AT517251A2 - Method for creating maps - Google Patents

Abstract

In order to be able to create a smooth characteristic map from known data, wherein predetermined boundary conditions are also to be taken into account in the mapping, it is provided that a target function (Z) is defined, into which an error (F) and an energy functional (ε) are received a function of the output quantity (ui) and of the known output quantities {u * in} i = 1,..., q; n = 1,..., N is used as the error (F) and a function is used as the energy functional (ε) which Terme as squares of the kth derivatives of the output quantities (ui) according to the input quantities (x), with k = floor (d / 2 + 1), and the objective function (Z) with respect to the output quantity (ui) and taking into account boundary conditions (h) is optimized.

Description

Method for creating maps

The present invention describes a method for generating i maps,

wherein each characteristic map is a d-dimensional characteristic map of d input variables x = [x! ..... xd] T and in each case one output variable u, and N input variables and the associated output variables are known for each characteristic diagram.

The calibration of modern internal combustion engines requires flexible methods for the mapping of maps. Maps are known to be d-dimensional tables of values that model the relationship and interdependencies of the physical dimensions of a process. Maps are often calculated from existing measurements. It is also known to resort to models of the physical process for the mapping. When creating the characteristic map, certain boundary conditions for the d dimensions must be adhered to as a rule. Similarly, it is often required that the maps are smooth, so do not involve value jumps. This also results in different applications of such maps.

Frequently, with maps, certain state quantities, e.g. Torque and speed of the internal combustion engine, to certain control variables, such as. a throttle position or exhaust gas recirculation adjustment, shown to meet certain targets, such as e.g. an emission quantity (e.g., NOx) or a consumption amount (e.g., fuel consumption). A map generally represents a control variable as a function of the n state variables, which is then also referred to as an n-dimensional map. The assessment is made on the basis of real measurements, for example by test runs on the internal combustion engine on a test bench, or also using a model of a target as a function of the state variables and the manipulated variable (for example NOx emission as a function of rotational speed and torque at different throttle positions). Due to the effort but not the entire map can be measured, but there are individual measurements made and it will be created from the data obtained the maps. The problem of the mapping of such maps can then be formulated as an optimization problem in which an objective function (for the targets) is optimized for the available data as a function of the state variables and the control variables, and taking into account boundary conditions for the state variables and / or control variables. A problem of such a parametric optimization for mapping is that the maps created with it are not smooth and therefore usually a post-processing (smoothing) of the maps is necessary.

Maps are also used to describe the dependencies of model sizes of a model of a physical process in the form of value tables. A model of a physical process can consist of several submodels (subcharacteristics), whereby a subcharacteristic field can in turn be dependent on other subcharacteristics in any way. These dependencies are then optimally adapted to existing data (such as measured values) in the form of value tables. Here it is obvious that the individual sub-maps can not be created independently. For this purpose, global solution strategies are needed, whereby from the viewpoint of the solution efficiency it is necessary that the number of parameters (dimensions) for each sub-map is as small as possible. Here, too, certain boundary conditions must be observed. For technical, physical systems or processes, it is important to be able to take boundary conditions into account in the mapping, since physical variables are always subjected to certain boundary conditions. For example, an internal combustion engine can only be operated up to a certain maximum speed and up to a maximum torque. Characteristic maps for such technical, physical systems or processes are also mostly subject to the requirement of the smoothness of the created characteristic field, since value jumps do not occur in the real technical, physical system or process. Apart from that, value jumps in a control variable from a control point of view are also undesirable because this can lead to an undesirable step response, to instability, of the system or process.

It is therefore an object of the present invention to provide a method with which a smooth characteristic map can be created from known data, wherein predetermined boundary conditions can also be taken into account in the mapping.

This object is achieved according to the invention by defining an objective function into which an error and an energy functional are received, a function of the output variable Ui and of the known output variables being used as the error, and a function is used as the energy functional, the terms being squares of the k -th derivatives of the output quantities u, according to the input variables x, with k = [d / 2 + lJ, contains and the objective function with respect to the output quantities u, and is optimized taking into account boundary conditions.

The required smoothness of the maps is inventively modeled using the energy functional. Two-dimensional maps correspond to the physical model of a plate with a small deflection. Forces are applied to this plate in the direction of the data, the deflection of the plate is damped by the stiffness of the plate. According to d'Alembert's principle, the plate (the map) minimizes the bending energy. For higher dimensional maps, only the energy functional has to be adapted to the higher dimension.

The optimization problem is solved particularly easily with methods of the finite elements, in which the output quantities u, as a linear combination

from with the weights

weighted known basis functions

nahä hert is.

As objective function is preferred

used.

In an advantageous embodiment, an optimization problem of the form ηύηξξ (α, · Β + Ι.-1.τ) ξ, -ξμυ,

/ζίχ.,ζϊ; (x.)) <0 V j = \, ..., M \ J 'J, with boundary conditions h (x, u (x)), where U, an output vector

, By an energy matrix

and L writes a data matrix.

In an alternative embodiment, under certain conditions, an equation (a.-B + L-lI) ξ. = LU. £ = \ £ £ f system '1> 1 1 according to the weights ^ ί ^ ΐ' ··· 'τ> <Μ j, which is a particularly efficient solution method for creating the maps.

The subject invention will be explained in more detail below with reference to Figures 1 to 4, which show by way of example, schematically and not by way of limitation advantageous embodiments of the invention. It shows

1 shows the result of a map according to the invention in a first application,

1 shows the result of a map according to the invention in a second application,

3 shows a third application and

4 shows the result of a characteristic diagram according to the invention for the third application.

A map is understood to be a table of values which maps a d-dimensional input variable vector (or state vector) x to an output variable (or manipulated variable) Ui. This is also called a d-dimensional map. In further

Sequence will be spoken only of input variables and output variables, even if the map rather describes the case of a state variable manipulated variable relationship. As a rule, there are several characteristic diagrams for q output quantities u = [ui, uq] T. The input variables define, for example, the operating point of the internal combustion engine as torque and rotational speed. For the characteristic map position according to the invention, an analogy to a thin plate is produced. For the two-dimensional case, a map is considered to be a two-dimensional plate that attacks forces that bend the plate toward the data. The plate counteracts the bending due to their rigidity. This approach can be generalized to any dimension. The smoothness of the bent plate may be in the form of a function of energy, e.g. the bending energy to be measured. The bending energy for a two-dimensional plate is e.g. in Ventsel, E., et al., "Thin plates and shells: theory: analysis, and applications.", 2001 CRC press. Iske, A., "Multiresolution methods in scattered data modeling", 2004, Vol. 37, Springer Science &amp; Business Media describes the generalization of bending energy for higher-dimensional cases. If the bending energy for a thin plate is transferred to the objective application of a characteristic field, an energy functional ε results in the form

Where Ω is the d-dimensional space of the input quantities x. u, is the ith output. The energy functional ε is written in the known multi-index notation with a differential operator DK for a multidimensional derivation. Kappa goes through all indices with magnitude k, and is well-defined in terms of multi-index notation: κ = [Kv ..., Kd] with

, For example, the energy functional ε for a 2- or 3-dimensional map (d = 2 or 3) would contain the 2nd derivatives (k = 2), whereby also partial d2u d2u

Are included, e.g. -Γ and-. dxi dxjch ^

The energy functional thus contains terms which are characteristic of the bending energy of a (generalized multi-dimensional) plate, namely squares of the kth derivatives (in the multidimensional case also as partial derivatives) of the output quantities u, according to the input quantities x, which are integrated via the input variables , The energy functional ensures a smooth function. Of course, the energy functional has no physical significance for a map. Nevertheless, the term energy functional is maintained here as well.

The optimization often takes place in such a way that a target function Z is optimized as a function of other variables. As the objective function Z, an error F, e.g. a quadratic error, used between existing (measured) data and real (calculated) data. If ixl) n = i, .., n is the N available (for example, measured) input quantities, and {u * n} i = i, ..., q; n = outputs that belong to the N available (for example, measured) then gives an error F, a manipulated variable u, as

According to the invention, an objective function Z is optimized in which the energy functional e (Uj) of each characteristic field and also an error F, between the characteristic field and the physical process are received, ie Z = f (((u,), F (u,)), for all i. The mistake can be thought of as analogous to the thin plate as the forces bending the plate in the direction of the data.

This results in an optimization problem in the general form

which can be implemented as follows.

With an error F, as a quadratic error results then

This optimization problem has to be solved to determine the characteristic map. Therein α denotes a stiffness parameter (based on the stiffness of a plate), which is fixed in advance in the simplest case as a constant, and h the boundary conditions.

By using an energy function, the smoothness of the characteristic field can be ensured even during optimization. A post-processing of the map to smooth the map is no longer necessary. At the same time, this approach also makes it possible to consider boundary conditions right from the optimization stage. For such optimization problems, there are several known approaches, but based on the fact that m ,. (je *) as a measured value or as a calculation value, e.g. from a model of the physical process known. In the above optimization problem, however, both the energy functional s (Ui) and the error term contain the function Ui (x), which is not known. In order to solve the optimization problem, finite element methods are also used according to the invention.

For this, the function u, (x) is given by a linear combination of weighted basis function

M φ = φΜ] Τ, ie ύί (χ) = ^ ξ..φ] (χ) with the weights ξι = [ξη, ..., ξιΜ] τ. 7 = 1

In finite element methods, M support points are defined, which are distributed over the state space Ω, preferably uniformly distributed. In principle arbitrary functions can be used as basic functions φ. Preferably, the hat functions often used in finite-element methods are used as basis functions, wherein for each support point M a hat function is defined which is one at the support point M and otherwise, for all other support points, is zero. Inserting this approach into the above optimization problem results in the optimization problem after a few transformations

Here U denotes an output variable vector JJt = [un, ..., uiN ~ ^, By an energy matrix

(again in multi-index notation) and L a data matrix

This optimization problem may then be after the weights

can be solved, whereby the maps for the output quantities u, from the finite element method approach can be calculated.

If the boundary conditions for the individual output variables are independent of one another, the general optimization problem with respect to the manipulated variables defined above can be decoupled. This means that each characteristic map for the manipulated variables u, alone and released from the other maps may be treated. This results in a considerable simplification in the solution of the optimization problem. If the constraints of the individual maps are independent and do not preempt inequality constraints, the components may be; are independently calculated as solutions of Euler-Lagrange equations taking the following form

Where Ak = AAkA denotes the kth iterated Laplace operator. This equation is a poly harmonic partial differential equation that can be solved with appropriate constraints. The obvious boundary conditions are

although other boundary conditions can be applied. In it 5Ω describes the edge of the space of the input quantities Ω. Particularly advantageous can instead of dAklu. i -L = 0 also Dirichlet conditions uA = g, with an arbitrary function g. Thus, in particular, the edge regions of the maps can be arbitrarily specified.

Using the finite element methods with the mapping functions as described above, the optimization problem can then be rewritten

or equivalent to

(2)

Thus, to determine the weights ξί no more optimization problem must be solved, but the weights, and subsequently the maps, can be directly calculated by solving a Μ x M system of linear equations.

FIG. 1 shows the result of an optimization according to the invention with target function with energy functional e (Ui) and with solution by means of finite element methods. In this case, a map 1 for the single output variable u was determined as a function of three input quantities x2, x3. In the example shown, the input variables are the supercharging pressure setpoint of a turbocharger (xi), the engine speed (x2), ambient pressure (x3) and the output variable u is the duty cycle of the supercharger pressure regulator.

The map 1 was thus created as a three-dimensional (d = 3) map, where it is shown in Figure 1 as a two-dimensional map 1 by the map 1 is shown for different ambient pressures (several maps one above the other in Figure 1). As data for the characteristic map position there were n measured data points DPn (u *, x * n, x * 2n, x] n), which are stored in

Fig.1 are shown as dots. From the above optimization problem (1) the map 1 was then determined.

A second application example of the method according to the invention will be explained with reference to FIG. This is based on an existing model, for example an emission model, which calculates a specific model value, for example the NOx emissions of an internal combustion engine, as a function of input variables x and the output variable. In the exemplary embodiment shown, an emission quantity is calculated using an emission model comprising two input variables Xi, x2 (such as rotational speed and torque) and an output variable u (such as air / fuel ratio). Based on the model, for certain input quantities Xi, x2, the output u can be varied in order to find the optimal output u, which minimizes the model value. This can be done for all N support points. This results in an optimized map with respect to the model value at the N interpolation points (top left diagram in FIG. 2), with the input variables x 1, x 2 and the respectively associated output variable u (points in the diagram). Since each input variable x and each output variable u has a permissible value range, the permissible value range 2 can also be represented in the characteristic field 1 (dashed line in FIG. 2).

With the calculation of the optimal model values on the basis of a model, however, a smooth characteristic diagram with respect to the input quantities x.sub.x2 and the output quantity u is generally not obtained. As a result, one would have to smooth out the map generated thereby later.

In order to avoid this, the input quantities x.sub.x2 and the output quantity u at the N interpolation points which were used for the calculation of the characteristic field optimized with respect to the model value can be used as existing data {w * B}; = i,...,? «= I, ..., iV and {jc *} b = U; JV be used for the characteristic map position according to the invention. Thus, as described above, a smooth map 2 can be calculated. The result is shown in Fig.2 top right. 2 are represented as two-dimensional diagrams in which a state variable Xi is plotted on one axis and the other state variable x2 was projected onto the two-dimensional map for different fixed values. The lines in the characteristic diagrams of FIG. 2 thus represent lines of constant state variables x 2. Of course, a representation as a three-dimensional diagram would also be possible.

As can be seen, however, violations of the permissible value range may result. To avoid this, the boundary conditions can be optimized.

For example, using optimization after the optimization problem (1), one could use the following constraints

In it 5Ω describes the edge of the space of the input quantities Ω. This makes use of the present circumstance that for larger xr values the manipulated variable u can be set to zero (FIG. 2, top left). And you get the map below left in Fig.2. The diagram at the bottom right in FIG. 2 shows the solution when using the map determination after the optimization problem (2), with the following boundary conditions being applied

With the figures 3 and 4, another possible application of the map of the invention is described. In practice, it often happens that the relationship between model input quantities Up and a model output variable Y exists as model 3 from a plurality of interdependent submodels 4a, 4b, 4c. It is thus known how the submodels 4a, 4b, 4c influence one another and how the model output variable Y is determined. Such a model is shown by way of example in FIG. In this example, four model inputs U1, U2, U3, U4 are provided, from which a model output Y is calculated. The model input variables U1, U2, U3, U4 are processed in the submodels 4a, 4b, 4c, the submodels partially using the outputs of other submodels as input to the respective submodel. For example, the output ua of the partial model 4a is a model input xbi of the partial model 4b. Such submodels 4 can be quite complex and computationally expensive. However, it is also conceivable that the submodels 4 themselves are not known at all, but because of physical relationships only the functional dependencies in the model 3 (ie the model structure) are known, ie how a submodel influences another submodel. Therefore, it is sometimes desirable to map the submodels 4 as maps, which allows the model output Y to be determined very quickly and easily as a function of the model input U. Since the submodels 4 are dependent on each other, but the individual maps for the submodels 4 can not be determined individually, but it must all maps are determined simultaneously.

For this purpose, one starts from known data which maps a model input U (here U1, U2, U3, U4) onto a model output Y. For example, this data may be obtained from measurements on a real physical process modeled by the model. For each existing N combinations of model input U and associated model output Y, the output variables ua, ub, uc of the submodels 4a, 4b, 4c are now varied at the model input U until the known model output Y is optimally approximated. For this purpose, suitable dissolution methods can be used, but it does not depend on the concrete Löseverfahen. Thus, for each submodule 4a, 4b, 4c, the known output quantities {uin} i = 1 q, n = u N are obtained for the characteristic map position according to the invention. The corresponding input variables {x *} "= 1 ^ result from the model structure, whereby the output quantities {uin} i = 1 q; n = h N of a submodel are partially used as input variables in another submodel. Thus, the wanted maps 1a, 1b, 1c can be determined as described above, for example according to the optimization problem (1) or (2). The output quantities ua, Ub, uc are here to be regarded as functions of the respective inputs xa, xb, xc, and therefore can not simply be varied directly by a solver. However, if these functions are represented as maps, it is known from the previous methods that these maps can be represented or parametrized by a few known data u *, x *. The solvers used thus vary the parameter vectors u *, x * until the observed measured data Y match the model output. The output quantities ua, ub, uc result then as maps through the optimal parameter vectors u *, x *.

The result is shown by way of example in FIG. 4, in which the characteristic diagrams 1a, 1b, 1c produced by the method according to the invention are represented as two-dimensional diagrams.

Claims (5)

claims 1. A method for generating i maps, each map is a d-dimensional map of d input variables x = [x1 ..... xd] T and each an output variable (u,) and for each map N input variables {x * } B = 1 ^ and the associated output quantities {u * n} i = \, ..., q; n = \, ..., N are known, characterized in that an objective function (Z) is defined in which an error (F) and an energy functional (ε), where as error (F) a function of the output variable (Uj) and the known output variables {u * "} i = l" = K ^ N ver is used and as energy functional ( ε) a function is used which contains terms as squares of the k-th derivatives of the output quantities (u,) according to the input quantities (x), with k = [d / 2 + lJ, and that the objective function (Z) with respect to the output quantity (u,) and under consideration of boundary conditions (h) is optimized. 2. The method according to claim 1, characterized in that the output variables (Ui) as a linear combination of weighted with the weights ξι = [ξη, ..., ξ ^ ge known basis functions φ = [φι, ..., φΜ] τ is approximated. 3. The method according to claim 1, characterized in that as a target function (Z) is used. 4. The method according to claim 2 and 3, characterized in that an optimization problem of the form with boundary conditions h (x, u (x)), where U, an output vector By an energy matrix and Leas data matrix L = [(ρ (χ *), ..., φ (χ ^)]. 5. The method according to claim 2 and 3, characterized in that a system of equations (α; · B + L Ü = LUi to the weights ξ! = [Ξη, ..., ξΙΜΥ is resolved.