, which is also referred to in the following as the “load angular velocity” ω_D. The load angular velocity ω_Dcan be detected, for example, by means of a rotary encoder 7. In the method according to the invention, the specimen angular velocity ω_ρis not available as a measured variable, and also the shaft torque T_STtransmitted by the connection shaft 3 is often not available in the form of a measured variable, which is indicated by the dashed signal lines Y in FIG. 1. Angular velocities ω and rotational speeds n of the test bench arrangement 4, for example specimen angular velocity ω_ρand specimen rotational speed n_ρ, are in this case connected to one another by the known expression ω=πn/30 and are to be considered synonymous with regard to their effect on the method according to the invention. In order to make the following text and the mathematical description of the above-mentioned models consistent, in the following, angular velocities will predominantly be used, without limiting the generality.

The first step of the method according to the invention provides the specification of a first system of differential equations for modeling the rotational behavior, hereinafter also referred to as oscillation behavior, of the test bench arrangement 4. For this purpose, a rotary multi-body system is preferably applied, the simplest embodiment of which is a two-mass oscillator. FIG. 2 shows such a two-mass oscillator. In addition to the test bench variables that are decisive for the oscillation behavior, of shaft torque T_ST, air gap torque T_Lgenerated by the loading machine 2, specimen torque T_P, loading machine angular velocity ω_ρ, and specimen angular velocity ω_ρ, FIG. 2 shows the shaft stiffness c and the shaft damping d for the description of the mechanical connection shaft 3 between the specimen 1 and the loading machine 2, and the specimen moment of inertia J_ρand the loading machine moment of inertia J_D. These parameters can be assumed to be known. On the basis of the two-mass oscillator shown in FIG. 2, a well known first system of differential equations can be used to describe the rotational behavior of the test bench arrangement 4 in the form of

$\frac{d}{dt} \underset{x}{\underset{︸}{[\begin{matrix} Δφ \\ ω_{D} \\ ω_{P} \end{matrix}]}} = \underset{A}{\underset{︸}{(\begin{matrix} 0 & 1 & - 1 \\ - \frac{c}{J_{D}} & - \frac{d}{J_{D}} & \frac{d}{J_{D}} \\ \frac{c}{J_{P}} & \frac{d}{J_{p}} & - \frac{d}{J_{P}} \end{matrix})}} [\begin{matrix} Δφ \\ ω_{D} \\ ω_{P} \end{matrix}] + \underset{b_{L}}{\underset{︸}{(\begin{matrix} 0 \\ \frac{- 1}{J_{D}} \\ 0 \end{matrix})}} T_{L} + \underset{b_{P}}{\underset{︸}{(\begin{matrix} 0 \\ 0 \\ \frac{1}{J_{P}} \end{matrix})}} T_{P}$

with the output equation

$ω_{D} = \underset{c_{A}}{\underset{︸}{(\begin{matrix} 0 & 1 & 0 \end{matrix})}} [\begin{matrix} Δφ \\ ω_{D} \\ ω_{P} \end{matrix}]$

where Δφ stands for the difference rotational angle between the test specimen 1 and loading machine 2. The variables x, A, b_L, b_ρand c_Ain this case stand, in corresponding order, for the state vector, the dynamic matrix, the first and second input vector, and for the output vector of the first system of differential equations, and result from the modeling. In the present case, the output equation describes the angular velocity ω_Dof the loading machine 2, which is often customary, since the associated load rotational speed is typically available as a measured variable. Different further embodiments of the output equation and thus of the output variable are conceivable by an alternative choice of the output vector.

Like each model of a real physical system, also the first system of differential equations according to the invention represents a simplified, modeled abstraction of the real behavior of the modeled test bench arrangement 4. Thus, in the formulation of the model, the mass inertia of the elastic connecting shaft can be added in equal proportions to the inertia J_Dand to the inertia J_P. In addition, it is assumed that all system components are linear elements. Possible nonlinear features of the test bench arrangement 4, such as shaft play, nonlinear rigidity, or static friction, are accordingly not taken into account. The actuator dynamics introduced by the loading machine 2, which are usually expressed by a combination of dead time and PT1 behavior during the conversion of the control variable T_Dinto an air gap torque T_L, are also disregarded. If the actuator dynamics are disregarded, the air gap torque T_Lin particular corresponds the control variable T_D. By taking into account the currently disregarded features, further, more complex models of the test bench arrangement 4 could also be specified directly, in the form of system of differential equations s. However, the first system of differential equations shown is sufficient for describing the method according to the invention. For the further considerations, the model parameters are assumed to be known. Based on the model indicated, the formulation of the general object of the present invention can be specified for the embodiment shown and for better understanding, namely as the determination of a method for estimating the variables ω_ρand T_ρ.

As mentioned, in order to achieve this object, a second system of differential equations for describing the specimen torque T_ρis specified. The generated specimen torque T_Pis modeled using a second system of differential equations, the direct component of the specimen torque T_ρbeing modeled on the basis of a first subsystem of the second system of differential equations, and m harmonics of the specimen torque T_ρbeing modeled on the basis of a number m of further oscillating subsystems of the second system of differential equations, in the form of damped and/or undamped oscillators.

In the specific embodiment, three (m=3) harmonics are considered, as a result of which the second system of differential equations assumes the form

$\frac{d}{dt} \underset{z}{\underset{︸}{[\begin{matrix} z_{0} \\ z_{1, 1} \\ z_{1, 2} \\ z_{2, 1} \\ z_{2, 2} \\ z_{3, 1} \\ z_{3, 2} \end{matrix}]}} = \underset{S}{\underset{︸}{(\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - ζ_{1} & 0 & 0 & 0 & 0 \\ 0 & ζ_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - ζ_{2} & 0 & 0 \\ 0 & 0 & 0 & ζ_{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - ζ_{3} \\ 0 & 0 & 0 & 0 & 0 & ζ_{3} & 0 \end{matrix})}} [\begin{matrix} z_{0} \\ z_{1, 1} \\ z_{1, 2} \\ z_{2, 1} \\ z_{2, 2} \\ z_{3, 1} \\ z_{3, 2} \end{matrix}]$

with the output equation

$T_{P} = \underset{c_{s}}{\underset{︸}{(\begin{matrix} 1 & 1 & 0 & 1 & 0 & 1 & 0 \end{matrix})}} [\begin{matrix} z_{0} \\ z_{1, 1} \\ z_{1, 2} \\ z_{2, 1} \\ z_{1, 2} \\ z_{3, 1} \\ z_{3, 2} \end{matrix}]$

Here too, analogously to the first system of differential equations, variables z, Sand c_sare specified, which, in corresponding order, denote the state vector with a number of 2m+1 state variables z, the dynamic matrix, and the output vector of the second system of differential equations. In this case, using the state variable z₀, which adheres to the differential equation

$\frac{d}{d t} z_{0} = 0$

the direct component of the specimen torque T_Pis described.

The remaining state variables z_k,1, z_k,2are always to be viewed in pairs of two, each of which pairs is to be attributed the k-th oscillating subsystem. In this case, the control variable k is used here to indicate the oscillating subsystems and the associated variables. Specifically, the oscillating subsystems are understood to be those parts of the system of differential equations which are each formed by mutually influencing state variables of the state vector. Accordingly, for each of these oscillating subsystems, for the state variables, z_k,1and z_k,2the following vector-valued differential equation applies

$\frac{d}{d t} [\begin{matrix} z_{k, 1} \\ Z_{k, 2} \end{matrix}] = (\begin{matrix} 0 & - ζ_{k} \\ ζ_{k} & 0 \end{matrix}) [\begin{matrix} z_{k, 1} \\ Z_{k, 2} \end{matrix}],$

which is known to correspond to a harmonic oscillator of the angular frequency ζ_k.

The eigenvalues of this subsystem, at ±jζ_k, are on the imaginary axis of the complex plane, and cannot leave it, even in the case of variations of the angular frequency ζ_k. An expansion possibility of this oscillator is provided by taking into account damping by means of a suitable damping factor. Such a modification would result in a damped oscillator having the differential equation

$\frac{d}{dt} [\begin{matrix} z_{k, 1} \\ z_{k, 2} \end{matrix}] = (\begin{matrix} - η_{k} & - ζ_{k} \\ ζ_{k} & - η_{k} \end{matrix}) [\begin{matrix} z_{k, 1} \\ z_{k, 2} \end{matrix}],$

the eigenvalues of which in the case of a positive and real damping factor η_k>0 generally lie in the left-hand half of the complex plane. The property that the eigenvalues of such an oscillator can never come to lie in the right-hand half of the complex plane is advantageous in particular in cases in which the second system of differential equations for modeling the specimen torques is T_ρis adapted to a variable of the test bench arrangement 4 which changes during the operation of the test bench, for example to the angular velocity ω_ρ.

Starting from an initial state different from zero, the two variables z_k,1and z_k,2each realise a sinusoidal profile which is phase-shifted by 90 degrees relative to one another. If one of these two state variables is selected, a single harmonic of a specimen torque can thus consequently be T_ρand an entire harmonic specimen torque T_ρcan be described by a sum of multiple harmonics determined in this way.

From the structure of the output vector c_s, it can be seen that, to form the output variable T_ρ, the state variable z₀and precisely one associated state variable of each of the oscillating subsystems are summed. FIG. 3 shows, in this connection, a possible profile of a specimen torque T_ρformed in this way.

In order to parameterize the angular frequencies ζ_k, for example integer multiples of the load angular velocity ω_Dare used, which leads to the relationship

$ζ_{k} = k ω_{D}$

The angular frequencies ζ_kthus become functions of the load angular velocity ω_D, which can be used to adapt the angular frequencies ζ_k, during operation, to a possibly changing load angular velocity ω_D. Instead of the load angular velocity ω_D, however, other rotational variables of the test bench arrangement 4 can also be used for parameterizing the angular frequencies ζ_k. Moreover, it is also possible to set the angular frequencies ζ_konce, at the beginning of the operation, in the manner described, but subsequently no longer to change them and accordingly keep them constant. In electric motor test benches, it is often advantageous, when selecting the frequencies ζ_k, to take into account an integer pole pair number p, which results in

$ζ_{k} = kp ω_{D}$

and can contribute to a reduction in the resulting model order.

It should be noted at this point that the selection of the frequencies ζ_kcan also be carried out in a completely different manner, for example by using crooked ratios between the frequencies ζ_kand the load angular velocity ω_D. The frequencies ζ_kcan, however, also be easily selected by a person skilled in the art.

An extension, mentioned at the outset, of the second system of differential equations, for increasing the model accuracy, can easily take place by adding further oscillators, as a result of which the number m would be increased. It should also be noted that successive harmonics need not necessarily be used. For example, only odd harmonics or only even harmonics, or any desired selection of harmonics, can be used.

For the actual determination of estimates for ω_ρand T_ρ, the first and the second differential equations are combined to form an overall system. For more detailed description of this step, firstly, based on the previously introduced abbreviations for the system parameters, the first system of differential equations is written in the compact form

$\begin{matrix} \frac{d}{d t} x = A x + b_{L} T_{L} + b_{P} T_{P} \\ ω_{D} = c_{A} z \end{matrix}$

and the second system of differential equations is written in the compact form

$\begin{matrix} \frac{d}{d t} z = S z \\ T_{P} = c_{s} z \end{matrix}$

If the input T_ρof the first system of differential equations is replaced by the output T_ρof the second system of differential equations, the overall system

$\begin{matrix} \frac{d}{dt} [\begin{matrix} x \\ z \end{matrix}] = (\begin{matrix} A & b_{P} c_{S} \\ 0_{x} & S \end{matrix}) [\begin{matrix} x \\ z \end{matrix}] + (\begin{matrix} b_{L} \\ 0_{z} \end{matrix}) T_{L} \\ ω_{D} = (\begin{matrix} c_{A} & 0_{z} \end{matrix}) [\begin{matrix} x \\ z \end{matrix}], \end{matrix}$

results, where 0_xand 0_zstand for zero matrices of a suitable dimension. A wide variety of approaches to design a state observer for estimating the state vector

$[\begin{matrix} x \\ z \end{matrix}]$

exist for such a linear system.

In a preferred manner, however, the well-known Luenberger observer can be used, which, however, is in no way to be understood as limiting. As is known, a Luenberger observer comprises a copy of the system of differential equations, which describes the dynamics of the variables to be estimated, and a correction term, which, in the specific case, corresponds to the observer differential equation

$\frac{d}{d t} [\begin{matrix} \hat{x} \\ \hat{z} \end{matrix}] = (\begin{matrix} A & b_{P} c_{S} \\ 0_{x} & S \end{matrix}) [\begin{matrix} \hat{x} \\ \hat{z} \end{matrix}] + (\begin{matrix} b_{L} \\ 0_{z} \end{matrix}) T_{L} + K (ω_{D} - (\begin{matrix} c_{A} & 0_{z} \end{matrix}) [\begin{matrix} \hat{x} \\ \hat{z} \end{matrix}])$

Herein, {circumflex over (x)} and {circumflex over (z)} stand for the estimates of the state vectors x and z to be determined. In this case, the vectorial correction gain K is usually selected in such a way that the dynamic matrix Â of the differential equation of the estimation error

$e = [\begin{matrix} x \\ z \end{matrix}] - [\begin{matrix} \hat{x} \\ \hat{z} \end{matrix}],$

assumes predetermined eigenvalues, which represents a procedure that is well known in control technology. The dynamic matrix of the differential equation of the estimation error e results from the above equation directly as

A
^

=

(

(

A

b
P

⁢

c
S

0
x