DE102020116218A1 - Method and device for operating a technical system - Google Patents

Abstract

According to the invention, a method for the approximate solution of an instantaneous control problem is provided. The basic procedure is based on formulating a moment-generating and / or cumulant-generating functional that is developed into a power series of a defined order. The procedure for determining the manipulated variable is to break off the respective power series at a lower order and to solve the resulting equations according to the difference between the actual value of the manipulated variable and its setpoint. This solution can then be used in each case to evaluate the remainder of the respective power series, in which case the value that has the respective smaller remainder is used for regulation. The great advantage of the procedure according to the invention is that comparatively simple expressions can be used for regulation without having to accept any loss of accuracy, i.e. H. According to the invention, a high control quality, as expected from numerical approaches, is combined with the ability to implement this on embedded systems with low computing power. According to the invention, a method for the approximate solution of a dynamic control problem is also provided. A corresponding device is also proposed. Such a device is in particular part of a vehicle.

Description

The present invention relates to a method and a device for operating a technical system having the features according to the patent claims.

According to the DE19634923A1 is a method for the regulation of non-linear technical processes / systems by linearization with a deviation observer state of the art. A model is formulated for an underlying technical system or a technical process, which relates the variable to be controlled with the existing manipulated variable.

The suitability of such methods in connection with control problems in embedded systems is always problematic, since these methods do not allow numerical solutions and are dependent on analytical functions.

It is the object of the present invention, proceeding from the prior art, to provide an improved method and an associated device for operating a technical system.

This object is achieved by a method and a device having the features according to the patent claims.

According to the invention, a method for the approximate solution of an instantaneous control problem is first provided. The basic procedure is based on formulating a moment-generating and / or cumulant-generating functional that is developed into a power series of a defined order. The procedure for determining the manipulated variable is to break off the respective power series at a lower order and to solve the resulting equations according to the difference between the actual value of the manipulated variable and its setpoint. This solution can then be used in each case to evaluate the remainder of the respective power series, in which case the value that has the respective smaller remainder is used for regulation. A central finding here is given by the fact that a finite power series of a generating functional leads to an infinite partial summation of the other generating functional and therefore goes far beyond a purely perturbative approach. The great advantage of the procedure according to the invention is that comparatively simple expressions can be used for regulation without having to accept any loss of accuracy, i.e. H. According to the invention, a high control quality, as expected from numerical approaches, is combined with the ability to implement this on embedded systems with low computing power. A linear approximation of the cumulant-generating functional is often sufficient, but conversely it would correspond to an infinite sub-series of the torque-generating functional. Ultimately, the corresponding derivatives of the model function must be known to evaluate the power series. This can be implemented either analytically or using characteristics.

In summary, the approach according to the invention is particularly suitable for control problems in embedded systems that do not allow numerical solutions and are dependent on analytical functions. According to the invention, a method for the approximate solution of a dynamic control problem is also provided. A corresponding device is also proposed. Such a device is in particular part of a vehicle.

In the further course, an application of the method according to the invention is shown using the example of an instantaneous (control and / or) regulation problem.

The method according to the invention can be used in technical systems and / or technical processes of any kind, that is to say for their operation.

With regard to the definition of the term “technical system”, see G. Ropohl, for example: General technology: A system theory of technology. Munich / Vienna 1979 referenced. The features mentioned there, which characterize a technical system, are hereby expressly included in the present description. With regard to the definition of the term “technical process”, refer to the DIN IEC 60050-351 referenced. The features mentioned there, which characterize a technical process, are hereby expressly included in the present description.

Since the control problem initially considered in connection with the operation of a technical system / process is an instant control problem, a functional relationship between a manipulated variable y and a variable to be regulated or controlled can be found, so that applies $$f\left(x\right)=Z\left(y\right)$$

It is assumed here that the manipulated variable function Z (y) can be differentiated infinitely often. Both ƒ and Z can depend on additional other quantities. There is no separate notation for this for reasons of clarity.

This is an instant control problem insofar as a change in y causes an instant change in Z and therefore in ƒ and x. A transfer over time or runtime effects are not explicitly taken into account. The quantities x, y and other quantities occurring in the system can of course be implicitly time-dependent. In this context, it is only important that relationship (1) is at least approximately valid.

The instantaneous control problem is solved when it is possible to find an inverse function Z ^{-1 such} that $$y=Z^{-1}\left[f\left(x\right)\right].$$

As a rule, such an inverse function can only be determined analytically precisely in special cases. For this reason, one often makes do with linearizing the system equations around operating points. Such approaches are also used in dynamic control approaches. The disadvantage of linearized solutions is that the individual operating points must be evaluated in a dedicated manner and the controller used must be designed. In many cases, moreover, the system equations are so non-linear that linearization is only possible with difficulty and must be carried out using large data structures (characteristic maps) that require a high degree of storage volume. For this reason, non-linear solution methods are often used to find an approximate solution of the shape (2). The method according to the invention represents such an approach.

Y _is an actual value to the manipulated variable to a corresponding value of the control variable x _is obtained. Now giving way but x _is the corresponding setpoint x _to off, it can be reversed concluded that a more optimal value for the manipulated variable y can be found, in the sense that this value will disappear the difference between desired and actual values of the controlled variable. This more optimal value for the manipulated variable y is referred to in the following as y _soll (setpoint of the manipulated variable). Furthermore, a difference between the actual value of the manipulated variable y _ist and the setpoint of the manipulated variable y set _is defined as $$\xi :=y_{\text{should}}-y_{\text{is}}$$

On the basis of a sufficient radius of convergence, Z can then be expanded into a power series in ζ according to the present invention. For this purpose, according to the invention, the quotient is formed from the reference variable function ƒ (x _soll ) and the actual manipulated variable function Z (y _ist ) on the basis of equation (1). Accordingly, applies or this delivers $$\frac{f\left(x_{\text{should}}\right)}{Z\left(y_{\text{is}}\right)}=\frac{Z\left(y_{\text{is}}+\xi \right)}{Z\left(y_{\text{is}}\right)}=:G_{\mathrm{S.}}\left[x_{\text{should}}\left(y_{\text{is}},\xi \right),y_{\text{is}}\right]$$

In practice, the variable to be controlled, namely the setpoint controlled variable function or reference variable function ƒ (x _soll ), is related to the existing manipulated variable, namely the actual manipulated variable function Z (y _ist ).

According to the invention, the quantity G _S (y _ist , ζ), which we will refer to as the torque-generating functional of the instantaneous control problem with regard to the following, is thus defined. A power series expansion up to order N yields $$G_{\mathrm{S.}}\left[x_{\text{should}}\left(y_{\text{is}},\xi \right)y_{\text{is}}\right]=\frac{1}{Z\left(y_{\text{is}}\right)}{\displaystyle \sum _{j=0}^N\frac{d^{j}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^j}\frac{{\xi }^j}{j!}}+\frac{1}{Z\left(y_{\text{is}}\right)}\frac{d^{N+1}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!}.$$

das Restglied der Potenzreihe zur entwickelten Ordnung dar. Wie zu erkennen ist, besteht eine mathematische Äquivalenz zwischen Gleichung (5) und einer momentenerzeugenden Funktion einer Wahrscheinlichkeitsverteilung, welche in den statistischen Wissenschaften häufig zum Einsatz kommt. Formal kann die Größe $$m_j:=\frac{1}{Z\left(y_{\text{ist}}\right)}\frac{{\text{d}}^{j}Z\left(y_{\text{ist}}\right)}{\text{d}{y}_{\text{ist}}^j}$$

als j-tes Moment der Stellgrößenfunktion Z(y) bezeichnet werden.The last summand is here $${\mathrm{R.}}_{G_{\mathrm{S.}}}=\frac{1}{Z\left(y_{\text{is}}\right)}\frac{{\text{d}}^{N+1}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!}$$

represents the remainder of the power series for the developed order. As can be seen, there is a mathematical equivalence between equation (5) and a moment-generating function of a probability distribution, which is often used in statistical sciences. Formally, the size $$m_j:=\frac{1}{Z\left(y_{\text{is}}\right)}\frac{{\text{d}}^{j}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^j}$$

can be referred to as the jth moment of the manipulated variable function Z (y).

Furthermore, it is possible according to the invention, the ratio of target controlled variable function f (x _soll) and Istregelgrößenfunktion ƒ _(x) to form on the basis of equation (1) $$G_{\mathrm{R.}}\left[x_{\text{should}}\left(y_{\text{is}},\xi \right),y_{\text{is}}\right]:=\frac{f\left(x_{\text{should}}\right)}{f\left(x_{\text{is}}\right)}=\frac{Z\left(y_{\text{is}}+\xi \right)}{f\left(x_{\text{is}}\right)}.$$

wobei bei der Entwicklung der Potenzreihe Gleichung (1) verwendet wurde, so dass vermittels ƒ(x_soll) = Z(y_soll) die Stellgrößenfunktion zur Entwicklung bereitsteht. Auch in Gleichung (8) stellt der letzte Summand $$R_{G_R}=\frac{1}{f\left(x_{\text{ist}}\right)}\frac{d^{N+1}Z\left(y_{\text{ist}}\right)}{\text{d}{y}_{\text{ist}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!}$$

das Restglied der Potenzreihe zur entwickelten Ordnung dar. D. h. erfindungsgemäß wird Gs gemäß Gleichung (5) als momentenerzeugendes Funktional in Stellgrößendarstellung und G_R gemäß Gleichung (8) als momentenerzeugendes Funktional in Regelgrößendarstellung bezeichnet (daher die im Folgenden auch verwendete Formulierung „Momentenmethode“). Vom mathematischen Standpunkt sind beide Darstellungen äquivalent. Allerdings können bei Modellungenauigkeiten und Laufzeiteffekten tatsächliche relative Verschiebungen zwischen beiden Formulierungen auftreten.In practice, the variable to be controlled, namely the setpoint controlled variable function or reference variable function ƒ (x _soll ), is related to the existing manipulated variable, namely the actual manipulated variable function Z (y _ist ), which, however, is represented by the actual controlled variable function ƒ (x _ist ). In any case, the torque-generating functional G _{R is} ready. Analogously to equation (5) results $$G_{\mathrm{R.}}\left[x_{\text{should}}\left(y_{\text{is}},\xi \right),x_{\text{is}}\right]=\frac{1}{f\left(x_{\text{is}}\right)}{\displaystyle \sum _{j=0}^N\frac{d^{j}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^j}\frac{{\xi }^j}{j!}+\frac{1}{f\left(x_{\text{is}}\right)}\frac{d^{N+1}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!}},$$

where equation (1) was used when developing the power series, so that the manipulated variable function is available for development by means of ƒ (x _soll ) = Z (y _soll ). The last term in equation (8) also represents $${\mathrm{R.}}_{G_{\mathrm{R.}}}=\frac{1}{f\left(x_{\text{is}}\right)}\frac{d^{N+1}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!}$$

represents the remainder of the power series to the developed order. D. h. According to the invention, Gs according to equation (5) is referred to as the torque-generating functional in the manipulated variable representation and G _R according to equation (8) as the torque-generating functional in the controlled variable representation (hence the phrase “torque method” also used below). From a mathematical point of view, both representations are equivalent. However, in the case of model inaccuracies and runtime effects, actual relative shifts between the two formulations can occur.

In summary, according to the invention, the torque-generating functional is created in the manipulated variable representation Gs according to equation (4) and then developed into a power series of a defined order N according to equation (5). Thus, if N is sufficiently small, an expression is available that allows the torque-generating functional to be resolved in the manipulated variable representation according to ζ and thus to solve the control problem approximately. The accuracy achieved here depends on the radius of convergence of the power series and can ultimately be estimated by calculating the associated remainder term. If this turns out to be small compared to the sum terms of the power series, then within the method there is a reliable value for ζ.

In summary, according to the invention, the torque-generating functional is also created in the controlled variable representation G _R according to equation (7) and then developed into a power series of defined order N according to equation (8). Thus, if N is sufficiently small, an expression is available that allows the torque-generating functional to be resolved in the manipulated variable representation according to ζ and thus to solve the control problem approximately. The accuracy achieved here depends on the radius of convergence of the power series and can ultimately be estimated by calculating the associated remainder term. If this turns out to be small compared to the sum terms of the power series, then within the method there is a reliable value for ζ.

If Z (y) is given and ƒ (x) is known as a measured or derived variable, the solution of the instantaneous control problem consists in solving one of the equations (5) or (8), neglecting the remainder of the term, and using equation (3) receive the updated value for the manipulated variable. In principle, the fundamental solvability of equations up to the fourth degree sets a limit, unless Z has special properties.

Nevertheless, on the one hand, this results in solutions for the control problem that are superior to pure linearization in terms of accuracy due to the higher N order taken into account. On the other hand, a separate operating point-dependent controller design can be dispensed with if only the manipulated variable function Z is known. In many cases one restricts oneself to a quadratic approximation due to the high complexity of solutions of rational equations with orders N greater than two. The method according to the invention is to be demonstrated by way of illustration using this approximation. Of course, higher orders N can also be considered. The corresponding application is then of course correspondingly more complicated.

If one approximates the moment-generating functional G _x , x ∈ {R, S} up to the second moment, one obtains $$\xi =-\frac{\frac{\text{d}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}}}{\frac{{\text{d}}^{2}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^2}}\left[1-\sqrt{1-2\left(1-G_X\right)\frac{Z\left(y_{\text{is}}\right)\frac{{\text{d}}^{2}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^2}}{{\left(\frac{\text{d}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}}\right)}^2}}\right].$$

Here, gx denotes the value of the torque-generating functional Gx according to the defining equation (4) or equation (7) at the respective operating point, ie g _s = G _s [x _target , y _is ], or g _R = G _R [x _target , x _is ].

From equation (9) with equation (3) it follows for the setpoint of the manipulated variable $$y_{\text{should}}=y_{\text{is}}-\frac{\frac{\text{d}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}}}{\frac{{\text{d}}^{2}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^2}}\left[1-\sqrt{1-2\left(1-G_X\right)\frac{Z\left(y_{\text{is}}\right)\frac{{\text{d}}^{2}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^2}}{{\left(\frac{\text{d}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}}\right)}^2}}\right].$$

einen Schwellwert C_S << 1 unterschreiten muss. Wird dieser Grenzwert überschritten, zeigt die erfindungsgemäße Methode hiermit an, dass die verwendete Approximation an ihre Grenzen stößt und stattdessen ein alternativer Regelungsalgorithmus verwendet werden sollte. Typischerweise wird in diesen Sonderfällen ein reines Steuerungsverfahren angewendet, welches die entsprechende Stellgröße in einen Bereich bringt, der es ermöglicht, die verwendeten Approximationen anzuwenden. Alternativ kann aber eben auch die in dem erfindungsgemäßen Verfahren enthaltene Kumulantenmethode (siehe unten) verwendet werden, sofern diese anwendbar ist.Like any approximate approach, the method according to the invention has a limited area of application. According to the invention, however, a particular advantage consists in the fact that it can be recognized whether this area of application is being left and thus other common methods must be used. The criterion used for this is obtained from the remainder (cf. equations (5.a) and (8.a)). If this is sufficiently small, it can be concluded that the power series expansion used in each case represents a permissible approximation. For this purpose, the sum term of the highest order N considered is compared with the remainder, so that the ratio $$\mathrm{C.}:=\frac{\xi }{N+1}\frac{\left|\frac{{\text{d}}^{N+1}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\right|}{\left|\frac{{\text{d}}^{N}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^N}\right|},$$

must fall below a threshold value C _S << 1. If this limit value is exceeded, the method according to the invention hereby indicates that the approximation used has reached its limits and an alternative control algorithm should be used instead. Typically, in these special cases, a pure control method is used, which brings the corresponding manipulated variable into a range that makes it possible to apply the approximations used. Alternatively, however, the cumulative method (see below) contained in the method according to the invention can also be used, provided that it is applicable.

Conversely, since the physical behavior of both the manipulated variable and the controlled variable function is known, a decision can be made even before the corrected control value y _soll ("y _ist - ζ" according to equation (10)) is evaluated based on the currently available values of gx whether the torque method according to the invention will be applicable or whether an alternative method, for example an alternative control algorithm or a pure control method, should be used.

The facts about the control process, which is based on the manipulated variable representation, are presented in 1 illustrated. This represents an action plan in which it is shown that a variable x or x _{ist to} be influenced results from the action of a manipulated variable y _soll or y _ist on a route influenced by disturbance variables u.

According to 1 corresponds _to the current actual value of the manipulated variable y _is the target value of the manipulated variable y in accordance with the respective previous computation step. In 1 these two sizes are only equated for reasons of clarity.

According to the action plan in 1 (Manipulated variable display) is carried out according to the invention in a specific operating point of the technical system, the formation of the quotient of the reference controlled variable function f (x _soll) and Iststellgrößenfunktion Z _(y) based on equation (1). I.e. A relationship is established between (a model if the target behavior is fulfilled) and (a model if the actual behavior of the underlying technical system is given).

As a result, the torque-generating functional G _S or its value gs is available at a specific operating point (equation (4)). Depending on the torque-generating functional Gs or its value gs at a specific operating point, the difference ζ is calculated using the solution of the N-th degree polynomial, for example, using a solver, as demonstrated for example for N = 2 in equation (9) has been. Since the zero point calculation for polynomials up to degree four is known, this solution can be implemented in advance in the solver for all N ≤ 4, as can the required derivatives. By applying the difference ξ determined in this way with the actual value of the current manipulated variable y _ist , which corresponds to the setpoint of the manipulated variable y _soll according to the previous calculation step, a current setpoint of the manipulated variable y _{soll is} formed, which now acts on the route so that the variable x or x _is to be influenced (controlled variable) by the action of the manipulated variable y _soll or y _ist on the system, the reference variable x _soll corresponds to or adapts to this specification.

As in 2 shown (controlled variable display) is carried out, according to the invention in a specific operating point of the technical system, the formation of the quotient of the reference controlled variable function ƒ (x _soll) and Istregelgrößenfunktion ƒ _(x) based on equation (1). I.e. a relationship is thus also formed between the model when the target behavior is fulfilled and the model when the actual behavior of the underlying technical system is given.

As a result, the torque-generating functional G _R or its value g _{R is available} at the specific operating point (equation (7)). Depending on the torque-generating functional G _R or its value g _R at the specific operating point, the difference ζ is calculated, for example by means of a solver, based on the solution of the polynomial of the Nth degree, as is the case, for example, for N = 2 in equation (10) was demonstrated. Since the zero point calculation for polynomials up to degree four is known, this solution can be implemented in advance in the solver for all N ≤ 4, as can the required derivatives. By applying the difference ξ determined in this way with the actual value of the current manipulated variable y _ist , which corresponds to the setpoint of the manipulated variable y _soll according to the previous calculation step, a current setpoint of the manipulated variable y _{soll is} formed, which now acts on the route so that the variable x or x _ist (controlled variable) to be influenced by the action of the manipulated variable y _soll or y _ist on the path corresponds to the reference variable x _soll or is adapted to this specification.

So far, the torque method according to the invention is first described or defined.

According to the invention, based on the mathematical equivalence between the method already described and the representation of a probability distribution by the corresponding moments, the formulation of a cumulant-generating functional is also proposed. In the following, we will refer to this procedure for short as the cumulative method.

d. h. das kumulantenerzeugende Funktional Fs bzw. F_R ist der natürliche Logarithmus des momentenerzeugenden Funktionals Gs bzw. G_R. Eine Potenzreihenentwicklung liefert sodann $$F_X={\displaystyle \sum _{j=1}^N\frac{d^j\text{ln}Z\left(y_{\text{ist}}\right)}{\text{d}{y}_{\text{ist}}^j}\frac{{\xi }^j}{j!}}+\frac{d^{N+1}\text{ln}Z\left(y_{\text{ist}}\right)}{\text{d}{y}_{\text{ist}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!},$$

wobei der Ausdruck $$k_j:=\frac{{\text{d}}^j\text{ln}Z\left(y_{\text{ist}}\right)}{\text{d}{y}_{\text{ist}}^j}$$

als die j-te Kumulante des instantanen Regelungsproblems bezeichnet wird. Der in Gleichung (13) nicht unter dem Summenzeichen inkludierte Term $$R_{F_X}=\frac{d^{N+1}\text{ln}Z\left(y_{\text{ist}}\right)}{\text{d}{y}_{\text{ist}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!}$$

entspricht dabei dem Restglied der Potenzreihe, d. h. auch hierin stellt der letzte Summand das Restglied der Potenzreihe zur entwickelten Ordnung dar.The function generating the cumulative is defined for the control and manipulated variable representation by $${\mathrm{F.}}_X:=\text{ln}{G}_X,X\in \left\{\mathrm{R.},\mathrm{S.}\right\}$$

ie the cumulant-generating functional Fs or F _R is the natural logarithm of the torque-generating functional Gs or G _R. A power series expansion then delivers $${\mathrm{F.}}_X={\displaystyle \sum _{j=1}^N\frac{d^j\text{ln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^j}\frac{{\xi }^j}{j!}}+\frac{d^{N+1}\text{ln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!},$$

where the expression $$k_j:=\frac{{\text{d}}^j\text{ln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^j}$$

is called the jth cumulant of the instantaneous control problem. The term not included under the sum symbol in equation (13) $${\mathrm{R.}}_{{\mathrm{F.}}_X}=\frac{d^{N+1}\text{ln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!}$$

corresponds to the remainder of the power series, ie here too the last summand represents the remainder of the power series for the developed order.

In analogy to the generating function of the moments (see description of the moment method above), equation (13) for the given order N can now be solved for ξ. The same properties with regard to the solvability of the approximations selected apply accordingly here. In analogy to the exemplary solution from the last section, an approximation up to the second order (i.e. N = 2) results $$\xi =-\frac{\frac{\text{dln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}}}{\frac{{\text{<figure-callout id="2" label="d" filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0107.png" state="{{state}}">d</figure-callout>}}^2\text{ln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^2}}\left[1-\sqrt{1+2{\mathrm{<figure-callout\; id="2"\; label="f\; X\; d"\; filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0107.png"\; state="\{\{state\}\}">f</figure-callout>}}_X\frac{\frac{{\text{d}}^{}}{}}{}\frac{\frac{{}^2\text{ln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^2}}{{\left(\frac{\text{dln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}}\right)}^2}}\right].$$

Gegenüber der Funktion Gx kann die Reihenentwicklung für F_X ein unterschiedliches Konvergenzverhalten aufweisen. Über eine Restgliedabschätzung kann hierbei eine Aussage darüber getroffen werden, welche der beiden Approximationen verlässlicher ist (wie im weiteren Verlauf noch konkret beschrieben wird) und welche ausgewählt werden soll für den Betrieb des technischen Systems. Insofern kann eine zusätzliche Auswertung via Gleichung (13) nicht nur als alternativer Zugang gewertet werden. Zugleich kann darüber befunden werden, welche der beiden Methoden in der jeweils betrachteten Betriebssituation eine exaktere Approximation darstellt. Ein Vergleich der Gleichungen (10) und (16) zeigt hierbei, dass beim Übergang auf die Kumulantenmethode keine neuerliche Rechnung ausgeführt werden muss, sondern, dass das jeweilige Moment (in diesem Fall m₁ und m₂) lediglich durch die jeweilige Kumulante (in diesem Fall k₁ und k₂) zu ersetzen ist. Diese Eigenschaft gilt allgemein und ist nicht auf den Fall N = 2 beschränkt. Ferner erlaubt der aus der Statistik bekannte Zusammenhang zwischen Momenten und Kumulanten, der aufgrund der vollständigen mathematischen Analogie der hier gezeigten Methoden volle Gültigkeit hat, die Bestimmung der Kumulanten durch entsprechende Kombination der Momente. Insofern erfordert die Bestimmung von ζ mittels der Kumulantenmethode nur wenig zusätzliche Rechenleistung, was das vorgestellte Verfahren insbesondere für eingebettete Systeme geringer Rechenkapazität geeignet macht.Here, in analogy to equation (9), ƒ _X denotes the value of the cumulant-generating function Fx according to the defining equation (13) at the respective operating point. Using equation (3), the setpoint is the manipulated variable in this representation $$y_{\text{should}}=y_{\text{is}}-\frac{\frac{\text{dln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}}}{\frac{{\text{<figure-callout id="2" label="d" filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0107.png" state="{{state}}">d</figure-callout>}}^2\text{ln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^2}}\left[1-\sqrt{1+2{\mathrm{<figure-callout\; id="2"\; label="f\; X\; d"\; filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0107.png"\; state="\{\{state\}\}">f</figure-callout>}}_X\frac{\frac{{\text{d}}^{}}{}}{}\frac{\frac{{}^2\text{ln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^2}}{{\left(\frac{\text{dln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}}\right)}^2}}\right].$$

Compared to the function Gx, the series expansion for F _{X can show} a different convergence behavior. A statement about which of the two approximations is more reliable (as will be specifically described below) and which should be selected for the operation of the technical system can be made here via a residual term estimate. In this respect, an additional evaluation via equation (13) cannot only be assessed as an alternative approach. At the same time, it can be determined which of the two methods represents a more exact approximation in the operating situation under consideration. A comparison of equations (10) and (16) shows that when switching to the cumulative method, no new calculation has to be carried out, but that the respective moment (in this case m ₁ and m ₂ ) is only replaced by the respective cumulative (in in this case k ₁ and k ₂ ) is to be replaced. This property applies in general and is not restricted to the case N = 2. Furthermore, the relationship between moments and cumulants, known from statistics, which is fully valid due to the complete mathematical analogy of the methods shown here, allows the cumulants to be determined by appropriate combination of the moments. In this respect, the determination of ζ by means of the cumulative method requires only a little additional computing power, which makes the presented method particularly suitable for embedded systems with little computing capacity.

The accumulation method according to the invention is thus also described or defined. From a mathematical point of view, one expects a better convergence behavior of the moment method compared to the cumulative method if G _X with ξ is disproportionately low. In the opposite case of a disproportionate course, the cumulative method will represent a better approximation than the moment method. According to the invention, this circumstance is made use of in the respective operating point of the technical system in that that power series with the faster convergence behavior is selected and, based on this, is resolved according to the difference ξ. In particular, there is thus the potential to avoid a high order N in the development, as a result of which simpler expressions can be used in the evaluation without having to accept losses in the control quality.

How according to 3 Shown in the control variable representation, according to the invention, the torque method and the cumulative method can be combined for this purpose for operating or controlling and / or regulating a technical system.

A selection can be made as to whether the difference ξ according to equation (3), which is used to form a manipulated variable y set or y _is used to form a manipulated variable y that _is influenced by disturbance variables u, corresponds to a difference ζ _Mom that is determined by means of a first solver (SolverMom), ie using the moment method / a first method for providing the difference ξ or ζ _{Mom is} provided or a difference ζ _Kum corresponds to that using a second solver (SolverKum), ie using the cumulative method / using a second method to provide the difference ξ or ζ _cum .

For this purpose, the quantities ζ _Mom or ζ _Kum , as described above, are determined in each case within the framework of the torque or the cumulant method for the operating point by means of a power series of order N. The quantities ζ _Mom and ζ _Cum that are thus available are then transferred to the selection hierarchy. In this hierarchy, the evaluation of the respective remaining links of the associated with the smaller absolute residual limb in the calculation of ζ _Mom, or ζ _Kum power series used, where ζ is with that value ζ _Mom or ζ _Kum equated now be made. In other words, the following rule is used: $$\xi =\{\begin{array}{c}{\xi }_{\text{Mom}},\text{if}\left|\frac{1}{f\left(x_{\text{is}}\right)}\frac{d^{N+1}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!}\right|\le \left|\frac{d^{N+1}\text{ln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\frac{{\xi }_{\text{Cum}}^{N+1}}{\left(N+1\right)!}\right|\\ {\xi }_{\text{Cum}},\text{if}\left|\frac{1}{f\left(x_{\text{is}}\right)}\frac{d^{N+1}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!}\right|>\left|\frac{d^{N+1}\text{ln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\frac{{\xi }_{\text{Cum}}^{N+1}}{\left(N+1\right)!}\right|\end{array}.$$

By applying the determined difference der with the actual value of the current manipulated variable, which corresponds to the setpoint of the manipulated variable y _soll according to the previous calculation step, a current setpoint of the manipulated variable y _{soll is} formed, which now acts on the route so that the to Influencing variable x or x _ist (controlled variable) through the action of the manipulated variable y _soll or y _ist on the path corresponds to the reference variable x _soll or is adapted to this specification.

As already noted in the presentation of the cumulative method, the calculation of ζ _Cum and the residual term estimate made in the selection do not require a separate calculation of the differentiations shown (cf. equations (13) and (14)). Rather, the cumulants can be obtained by combining the moments. A high order N can thus be avoided during development through a targeted selection between the two methods, which means that comparatively simple expressions can be used in the evaluation without having to accept losses in the control quality, as described above. The computing effort to be performed by the computing machine can hereby be kept low. In particular, the combination according to the invention of the two methods often results in an applicability in operational areas which, as a rule, could not be represented by one of the two methods alone. By calculating the cumulative values from the moments, this gain is bought at the cost of a very small additional computational effort.

A typical application for instantaneous control problems is in the area of thermal management using indirect temperature control systems. The frequently occurring case that a working medium (usually intake air or exhaust gas) is to be tempered to a certain value is realized by a pump whose speed can be specified depending on the operating conditions. A control algorithm that can be used in connection with the method presented here results from the method of the dimensionless temperature change P, see also DE 10 2016 124 652 B3 . For a heat exchanger, this is defined by $$P=\frac{{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0108.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{1,}u}-{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0108.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{1,}\mathrm{<figure-callout\; id="1"\; label="d\; T"\; filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0108.png"\; state="\{\{state\}\}">d</figure-callout>}}}{T_{}}.$$

Here, T _{1, u} , and T _{1, d} each denote the temperature of the working medium upstream and downstream of the heat exchanger. The temperature of the temperature control medium (coolant) upstream of the component is given by T _{2, u} .

ab, also $$P=P\left({\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2\right).$$

As is known, the dimensionless temperature change essentially depends on the heat capacity flows (with c _{p, j} as heat capacity and ṁ _j as mass flow) $${\mathrm{\Gamma }}_j:=c_{p,j}{\dot{m}}_j\text{With}j\in \left\{1.2\right\}$$

off, so $$P=P\left({\mathrm{<figure-callout\; id="1"\; label="\Gamma "\; filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0108.png"\; state="\{\{state\}\}">\Gamma </figure-callout>}}_1,{\mathrm{\Gamma }}_2\right).$$

If the temperature of the working medium is now to be regulated, then by substituting T _{1, d} → T _{1, d, des} with otherwise the same variables, a nominal value for the dimensionless temperature change is obtained according to $$P_{\text{of}}=\frac{{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0108.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{1,}u}-{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0108.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{1,}d,\text{of}}}{{\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0108.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{1,}u}-{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0107.png"\; state="\{\{state\}\}">T</figure-callout>}}_{\mathrm{2,}u}}$$

Using equation (19), this results in that the transformation P → P _{of the} inverted Γ ₂ Γ → _{2, the} implied. As a rule, the functional relationship between the dimensionless temperature change and the coolant heat capacity flow is so complicated that a direct inversion of the form Γ _{2, des} = Γ _{2, des} (Γ ₁ , P _des ) is not easily possible. For this reason, the approach according to the invention should be applied to this problem.

wobei f(x)=e^x die Exponentialfunktion repräsentiert. Die effektive Wärmeleitzahl ist dabei gegeben durch γ = (1 - Θ[Γ₂ - Γ_2,c)γ_l + Θ[Γ₂ - Γ_2,c]γ_t. Es wird also mittels der Heaviside-Funktion $$\mathrm{\Theta }\left(x\right)=\{\begin{array}{c}0\forall x<0\\ 1\forall x\ge 0\end{array}$$

entschieden, ob laminare (γ₁) oder turbulente (γ_t) Kühlmittelströmung vorliegt. Das Kriterium hierfür ist, ob der Wärmekapazitätsstrom unterhalb oder oberhalb des kritischen Wertes Γ_2,c liegt. Im laminaren Fall kann die effektive Wärmeleitzahl beschrieben werden durch $${\gamma }_l=a_{00}+a_{10}{\mathrm{\Gamma }}_1+a_{01}{\mathrm{\Gamma }}_2+a_{20}{\mathrm{\Gamma }}_1^2+a_{02}{\mathrm{\Gamma }}_2^2+a_{11}{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2+a_{30}{\mathrm{\Gamma }}_1^3,$$

wohingegen für turbulente Kühlmittelströmung $${\gamma }_t=\frac{\alpha {\mathrm{\Gamma }}_1^m{\mathrm{\Gamma }}_2^m}{\beta {\mathrm{\Gamma }}_2^m+{\mathrm{\Gamma }}_1^m}$$

gilt. Hierbei fungieren die Größen a_j, sowie α, β und m als freie Parameter. Wir bilden die momentenerzeugende Funktion $$G\left[\xi \right]=\frac{P\left({\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2+\xi \right)}{P\left({\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2\right)}.$$

In 4 ist der Verlauf der momentenerzeugenden Funktion mit ξ für einen definierten Betriebspunkt gezeigt (Veranschaulichung der momentenerzeugenden Funktion G in Gleichung (21)). Aufgrund des unterproportionalen Anstieges in der Nähe dieses Betriebspunktes ist anzunehmen, dass die Bestimmung von ζ über G exaktere Resultat liefert als über die kumulantenerzeugende Funktion $$F=\text{ln}G$$

To validate the method according to the invention, a heat exchanger was used whose dimensionless temperature change can be expressed by the heat capacity flows using the following functional relationship: $$P=\frac{1-{\text{e}}^{\gamma }\left({\mathrm{\Gamma }}_2^{-1}-{\mathrm{\Gamma }}_1^{-1}\right)}{1-\frac{{\mathrm{\Gamma }}_1}{{\mathrm{\Gamma }}_2}{\text{e}}^{\gamma }\left({\mathrm{\Gamma }}_2^{-1}-{\mathrm{\Gamma }}_1^{-1}\right)},$$

where f (x) = e ^{x represents} the exponential function. The effective coefficient of thermal conductivity is given by γ = (1 - Θ [Γ ₂ - Γ _{2, c} ) γ _l + Θ [Γ ₂ - Γ _{2, c} ] γ _t . So it is done using the Heaviside function $$\mathrm{\Theta }\left(x\right)=\{\begin{array}{c}0\forall x<0\\ 1\forall x\ge 0\end{array}$$

decided whether the coolant flow was laminar (γ ₁ ) or turbulent (γ _t ). The criterion for this is whether the heat capacity flow is below or above the critical value Γ _{2, c} . In the laminar case, the effective coefficient of thermal conductivity can be described by $${\gamma }_l=a_{00}+a_{10}{\mathrm{\Gamma }}_1+a_{01}{\mathrm{\Gamma }}_2+a_{\mathrm{20th}}{\mathrm{\Gamma }}_1^2+a_{02}{\mathrm{\Gamma }}_2^2+a_{11}{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_2+a_{\mathrm{30th}}{\mathrm{\Gamma }}_1^3,$$

whereas for turbulent coolant flow $${\gamma }_t=\frac{\alpha {\mathrm{\Gamma }}_1^m{\mathrm{\Gamma }}_2^m}{\beta {\mathrm{\Gamma }}_2^m+{\mathrm{\Gamma }}_1^m}$$

applies. The quantities a _j as well as α, β and m act as free parameters. We form the moment generating function $$G\left[\xi \right]=\frac{P\left({\mathrm{<figure-callout\; id="1"\; label="\Gamma "\; filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0108.png"\; state="\{\{state\}\}">\Gamma </figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="\Gamma "\; filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0107.png"\; state="\{\{state\}\}">\Gamma </figure-callout>}}_2+\xi \right)}{P\left({\mathrm{<figure-callout\; id="1"\; label="\Gamma "\; filenames="DE102020116218A1\_0106.png,DE102020116218A1\_0108.png"\; state="\{\{state\}\}">\Gamma </figure-callout>}}_1,{\mathrm{\Gamma }}_2\right)}.$$

In 4th the curve of the torque-generating function is shown with ξ for a defined operating point (illustration of the torque-generating function G in equation (21)). Due to the disproportionate increase in the vicinity of this operating point, it can be assumed that the determination of ζ via G delivers more exact results than via the cumulant-generating function $$\mathrm{F.}=\text{ln}G$$

definiert als $${\Delta }_{\mathrm{\Xi }}^{\left(N\right)}:=\left|\mathrm{\Xi }\left[\xi \right]-{\mathrm{\Xi }}_N\left[\xi \right]\right|$$

wobei Ξ_N die Potenzreihenentwicklung bis zur N-ten Ordnung darstellt. Weiter definieren wir $${\Delta }_{FG}^{\left(MN\right)}={\Delta }_F^{\left(M\right)}-{\Delta }_G^{\left(N\right)}.$$

M und N bedeuten in diesem Zusammenhang die jeweils höchste berücksichtigte Ordnung bei der Auflösung nach ζ_Mom, beziehungsweise ζ_Kum. Diese Ordnungen müssen nicht zwingend gleich sein, so kann man beispielsweise ζ_Kum aus dem kumulantenerzeugenden Funktional in linearer Ordnung berechnen, während man die Größe ζ_Mom aus der quadratischen Approximation des momentenerzeugenden Funktionals erhalten könnte. Dann wäre M=1 und N=2. Gleichung (23) drückt dabei aus, ob das Restglied von F in M-ter Ordnung größer ausfällt als das Restglied von G in N-ter Ordnung. Wenn die Antwort ja ist, wird man die Lösung ζ_Mom heranziehen und ζ = ζ_Mom setzen. Im umgekehrten Fall bedient man sich der Lösung aus F. Insofern handelt es sich bei M und N in Gleichung (23) um die jeweilige Ordnung, anhand der die Restglieder verglichen werden.This assumption is confirmed if one compares the respective absolute deviation in the expansions of the power series. This expression is for the respective generating function $$\mathrm{\Xi }\in \left\{\mathrm{F.},G\right\}$$

defined as $${\Delta }_{\mathrm{\Xi }}^{\left(N\right)}:=\left|\mathrm{\Xi }\left[\xi \right]-{\mathrm{\Xi }}_N\left[\xi \right]\right|$$

where Ξ _{N represents} the power series expansion up to the Nth order. We further define $${\Delta }_{\mathrm{F.}G}^{\left(\mathrm{M.}N\right)}={\Delta }_{\mathrm{F.}}^{\left(\mathrm{M.}\right)}-{\Delta }_G^{\left(N\right)}.$$

In this context, M and N mean the highest order taken into account when resolving according to ζ _Mom and ζ _Kum , respectively. These orders do not necessarily have to be the same, for example ζ _Cum can be calculated from the cumulant-generating functional in linear order, while the quantity ζ _{Mom could be obtained} from the quadratic approximation of the torque-generating functional. Then M = 1 and N = 2. Equation (23) expresses whether the remainder of F in the M-th order is larger than the remainder of G in the N-th order. If the answer is yes, you will use the solution ζ _Mom and set ζ = ζ _Mom . In the opposite case, use the solution from F. In this respect, M and N in equation (23) are the respective order on the basis of which the remaining terms are compared.

folgt, dass die Potenzreihe für das momentenerzeugende Funktional beziehungsweise für das kumulantenerzeugende Funktional für die Ordnung N schneller konvergiert. Wie in 5 für M=N=2 gezeigt, weist die Potenzreihenentwicklung für die untersuchten Betriebspunkte beim momentenerzeugenden Funktional klar das schnellere Konvergenzverhalten auf.For ${\Delta }_{\mathrm{F.}G}^{\left(NN\right)}>0\text{or}\text{.}{\Delta }_{\mathrm{F.}G}^{\left(NN\right)}<0$

it follows that the power series for the moment-generating functional or for the cumulant-generating functional for order N converges faster. As in 5 shown for M = N = 2, the power series expansion for the investigated operating points clearly shows the faster convergence behavior for the torque-generating functional.

und $$F={\displaystyle \sum _{j=1}^N{k}_j}\frac{{\xi }^j}{j!}+k_{N+1}\frac{{\xi }^{N+1}}{\left(N+1\right)!}$$

unter Vernachlässigung des Restgliedes nach ζ aufzulösen. Das so erhaltene Resultat wird sodann in die jeweiligen betragsmäßigen Restterme $$R_G=\left|m_{N+1}\frac{{\xi }^{N+1}}{\left(N+1\right)!}\right|$$

Und $$R_F=\left|k_{N+1}\frac{{\xi }^{N+1}}{\left(N+1\right)!}\right|$$

eingesetzt und die Güte des jeweiligen Ansatzes ausgewertet.In other words, the basic strategy of the approach according to the invention is to use the respective equations $$G=1+{\displaystyle \sum _{j=1}^N{m}_j\frac{{\xi }^j}{j!}}{m}_{N+1}\frac{{\xi }^{N+1}}{\left(N+1\right)!}$$

and $$\mathrm{F.}={\displaystyle \sum _{j=1}^N{k}_j}\frac{{\xi }^j}{j!}+k_{N+1}\frac{{\xi }^{N+1}}{\left(N+1\right)!}$$

to be solved by neglecting the remaining term according to ζ. The result obtained in this way is then converted into the respective residual terms $${\mathrm{R.}}_G=\left|m_{N+1}\frac{{\xi }^{N+1}}{\left(N+1\right)!}\right|$$

And $${\mathrm{R.}}_{\mathrm{F.}}=\left|k_{N+1}\frac{{\xi }^{N+1}}{\left(N+1\right)!}\right|$$

used and evaluated the quality of the respective approach.

Finally, that ξ with the smaller absolute remainder R _G , R _{F is} selected and processed further within the control function or regulation function, see 6th , so that the optimal solution of order N is always selected (which is in each case in the "Selection" block in the 3 and 7th is performed). From the considerations made above, it follows that in the vicinity of the operating point under consideration this would usually lead to a selection of the solution from equation (24).

So far, an instantaneous control problem was initially considered and a control strategy based on generating functionals for instantaneous systems was presented. In principle, however, the applicability of this approach is not restricted to such systems. If it is possible to find a suitable torque-generating functional for a dynamic system, the deviation between the setpoint and the actual manipulated variable can be resolved using a power series expansion in the now known manner.

worin x_ist den Istwert der Regelgröße und y_ist den Istwert der Stellgröße darstellen. Wie auch beim instantanen System können weitere Abhängigkeiten auf die Dynamik Einfluss nehmen. Es soll aber auch hier eine gesonderte Notation hierfür unterbleiben. Die Funktion h sei unendlich oft differenzierbar. Wir nehmen an, dass das Regelungsproblem eine eindeutige Lösung besitzt, dass also genau ein y_ist existiert, welches den Zustand x_ist dem Sollwert x_soll angleicht. Geben wir die Stellgröße, welche das System auf den Sollwert x_soll einregelt mit y_soll (Sollwert der Stellgröße) an und die zu einem bestimmten Zeitpunkt t vorliegende Stellgröße mit y_ist, so können wir die Differenz dieser beiden Größen (ebenfalls) durch ξ ausdrücken gemäß $$y_{soll}\left(t\right)=y_{ist}\left(t\right)+\xi \left(t\right).$$

Ist das System vollständig eingeregelt, so folgt aus Gleichung (28) $${\dot{x}}_{soll}=h\left(x_{soll},y_{soll}\right).$$

Bildet man die Differenz aus den Gleichungen (30) und (28), folgt hieraus $$\dot{\zeta }=h\left(x_{soll},y_{soll}\right)-h\left(x_{ist},y_{ist}\right),$$

mit der Regelabweichung $$\zeta :=x_{soll}-x_{ist}.$$

Then there is also the possibility of deriving a cumulant-generating functional from the torque-generating functional. The corresponding possibility of also using this expression as a basis for the regulation exists in complete analogy to the instantaneous case. Finding a torque-generating functional for a dynamic system with a degree of freedom of regulation should be in Are shown below. For this we consider a system which is described by the following differential equation $${\dot{x}}_{ist}=H\left(x_{ist},y_{ist}\right)$$

where x _is the actual value of the controlled variable and y _is the actual value of the manipulated variable. As with the instantaneous system, further dependencies can influence the dynamics. However, a separate notation for this should also be omitted here. Let the function h be infinitely differentiable. We assume that the control problem has a unique solution that exactly one _y exists which x _is the state of the target value x _to equalize. We provide the manipulated variable, which the system to the set value x _to adjusts y _to (target value of the manipulated variable), and which _is at a certain time t present manipulated variable y, we can express the difference between these two sizes (also) by ξ according to $$y_{sOll}\left(t\right)=y_{ist}\left(t\right)+\xi \left(t\right).$$

If the system is fully regulated, then from equation (28) $${\dot{x}}_{sOll}=H\left(x_{sOll},y_{sOll}\right).$$

If one forms the difference from equations (30) and (28), it follows $$\dot{\zeta }=H\left(x_{sOll},y_{sOll}\right)-H\left(x_{ist},y_{ist}\right),$$

with the control deviation $$\zeta :=x_{sOll}-x_{ist}.$$

For the following we define the moment-generating functional $$G\left[x_{sOll},x_{ist};y_{ist},\dot{\zeta }\right]=\frac{\dot{\zeta }}{H\left(x_{sOll},y_{sOll}\right)-H\left(x_{ist},y_{ist}\right)}.$$

The approach here is to use this ratio, i. H. to form the torque-generating functional according to equation (32), from the time derivative ζ̇ of the control deviation ζ and a corresponding value, so that in a subsequent development of the torque-generating function G (equation (32)) into a power series, the expressions then obtained are the Take shape of moments.

In practice, the expressions in the denominator of equation (32) result from the fact that a distinction between “target” and “actual” is maintained in size x, whereas the calculation _is referenced to y ist, since ultimately developed around point y _ist shall be. The denominator given in equation (32) is chosen or used so that the terms that result from the expansion of the power series take on the mathematical form of moments.

In the further course, after equation (31) has been inserted into equation (32), expansion into a power series around ξ = 0 is carried out using equation (29).

mit dem j-ten Moment der Stellgrößenfunktion $$m_j:=\frac{1}{h\left(x_{soll},y_{istl}\right)-h\left(x_{ist},y_{ist}\right)}{\frac{d^j\left[h\left(x_{soll},y_{ist}+\xi \right)-h\left(x_{ist},y_{ist}\right)\right]}{d{\xi }^j}|}_{\xi =0}.$$

In Gleichung (33) stellt wiederum der letzte Summand $$R=\frac{m_{N+1}}{\left(N+1\right)!}{\xi }^{N+1}$$

das Restglied dar. Man erkennt die mathematische Analogie zwischen Gleichung (33) und dem korrespondierenden Ausdruck für das instantane Regelungsproblem (vgl. Gleichung (24)). Das Nämliche gilt demnach auch in Bezug auf die momentenerzeugende Funktion, wie sie aus der mathematischen Statistik bekannt ist.The torque-generating function results from this $$G\left[x_{sOll},x_{ist};y_{ist};\zeta \right]=1+{\displaystyle \sum _{j=1}^N\frac{m_j}{j!}}{\xi }^j+\frac{m_{N+1}}{\left(N+1\right)!}{\xi }^{N+1},$$

with the jth moment of the manipulated variable function $$m_j:=\frac{1}{H\left(x_{sOll},y_{istl}\right)-H\left(x_{ist},y_{ist}\right)}{\frac{d^j\left[H\left(x_{sOll},y_{ist}+\xi \right)-H\left(x_{ist},y_{ist}\right)\right]}{d{\xi }^j}|}_{\xi =0}.$$

In equation (33) again represents the last summand $$\mathrm{R.}=\frac{m_{N+1}}{\left(N+1\right)!}{\xi }^{N+1}$$

the remainder. One recognizes the mathematical analogy between equation (33) and the corresponding expression for the instantaneous control problem (cf. equation (24)). This also applies to the torque-generating function as it is known from mathematical statistics.

• - Die prinzipielle Möglichkeit, dass gelten kann G[·] ≤ 0, stellt keine Einschränkung der erfindungsgemäßen Methode dar, wenn der komplexe Logarithmus und zwar stets derselbe Zweig desselben verwendet wird. Allerdings ergibt sich hierbei ein von Null verschiedener Imaginärteil, der nicht durch eine endliche Summe reeller Terme dargestellt werden kann. Insofern wollen wir die kumulantenerzeugende Funktion ausschließlich für den Fall G[·] > 0 definieren. Da der eingeregelte Zustand durch G[·] = 1 charakterisiert ist, steht allerdings ohnehin zu vermuten, dass eine Entwicklung für G[·] ≤ 0 eine große Anzahl an Termen benötigt, um zu einer guten Abschätzung für ξ zu kommen.
• - Tatsächlich ist das instantane Regelungsproblem als Grenzfall zum dynamischen Regelungsproblem anzusehen. Anschaulich kann man dies so verstehen, dass im Grenzfall einer langsam veränderlichen Zustandsvariablen der instantane Regelungsfall erfüllt ist. Mathematisch macht man sich diesen Sachverhalt dadurch klar, dass in diesem Grenzfall stets gilt x_ist << 1. Hiermit folgt aus Gleichung (28) h(x_ist,y_ist) ≈ 0. Sofern die Funktion h nun additiv separabel bezüglich x_ist und y_ist ist, schreiben wir

$$h\left(x,y\right)=:f\left(x\right)-Z\left(y\right)=0.$$

Thus the dynamic control problem is reduced to the determination of ξ. Mathematically, the resulting expression is equivalent to the instantaneous case. The definition of the moment-generating function is of course also accompanied by the possibility of defining the cumulative-generating function and with it the entire formalism (moment method and cumulative method) that has already been discussed in the present description (see above) in connection with the instantaneous control problem, see also 7th . The following comments should be made on this:

• The possibility in principle that G [·] 0 does not represent a restriction of the method according to the invention if the complex logarithm and, in fact, always the same branch thereof is used. However, this results in an imaginary part different from zero, which cannot be represented by a finite sum of real terms. In this respect, we want to define the function generating the cumulative exclusively for the case G [·]> 0. Since the regulated state is characterized by G [·] = 1, it can be assumed that a development for G [·] ≤ 0 requires a large number of terms in order to arrive at a good estimate for ξ.
• - In fact, the instantaneous control problem is to be seen as a borderline case to the dynamic control problem. This can be clearly understood in such a way that in the borderline case of a slowly changing state variable, the instantaneous control case is fulfilled. Mathematically, this fact makes you look thereby clear that in this limit always applies _x << 1. This follows from equation (28) h _{(x, y)} ≈ 0. If the function h now additively separable with _respect to x and y _is is, we write

$$H\left(x,y\right)=:f\left(x\right)-Z\left(y\right)=0.$$

This brings us back to the definition equation (cf. equation (1)) for the instantaneous control problem.

As in 7th Shown in an action plan, a variable x or x _ist to be influenced results from the action of a manipulated variable y _soll or y _ist on a path influenced by disturbance variables u.

According to 7th _{is intended} corresponds to the (current) actual value of the manipulated variable y _is the target value of the manipulated variable y in accordance with the respective previous computation step. In 7th these two sizes are only equated for reasons of clarity.

According to the action plan in 7th According to the invention, the deviation ζ: = x _soll - x _ist between the setpoint x _soll and the actual value of the controlled variable x _ist (i.e. the control deviation) _is initially formed at a specific operating point (of the dynamic technical system). The time derivative ζ̇ of the deviation ̇ζ (i.e. the control deviation) is then provided by a differentiation (i.e. dζ / dt).

In the further course of the process, the torque-generating functional G or its value g is formed at the specific operating point according to equation (32). In analogy to 3 In the present (dynamic) case, the torque method and the cumulative method can also be combined for operation or control and / or regulation of a technical system.

A selection can be made as to whether the difference ξ according to equation (29), which is to _{be used} or y _is used to form a manipulated variable y acting on a path influenced by disturbance variables u. is used, corresponds to a difference ζ _Mom , which is provided by means of a first solver (SolverMom), ie by means of the moment method / a first method for providing the difference ξ or ζ _Mom , or corresponds to a difference ζ _Kum , which is provided by means of a second solver ( SolverKum), ie using the cumulative method / using a second method for providing the difference Bereitstellung or ζ _Kum .

worin k_N+1 die (N + 1)-ste Kumulante des dynamischen Regelungsproblems darstellt. Die Berechnung der Kumulanten erfolgt analog zum instantanen Fall und ist durch $$k_l={\frac{{\partial }^l\text{ln }G}{\partial {\xi }^l}|}_{\xi =0}$$

gegeben. Das (N + 1)-te Moment berechnet sich wie in Gleichung (34) demonstriert. Durch eine Beaufschlagung der so ermittelten Differenz ξ mit dem tatsächlichen Wert der aktuellen Stellgröße, welche dem Sollwert der Stellgröße y_soll gemäß dem vorherigen Rechenschritt entspricht, wird ein aktueller Sollwert der Stellgröße y_soll gebildet, der nunmehr auf die Strecke einwirkt, so dass die zu beeinflussende Größe x bzw. x_ist (Regelgröße) durch Einwirkung der Stellgröße y_soll bzw. y_ist auf die Strecke der Führungsgröße x_soll entspricht bzw. sich dieser Vorgabe angleicht.For this purpose, the quantities ζ _Mom or ζ _Kum , as described above, are first used in the context of the torque or cumulative method for the operating point by means of a power series of order N (strictly speaking, the respective orders of the The moment and the cumulative method differ). The quantities ζ _Mom and ζ _Cum that are thus available are then transferred to the selection hierarchy. In this hierarchy, the evaluation of the respective remaining links of the associated with the smaller absolute residual limb in the calculation of ζ _Mom, or ζ _Kum power series used, where ζ is with that value ζ _Mom or ζ _Kum equated now be made. In other words, the following rule is used: $$\xi =\{\begin{array}{c}{\xi }_{\text{Mom}},\text{if}\left|\frac{m_{N+1}}{\left(N+1\right)!}{\xi }^{N+1}\right|\le \left|\frac{k_{N+1}}{\left(N+1\right)!}{\xi }_{\text{Cum}}^{N+1}\right|\\ {\xi }_{\text{Cum}},\text{if}\left|\frac{m_{N+1}}{\left(N+1\right)!}{\xi }^{N+1}\right|>\left|\frac{k_{N+1}}{\left(N+1\right)!}{\xi }_{\text{Cum}}^{N+1}\right|\end{array},$$

where k _{N + 1 represents} the (N + 1) th cumulant of the dynamic control problem. The calculation of the cumulants is analogous to the instantaneous case and is through $$k_l={\frac{{\partial }^l\text{ln}G}{\partial {\xi }^l}|}_{\xi =0}$$

given. The (N + 1) th moment is calculated as demonstrated in equation (34). By applying the determined difference der with the actual value of the current manipulated variable, which corresponds to the setpoint of the manipulated variable y _soll according to the previous calculation step, a current setpoint of the manipulated variable y _{soll is} formed, which now acts on the route so that the to Influencing variable x or x _ist (controlled variable) through the action of the manipulated variable y _soll or y _ist on the path corresponds to the reference variable x _soll or is adapted to this specification.

As noted in the instantaneous case, the calculation of ζ _Cum and the residual term estimate made in the selection do not require a separate calculation of the differentiations shown. Rather, the cumulants can be obtained by combining the moments. A high order N can thus be avoided during development through a targeted selection between the two methods, which means that comparatively simple expressions can be used in the evaluation without having to accept losses in the control quality, as described above. The computing effort to be performed by the computing machine can hereby be kept low. In particular, the combination according to the invention of the two methods often results in an applicability in operational areas which, as a rule, could not be represented by one of the two methods alone. By calculating the cumulative values from the moments, this gain is bought at the cost of a very small additional computational effort.

die momentenerzeugende Funktion in Stellglieddarstellung (Hier ist beispielhaft nur die Stellgrößendarstellung veranschaulicht.) berechnet werden kann $$G_S\left(y_{ist},\xi \right):=\frac{Z\left(y_{ist}+\xi \right)}{Z\left(y_{ist}\right)},$$

wobei der Wert g_S der Funktion im betrachteten Betriebspunkt nach Voraussetzung bekannt ist. Sodann entwickelt man die momentenerzeugende Funktion in eine Potenzreihe nach ζ. Hiermit ergibt sich letztlich $$G_S=\frac{1}{Z\left(y_{ist}\right)}{\displaystyle \sum _{j=0}^N\frac{d^{j}Z\left(y_{ist}\right)}{d{y}_{ist}^i}\frac{{\xi }^j}{j!}+R_{G,N,S}{\xi }^{N+1}.}$$

In summary, the principle functionality of the combined moment-cumulative method consists in that from the functional relationship $$f\left(x\right)=Z\left(y\right)$$

the torque-generating function in the actuator representation (only the manipulated variable representation is shown here as an example) can be calculated $$G_{\mathrm{S.}}\left(y_{ist},\xi \right):=\frac{Z\left(y_{ist}+\xi \right)}{Z\left(y_{ist}\right)},$$

where the value g _{S of} the function at the operating point under consideration is known according to the assumption. The moment-generating function is then expanded into a power series according to ζ. This ultimately results $$G_{\mathrm{S.}}=\frac{1}{Z\left(y_{ist}\right)}{\displaystyle \sum _{j=0}^N\frac{d^{j}Z\left(y_{ist}\right)}{d{y}_{ist}^i}\frac{{\xi }^j}{j!}+{\mathrm{R.}}_{G,N,\mathrm{S.}}{\xi }^{N+1}.}$$

The solution to the control problem now consists in breaking off the power series at an order N at which the resulting equation can still be solved for ζ.

The mathematical structure of the power series expansion shows a formal analogy to that of a moment-generating function, as it is often used in mathematical statistics. Because of this, it is suggested to define a cumulant-generating function F: = In G in addition to the torque-generating function. A power series expansion can also be carried out for this. The idea behind this approach is that the series expansion for the cumulant-generating function exhibits a convergence behavior which deviates from that of the torque-generating function. Based on the comparison of the remaining terms, it can then be decided which power series actually represents a better approximation in the operating point under consideration. Thus, the application range of the control method is expanded compared to a simple power series approach, since the reliability of the method requires the convergence of only one method.

beziehungsweise $$m_l=B_l\left(k_1,k_2,\dots ,k_l\right),$$

worin B die jeweiligen vollständigen Bell-Polynome darstellt.From a computational point of view, the cumulants k and moments m can also be algebraically converted into one another according to $$k_l={\displaystyle \sum _{j=1}^l{\left(-1\right)}^{j+1}\left(j-1\right)!{\mathrm{B.}}_{l,j}\left(m_1,m_2,...,m_l\right),}$$

respectively $$m_l={\mathrm{B.}}_l\left(k_1,k_2,...,k_l\right),$$

where B represents the respective complete Bell polynomials.

Further findings can be formulated as follows. From a mathematical point of view, the different convergence behavior of the power series is due to the fact that different functions are developed. This has to do with the degree of non-linearity of the system. One can clearly see this using the example that the order of the power series is chosen as N = 1. If the function Z (y) now shows, for example, a strongly disproportionate behavior at the operating point under consideration (solid line), a comparatively "steep" function is approximated by a straight line (dashed) (see linear approximation of the function G (ζ) schematically in 8a using the function Z (y) = exp [y + y ² ] at the operating point y = 1). On the other hand, taking the logarithm when forming the cumulant generating function mitigates the strong overproportionality (see the linear approximation of the function F (ζ) schematically 8b) .

If the function under consideration is instead strongly disproportionate, then taking the logarithm will reinforce this tendency. For this reason, one expects a better approximation in the context of the torque generating function than for the cumulant generating function. This is in each case schematically for the function Z (y) = ln [y + y ² ] 9a and 9b illustrated.

A generalization of the torque-generating and cumulant-generating functions or a possible generalization of the method discussed above consists in the following approach.

Instead of the torque generating function G, we define the generating function H ⁽⁰⁾ zeroth “level” $$H^{\left(0\right)}=\left(\xi \right):=G\left(\xi \right)-1.$$

In this notation, the cumulant-generating function would then correspond to the following expression $$\mathrm{F.}\left(\xi \right)=\text{ln}\left[1+H^{\left(0\right)}\left(\xi \right)\right].$$

Therefore we define the right side of this equation in the new notation. From this it follows $$H^{\left(-1\right)}\left(\xi \right)=\text{ln}\left[1+H^{\left(0\right)}\left(\xi \right)\right].$$

But of course there is no need to limit yourself to the “levels” (0) and (-1). Instead, you can write recursively $$H^{\left(n\right)}\left(\xi \right)=\text{ln}\left[1+H^{\left(n+1\right)}\left(\xi \right)\right].$$

und $$m_l^{\left(n+1\right)}=B_l\left(m_1^{\left(n\right)},m_2^{\left(n\right)},\dots ,m_l^{\left(n\right)}\right),$$

worin die Größe $m_l^{\left(n\right)}$

das I-te Moment der erzeugenden Funktion (n)-ter Stufe darstellt.As can be shown, the relationships (38) and (39) also exist between adjacent levels n and n + 1. In summary, these relationships can be written through $$m_l^{\left(n\right)}={\displaystyle \sum _{j=1}^l{\left(-1\right)}^{j+1}\left(j-1\right)!{\mathrm{B.}}_{l,j}\left(m_1^{\left(n+1\right)},m_2^{\left(n+1\right)},...,m_l^{\left(n+1\right)}\right)}$$

and $$m_l^{\left(n+1\right)}={\mathrm{B.}}_l\left(m_1^{\left(n\right)},m_2^{\left(n\right)},...,m_l^{\left(n\right)}\right),$$

wherein the size $m_l^{\left(n\right)}$

represents the I-th moment of the generating function (n) -th stage.

eine Potenzreihenentwicklung mit jeweils unterschiedlichem Konvergenzverhalten vorgenommen und die jeweils entstehende Funktion nach ζ aufgelöst werden.Thus, for every function H ⁽ⁿ⁾ (ζ) with $n\in \mathbb{Z}\_\text{U}\left\{0\right\}$

a power series expansion with different convergence behavior is carried out and the resulting function is resolved according to ζ.

A further generalization of the procedure is to reverse the logarithm formation, so that equation error! Reference source not found. generally to be declared valid for n ∈ ℤ, whereby the relations (38) and (39) are always valid between adjacent levels.

der erzeugenden Funktion H⁽ⁿ⁾ n-ter Stufe erhalten, so dass ebenjenes n zu wählen ist, welches im Betriebspunkt mit dem minimalen zweiten Moment korrespondiert.A generalization of the solution of a control problem (or as already mentioned above) we get with the generalization of the moment and the cumulative generating functions of infinitely many power series expansions, all of which have different convergence behavior. Now one would like to take advantage of this fact by identifying the stage that has the fastest convergence behavior at the operating point before the calculation of ζ. The guideline here is the knowledge that the polynomial approximation works better the sooner the function curve of the generated function under consideration corresponds to that of a straight line. In other words, we are looking for the generating function of that stage n which has the smallest curvature at the operating point under consideration. The curvature can be seen from the respective second moment $H_2^{\left(n\right)}$

of the generating function H ⁽ⁿ⁾ n-th stage, so that the same n is to be selected which corresponds to the minimum second moment at the operating point.

so dass sich rekursiv $$h_2^{\left(n\right)}=h_2^{\left(0\right)}+n{\left(h_1^{\left(0\right)}\right)}^2$$

ergibt.Furthermore, from relation (38) we get the following equation between the second moments of the generating functions of neighboring levels: $$H_2^{\left(n\right)}=H_2^{\left(n-1\right)}+{\left(H_1^{\left(0\right)}\right)}^2,$$

so that is recursive $$H_2^{\left(n\right)}=H_2^{\left(0\right)}+n{\left(H_1^{\left(0\right)}\right)}^2$$

results.

wobei natürlich zu beachten ist, dass die so angegebene Lösung eine reelle Zahl darstellt, wohingegen die Stufe durch eine ganze Zahl repräsentiert wird.The second moment of the nth level is accordingly represented by a straight line in n, the slope of which is given by the first moment and the ordinate section of which is given by the second moment of the zeroth level. The minimum curvature is then obtained for that n for which equation (46) has a zero. So it applies $$n_{\mathrm{M.}in}=-\frac{H_2^{\left(0\right)}}{{\left(H_1^{\left(0\right)}\right)}^2},$$

Of course, it should be noted that the solution given in this way represents a real number, whereas the level is represented by an integer.

For this reason, the step with the smallest curvature is determined by the ceil or floor function $$n_{\mathrm{M.}in}=\{\begin{array}{c}\lfloor -\frac{H_2^{\left(0\right)}}{{\left(H_1^{\left(0\right)}\right)}^2}\rfloor \text{For}\left|\lfloor \frac{H_2^{\left(0\right)}}{{\left(H_1^{\left(0\right)}\right)}^2}\rfloor -\frac{H_2^{\left(0\right)}}{{\left(H_1^{\left(0\right)}\right)}^2}\right|\le \left|\lceil \frac{H_2^{\left(0\right)}}{{\left(H_1^{\left(0\right)}\right)}^2}\rceil -\frac{H_2^{\left(0\right)}}{{\left(H_1^{\left(0\right)}\right)}^2}\right|\\ \lfloor -\frac{H_2^{\left(0\right)}}{{\left(H_1^{\left(0\right)}\right)}^2}\rfloor \text{For}\left|\lfloor \frac{H_2^{\left(0\right)}}{{\left(H_1^{\left(0\right)}\right)}^2}\rfloor -\frac{H_2^{\left(0\right)}}{{\left(H_1^{\left(0\right)}\right)}^2}\right|>\left|\lceil \frac{H_2^{\left(0\right)}}{{\left(H_1^{\left(0\right)}\right)}^2}\rceil -\frac{H_2^{\left(0\right)}}{{\left(H_1^{\left(0\right)}\right)}^2}\right|\end{array}.$$

After the level of minimum curvature has been determined in this way, the associated power series can be terminated at an order N, which analytically allows a resolution according to ζ and the new manipulated variable can be calculated with this. In this case, the associated power series is the one that shows the fastest convergence of all sub-series (SKAT) and therefore represents the best approximation to the actually nonlinear function Z (y) in the order under consideration, so that the quality of the solution of the simple moments found with it -Cumulant method is superior.

Finally, in order to minimize fluctuations, the power series of adjacent stages can be used to solve the problem, so that a subsequent residual term estimate can be used to find the operating point-dependent solution of the control problem. If this is not done, the SKAT method shows a higher control quality with less computational effort (only a power series equation has to be solved for ζ. In contrast, the determination of n _Min is numerically favorable). If the verification is also carried out in accordance with the remaining term estimate, the computational effort increases, but the method is then in any case of higher control quality and more stable against fluctuations.

QUOTES INCLUDED IN THE DESCRIPTION

This list of the documents listed by the applicant was generated automatically and is included solely for the better information of the reader. The list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions.

Patent literature cited

• DE 19634923 A1 [0002]
• DE 102016124652 B3 [0049]

Non-patent literature cited

• DIN IEC 60050-351 [0010]

Claims (12)

Method for operating a technical system, - whereby a reference variable (x _soll ) is specified and a variable to be influenced (x _ist ) results from the action of a manipulated variable (y _soll , y _ist ) on a route influenced by disturbance variables (u), - with a torque-generating function (G _S , G _R ) at a specific operating point of the technical system based on a setpoint controlled variable function f (x _soll ) and an actual manipulated variable function Z (y _ist ) or on the basis of a setpoint controlled variable function f (x _soll ) and an actual controlled variable function ƒ (x _ist ) ) taking into account a difference (ζ) between the actual value of a manipulated variable (y _ist ) and the setpoint of a manipulated variable (y _soll ) is formed, - based on the development of the torque-generating functional (Gs, G _R ) in a power series, up to one Order N a polynomial is provided that is resolved for the difference (ξ) so that the difference (ξ) is provided, - where depending on the difference (ξ) un d according to the actual value of the manipulated variable (y _ist ) $$\xi =y_{sOll}-y_{ist}$$

a current setpoint of the manipulated variable (y _soll ) is formed and the path is acted upon depending on the current setpoint of the manipulated variable (y _soll ) so that the variable to be influenced (x _ist ) corresponds to or is equal to the reference variable (x _soll ) Adjusts default. Procedure according to Claim 1 , - where a reference variable (x _soll ) is specified and a variable (x _ist ) to be influenced results from the action of a manipulated variable (y _soll , y _ist ) on a path influenced by disturbance variables (u), - where in a specific operating point of the technical system, the formation of the quotient from a target controlled variable function ƒ (x _soll ) and an actual manipulated variable function Z (y _ist ) takes place, so that the torque-generating functional (Gs) according to the equation $$\frac{f\left(x_{\text{should}}\right)}{Z\left(y_{ist}\right)}=\frac{Z\left(y_{ist}+\xi \right)}{Z\left(y_{ist}\right)}=:G_{\mathrm{S.}}\left[x_{\text{should}}\left(y_{\text{is}},\xi \right),y_{\text{is}}\right]$$

is available, where (ζ) is the difference between the actual value of the manipulated variable (y _is ) and the nominal value of the manipulated variable (y _soll ), - where based on the development of the torque-generating functional (G _S ) in a power series, up to an order N. polynomial $$G_{\mathrm{S.}}\left[x_{sOll}\left(y_{\text{is}},\xi \right),x_{\text{is}}\right]=\frac{1}{Z\left(y_{\text{is}}\right)}{\displaystyle \sum _{j=0}^N\frac{d^{j}Z\left(y_{\text{is}}\right)}{{\text{dy}}_{\text{is}}^j}}\frac{{\xi }^j}{j!}+\frac{1}{Z\left(y_{\text{is}}\right)}\frac{{\text{d}}^{N+1}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!}$$

standing, the precipitate is dissolved with neglect of the residual limb to the difference (ξ) such that the difference (ξ) is provided, - wherein in dependence on the difference (ξ) and the actual value of the manipulated variable _(y), according to $$\xi =y_{sOll}-y_{ist}$$

a current setpoint of the manipulated variable (y _soll ) is formed and the path is acted upon depending on the current setpoint of the manipulated variable (y _soll ) so that the variable to be influenced (x _ist ) corresponds to or is equal to the reference variable (x _soll ) Adjusts default. Procedure according to Claim 1 , - where a reference variable (x _soll ) is specified and a variable (x _ist ) to be influenced results from the action of a manipulated variable (y _soll , y _ist ) on a path influenced by disturbance variables (u), - where in a specific operating point of the technical system, the formation of the quotient from a target controlled variable function f (x _soll ) and an actual controlled variable function ƒ (x _ist ) takes place, so that the torque-generating functional (G _R ) according to the equation $$G_{\mathrm{R.}}\left[x_{sOll}\left(y_{\text{is}},\xi \right),x_{\text{is}}\right]:=\frac{f\left(x_{\text{should}}\right)}{f\left(x_{\text{is}}\right)}=\frac{Z\left(y_{\text{is}}+\xi \right)}{f\left(x_{\text{is}}\right)}$$

is available, where (ζ) is the difference between the actual value of the manipulated variable (y _is ) and the setpoint of the manipulated variable (y _soll ), - where based on the development of the torque-generating functional (G _R ) in a power series up to an order N is a polynomial $$G_{\mathrm{R.}}\left[x_{sOll}\left(y_{\text{is}},\xi \right),x_{\text{is}}\right]=\frac{1}{f\left(x_{\text{is}}\right)}{\displaystyle \sum _{j=0}^N\frac{d^{j}Z\left(y_{\text{is}}\right)}{{\text{dy}}_{\text{is}}^j}}\frac{{\xi }^j}{j!}+\frac{1}{f\left(x_{\text{is}}\right)}\frac{{\text{d}}^{N+1}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!},$$

standing, the precipitate is dissolved with neglect of the residual limb to the difference (ξ) such that the difference (ξ) is provided, - wherein in dependence on the difference (ξ) and the actual value of the manipulated variable _(y), according to $$\xi =y_{sOll}-y_{ist}$$

a current setpoint of the manipulated variable (y _soll ) is formed and the path is acted upon depending on the current setpoint of the manipulated variable (y _soll ) so that the variable to be influenced (x _ist ) corresponds to or is equal to the reference variable (x _soll ) Adjusts default. Method for operating a technical system, whereby - a reference variable (x _soll ) is specified and a variable to be influenced (x _ist ) results from the effect of a manipulated variable (y _soll , y _ist ) on a route influenced by disturbance variables (u), - in a specific operating point of the technical system, the formation of a deviation (ζ) between the reference variable (x _soll) and influencing variable _(x) is carried out, - the time derivative (dζ / dt) of the deviation (ζ) is provided, - the formation of the torque-generating functional (G) according to the equation $$G\left[x_{sOll},x_{ist};y_{ist,}\dot{\zeta }\right]=\frac{\dot{\zeta }}{H\left(x_{sOll},y_{ist}\right)-H\left(x_{ist},y_{ist}\right)}$$

takes place, - the torque-generating functional (G) using $$y_{sOll}\left(t\right)=y_{ist}\left(t\right)+\xi \left(t\right)$$

is expanded into a power series of the Nth order around ζ = 0, where (ζ) is the difference between the actual value of the manipulated variable (y _ist ) and the nominal value of the manipulated variable (y _soll ), so that $$G\left[x_{sOll},x_{ist};y_{ist};\zeta \right]=1+{\displaystyle \sum _{j=1}^N\frac{m_j}{j!}{\xi }^j+\frac{m_{N+1}}{\left(N+1\right)!}{\xi }^{N+1}},$$

is ready, - the power series obtained, neglecting the remainder of the term, is resolved for the difference (ξ), so that the difference (ξ) is made available, - as a function of the difference (ξ) and the actual value of the manipulated variable (y _ist ), according to $$\xi \left(t\right)=y_{sOll\left(t\right)}-y_{ist\left(t\right)}$$

a current setpoint of the manipulated variable (y _soll ) is formed and the path is acted upon depending on the current setpoint of the manipulated variable (y _soll ) so that the variable to be influenced (x _ist ) corresponds to or is equal to the reference variable (x _soll ) Adjusts default. Method for operating a technical system, - whereby a reference variable (x _soll ) is specified and a variable to be influenced (x _ist ) results from the action of a manipulated variable (y _soll , y _ist ) on a route influenced by disturbance variables (u), - with a torque-generating function (G _S , G _R ) at a specific operating point of the technical system based on a setpoint controlled variable function f (x _soll ) and an actual manipulated variable function Z (y _ist ) or on the basis of a setpoint controlled variable function f (x _soll ) and an actual controlled variable function ƒ (x _ist ) ) taking into account a difference (ζ) between the actual value of the manipulated variable (y _ist ) and the setpoint of the manipulated variable (y _soll ) is formed, - where the natural logarithm of the torque-generating functional (G _S , G _R ) is the cumulative-generating functional (F _S , F _R ), - where, based on the expansion of the cumulant-generating functional (F _S , F _R ) into a power series up to an order N, a polynomial is provided rd, which is resolved according to the difference (ξ) so that the difference (ξ) is provided, - depending on the difference (ξ) and the actual value of the manipulated variable (y _ist ), according to $$\xi =y_{sOll}-y_{ist}$$

a current setpoint of the manipulated variable (y _soll ) is formed and the path is acted upon depending on the current setpoint of the manipulated variable (y _soll ) so that the variable to be influenced (x _ist ) corresponds to or is equal to the reference variable (x _soll ) Adjusts default. Procedure according to Claim 5 , - where a reference variable (x _soll ) is specified and a variable (x _ist ) to be influenced results from the action of a manipulated variable (y _soll , y _ist ) on a route influenced by disturbance variables (u), - in a specific operating point of the technical system the formation of the quotient of a desired controlled variable function f (x _soll) and a Iststellgrößenfunktion Z _(y) is carried out so that the torque-generating functional (Gs) according to the equation $$\frac{f\left(x_{\text{should}}\right)}{Z\left(y_{\text{is}}\right)}=\frac{Z\left(y_{\text{is}}+\xi \right)}{Z\left(y_{\text{is}}\right)}=:G_{\mathrm{S.}}\left[x_{\text{should}}\left(y_{\text{is}},\xi \right),y_{\text{is}}\right]$$

ready, wherein (ζ) is the difference between the actual value of the manipulated variable _(y) and the target value of the manipulated variable (y _soll), - the natural logarithm of the moment generating functional (Gs), the kumulantenerzeugende functional (Fs) according to the equation $${\mathrm{F.}}_{\mathrm{S.}}:=\text{ln}{G}_{\mathrm{S.}}$$

- where, based on the expansion of the cumulant-generating functional (F _S ) into a power series up to order N, a polynomial $${\mathrm{F.}}_{\mathrm{S.}}={\displaystyle \sum _{j=1}^N\frac{d^j\text{ln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^j}}\frac{{\xi }^j}{j!}\frac{d^{N+1}\text{ln}Z\left(y_{\text{is}}\right)}{\text{d}{y}_{\text{is}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!},$$

standing, the precipitate is dissolved with neglect of the residual limb to the difference (ξ) such that the difference (ξ) is provided, - wherein in dependence on the difference (ξ) and the actual value of the manipulated variable _(y), according to $$\xi =y_{sOll}-y_{ist}$$

a current setpoint of the manipulated variable (y _soll ) is formed and the path is acted upon depending on the current setpoint of the manipulated variable (y _soll ) so that the variable to be influenced (x _ist ) corresponds to or is equal to the reference variable (x _soll ) Adjusts default. Procedure according to Claim 5 , - where a reference variable (x _soll ) is specified and a variable (x _ist ) to be influenced results from the action of a manipulated variable (y _soll , y _ist ) on a path influenced by disturbance variables (u), - where in a specific operating point of the technical system, the formation of the quotient from a target controlled variable function f (x _soll ) and an actual controlled variable function ƒ (x _ist ) takes place, so that the torque-generating functional (G _R ) according to the equation $$G_{\mathrm{R.}}\left[x_{sOll}\left(y_{\text{is}},\xi \right),x_{\text{is}}\right]:=\frac{f\left(x_{\text{should}}\right)}{f\left(x_{\text{is}}\right)}=\frac{Z\left(y_{\text{is}}+\xi \right)}{f\left(x_{\text{is}}\right)}$$

ready, wherein (ζ) is the difference between the actual value of the manipulated variable _(y) and the target value of the manipulated variable (y _soll), - the natural logarithm of the moment generating functional (G _R), the kumulantenerzeugende functional (F _R) according to the equation $${\mathrm{F.}}_{\mathrm{R.}}=\text{ln}{G}_{\mathrm{R.}}$$

is, - where a polynomial is based on the expansion of the cumulant-generating functional (F _R ) into a power series of the N-th order $${\mathrm{F.}}_{\mathrm{R.}}={\displaystyle \sum _{j=1}^N\frac{d^j\text{ln}Z\left(y_{ist}\right)}{d{y}_{\text{is}}^j}\frac{{\xi }^j}{j!}}+\frac{d^{N+1}\text{ln}Z\left(y_{ist}\right)}{d{y}_{\text{is}}^{N+1}}\frac{{\xi }^{N+1}}{\left(N+1\right)!},$$

is ready, which is dissolved, ignoring the residual limb to the difference (ξ) such that the difference (ξ) is provided, - wherein, depending on the difference (ξ) and the actual value of the manipulated variable _(y) according to $$\xi =y_{sOll}-y_{ist}$$

a current setpoint of the manipulated variable (y _soll ) is formed and the path is acted upon depending on the current setpoint of the manipulated variable (y _soll ) so that the variable to be influenced (x _ist ) corresponds to or is equal to the reference variable (x _soll ) Adjusts default. Method for operating a technical system, whereby - a reference variable (x _soll ) is specified and a variable to be influenced (x _ist ) results from the effect of a manipulated variable (y _soll , y _ist ) on a route influenced by disturbance variables (u), - in a specific operating point of the technical system, the formation of a deviation (ζ) between the reference variable (x _soll) and influencing variable _(x) is carried out, - the time derivative (dζ / dt) of the deviation (ζ) is provided, - the formation of the torque-generating functional (G) according to the equation $$G\left[x_{sOll},x_{ist};y_{ist},\dot{\zeta }\right]=\frac{\dot{\zeta }}{H\left(x_{sOll},y_{ist}\right)-H\left(x_{ist},y_{ist}\right)}$$

takes place, - the natural logarithm of the torque-generating functional (G) is the cumulant-generating functional (F), - the cumulant-generating functional (F) using $$y_{sOll}\left(t\right)=y_{ist}\left(t\right)+\xi \left(t\right)$$

is developed into a power series around ξ = 0, where (ξ) is the difference between the actual value of the manipulated variable (y _ist ) and the nominal value of the manipulated variable (y _soll ), so that the function generating the cumulative is available, - the cumulative-generating function is resolved according to the difference (ξ) so that the difference (ξ) is made available, - depending on the difference (ξ) and the actual value of the manipulated variable (y _ist ), according to $$\xi \left(t\right)=y_{sOll\left(t\right)}-y_{ist\left(t\right)}$$

a current setpoint of the manipulated variable y _soll ) is formed and the path is acted upon depending on the current setpoint of the manipulated variable (y _soll ) so that the variable to be influenced (x _ist ) corresponds to the reference variable (x _soll ) or is based on this specification aligns. Method for operating a technical system, the method according to Claim 1 or 2 or 3 or 4th first procedures for providing the difference (ξ, ξ _Mom ) are and the procedures according to Claim 5 or 6th or 7th or 8th Second methods for providing the difference (ξ, ξ _Kum ) are, with a selection being made as to whether the difference (ξ, ξ _Mom ) according to the first method is used to form the current setpoint of the manipulated variable (y _soll ) for acting on the route or the difference (ξ, ξ _Cum ) is used according to the second method, whereby the selection can be made depending on the order N in which the respective power series is developed and for which power series after resolution according to the differences (ξ, ξ _Mom ) and (ξ, ξ _Cum ) the smaller residual term results. Method for operating a technical system, wherein the method according to Claims 1 to 9 is based on a general principle with the following features: - where instead of the torque-generating function (G) a function (H ⁽⁰⁾ ) is defined by $$H^{\left(0\right)}\left(\xi \right):=G\left(\xi \right)-\mathrm{1,}$$

- where accordingly the cumulant-generating function is given by the following expression $$\mathrm{F.}\left(\xi \right)=\text{ln}\left[1+H^{\left(0\right)}\left(\xi \right)\right]$$

corresponds - so that from this $$H^{\left(-1\right)}\left(\xi \right)=\text{ln}\left[1+H^{\left(0\right)}\left(\xi \right)\right]$$

results in - where, in order not to be restricted to the levels (0) and (-1), is recursive $$H^{\left(n\right)}\left(\xi \right)=\text{ln}\left[1+H^{\left(n+1\right)}\left(\xi \right)\right]$$

results, - where the cumulants correspond to level n + 1 and the moments to level n, so that for each function $$H^{\left(n\right)}\left(\xi \right)\text{With}n\in \mathbb{Z}\_\text{U}\left\{0\right\}$$

a power series expansion with different convergence behavior is carried out and the function that arises in each case can be resolved according to the difference (ξ), - whereby a selected function is resolved according to the difference (wobei), - whereby depending on a difference (ξ), which is based on a selected Function has been formed, a current setpoint of the manipulated variable (y _soll ) is formed and the path is acted on depending on the current setpoint of the manipulated variable (y _soll ) so that the variable to be influenced (x _ist ) of the reference variable (x _soll ) corresponds to or approximates this requirement. Procedure according to Claim 10 , where another generalization is to reverse the logarithm and the equation $$H^{\left(n\right)}\left(\xi \right)=\text{ln}\left[1+H^{\left(n+1\right)}\left(\xi \right)\right]$$

generally to be declared valid for n ∈ ℤ, - with the generalization of the moment-generating and cumulant-generating functions, an infinite number of power series expansions, all of which have different convergence behavior, are obtained, - where the level n is identified before the calculation of the difference (ξ) which has the fastest convergence behavior at the operating point, - taking into account the knowledge that a polynomial approximation works better, the sooner the function curve of the generated function under consideration corresponds to that of a straight line, the generating function of that level n is searched for in the Operating point has the smallest curvature, - the curvature resulting from the respective second moment $\left(H_2^{\left(n\right)}\right)$

of the generating function (H ⁽ⁿ⁾ ) n-th stage, so that n is to be chosen which corresponds to the minimum second moment at the operating point, - whereby this means that moments of generating functions of neighboring stages can be algebraically converted into one another , the following relationship between the second moments of the generating functions of the same neighboring stages $$H_2^{\left(n\right)}=H_2^{\left(n-1\right)}+{\left(H_1^{\left(0\right)}\right)}^2$$

results so that recursively $$H_2^{\left(n\right)}=H_2^{\left(0\right)}+n{\left(H_1^{\left(0\right)}\right)}^2$$

- where the second moment of the n-th stage is represented by a straight line in n, the slope of which is given by the first moment and the ordinate section of which is given by the second moment of the zeroth stage and the minimum curvature results for the n for which the equation $$H_2^{\left(n\right)}=H_2^{\left(0\right)}+n{\left(H_1^{\left(0\right)}\right)}^2$$

has a zero, - where $$n_{\mathrm{M.}in}=\frac{H_2^{\left(0\right)}}{{\left(H_1^{\left(0\right)}\right)}^2},$$

applies, - it should be noted that the solution given in this way represents a real number, whereas the level is represented by an integer. Device that is set up, a method according to one of the Claims 1 to 11 execute, wherein the device can be part of a vehicle.

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