Generalized total least squares for identification of electromagnetic parameters of an induction motor

This paper proposes an algorithm for electromagnetic parameters with errors in variables. The estimates obtained by the ordinary least squares (OLS) are biased due to errors in the variables. It is shown that even if there are errors in all variables with the same variance, the problem is reduced to generalized total least squares (GTLS). To find a solution to the generalized total least squares problem, an augmented system of equations is used that is equivalent to a biased normal system. This approach improves the conditionality of the system of equations being solved in comparison with the biased normal system and requires less memory compared to the solution based on the right singular vector. The simulation results show that the GTLS estimates are much more accurate than OLS estimates. Based on the proposed approach, recursive algorithms can be created for evaluating the parameters of an asynchronous electric motor online.


Introduction
The parameters of the induction motor can vary greatly over time, the active rotor and stator resistance in different modes can change by half [1]. The reasons for changing the parameters may vary: environmental conditions, change in operating modes, aging, and wear of the electric motor, in addition, the parameters of specific specimens differ within the error from the parameters given in the passport.
Changes in the parameters of an induction motor can significantly degrade the quality of control. In this regard, the problem of experimental determination of the characteristics of an asynchronous electric motor is urgent. There are two ways to improve the accuracy of the estimated parameters: 1. The use of measuring instruments of higher accuracy, as well as the protection of measuring instruments from external noise, leads to a significant increase in cost and increases the complexity of the implementation of the induction motor control system. 2. Application of noise-proof algorithms for identifying the parameters of induction motors. Today, methods for identifying the parameters of induction motor based on equivalent circuits are being actively developed. Estimations of induction motor parameters based on ordinary least squares and their recursive modifications are considered in [2], [3]. The application of the total least squares method and its recursive modifications was considered in [3], [4], [5], [6].
In [7], a method for estimating electromagnetic parameters based on the generalized total least squares method is proposed. In this paper, left-hand differences are used to approximate the derivatives. This approach can introduce a significant error in the estimation of derivatives and require a very small sampling step. Also, the estimation of K parameters of an asynchronous motor is associated with poor conditionality of the problem (Problem K2 [2], [3], [5]). The minimization of the generalized Rayleigh ratio has low numerical stability.
In [4], a method of generalized total least squares is proposed for estimating the parameters of an induction motor, but the parameterization of the model of an induction motor has a higher dimension than K parameters. This paper proposes a generalized total least squares method for estimating K parameters of an induction motor for K parameters, as well as its effective numerical implementation based on an augmented system of equations for an equivalent biased normal system.

Problem Statement
The mathematical model for the asynchronous traction motor can be expressed in the fixed coordinates ( , ,0) α β as the following equations: p is the number of pole pairs; ω is the rotor rotation speed; J is the equivalent moment of inertia for the motor; M is an electromagnetic moment; c M is a motion resistance moment; and and γ β are certain coefficients that depend on the induction and active resistance of the motor. It is noteworthy that the equation parameters are dependent, and to unequivocally identify the asynchronous motor, one only needs to determine s R , s L , σ , and r T . The identification of parameters requires excluding the unmeasurable projections of rotor flux linkages from the equations. Assuming that 0 d dt ω = after the transformations, we obtain [5]: As the parameters are identified digitally, it is convenient to move from differential equations to difference equations.
are noisy values for stator-current projections, respectively; Thus, the problem of identifying parameters can be formulated as one of finding coefficient estimates for 1 K , 2 K , 3 K , 4 K , and 5 K in equations (9), (10) according to noisy observations (10)-(15).

Identification Criterion
Let us assume that the following conditions are satisfied: where Ε is an expectation operator: In [7], for identification with uncorrelated observational noises, a criterion was proposed for finding an estimate of the vector of parameters K .

H H
The Cholesky decomposition for the matrix is H : where m is an arbitrary positive factor, the choice of factor is considered in [9]; min σ is minimal singular values of a matrix ( ) 1 , − ΦL Y , 5 I , 3 I , 2 N I are unit matrices of corresponding dimensions, 1 j = − .

Simulation Results
The algorithm based on criterion (21) was compared with the ordinary least squares, total least square. It was assumed that currents and voltages were measured with noise. The noise was modeled by using independent Gaussian random variables with zero means.
The noise-to-signal ratio for the standard deviations of currents, voltages, and their derivatives is 0.001.

Conclusion
The paper proposes a method for estimating the electromagnetic parameters of an induction motor based on generalized total least squares. To find a solution to the generalized total least squares problem, an augmented system of equations is used that is equivalent to a biased normal system. This approach improves the conditionality of the system of equations being solved in comparison with the biased normal system and requires less memory compared to the solution based on the right singular vector.
The simulation results show that the GTLS estimates are highly accurate than OLS estimates.