WO2000034904A2

WO2000034904A2 - Method and arrangement for designing a technical system

Info

Publication number: WO2000034904A2
Application number: PCT/DE1999/003877
Authority: WO
Inventors: Stefan SCHÄFFLER; Klaus Weinzierl
Original assignee: Siemens Aktiengesellschaft
Priority date: 1998-12-04
Filing date: 1999-12-03
Publication date: 2000-06-15
Also published as: WO2000034904A3

Abstract

The invention permits the design of a technical system comprising two target functions by detecting gradients for both target functions. An area around the bisector between the gradients determines the new search direction. An iterative determination of the search direction produces a path to an efficient working point which is suitable for designing the technical system.

Description

description

Process and arrangement for the design of an external technical system

The invention relates to a method and an arrangement for designing a technical system.

For a complex technical system, significant sizes such as costs for the manufacture or efficiency of the system are of interest in an early planning phase or during operation. A dependency of each of these variables on a predetermined set n of parameters influencing them (operating parameters), summarized in a parameter vector x of dimension n, is detected by a target function.

If there are several objective functions with competing objectives, it must be ensured that the parameters simultaneously represent a solution with sufficient good for the given objective functions.

The object of the invention is to provide a method and an arrangement for designing a technical system, parameters of the system being determined which are optimal with regard to two predetermined target functions.

This object is achieved according to the features of the independent claims. Further developments of the invention also result from the dependent claims.

To solve the task, a procedure is specified that enables the design of a technical system. The technical system comprises two target functions fη_ and ± _{2 r} , each target function having a predetermined set of n parameters

(xi, X2, ■ ■ -, x _n ⁾ = * r (1) summarized in a parameter vector x, can be influenced. Gradients of the parameter vector x are determined for the two target functions. An approximately bisector (dot product greater than zero) for directions determined by the gradients is determined and a next parameter vector is determined in a predetermined range around the direction of the bisector. The next parameter vector is used to design the technical system.

In particular, the "approximately" bisector is determined by the relationship

z ^T • gj_> 0,

where z denotes the search direction and gi (i = 1, 2, ..., k) the gradients of the target functions.

Optionally, a new design of the technical system or an adaptation of an existing technical system can be carried out on the basis of the parameter vector determined using the described method. In both cases it is a draft (once as a regeneration and once for adaptation) in the sense of the present explanations.

A further development is that the described method is used iteratively as long as no termination condition is met. In the case of the iteration, the next parameter vector is set equal to the parameter vector x and branched to the step in which the gradients of the

Objective functions can be determined. This ensures that, along a path that comprises several parameter vectors in accordance with the iterations of the method, the next parameter vector from iteration step to iteration step guarantees a qualitatively improved design (a higher quality) of the technical system compared to its predecessor. The quality for the design of the technical system is evaluated based on the target functions fi and ± 2. Each value assignment of the parameters (referred to as the value of the parameter vector x) gives a value for the quality for each target function. Accordingly, the two objective functions can be understood as competing with one another. A high quality of one objective function usually corresponds to a low quality of the other. Examples of target functions, especially competing target functions, are:

f _l = plant efficiency, ± 2 = plant costs

or

f _l = product throughput,

± 2 = probability of failure

For a successful design, i.e. a successful compromise solution between the competing target functions, values for the parameter vector are determined, which are each efficient. An efficient value assignment of the parameter vector means that no parameter of the

Parameter vector can be changed without there being a deterioration in the quality of at least one target function. Such a value of the parameter vector is called an efficient point or a pareto-optimal point.

A technical system can be a process engineering system or another system that is to be designed or set with regard to (preferably several) parameters. In particular, the parameters of the parameter vector x can be design parameters or operating parameters of the technical system. Operating parameters indicate possible adjustable sizes, while design parameters describe, in particular, physical dimensions of the technical system and cannot be adapted or changed at all or only with great effort during operation.

In a further embodiment, the technical system is implemented or set on the basis of the determined parameters. This has the advantage that the parameters in a parameter vector, which was determined by means of the invention, identify a stable operating point and the setting of the system to this operating point ensures permanent, safe operation of the system / system.

An embodiment of the invention is that the

Abort condition is given as soon as a predetermined number of iterations has been carried out. In this case it is ensured that the method terminates after a certain time and that the parameter vector last determined provides a suitable approximation for the efficient point sought.

Another possibility for the termination condition is given if the two gradients of the target functions determined point in opposite directions. In this case, an efficient point has been found, further iterations do not lead to an improvement in the quality for both target functions.

The termination conditions can be combined with one another as part of an additional development: If the gradients point in opposite directions, the method terminates, otherwise the iterations are continued up to the predetermined maximum number of iterations.

In particular, the method described provides an efficient point (parameter vector) after a pass with possibly numerous iterations. To make several efficient To get points, the determination of the gradients of the two target functions is overlaid with a stochastic variable. As a result, several different efficient points result when the method is used repeatedly. It is advantageous here that the globally efficient point is determined with a high degree of probability by the superposition with the stochastic variable. Locally efficient points are overcome by the random size, by scattering the random size also examining the surroundings of this supposedly efficient point for further improvement. This leads to a high degree of certainty that there is a further possibility for improvement in the vicinity of a locally efficient point, along which the path to the globally efficient point is pursued.

Furthermore, it is advantageous that the stochastic superimposition makes it possible to determine several different efficient points that lie along a line in the n-space (n: dimension of the parameter space) and thus, with a sufficiently frequent repetition of the method, this line with efficient points becomes clear to mark.

As part of a detailed design of the technical system, e.g. a technical system, sometimes a setting according to the specifications of an determined efficient point (parameter vector) cannot be implemented, which is why an alternative solution, that is to say another efficient point, must be used. Based on the configuration described above, the invention enables the automatic generation of a number of alternative solutions, each of which represents an efficient implementation with regard to the setting or design of the technical system or the technical system.

A further development consists in that in a predetermined area along the bisector by a predetermined one Move length and the next parameter vector is determined. The direction of the bisector thus gives the path to the efficient point. In particular, the next parameter vector is along the direction

x = ci • Vfι_ (x) + C2 ^• f2 (x) (2)

is determined, where cl denotes a first predetermined constant, C2 a second predetermined constant, f] _ the first objective function, f2 the second objective function. Furthermore, the constants ci and C2 can be given by

0.5 ^c 2

| v ₂ ( ^χ j

In particular, equation (2) is maximized, with no

Limitation of generality can also be minimized (after multiplication with "-1").

Another embodiment of the invention is that the stochastic size is given by

ε B _t , (5)

where ε denotes a constant that can be specified for scaling and Bt denotes a random number. Equation (2) thus results taking into account the stochastic size

x = ci • Vfι (x) + C2 • Vf (x) + ε • B _t (6). Finally, it is a further development that the described method is carried out with a predetermined set of different start values for the parameter vector x.

To achieve the object, an arrangement for designing a technical system is also specified, which system can be described by two predefined target functions, each target function being influenced by a predefined set of n parameters and a value assignment of the n parameters being combined in a parameter vector x. The arrangement comprises a processor unit which is set up in such a way that

a) for the parameter vector x gradients of both target functions can be determined;

b) an angle bisector to directions determined by the gradients can be determined;

c) a next parameter vector can be determined in a predetermined range around the direction of the bisector of the angle;

d) the next parameter vector can be used to design the technical system.

This arrangement is particularly suitable for carrying out the method according to the invention or one of its developments explained above.

Embodiments of the invention are illustrated and explained below with reference to the drawing.

Show it

Fig.l a sequence of a method for designing a technical system; Fig.2 termination conditions for the method of Fig.l;

3 is a sketch that shows a parameter space with target functions;

4 shows a sketch which illustrates the determination of an efficient point in a parameter space;

Fig.5 is a sketch showing a local view on the

Search for a parameter vector of improved quality;

6 shows a processor unit,

In Fig.l is a block diagram showing steps of a method for designing a technical system.

The technical system can be described by two

Target functions fi and f2, each of which depends on a predetermined set n parameters, which are combined in a parameter vector x. A value assignment of the parameters is referred to as the value of the parameter vector x. This value of the parameter vector x represents a possible assignment of the

Parameters xi, X2, •••, x _n represent. The parameters are preferably design parameters or operating parameters (operating points) of the technical system, in particular a technical system. To determine a value assignment for the parameter vector x, which is an efficient point with regard to the design or operation of the technical system (see explanation above), the procedure is as follows:

In a step 101, a start value for the parameter vector x is specified, referred to here as the parameter vector Xi Gradients of the two target functions determined for the parameter vector xi. On the basis of the directions specified by the gradients, an angle bisector between these directions is determined in a step 103 and traversed in a predetermined range around this angle bisector with a predetermined step width along the direction in order ^to determine a next parameter vector xi ₊ 1 (cf. step 104) . It is checked in a step 105 whether an abort condition is fulfilled and, if this should be the case, the process branches to step 106 and the method ends.

Otherwise, that is, if the termination condition is not met, a branch is made to a step 107 and the next parameter vector x_i _{+ l is used} as the parameter vector Xi and branched back to step 102. The method is iterated, the quality of the resulting parameter vector being improved for each iteration step.

2 shows possible termination conditions which are preferably used in combination in the method according to FIG. 1 (there in step 105). Block 201 indicates that the termination conditions explained below are contained in block 105 from FIG. In a step 202, a query is made as to whether the determined gradients point in opposite directions. If this is the case, a branch is made to step 205 (or step 106) and the method is ended, otherwise a branch is made to step 203. In step 203, it is checked whether a predetermined number of maximum iteration steps to be carried out has already been reached and, if this has already happened, the method is ended in steps 205 and 106, respectively. Otherwise, the method is continued (cf. step 204 or 107).

The following Figures 3 to 5 illustrate the search for the efficient point using the example of a two-dimensional (n = 2) space. The parameter vector x accordingly comprises two components xi and X2, which span one level. In this xι, X2 ~ level, the The sketches described below clearly explain the relationship.

3 shows a sketch which shows a two-dimensional xi, X2 ~ parameter space with the two target functions fi and f ₂ . A curve course P shows the set of efficient points, a point on this line is given as an example with xp- In this point x _p are the gradients 301 and 302 as

each with a certain length. As already described above, an efficient point is characterized in that the two gradients point in opposite directions. This is the case for all points on curve P. If you start looking for an efficient point at a starting point x_0 and the two gradients 303 and 304 of the target functions are determined in this starting point xo, a region 305 is found

(hatched in FIG. 3) pointing the way for a parameter vector which is improved compared to the starting value xo. Accordingly, a necessary condition for a point x on the curve P can be formulated:

3α e [0,1] with • Vfι_ (x _p ) + (l - α) • Vf ₂ (x _p ) = 0 (:

At least one of the two gradients disappears at an efficient point on the curve P or both gradients point in opposite directions.

4 illustrates the path, beginning at a starting point xo hi- ⁿ to an efficient point Xp. A hatched area 401 indicates in which direction, starting from the starting point XQ, an improvement in the quality for at least one objective function is expected. Accordingly, the efficient point x_p is sought in the direction of the hatched area. At this point we would like to point out the values for the target functions in the xι, X2 ~ level. Several lines are drawn for fι = const and f2 = const in Fig.4. These lines can be interpreted as "contour lines", the gradient (for the respective) target function points in the direction of the maximum or minimum for this target function. It is easy to see that the two goal functions, for example the goal cost and the goal

Competing safety of a technical system: A system built with the lowest cost will not guarantee the highest level of safety and vice versa. The efficient point represents a compromise between these two goals. The parameters (here: xi and x) are those sizes that influence both the safety and the costs of the technical system. From these explanations it also appears that the search for an efficient point can be formulated both as a minimization problem or as a maximization problem, depending on the specification.

Figure 5 shows how to find a parameter vector (or point) that is better in terms of good. FIG. 5 shows a local view of the relationships shown in FIGS. 3 and 4. The following describes how the improved parameter vector is found in the hatched area 305 and 401.

On a line 501 all values of the target function fi are constant, on a line 502 all values of the

Target function f2 constant. Both lines 501 and 502 are shown in broken lines in FIG. A gradient 503 of the objective function fi has a length different from a gradient 504 of the objective function f2. Starting from the parameter vector 505, a subsequent parameter vector 506 is determined according to the following relationship: 0.5 0.5 x = Vfι (x) + Vf ₂ (x)

| Vfι (x ^ || Vf ₂ (χ ^

5 corresponds to the bisector between the gradients standardized in length. This bisector lies exactly in the dashed area 507. The solution to the right in equation (8) by means of conventional numerical methods that can be run on computers results in a candidate for an efficient point, that is to say an optimal assignment of its parameters for the design of the technical system. Such numerical methods are all methods that allow numerical treatment of initial value problems.

With regard to the determination of several efficient points, reference is made to the solution of the right side of equation (6) by means of numerical methods. Using the values of equation (8), we get:

x = "_ ° ' ⁵ ," Vfι (x) + r, - ^ Vf ₂ (x) + ε • B _t (9).

| Vfι (x ¹ | Vf ₂ (χ

The proportion of the random number "ε • B ^" results in a lot of efficient points when solving the right side of equation (9), all of which represent optimal alternative solutions for a design of the technical system. This results (at least in part) in the curve of line P, which includes all the efficient points of the respective technical system with the specified target functions.

A processor unit PRZE is shown in FIG. The processor unit PRZE comprises a processor CPU, a memory SPE and an input / output interface IOS, which is used in different ways via an interface IFC: an output is visible on a monitor MON and / or on a printer via a graphic interface PRT spent. An entry is made using a mouse MAS or a keyboard TAST. The processor unit PRZE also has a data bus BUS, which ensures the connection of a memory MEM, the processor CPU and the input / output interface IOS. Additional components can also be connected to the data bus BUS, for example additional memory, data memory (hard disk) or scanner.

Claims

claims

1. procedure for the design of a technical system,

a) in which the technical system has two predetermined target functions;

b) in which each objective function is influenced by a predetermined set of n parameters, with a value assignment of the n parameters in one

Parameter vector x is summarized;

c) in which gradients of both target functions are determined for the parameter vector x;

d) in which an angle bisector to directions determined by the gradients is determined;

e) in which a next parameter vector is determined in a predetermined range around the direction of the bisector;

f) in which the next parameter vector is used to design the technical system.

2. The method of claim 1, in which, as long as an abort condition is not met, after step e) the next parameter vector is set as the parameter vector x and a branch is made to step c).

3. The method of claim 2, wherein the termination condition is given when the two gradients point in opposite directions.

4. The method according to claim 2, wherein the termination condition is given when a predetermined number of iterations has been carried out.

5. The method according to any one of claims 1 to 4, wherein the parameter vector as a parameter comprises operating points and / or design parameters of the technical system.

6. The method according to any one of claims 1 to 5, in which move in the predetermined area along the bisector by a predetermined length and thereby the next parameter vector is determined.

7. The method according to any one of the preceding claims, wherein the next parameter vector along a direction

Xx = ci ^■ Vfι (x) + c ₂ • Vf ₂ (x)

is determined, ci denoting a first predefined constant, C2 a second predefined constant, fl the first target function, f2 the second target function.

8. The method according to claim 7, wherein the constants oχ and C2 are given by

0.5 ci and

I ^ G I

9. The method according to any one of the preceding claims, in which the technical system is implemented or adapted on the basis of the parameters obtained.

10. The method according to any one of the preceding claims, in which the gradients are additionally superimposed with a stochastic size in step c).

11. The method according to claim 10, wherein the stochastic size is given by

ε B _t ,

where ε denotes a constant that can be specified for scaling and B denotes a random number.

12. The method according to claim 10 or 11, which is carried out for a predetermined set of different parameter vectors x.

13. Arrangement for the design of a technical system which has two predetermined target functions, each target function being influenced by a predetermined set of n parameters and a value assignment of the n parameters being combined in a parameter vector x, with a processor unit which is set up in such a way that

c) a next parameter vector in a predetermined range around the direction of the bisector is determinable;

the next parameter vector can be used to design the technical system.