2.1. Problem Statement
T-form circuit models are more complex than necessary. They can be transformed into simpler models with no loss of information or accuracy [
28]. This configuration has been denoted as the Г form, from the structure of its two inductances. Modeling a Γ-equivalent circuit will make a locked rotor for Ω = 0.
The equivalent circuit shown in
Figure 1 does not describe the skin effects typical of real induction motors. To simulate skin effects, models with non-integer-order derivatives can be presented. Let us replace the resistor
and the inductance
with the complex resistance
. The equivalent circuit is shown in
Figure 2.
The equivalent impedance of an induction motor is defined as:
Various models with derivatives of non-integer order are known.
is the one-derivative black box mode [
23]:
or the two-derivative model [
24]:
Furthermore, there is a three-parameter model [
14]:
The relationship between stator current and stator voltage is defined as:
where
is the stator voltage and
is the stator current.
In real conditions, currents and voltages are always measured with noise:
where
and
are additive zero-mean white Gaussian noises corrupting the voltage signal and the current signal, and they are assumed not to be correlated with the voltage signal and the current signal.
The problem with identifying an induction motor is the estimation of the vector of unknown parameters
from the measured values of the stator current
and stator voltage
. The parameter vector of impedance
(1) for the model
(2) is:
For the model
(3), the parameter vector of impedance
is:
For the model
(4), the parameter vector of impedance
is:
2.2. GTLS Algorithm for Identification of Induction Motor
Let us express the value of the impedance in terms of the physical parameters of the equivalent circuits. The impedance value
(1) for model
(2) is:
The impedance value
(1) for model
(3) is:
The impedance value
(1) for model
(4) is:
Equation (5) in the time domain for impedances (2)–(4) is defined as:
where
is the Grünwald–Letnikov fractional operator,
is the sampling period, and
is Newton’s binomial generalized to fractional orders.
Equations (13)–(15) in matrix form are described as:
Equation (13) is described as:
, , , , .
Equation (14) is described as:
, , , , ,
Equation (15) is described as:
, , , , .
For noisy current
and voltage
, Equation (16) is described as:
where
and .
Calculating the fractional derivative from noisy data is a serious problem in identifying a fractional system and leads to large errors. Therefore, the signals must be processed by the state variable filter (SVF) proposed in [
29]. The SVF is defined by the following equation:
where the order
is an integer chosen such that
and
denote the filter cut-off frequency. The choice of the number
is a compromise between filter complexity and filtering quality. However, increasing the order for large
produces a very slight increase in the filtering quality.
The filtered input and output signals
and
are determined as follows:
Using the filtered input and output signals, Equation (17) can be reformulated as:
Equation (20), in discrete time, is described as:
It is assumed that the fractional order is already known; our goal is to estimate only the fractional differential equation coefficients. We will use generalized total least squares for this. The solving of generalized total least squares is reduced to finding the minimum of the objective function:
where
is the Euclidian norm,
is the diagonal matrix of noise variances, and
where
.
Total least squares regression assumes that the noise variance is the same in all columns and in the right side. This assumption is not satisfied for Equation (22). Calculating the exact value of the noise variances of each column is a very difficult task. We will assume that the use of the SVF filter makes it possible to achieve an approximately equal signal-to-noise ratio in each column. Then, normalization will make it possible to obtain approximately equal noise variances in each column. The standard deviations for each column can be defined as:
where
is
j-th column of the matrix
.
The generalized total least squares problem (22) can be reduced to the total least squares problem [
30]:
where
.
The minimum of function (24) can be found as a solution to the biased normal system of equations [
31]:
where
is the minimal singular values of matrices
and
is the identity matrix.
An augmented symmetric system of equations used to solve total least squares (16) [
32] can be expressed as follows:
An inverse change of variable can be performed as follows: .
Let us determine the parameter estimates
from the estimates
. For impedance model (2), this would be carried out as follows:
For impedance model (3), this would be carried out as follows:
For impedance model (4), this would be carried out as follows:
If the order of differentiation is unknown, as is often the case in practice, order estimation must be considered along with transfer function coefficient estimation. The use of generalized total least squares is possible when the fractional order is known a priori. This section describes an algorithm for extending the identification method presented above for the case when the order of differentiation is unknown. The algorithm is based on a combination of a generalized least squares method for estimating coefficients and a nonlinear algorithm for optimizing the order of differentiation. The parameter identification problem is presented as a functional minimization. Therefore, the main goal of this approach is to reduce the residual error with respect to
.
The objective function (31) depends on one parameter. From a priori knowledge, it follows that for the impedance model (2) and for models (2) and (3). The minimum of a function can be found by one of the standard methods for optimizing a function of one variable.