the first and second exhaust gas sensors based on outputs from the first exhaust gas sensor up to the present and based on to-date outputs from the second exhaust gas sensor. At this time, the correction quantity calculation means has means for determining the air-fuel ratio of the internal combustion engine to calculate the concentration of the component on the basis of the condition before predicting device to compare the estimated concentration of the component of the exhaust gas downstream of the catalyst with the predetermined suitable value.

Furthermore, from the DE 101 58 262 A1 a method for determining the composition of a gas mixture in a combustion chamber of an internal combustion engine with exhaust gas recirculation and a control system for controlling an internal combustion engine are known. In this case, fresh air is mixed with an exhaust gas from the internal combustion engine that is recirculated via the exhaust gas recirculation and the resulting gas mixture is fed to the combustion chamber of the internal combustion engine. It is provided that the composition of the gas mixture in the combustion chamber of the internal combustion engine is determined by determining corresponding state variables of the internal combustion engine using corresponding physically based models which simulate the behavior of the internal combustion engine with regard to the state variable to be determined.

The methodology of the DE 10 2009 007 808 A1 However, the known approach leads to a change in the model properties, for example to a difficult to understand detriment of the dynamics and the introduction of correction mass flows. In addition, a closed analytical description of the implemented model can no longer be drawn up, which is disadvantageous for further use of the model, for example for diagnostic purposes.

According to the invention, it is therefore proposed to stabilize the explicit Euler method using an additional term for the required fixed step size. This term is in particular an operating point-dependent amplification factor for the required fixed step size, with the explicit Euler method being stabilized by the shift in the eigenvalues of the system matrix associated with the amplification factor. In particular, the explicit Euler method used for integration is thereby approximated to the implicit Euler method, which is known to be A-stable and thus provides a stable, discrete implementation of a continuously stable model. This procedure enables a real-time capable and stable calculation of the model at all operating points. i.e. With the method according to the invention it is possible to calculate the states in a system consisting of at least two containers coupled by a separation point in a numerically stable manner, independently of the properties of the separation point. In contrast to the from DE 10 2009 007 808 A1 According to the known method, the numerical stabilization does not take place through an explicit correction/modification in the model, but through the introduction of an amplification factor into the integration rule of the explicit Euler method. Thus, the model description of the discretely implemented model does not deviate from the analytical and physically motivated description of the continuous model, as is necessary, for example, due to the introduction of correction mass flows. As in the case of the explicit Euler method, the implementation within the framework of a real-time implementation is also quasi-continuous, whereby the physical perspective from the continuous model is retained for the discrete implementation as well. The differential equation system resulting for a system consisting of a series connection of containers and separation points can be calculated numerically stable with the present method according to the invention, independently of the mass flows flowing through the separation points and the pressure conditions prevailing between the containers, as a time-discrete implementation with fixed steps. In this case, the introduction of further correction terms or model modifications, for example dynamic adaptation, model reduction or the like, going beyond the physical modeling of the system is not necessary. According to the invention, the present method advantageously enables the real-time capable implementation of a model for describing the operating behavior of an internal combustion engine, for example with an increment of 10 ms, in physical representation form, ie there is a quasi-continuous implementation. A corresponding implementation using the explicit Euler method according to the prior art would not allow a consistently stable calculation across all operating points. Although the implementation based on the implicit Euler method is expedient for reasons of stability, it cannot be implemented de facto for resource reasons due to the necessary iterative calculation. The present invention is characterized in particular by the fact that the analytical and time-continuous differential equations of the model also remain valid for the discrete implementation in the real-time environment. In addition, a separate application of stabilizing components in the model is not required. Accordingly, with the present method it is possible to transform a continuous model one-to-one into a discrete model without causing numerical instabilities.

example

Further advantageous refinements of the present invention can be found in the following exemplary embodiment and in the dependent patent claims. First of all, that is from the DE 10 2009 007 808 A1 known methods are described in detail in order to then explain the method according to the invention.

• 1: ein Modell der Ansauganlage einer Verbrennungskraftmaschine.

This shows:

• 1 : a model of the intake system of an internal combustion engine.

A well-known physical model approach to the control and regulation of an internal combustion engine describes the intake system. As in 1 shown, a simple model of the intake system describes a first container 1 and a second container 2, which are separated from one another by a throttle valve 3 or can be separated from one another by closing the throttle valve 3. Ambient air ṁ _LF flows to the container 2 via an air filter 4, as indicated by the arrow. Temperature T _2i and pressure p _2i prevail in container 2 and temperature T _1i and pressure p _{1i prevail} in container 1 . To simplify things, this isothermal model approach assumes that the temperatures T _1i and T _2i in containers 1 and 2 and the ambient temperature have the same values and are constant. In addition, it is assumed that the temperature T _1i , T _2i in both containers 1 and 2 is equal to the temperature of the inflowing ambient air and that there is no heat exchange with the surroundings. A mass flow ṁ _DK flows to the container 1 via the throttle valve 3 . A mass flow m _EV continues to flow out of container 1 to the combustion chambers of the internal combustion engine. Based on the general gas law, the state differential equation of the pressure p _1i in the container 1 can be represented as a function of the mass flows ṁ _DK and ṁ _EV according to equation (1) under the assumptions mentioned and defined variables. $${\dot{p}}_{1i}=\frac{R_{L\mathrm{and}ft}\cdot T_{1i}}{V_{1i}}\left({\dot{m}}_{DK}-{\dot{m}}_{EV}\right)$$

$$\psi \left(\frac{p_{1i}}{p_{2i}}\right)=\{\begin{array}{l}\sqrt{\frac{2\cdot \kappa }{\kappa -1}\left({\left(\frac{p_{1i}}{p_{2i}}\right)}^{\frac{2}{\kappa }}-{\left(\frac{p_{1i}}{p_{2i}}\right)}^{\frac{\kappa +1}{\kappa }}\right)}f\ddot{u}r\frac{p_{1i}}{p_{2i}}\ge {\left(\frac{2}{\kappa +1}\right)}^{\frac{\kappa }{\kappa -1}}\hfill \\ \sqrt{\kappa {\left(\frac{2}{\kappa +1}\right)}^{\frac{\kappa +1}{\kappa -1}}}f\ddot{u}r\frac{p_{1i}}{p_{2i}}<{\left(\frac{2}{\kappa +1}\right)}^{{}^{\frac{\kappa }{\kappa -1}}}\hfill \end{array}$$

The mass flow ṁ _DK via the throttle valve 3 can be represented as is generally known according to equation (2), where A corresponds to the flow area as a function of the throttle valve angle α _DK and Ψ to the flow function, which corresponds to the effect of the throttle valve 3 as a function of the pressure ratio across the throttle valve 3 describes. $${\dot{m}}_{DK}=A\left(a_{DK}\right)\frac{p_{2i}}{\sqrt{R_{L\mathrm{and}ft}{T}_{2i}}}\psi \left(\frac{p_{1i}}{p_{2i}}\right)$$

$$\psi \left(\frac{p_{1i}}{p_{2i}}\right)=\{\begin{array}{l}\sqrt{\frac{2\cdot k}{k-1}\left({\left(\frac{p_{1i}}{p_{2i}}\right)}^{\frac{2}{k}}-{\left(\frac{p_{1i}}{p_{2i}}\right)}^{\frac{k+1}{k}}\right)}f\ddot{\mathrm{and}}\mathrm{right}\frac{p_{1i}}{p_{2i}}\ge {\left(\frac{2}{k+1}\right)}^{\frac{k}{k-1}}\hfill \\ \sqrt{k{\left(\frac{2}{k+1}\right)}^{\frac{k+1}{k-1}}}f\ddot{\mathrm{and}}\mathrm{right}\frac{p_{1i}}{p_{2i}}<{\left(\frac{2}{k+1}\right)}^{{}^{\frac{k}{k-1}}}\hfill \end{array}$$

The derivation of the flow function Ψ can be found, for example, in the literature reference Merker, G. P.; Black, C. ; Stiesch, G. : Internal combustion engines. Simulation of combustion and pollutant formation. Teuber, 2004.

und n_Mot der Drehzahl der Kurbelwelle der Verbrennungskraftmaschine entspricht. Die beiden Koeffizienten c_EV,1 und c_EV,2 beschreiben ferner den im Wesentlichen linearen funktionalen Zusammenhang zwischen dem Druck p_1i im Behälter 1 und einer so genannten relativen Füllung, wobei die relative Füllung als Quotient der aktuellen Luftfüllung zu einer Luftfüllung unter bestimmten Normbedingungen, multipliziert mit 100%, beschrieben wird. $${\dot{m}}_{EV}=K_{Mot}\cdot n_{Mot}\cdot \left(c_{EV\mathrm{,1}}\cdot p_{1i}+c_{EV\mathrm{,2}}\right)$$

The mass flow ṁ _EV that flows out of the container 1 to the combustion chambers can also be described according to equation (3), where K _Mot is an engine-specific constant with the unit $\frac{min\cdot kG}{\%\cdot s}$

and n _Mot corresponds to the rotational speed of the crankshaft of the internal combustion engine. The two coefficients c _EV,1 and c _EV,2 also describe the essentially linear functional relationship between the pressure p _1i in the container 1 and a so-called relative filling, the relative filling being the quotient of the current air filling to an air filling under certain standard conditions , multiplied by 100%. $${\dot{m}}_{EV}=K_{MOt}\cdot n_{MOt}\cdot \left(c_{EV\mathrm{,1}}\cdot p_{1i}+c_{EV\mathrm{,2}}\right)$$

In addition, container 2 is considered according to equation (4). $${\dot{p}}_{2i}=\frac{R_{L\mathrm{and}ft}{T}_{2i}}{V_{2i}}\left({\dot{m}}_{Lf}-{\dot{m}}_{DK}\right)$$

The container 2 is emptied via the throttle valve 3 and is delimited by a linear flow resistance against the pressure of the ambient air p _Amb , the flow resistance being interpreted as a linearized representation of the air filter 4 according to equation (5). $${\dot{m}}_{Lf}=c_{Lf}\left(p_{umG}-p_{2i}\right)$$

The coefficient c _LF describes the flow resistance that is caused by the air filter 4 .

das mit der Anfangsbedingung gemäß (5.2) $$x\left(t_0\right)=x_0,x_0\in {\Re }^n$$

beschrieben werden kann. Dafür werden (5.1) und (5.2) zunächst in die Integralform gemäß (5.3) überführt. $$x\left(t\right)=x_0+{\displaystyle {\int }_{t_0}^{t}f\left(x\left(s\right)\right)ds,}t\in \left[t_0,T\right]$$

For further understanding, the explicit Euler method for the numerical solution of ordinary first-order differential equation systems of the form according to Equation (5.1) will be discussed first, $$\dot{x}\left(t\right)=f\left(x\left(t\right)\right),t\in \left[t_0,T\right]x\in {\Re }^n,f\in {\Re }^n$$

that with the initial condition according to (5.2) $$x\left(t_0\right)=x_0,x_0\in {\Re }^n$$

can be described. For this purpose (5.1) and (5.2) are first converted into the integral form according to (5.3). $$x\left(t\right)=x_0+{\displaystyle {\int }_{t_0}^{t}f\left(x\left(s\right)\right)\mathrm{i.e}s,}t\in \left[t_0,T\right]$$

The solution of (5.3) at time t ₀ +Δt corresponds to (5.4) shown in FIG. $$x\left(t_0+\Delta t\right)=x_0+{\displaystyle {\int }_{t_0}^{t_0+\Delta t}f\left(x\left(s\right)\right)\mathrm{i.e}s,}t\in \left[t_0,T\right]$$

$$x\left(t_0+\Delta t\right)=x_0+{\displaystyle {\int }_{t_0}^{t_0+\Delta t}f\left(x_0\right)ds=x_0+\Delta t\cdot f\left(x_0\right)}$$

Since the integrand cannot be integrated numerically in many cases, it is assumed that it is constant in the interval [t ₀ , t ₀ + Δt] for a sufficiently small integration step size Δt according to (5.5) or (5.6). $$f\left(x\left(s\right)\right)=f\left(x_0\right)$$

$$x\left(t_0+\Delta t\right)=x_0+{\displaystyle {\int }_{t_0}^{t_0+\Delta t}f\left(x_0\right)\mathrm{i.e}s=x_0+\Delta t\cdot f\left(x_0\right)}$$

From the equation for the first step (5.6) the recursion equation (5.7) for the approximated solution of (5.1) and (5.2) in the entire interval [t ₀ ,T] is derived. $$x\left(t_i+t\Delta \right)=x\left(t_i\right)+\Delta t\cdot f\left(x\left(t_i\right)\right)$$

For a better overview, the notation according to (5.8) is used below. $$x^{i+1}=x^i+\Delta t\cdot f\left(x^i\right)$$

The convergence of the explicit Euler method essentially depends on the chosen step size Δt. If there are pronounced changes in the states x within an integration step, the explicit Euler method is not able to map them. Not only the convergence of the method, but also the stability depends on the selected step size Δt. If the step size for calculating a specific proper motion in (5.1), (5.2) is too large, the error in the method will grow exponentially and the explicit Euler method will become unstable. The stability of the explicit Euler method in connection with the underlying model of the internal combustion engine is only verified at selected operating points of the system. If the explicit Euler method is stable in all operating points measured on the test bench and if these are chosen narrow enough, it is assumed that all operating point transitions can also be calculated stably and the method in the relevant working area works stably overall. The working area is understood to mean the space covered by the working points. The process stability itself is examined using the stability criterion for time-discrete systems in state representation.

wird nach dem Linearglied abgebrochen, vergleiche (5.10). $$f\left(x\right)\approx f\left(x_R\right)+{\frac{\partial f\left(x\right)}{\partial x}|}_{x=x_r}\left(x-x_R\right)$$

To describe the system dynamics, the change in the system states f(x(t)) in (5.1) is linearized at the operating point x _R . The designation working point means that the system is in a rest position at this point, so the states are constant. The series expansion of f(x) according to (5.9) $$f\left(x\right)=f\left(x_R\right)+{\frac{\partial f\left(x\right)}{\partial x}|}_{x=x_R}\left(x-x_R\right)+\frac{1}{2}{\frac{{\partial }^{2}f\left(x\right)}{\partial x{x}^T}|}_{x=x_R}\left(x-x_R\right){\left(x-x_R\right)}^T+...$$

is terminated after the linear term, compare (5.10). $$f\left(x\right)\approx f\left(x_R\right)+{\frac{\partial f\left(x\right)}{\partial x}|}_{x=x_{\mathrm{right}}}\left(x-x_R\right)$$

The linearization of (5.1) should only describe the system dynamics in the vicinity of x _R , since x _R is a rest position of the system (5.1), f(x _R ) = 0 applies, which simplifies (5.10) according to (5.11). $$f\left(x\right)\approx {\frac{\partial f\left(x\right)}{\partial x}|}_{x=x_{\mathrm{right}}}\left(x-x_R\right)$$

The linearized representation of (5.1) at the operating point x _R thus corresponds to the representation according to (5.12). $$\frac{\mathrm{i.e}}{\mathrm{i.e}t}\left(x\left(t\right)-x_R\right)={\frac{\partial f\left(x\right)}{\partial x}|}_{x=x_R}\left(x\left(t\right)-x_R\right)$$

gemäß (5.13) $$\dot{x}\left(t\right)={\frac{\partial f\left(x\right)}{\partial x}|}_{x=x_R}x\left(t\right)$$

beziehungsweise gemäß (5.14) $${\frac{\partial f\left(x\right)}{\partial x}|}_{x=x_R}={\left[\begin{array}{ccc}\frac{\partial f_1}{x_1}& \cdots & \frac{\partial f_1}{\partial x_n}\\ ⋮& & ⋮\\ \frac{\partial f_n}{\partial x_1}& \cdots & \frac{\partial f_n}{\partial x_n}\end{array}\right]|}_{X_R}={A|}_{X_R}$$

lautet. Damit kann die Gleichung des expliziten Euler-Verfahrens (5.8) gemäß (5.15) beziehungsweise gemäß (5.16) und mit Φ = (I_n + Δt · A) gemäß (5.17) geschrieben werden. $$x^{i+1}=x^i+\Delta t\cdot {A|}_{X_R}{x}^i$$

$$x^{i+1}=\left(I_n+\Delta t\cdot A\right)x^i$$

$$x^{i+1}=\Phi x^i$$

The shift x(t) - x _R is usually neglected in the notation, so that the representation of the system (5.12) linearized at the operating point with $\frac{\mathrm{i.e}}{\mathrm{i.e}t}{x}_R=0$

according to (5.13) $$\dot{x}\left(t\right)={\frac{\partial f\left(x\right)}{\partial x}|}_{x=x_R}x\left(t\right)$$

or according to (5.14) $${\frac{\partial f\left(x\right)}{\partial x}|}_{x=x_R}={\left[\begin{array}{ccc}\frac{\partial f_1}{x_1}& \cdots & \frac{\partial {\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">f</figure-callout>}}_1}{\partial x_n}\\ ⋮& & ⋮\\ \frac{\partial f_n}{\partial x_1}& \cdots & \frac{\partial f_n}{\partial x_n}\end{array}\right]|}_{X_R}={A|}_{X_R}$$

reads. Thus the equation of the explicit Euler method (5.8) can be written according to (5.15) or according to (5.16) and with Φ = (I _n + Δt · A) according to (5.17). $$x^{i+1}=x^i+\Delta t\cdot {A|}_{X_R}{x}^i$$

$$x^{i+1}=\left(I_n+\Delta t\cdot A\right)x^i$$

$$x^{i+1}=\Phi x^i$$

The system (5.17) is stable if all eigenvalues λ _Φ,k of the matrix Φ fulfill the condition according to (5.18), as can be taken from the literature Lunze, DIJ: Regeltechnik 2. Bd. 2. 2. Springer, 2002. $$\left|{\lambda }_{\Phi ,k}\right|<1\left(k=\mathrm{1,2,...,}n\right)$$

From this it follows that the eigenvalues λ _A,k of the system matrix A must satisfy the condition according to (5.19) for the stability of (5.15). $$\left|{\lambda }_{A,k}\cdot \Delta t+1\right|<1\left(k=\mathrm{1,2,...,}n\right)$$

With these inequalities (5.18, 5.19) the relationship between the system dynamics of the model and the sampling time required for a stable calculation is formulated. In order to make a statement as to whether a given system (5.1, 5.2) with the sampling time Δt at the rest point x _R or a certain environment around this point can be calculated stably with the explicit Euler method, the system is linearized at this working point, the system matrix A is calculated and based on the eigenvalues λ _A,k of the system matrix A it is decided whether the inequalities from (5.19) are fulfilled. From (5.19) it follows that the real eigenvalues λ _A,k of the system matrix A must be in the range according to (5.20) for a sampling time of 10 ms. $$-200\frac{1}{s}<{\lambda }_{A,k}<0\frac{1}{s}$$

If the eigenvalues are complex, (5.19) must be evaluated as complex.

über die Drosselklappe 3 nahe eins ist. Die Parameter des Modells und die Prozessgrößen in dem gemessenen Arbeitspunkt sind gemäß Tabelle (a) dargestellt. Tabelle (a) p_1i = 1.2921 · 10⁵ Pa T_1i = 300K V_1i = 2dm³ $n_{Mot}=1500\frac{1}{min}$

p_2i = 1.3133 · 10⁵ Pa T_2i = 300K V_2i = 1dm³ α_DK = 34% A(α_DK)= 3.5 · 10^-4 m² $R_{Luft}=287\frac{J}{K\cdot kg}$

$c_{LF}=1.5\cdot {10}^{-7}\frac{kg}{s\cdot Pa}$

K_Mot = 2.155 · 10^-7 c_EV,1 = 7.4051 · 10^-4 c_EV,2 = -4.8540 κ = 1.4 In order to describe the influence of the mass flow ṁ _DK via the throttle valve 3 on the dynamics of the respective model, all process variables are set to measurements that were determined at an operating point using an engine test bench and suitable measurement technology. In this case, an operating point is preferably selected at which the pressure ratio $\frac{p_{1i}}{p_{2i}}$

across the throttle valve 3 is close to one. The parameters of the model and the process variables in the measured operating point are shown according to table (a). table (a) p _1i = 1.2921 10 ⁵ Pa T _1i = 300K V _1i = 2dm ³ $n_{MOt}=1500\frac{1}{min}$

p _2i = 1.3133 10 ⁵ Pa _T2i = 300K V _2i = 1dm ³ α _DK = 34% A(α _DK )= 3.5 10 ^-4 m ² $R_{L\mathrm{and}ft}=287\frac{J}{K\cdot kG}$

$c_{Lf}=1.5\cdot {10}^{-7}\frac{kG}{s\cdot Pa}$

K _mot = 2,155 x 10 ^-7 c _EV,1 = 7.4051 10 ^-4 c _EV,2 = -4.8540 k = 1.4

$$=\frac{R_{Luft}{T}_{1i}}{V_{1i}}\cdot \frac{\partial }{\partial p_{1i}}{\dot{m}}_{DK}\left(p_{1i}\right)-\frac{R_{Luft}{T}_{1i}}{V_{1i}}\cdot \frac{\partial }{\partial p_{1i}}{\dot{m}}_{EV}\left(p_{1i}\right)$$

$$={\lambda }_{DK}+{\lambda }_{EV}$$

For a further description of the influence of the mass flow ṁ _DK via the throttle valve 3 on the dynamics of the respective model, the dynamics of the differential equation according to (1) will first be examined in the further course. For this purpose, according to (6, 6.1, 6.2), a system matrix A of equation (1) is formed by linearization. $$A=\lambda =\frac{\partial }{\partial p_{1i}}\left(\frac{R_{L\mathrm{and}ft}{T}_{1i}}{V_{1i}}\left({\dot{m}}_{DK}\left(p_{1i}\right)-{\dot{m}}_{EV}\left(p_{1i}\right)\right)\right)$$

$$=\frac{R_{L\mathrm{and}ft}{T}_{1i}}{V_{1i}}\cdot \frac{\partial }{\partial p_{1i}}{\dot{m}}_{DK}\left(p_{1i}\right)-\frac{R_{L\mathrm{and}ft}{T}_{1i}}{V_{1i}}\cdot \frac{\partial }{\partial p_{1i}}{\dot{m}}_{EV}\left(p_{1i}\right)$$

$$={\lambda }_{DK}+{\lambda }_{EV}$$

$${\lambda }_{EV}=-\frac{R_{Luft}{T}_{1i}}{V_{1i}}{K}_{Mot}\cdot n_{Mot}\cdot c_{EV\mathrm{,1}}$$

$${\lambda }_{EV}=-\text{10}\text{.31}\frac{1}{s}$$

The eigenvalue contribution λ _EV of the mass flow ṁ _EV through the intake valves can therefore be determined according to (7, 7.1, 7.2) based on equation (3). $${\lambda }_{EV}=-\frac{R_{L\mathrm{and}ft}{T}_{1i}}{V_{1i}}\cdot \frac{\partial }{\partial p_{1i}}\left(K_{MOt}\cdot n_{MOt}\left(c_{EV\mathrm{,1}}\cdot p_{1i}+c_{EV\mathrm{,2}}\right)\right)$$

$${\lambda }_{EV}=-\frac{R_{L\mathrm{and}ft}{T}_{1i}}{V_{1i}}{K}_{MOt}\cdot n_{MOt}\cdot c_{EV\mathrm{,1}}$$

$${\lambda }_{EV}=-\text{10}\text{.31}\frac{1}{s}$$

The amount of the eigenvalue contribution λ _EV is small and does not depend on the pressure p _1i in container 1 or on the pressure p _2i in container 2.

sehr stark vom Druckverhältnis über die Drosselklappe 3 ab, wie gemäß (8, 8.1, 8.2) dargestellt. $${\lambda }_{DK}=\frac{R_{Luft}{T}_{1i}}{V_{1i}}\cdot \frac{\partial }{\partial p_{1i}}{\dot{m}}_{DK}\left(p_{1i}\right)$$

$${\lambda }_{DK}=\frac{R_{Luft}{T}_{1i}}{V_{1i}}\cdot \frac{A\left({\alpha }_{DK}\right)}{\sqrt{R_{Luft}{T}_{2i}}}\psi \text{'}\left(\frac{p_{1i}}{p_{2i}}\right)$$

$${\lambda }_{EV}=-278.\text{32}\frac{1}{s}$$

In contrast, the eigenvalue contribution λ _DK of the mass flow ṁ _DK via the throttle valve 3 depends on the flow characteristic $\psi \left(\frac{p_{1i}}{p_{2i}}\right)$

depends very much on the pressure ratio across the throttle valve 3, as shown in accordance with (8, 8.1, 8.2). $${\lambda }_{DK}=\frac{R_{L\mathrm{and}ft}{T}_{1i}}{V_{1i}}\cdot \frac{\partial }{\partial p_{1i}}{\dot{m}}_{DK}\left(p_{1i}\right)$$

$${\lambda }_{DK}=\frac{R_{L\mathrm{and}ft}{T}_{1i}}{V_{1i}}\cdot \frac{A\left(a_{DK}\right)}{\sqrt{R_{L\mathrm{and}ft}{T}_{2i}}}\psi \text{'}\left(\frac{p_{1i}}{p_{2i}}\right)$$

$${\lambda }_{EV}=-278\text{32}\frac{1}{s}$$

theoretisch für Druckverhältnisse nahe eins gegen minus unendlich geht. Das bedeutet, dass auch der Eigenwertbeitrag λ_DK des Massenstroms ṁ_DK über die Drosselklappe 3 gegen minus unendlich geht.The reason for this is that the derivation of the flow rate characteristic $\psi \text{'}\left(\frac{p_{1i}}{p_{2i}}\right)$

theoretically goes to minus infinity for pressure ratios close to one. This means that the eigenvalue contribution λ _DK of the mass flow ṁ _DK via the throttle valve 3 also approaches minus infinity.

$${\lambda }_1=-856.31\frac{1}{s}{\lambda }_2=\text{ -11}\text{.16}\frac{1}{s}$$

If the model is considered according to equations (1) and (4), its dynamics are characterized by the eigenvalues λ ₁ and λ ₂ of the system matrix A according to (9, 9.1). $$A=\left[\begin{array}{cc}\frac{\partial }{\partial p_{1i}}{\dot{p}}_{1i}& \frac{\partial }{\partial p_{2i}}{\dot{p}}_{1i}\\ \frac{\partial }{\partial p_{1i}}{\dot{p}}_{2i}& \frac{\partial }{\partial p_{2i}}{\dot{p}}_{2i}\end{array}\right]=\left[\begin{array}{cc}-288.62& 282.97\\ 556.63& -578.85\end{array}\right]$$

$${\mathrm{<figure-callout\; id="1"\; label="\lambda "\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">\lambda </figure-callout>}}_1=-856.31\frac{1}{s}{\mathrm{<figure-callout\; id="2"\; label="\lambda "\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">\lambda </figure-callout>}}_2=\text{-11}\text{.16}\frac{1}{s}$$

It becomes clear that the model consisting of containers 1 and 2 and throttle valve 3 cannot be calculated using the explicit Euler method and a step size of 10 ms. This result can be transferred to a higher-level overall model.

In order to be able to calculate the model according to (1) and (4) with an increment of 10 ms, the integration rule for the classic Euler method (5.8) is expanded by one term according to the invention and thus approximated to the implicit Euler method. The procedure is explained in more detail below.

welches für alle stabilen Systeme gleichfalls eine stabile diskrete Implementierung unabhängig von der Schrittweite ermöglicht, kann mit Hilfe der Taylor-Reihenentwicklung $$ƒ\left(x^{i+1}\right)\approx ƒ\left(x^i\right)+\frac{\partial ƒ}{\partial x}{|}_{{\text{x}}^i}\left(x^{i+1}-x^i\right)$$

eine Näherungslösung in der Form $$x^{i+1}=x^i+\Delta t\cdot {\left(E-\Delta t\frac{\partial ƒ}{\partial x}{|}_{x^3}\right)}^{-1}\cdot ƒ\left(x^i\right)$$

ermittelt werden, wobei mit E die Einheitsmatrix und mit $${\frac{\partial ƒ\left(x\right)}{\partial x}|}_{x^i}={\left[\begin{array}{ccc}\frac{\partial ƒ}{\partial x_1}& \cdots & \frac{\partial {ƒ}_1}{\partial x_n}\\ ⋮& & ⋮\\ \frac{\partial {ƒ}_n}{\partial x_1}& \cdots & \frac{\partial {ƒ}_n}{\partial x_n}\end{array}\right]|}_{x^i}={A|}_{x^i}$$

die linearisierte Prozessdynamik gemäß Gleichung (5.14) beschrieben wird. Unter der Annahme einer gewissen Beschränktheit (u. a. der Erfüllung der Lipschitz-Bedingung) von Gleichung (11) liefert die Berechnungsvorschrift aus (12) ebenfalls eine numerisch stabile Lösung. Diese ist unabhängig von den Systemeigenschaften/der Systemdynamik und der Schrittweite der Berechnung. Dabei wird aus einem Vergleich von Gleichung (5.8) und Gleichung (12) deutlich, dass die Integrationsvorschrift für das explizite Euler-Verfahren erfindungsgemäß um eine arbeitspunktabhängige Verstärkung ${\left(E-\Delta t{\frac{\partial ƒ}{\partial x}|}_{x^i}\right)}^{-1}$

erweitert und letztendlich stabilisiert wird. Diese Tatsache wird durch die folgenden Ausführungen näher erläutert. Für die im Arbeitspunkt x_R linearisierte Darstellung des neuen Verfahrens aus Gleichung (12) folgt analog zur Herleitung von Gleichung (5.16) $$x^{i+1}=\left[I_n+\Delta t\cdot {\left(I_n-\Delta tA\right)}^{-1}\cdot A\right]x^i.$$

Starting from the system equation for the implicit Euler method $$x^{i+1}=x^i+\Delta t\cdot ƒ\left(x^{i+1}\right)$$

which also enables a stable discrete implementation for all stable systems, independent of the step size, can be developed with the help of the Taylor series expansion $$ƒ\left(x^{i+1}\right)\approx ƒ\left(x^i\right)+\frac{\partial ƒ}{\partial x}{|}_{{\text{x}}^i}\left(x^{i+1}-x^i\right)$$

an approximate solution in the form $$x^{i+1}=x^i+\Delta t\cdot {\left(E-\Delta t\frac{\partial ƒ}{\partial x}{|}_{x^3}\right)}^{-1}\cdot ƒ\left(x^i\right)$$

be determined, where E is the identity matrix and $${\frac{\partial ƒ\left(x\right)}{\partial x}|}_{x^i}={\left[\begin{array}{ccc}\frac{\partial ƒ}{\partial x_1}& \cdots & \frac{\partial {\mathrm{<figure-callout\; id="1"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_1}{\partial x_n}\\ ⋮& & ⋮\\ \frac{\partial {ƒ}_n}{\partial x_1}& \cdots & \frac{\partial {ƒ}_n}{\partial x_n}\end{array}\right]|}_{x^i}={A|}_{x^i}$$

the linearized process dynamics are described according to equation (5.14). Assuming that Equation (11) is limited to a certain extent (including the fulfillment of the Lipschitz condition), the calculation rule from (12) also provides a numerically stable solution. This is independent of the system properties/system dynamics and the increment of the calculation. A comparison of equation (5.8) and equation (12) makes it clear that the integration rule for the explicit Euler method according to the invention includes an operating-point-dependent gain ${\left(E-\Delta t{\frac{\partial ƒ}{\partial x}|}_{x^i}\right)}^{-1}$

expanded and ultimately stabilized. This fact is explained in more detail by the following statements. For the representation of the new method linearized at the operating point x _R from Equation (12), it follows analogously to the derivation of Equation (5.16): $$x^{i+1}=\left[I_n+\Delta t\cdot {\left(I_n-\Delta tA\right)}^{-1}\cdot A\right]x^i.$$

Compared to Equation (5.16), the associated modified system matrix results in $$A^{\prime }={\left(I_n-\Delta tA\right)}^{-1}\cdot A={\left(E-\Delta t{\frac{\partial ƒ}{\partial x}|}_{x_R}\right)}^{-1}\cdot {\frac{\partial ƒ}{\partial x}|}_{x_R}.$$

und die damit verbundenen Eigenwerte zu $${\lambda }_1=-\text{89}\text{,54}\frac{1}{S}{\lambda }_2=-\mathrm{10,04}\frac{1}{S}.$$

This change in the system matrix caused by the amplification represents the key to the stable numerical calculation of the system with a fixed sampling time. For clarification, the changed system matrix from equation (15) is calculated and evaluated in the working point considered as an example. With the values from table (a), A' results in: $$A\text{'}\left[\begin{array}{cc}-\mathrm{36:14}& 26.62\\ 52.36& -63.44\end{array}\right].$$

and the associated eigenvalues $${\lambda }_1=-\text{89}\text{,54}\frac{1}{\mathrm{<figure-callout\; id="2"\; label="S\; \lambda "\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">S</figure-callout>}}{\lambda }_{}{}_2=-10.04\frac{1}{S}.$$

A comparison with the condition from equation (5.20) provides the proof of stability.

und $${\dot{p}}_{2i}={ƒ}_2\left(p_{1i},p_{2i}\right)$$

eingeführt. Entsprechend der Rechenvorschrift aus Gleichung (12) folgt $$\left[\begin{array}{c}{p}_{1i}^{i+1}\\ p_{2i}^{i+1}\end{array}\right]=\left[\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}\right]+{\left\{\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]-\Delta t{\left[\begin{array}{cc}\frac{\partial {ƒ}_1}{\partial p_{1i}}& \frac{\partial {ƒ}_1}{\partial p_{2i}}\\ \frac{\partial {ƒ}_2}{\partial p_{1i}}& \frac{\partial {ƒ}_2}{\partial p_{2i}}\end{array}\right]|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right\}}^{-1}\left[\begin{array}{c}{ƒ}_1\left(p_{1i}^i,p_{2i}^i\right)\\ {ƒ}_2\left(p_{1i}^i,p_{2i}^i\right)\end{array}\right]$$

The calculation rule to be implemented according to equation (12) for the present application example is now derived below for illustrative purposes. Referring to the above application example with the differential equations (1) and (4), the simplified notation is used here $${\dot{p}}_{1i}={ƒ}_1\left(p_{1i},p_{2i}\right)$$

and $${\dot{p}}_{2i}={ƒ}_2\left(p_{1i},p_{2i}\right)$$

introduced. According to the calculation rule from Equation (12). $$\left[\begin{array}{c}{p}_{1i}^{i+1}\\ p_{2i}^{i+1}\end{array}\right]=\left[\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}\right]+{\left\{\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]-\Delta t{\left[\begin{array}{cc}\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_1}{\partial p_{1i}}& \frac{\partial {ƒ}_1}{\partial p_{2i}}\\ \frac{\partial {ƒ}_2}{\partial p_{1i}}& \frac{\partial {ƒ}_2}{\partial p_{2i}}\end{array}\right]|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right\}}^{-1}\left[\begin{array}{c}{ƒ}_1\left(p_{1i}^i,p_{2i}^i\right)\\ {ƒ}_2\left(p_{1i}^i,p_{2i}^i\right)\end{array}\right]$$

und $$p_{2i}^{i+1}=p_{2i}^i+\Delta t\frac{\left(1-\Delta t{\frac{\partial {ƒ}_1}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right){ƒ}_2\left(p_{1i}^i,p_{2i}^i\right)+\Delta t{\frac{\partial {ƒ}_2}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}{ƒ}_1\left(p_{1i}^i,p_{2i}^i\right)}{\left(1-\Delta t{\frac{\partial {ƒ}_1}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)\left(1-\Delta t{\frac{\partial {ƒ}_2}{\partial p_{2i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)-\left(-\Delta t{\frac{\partial {ƒ}_1}{\partial p_{2i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)\left(-\Delta t{\frac{\partial {ƒ}_2}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)}$$

die diskrete Berechnungsvorschrift als Grundlage für die echtzeitfähige Implementierung des Modells.If one solves the matrix inversion in equation (19), it follows with $$p_{1i}^{i+1}=p_{1i}^i+\Delta t\frac{\left(1-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_2}{\partial p_{2i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right){ƒ}_1\left(p_{1i}^i,p_{2i}^i\right)+\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_1}{\partial p_{2i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}{ƒ}_2\left(p_{1i}^i,p_{2i}^i\right)}{\left(1-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_1}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)\left(1-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_2}{\partial p_{2i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)-\left(-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_1}{\partial p_{2i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)\left(-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_2}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)}$$

and $$p_{2i}^{i+1}=p_{2i}^i+\Delta t\frac{\left(1-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_1}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right){ƒ}_2\left(p_{1i}^i,p_{2i}^i\right)+\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_2}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}{ƒ}_1\left(p_{1i}^i,p_{2i}^i\right)}{\left(1-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_1}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)\left(1-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_2}{\partial p_{2i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)-\left(-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_1}{\partial p_{2i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)\left(-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_2}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)}$$

the discrete calculation rule as the basis for the real-time capable implementation of the model.

und $$p_{2i}^{i+1}=p_{2i}^i+\Delta t\frac{\left(1-\Delta t{\frac{\partial {ƒ}_1}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right){ƒ}_2\left(p_{1i}^i,p_{2i}^i\right)+\Delta t{\frac{\partial {ƒ}_2}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}{ƒ}_1\left(p_{1i}^i,p_{2i}^i\right)}{\left(1-\Delta t{\frac{\partial {ƒ}_1}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)\left(1-\Delta t{\frac{\partial {ƒ}_2}{\partial p_{2i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)-\left(-\Delta t{\frac{\partial {ƒ}_1}{\partial p_{2i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)\left(-\Delta t{\frac{\partial {ƒ}_2}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)}$$

ergeben. Dabei wird zunächst das Ergebnis aus Gleichung (23) berechnet und anschließend in Gleichung (22) eingesetzt. Damit lässt sich ein Teil des Rechenaufwandes einsparen.At this point, solving the equation by inverting the matrix makes only limited sense for reasons of computational effort. Here, among other things, a solution of the system of equations according to the Gauss elimination method can be used, whereby the equations to be solved are $$p_{1i}^{i+1}=p_{1i}^i+\Delta t\frac{{ƒ}_1\left(p_{1i}^i,p_{2i}^i\right)+{\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_1}{\partial p_{2i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\left(p_{1i}^{i+1}-p_{2i}^{i+1}\right)}{1-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_1}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}}$$

and $$p_{2i}^{i+1}=p_{2i}^i+\Delta t\frac{\left(1-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_1}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right){ƒ}_2\left(p_{1i}^i,p_{2i}^i\right)+\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_2}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}{ƒ}_1\left(p_{1i}^i,p_{2i}^i\right)}{\left(1-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_1}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)\left(1-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_2}{\partial p_{2i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)-\left(-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_1}{\partial p_{2i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)\left(-\Delta t{\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="ƒ"\; filenames="DE102011014767B4\_0067.png"\; state="\{\{state\}\}">ƒ</figure-callout>}}_2}{\partial p_{1i}}|}_{\begin{array}{c}{p}_{1i}^i\\ p_{2i}^i\end{array}}\right)}$$

result. First, the result from equation (23) is calculated and then used in equation (22). This saves part of the computational effort.

wird dabei in Kennfeldern oder Kennlinien im Steuergerät abgelegt und zur Online-Berechnung herangezogen.In order to further drastically reduce the computing effort, an operating point-dependent offline calculation of the amplification matrix is also possible. The working point dependent matrix $$\mathrm{\Gamma }\left(x_R\right)={\left(E-\Delta t{\frac{\partial ƒ}{\partial x}|}_{x_R}\right)}^{-1}.$$

is stored in characteristic diagrams or characteristic curves in the control unit and used for online calculation.

Reference List

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throttle

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air filter

Claims (5)

Method for operating an internal combustion engine, wherein a model of the internal combustion engine serving to control and regulate the internal combustion engine is calculated using the explicit Euler method with a fixed increment, characterized in that the explicit Euler method uses a supplementary, operating-point-dependent term for the required fixed step size is stabilized by shifting the eigenvalues of the system matrix. procedure after Claim 1 , characterized in that the term is a gain factor such that the explicit Euler method used for integration is approximated to the implicit Euler method. procedure after patent claim 2 , characterized in that an approximate solution is determined based on the system equation for the explicit Euler method with the aid of the Taylor series expansion. procedure after patent claim 2 , characterized in that the system equation for the implicit Euler method has the following form $$x^{i+1}=x^i+\Delta t\cdot ƒ\left(x^{i+1}\right)$$

and the Taylor series expansion has the following form $$ƒ\left(x^{i+1}\right)\approx ƒ\left(x^i\right)+{\frac{\partial ƒ}{\partial x}|}_{x^i}\left(x^{i+1}-x^i\right)$$

and the approximate solution is found in the following form $$x^{i+1}=x^i+\Delta t\cdot {\left(E-\Delta t{\frac{\partial ƒ}{\partial x}|}_{x^i}\right)}^{-1}\cdot ƒ\left(x^i\right),$$

where with E the identity matrix and with $${\frac{\partial ƒ\left(x\right)}{\partial x}|}_{x^i}={\left[\begin{array}{ccc}\frac{\partial {ƒ}_1}{\partial x_1}& \cdots & \frac{\partial {ƒ}_1}{\partial x_n}\\ ⋮& & ⋮\\ \frac{\partial {ƒ}_n}{\partial x_1}& \cdots & \frac{\partial {ƒ}_n}{\partial x_n}\end{array}\right]|}_{x^i}$$

the linearized process dynamics is described. procedure after Claim 1 until 4 , characterized in that an amplification matrix is calculated offline and stored in the engine control unit in the form of operating-point-dependent parameters.

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