DE102009009328B4 - Method for reducing structural vibrations and control device for this purpose - Google Patents

Abstract

A control device is described with a plurality of controllers ( R _n ) for a number of controlled systems ( G _n ) corresponding to the number N of controllers ( R _n ), which are weighted by respective linearly dependent weighting factors (β _n ). The effective, from the plurality of controllers (R _n) interpolated controller (R) is given by the sum of a respective associated weighting factor (β _n) Weighted individual controller output signals of the controller (R _n). The effective of the plurality of control paths (G _n) interpolated controlled system (G) resulting from the sum of a respective associated weighting factor (β _n) Weighted individual plant output signals of the control lines (G _n). The sum of the weighting factors (β _n ) is equal to one. The linear dependent weighting factors (β _n ) describing a regular (N - 1) simplex are defined by transformation into a (N - 1) -dimensional hypercube completely described with linearly independent variables (δ _k ).

Description

The invention relates to a method for reducing structural vibrations with a control loop by means of vibration-influencing actuators, which are controlled by regulators, according to claim 1.

The invention further relates to a control device according to claim 4

The control device and the method for reducing structural vibrations based on the fundamental problem that in the control of structural oscillations often non-linear control systems are present. To solve the control problem with techniques based on a linear model approach, the behavior of the non-linear control systems is linearized at specific points, the so-called operating points. In each of these operating points, a controller is designed according to the local behavior. In order to find a controller for a condition that lies between certain operating points, a controller must be calculated by interpolation from three neighboring operating points. When changing the states of the system, the interpolated controller must also change, which is done by switching the controller. The switching regulators must be stable, d. H. the control loop must be stable for all the time profiles of the weighting factors β.

The weighting factors β form a plane or a space that can be described by the totality of the weighting factors β _n combined into a vector. The range of values of the weighting factors β _n is usually in the range of 0 ≦ β _n ≦ 1.

The control device usually has a control loop formed from the controllers, the weighting factors and the controlled systems, which forms a closed, feedback system and consists of the respectively weighted controlled systems and regulators as well as a feedback. The regulation takes place with a reference variable r . From the difference between the reference variable r (setpoint) and the controlled variable y (actual value) at the output of the weighted control path, a control difference is calculated, which is performed as an input to the controller R _n . The sum of the weighted output of the controller R _n forms the manipulated variable u , which acts on the controlled systems G _n and thus acts on the controlled variable y .

, für den gilt:

An interconnected of several controllers and control systems R _n G _n of interconnected control loop with the weighting factors β _n leads to an effective, interpolated controller

for which applies:

The same applies to the effective, interpolated route Ĝ

Furthermore, the sum of the weighting factors β _n , where n = 1 to N, is equal to one and the value range of the weighting factors β _{n is} in the range from 0 to 1.

For time-variant weighting factors β _n , a common method for determining the stability is the application of the so-called "small-gain theorem". For this purpose, the time-variant weighting factors β _{n are separated} from the other elements in the control loop, ie the weighting factors β _n are practically pulled out of the control loop. The remaining control loop without the weighting factors β _n is combined into an overall system M with the input vector i and output vector o . The weighting factors β _n are arranged diagonally in β and connected to the overall system M.

The small-gain theorem states that the control loop is stable for all time profiles of the weighting factors β _n if: || β || ₂ ≤ 1 / γ and || M || _∞ <γ where || ... || _{2 is} the H-2 norm of a matrix and || ... || _{∞ is} the H-infinite norm of a state-space model.

Since all weighting factors are β _n ≤ 1, γ = 1.

A problem of conventional stable control devices that fulfill the small-gain theorem is that there is still a significant stability reserve available in the overall system, which, however, can not be used.

From the US 2002/0099677 A1 An adaptive control system for controlling a factory is known. The system is particularly suitable for particularly non-uniform, non-linear systems to be controlled, for which a neural network is proposed there.

From the publication of Raisch, Jörg: Multi-variable control in the frequency domain, Technical University of Berlin, various approaches to multi-variable controller design based on frequency domain methods are described. The above-mentioned approaches have in common that the still available stability reserves can not be used in the overall system.

Object of the present invention is therefore to provide an improved control device and an improved method for reducing structural vibrations with a control loop, wherein the scheme is still stable using the conventional stability reserve.

The object is achieved by the method with the features of claim 1 and with the control device with the features of claim 4.

It was recognized that the second condition || M || _∞ <γ is very strict and gives only very conservative results. This condition causes the overall system to still have a significant stability reserve, although the small-gain theorem predicts the instability.

It is therefore proposed to replace the linearly dependent weighting factors, which are usually in the value range of 0 ≦ β _n ≦ 1 and are linearly dependent on each other, by linearly independent parameters δ _k in the value range of -1 ≦ δ _k ≦ 1, where the parameters δ _k completely describe an (N - 1) -dimensional hypercube.

Whereas the weighting factors β _n usually span a regular (N - 1) simplex in the value range of 0 ≤ β _n ≤ 1, transformation into the (N - 1) -dimensional hypercube spanning variable δ _k and exploitation by transformation obtained weighting factors a less conservative control system is created, which is still stable and meets the requirement of small-gain theorem on stability.

The linearly dependent weighting factors β _n are thus described by transformation of a (N - 1) -dimensional hypercube completely described with linearly independent variable δ _k into a regular (N-1) simplex.

The method and the control device can be used for example for controlling structural vibrations in robots, in which a switchover between different, linearized to respective operating points regulators takes place.

Advantageous embodiments are described in the subclaims.

The invention will be explained in more detail with reference to embodiments with the accompanying drawings. Show it;

1 Block diagram of a control loop with three interpolated links and controllers;

2 - control loop with weighting factors β _n removed from the control loop;

3 Block diagram of a combined control loop with external weighting factors β _n ;

4a - Representation of the regular (N - 1) -Simplex spanned by the weighting factors β _n with N = 3;

4b - representation of the (N - 1) -dimensional hypercube spanned by the variables δ _n with N = 3;

5 - Example of a transformed control loop with three controllers and links;

6 - combined control loop with external variables δ;

7 - combined control loop with a scaling matrix;

8th Flow chart of a procedure for checking the stability of the switching of three controllers;

9 - Representation of the divided workspace of a robot in the yz plane with ten working points;

10 - Course of the maximum singular value of the combined control loop using weighting factors β _n , of transformed weighting factors with the variables δ _k and of transformed and scaled weighting factors with the variables δ _k .

1 shows a block diagram of a control loop with three controllers R _n and three controlled systems G _n , which are each weighted by three weighting factors β _n , where n is an integer from 1 to 3. The difference between the reference variable r and the controlled variable y (negative feedback) at the output of the control loop is given as input to the three regulators R _n . The output of the controller R _n is multiplied by the respectively assigned weighting factors β _n and the weighted output values of the individual controllers obtained therefrom are added up in order to form the manipulated variable u . This control value u is given as an input to the control systems G _n , the control systems G _n being weighted by the same weighting factors β _n as the associated controllers R _n .

The weighted output signals of the control paths G _n are added up and the sum forms the controlled variable y at the output of the control loop, which is used, for example, to control an actuator and at the same time serves as a negative feedback.

gilt:

For the effective, interpolated controller

applies:

und 0 ≤ β_n ≤ 1. For the effective, interpolated route Ĝ, the following applies:

and 0 ≤ β _n ≤ 1.

For time-variant weighting factors β _n , the so-called small-gain theorem is used to determine the stability. For this theorem to be used, the time-variant weighting factors β _n must be separated from the other elements in the control loop. The weighting factors β _n are practically pulled out of the control loop, as in the 2 outlined.

The remaining control loop is combined into an overall system M with the input vector i and in the output vector o . The weighting factors β _n are arranged diagonally in a matrix β and connected to the system. The summarized control loop M with the external weighting factors β _n is in 3 outlined.

The small-gain theorem states that the control loop M is stable for all time profiles of the weighting factors β _n , if: || β || ₂ ≤ 1 / γ and || M || _∞ <γ, where || ... || _{2 is} the 2-norm of a matrix and || ... || _∞ the H _∞ norm is a state-space model. Since all weighting factors are β _n ≤ 1, γ = 1.

To make the statement of the small-gain theorem about the stability of, for example, three switching regulators less conservative, a transformation of the regular (N-1) simplex, with N = 3 for the three-dimensional case, in the value range of 0 ≤ β _n ≦ 1 describing weighting factors β _n in a (N-1) -dimensional hypercube by means of variables δ _k in the value range of -1 ≦ δ _k ≦ 1. In addition, a scaling occurs.

Due to the condition that the sum of all weighting factors β _n = 1, there is a linear dependence of the weighting factors β _n . By transforming the regular (N - 1) simplex into an (N - 1) -dimensional hypercube, the weighting factors β _{n are transformed} into a representation with the variables δ _k with k = 1 to N-1. The variables δ _k are valid linearly independent in the interval -1 ≤ δ _k ≤ 1 and describe the same relationship.

4a shows the representation of the β by the weighting factors _n spanned regular (N - 1) simplex of the three-dimensional case, while 4b shows the transformed representation of the (N - 1) -dimensional hypercube with N = 3 spanned by the linearly independent variable δ _k .

mit: –1 ≤ δ_k ≤ 1 und k = 1 bis (N – 1). The transformation rule for the weighting factors β _n is:

With: -1 ≤ δ _k ≤ 1 and k = 1 to (N-1).

e _n is the nth unit vector with the value 1 in the nth row and otherwise the values 0. The magnitude of the unit vector is therefore equal to one.

ẽ _n is the nth transformed unit vector whose magnitude is not one.

The transformed unit vectors ẽ _{n are} calculated recursively as follows:

The transformed controlled system is then:

The transformed total controller is then:

For the case N = 3, the weighting factors β _n thus result as follows:

This result follows from the transformed unit vectors

Then, for the transformed weighting factors β _n :

The interpolated controller then results in:

The interpolated controlled system is then calculated as: Ĝ = 1/4 [[1 - δ ₂ - δ ₁ (1 - δ ₂ )] G ₁ + [1 - δ ₂ + δ ₁ (1 - δ ₂ )] G ₂ + 2 (1 + δ ₂ ) G ₃ ]

5 shows the implementation of the transformed control loop in a block diagram showing the less conservative evaluated yet stable control loop.

If now the variables δ _k , ie the matrix Δ also separated and pulled out of the control loop, there is a summarized, in 6 sketched control loop M.

Testing this loop for stability using the small-gain theorem yields results that are less conservative than before.

mit E = Einheitsmatrix d_k ∊ |R, d_k > 0 erreichen. Die Größe n_u ist die Anzahl der Stellgrößen.A further reduction in conservatism can be achieved with the introduction of a scaling matrix

with E = unit matrix _dk ε | R, _dk > 0 to reach. The size n _u is the number of manipulated variables.

The scaling matrix D is in the control loop according to 6 brought in. This results in a combined control loop M with the scaling matrix D , which in 7 outlined.

The scaling matrix D does not change the actual control system. However: || DMD ^-1 || _∞ <|| M || _∞

With the help of this measure, the statement of the small-gain theorem can be further tightened and the conservatism lowered even further.

The introduction of a scaling matrix D is taken from a μ-robustness analysis. The calculation of the scaling matrix D occurs in the calculation of the value μ for the transmission matrix M (Jω _D ) with the angular frequency ω _D. The angular frequency ω _D is chosen such that it is the frequency at which the profile of the maximum singular value of the combined control loop M reaches its maximum.

8th gives a flowchart of the procedure of checking the stability of the switching of N-controllers.

In a first step a) the controllers and paths are assigned to the individual controllers R _n and G _n .

In step b), the combined control loop M is then formed.

In step c), the frequency of the maximum singular value is searched for, ie the frequency at which the profile of the maximum singular value of the combined control loop M reaches its maximum.

Then in step d) the scaling matrix D is calculated as described above.

In step e) it is then checked whether the condition || DMD ^-1 || _∞ <1 is.

If this is the case, it is determined in step f) that the controller switching is stable. Otherwise, the method is re-run by returning to step a).

If all combinations have been tested and no stable controller switching has been found, the result in step g) is that the switching in the triangle is unstable.

The controller structure and the method will be further explained below by means of a control of structural vibrations in robots. The vibration behavior of robots is not constant over their working space, but changes sometimes strongly. To describe the behavior, the work points are displayed in one level. At each operating point, the behavior of the robot is measured and stored in the form of the controlled system G _n . For each operating point n, a controller R _{n is} designed.

The workspace division of the robot is in the 9 shown as an example for the yz plane. The operating points are in the range of -0.2 to +0.2 m for both the Y and Z axes.

It can be seen that the working points always span triangles. With the transformed control loop described above, a stable switching of the controller between the operating points is possible.

10 allows an exemplary course of the maximum singular value of the combined control loop M to be recognized in the case of different control approaches. The solid line corresponds to the conventional approach with weighting factors β _n , which span a regular (N-1) simplex. The dashed line describes the transformed weighting factor approach using variables δ _k that span an (N-1) -dimensional hypercube. The dash-dot line shows the behavior with transformed and scaled weighting factors using the variable δ _k , which span the (N-1) -dimensional hypercube and are scaled.

The maximum value of a curve corresponds to the value of the H-∞ norm of the state space model of the combined control loop M. For the system to be stable, this value and therefore the entire curve must be below the value one.

The solid curve is the singular value curve of the combined control loop M , according to FIG 1 with weighting factors β _n in the value range of 0 ≤ β _n ≤ 1. The maximum value of more than six indicates that the system value is removed from stable switching.

After the transformation of the weighting factors β _n into the (N-1) -dimensional hypercubes using the variable δ _k in the value range of -1 ≦ δ _k ≦ 1 and a construction of the combined control loop M after 8th the dashed singular value curve results. The maximum of three indicates that the system is not stable. The calculation of a suitable scaling matrix D and the multiplication with the combined control loop M generate a singular value curve (dash-dot) which lies below one and certifies stability in the system during a switchover.

This in 10 The result of the stability test shown with the control loop as such is to be understood as meaning that the system has always been stable in the switchover for all three approaches. The first two approaches, however, have yielded conservative statements in order to recognize this, so that a stability reserve remained unused.

Claims (5)

A method for reducing structural vibrations with a control loop by means of vibration-influencing actuators, which are controlled by regulators (R _n ), wherein the control loop by a number N of controllers (R _n ) and one of the number N of controllers (R _n ) corresponding number of controlled systems (G _n ) is defined and the controllers (R _n ) and controlled systems (G _n ) are weighted by a number N of assigned, linearly dependent weighting factors (β _n ), wherein the stability assessment of the control loop is carried out on the basis of the criteria the sum of the weighting factors (β _n ) is equal to one, and that the weighting factors (β _n ) are transformed into new weighting factors (δ _n ) by transforming a number of variables in the value range from -1 to 1 according to the transformation rule

where β is the N-dimensional vector with the vector elements β _n , e _n unit vectors having the length one and ẽ _n transformed into the n-th dimension, the transformed unit vectors ẽ _n being the recursive rule

for n = 1 ... N - 1 and with δ ₀ and ẽ _{0 are set} equal to zero, - that || M || _∞ ≤ 1 where || M || corresponds to the control loop when the new weighting factors (δ _n ) are separated from the control loop. Method according to Claim 1, characterized by scaling of the control loop containing the transformed weighting factors (δ _n ). Method according to Claim 2, characterized by scaling the control loop M with a scaling matrix D according to the condition

with the scaling factors d _k ε | R |, d _k > 0 and the unit matrix E with the number n _u of manipulated variables. Control device for reducing structural vibrations with a plurality of regulators (R _n ) for a number of control paths (G _n ) corresponding to the number N of controllers (R _n ), the control device being adapted to carry out a method according to one of the preceding claims. Control device according to claim 4 for controlling non-linear controlled systems, characterized in that the controllers (R _n ) are designed to linearize the behavior of the controlled systems (G _n ) at respective operating points, and the control device for control in work areas around the respective operating points around is set up with at least one respective controller (R _n ) provided for the work area and for switching between the controllers (R _n ) at working range limits by regulation with an interpolation of controllers (R _n ) for the operating points adjacent to working range limits.

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