Introducing Improved Iterated Extended Kalman Filter (IIEKF) to Estimate the Rotor Rotational Speed, Rotor and Stator Resistances of Induction Motors

This paper introduces the Improved Iterated Extended Kalman Filter (IIEKF) for estimating the rotational speed, rotor resistance, and stator resistance of three-phase induction motors (IMs). Two state-space models for estimating the variables are presented. An optimal estimation of rotational speed is obtained by introducing a data fusion approach. The effectiveness of the IIEKF in comparison with the Extended Kalman Filter (EKF), using experimental data and in a wide range of operating conditions, is shown.


I. INTRODUCTION
The induction motor (IM) parameters may change due to the winding temperature fluctuations, flux saturation, and skin effect [1]. Temperature variation can affect the rotor and stator resistances, while it has no remarkable effect on inductances. Conversely, high current values cause saturation of inductances [2].
Many control strategies of IMs, like Field Oriented Control (FOC), require accurate values of IM parameters, therefore, several methodologies have been presented to estimate the IM parameters [3], [4]. In recent years, sensorless estimation of rotational speed and rotor flux has attracted considerable attention in the introduced control strategies [5], [6]. In addition to the control strategies, the estimation of IM parameters is one of the conventional methods for fault detection [7], [8], [9], [10].
Generally, there are a large number of studies on estimating IM parameters. These studies can be categorized into three main groups: (i) model reference adaptive system methods, The associate editor coordinating the review of this manuscript and approving it for publication was R. K. Saket .
In [16], an artificial neural network has been designed to speed estimated through the estimation of stator current with the purpose of direct torque control of three-phase IM. It is worth noting that such data-driven methods require a considerable volume of data at different operating conditions of the motor. This volume of data not only requires strong processors but also analyzing this data is a time-consuming process.
In [11], a Model Reference Adaptive System (MRAS) for online rotor time constant estimation has been introduced. In [17], a comprehensive review of speed estimation based on MRAS techniques has been conducted. This study showed that MRAS technique has not appropriate accuracy in the presence of measurement noise and uncertainties.
A wide range of literature has been published regarding parameter estimation of IMs based on observers [18], [19], [20], [21], [22], [23], [24], [25], [26]. Among methods in this category, the Kalman Filters (KFs) family uses information on both dynamical and statistical model parameters of IM to estimate optimal values [27]. Hence, unlike deterministic observers, the stochastic nature of the KF addresses the issues regarding the model uncertainties and measurement noise of IM [23].
In recent studies on the estimation and identification of IM parameters using KF and its extensions, speed estimation of IM for employing speed sensorless control has been considerable [23], [28], [29], [30], [31], [32], [33]. Table 1 presents a brief review of some studies of various types of KFs used to estimate IM parameters. According to Table 1, Extended KF (EKF) and Unscented KF (UKF) are two successful methods in the sensorless control strategy.
In [28], adaptive observers along with second-order KF were suggested to estimate flux and rotational speed. In [29], [34], and [35], the flux and speed were estimated using EKF. In [31], a braided EKF has been applied for sensorless control of IM under speed and load variations in the presence of measurement noise. Although EKF is employed for nonlinear processes, it basically uses a linearization approach to determine the current and covariance of the state [19]. Despite the wide use of this application, it has an obvious disadvantage in the case of filter instability due to the linearization, when sample time is not proper, affecting the Jacobian matrix and estimation results [19], [31]. It should be noted that in [29], [34], [35], [36], and [37], the speed is assumed as a constant parameter, affecting the estimation of transient speed. None of these studies estimate rotor resistance, changing during operation conditions, thereby affecting the estimation of parameters. However, in the case of [16] and [36] the effects of rotor resistance variation have been reflected. In [38], a single EKF using two extended IM models has been introduced as a BI-EKF algorithm to estimate load torque, rotor, and stator resistances.
In some studies about the controller design of IM, the estimation of rotor resistance has been considered [39], [41], [42]. In [39], using the direct FOC scheme, besides rotor flux and speed, rotor resistance was estimated using EKF. In [40] and [41], the authors estimated rotor resistance, speed, and rotor flux using the DTC scheme. In [42], a modified EKF has been used to decrease execution time for estimating the parameters of six-phase IM controlled by DTC.
This paper introduces an Improved Iterared EKF (IIEKF) to estimate rotor rotational speed, rotor, and stator resistances based on two extracted state-space models. Due to the nonlinear behavior of IM, IIEKF as a modified version of EKF has been introduced. The performance of IIEKF is studied under various machine operating conditions and load variations in a wide range of the rotational speed of IM. Also, the experimental results of employing IIEKF have been compared with the estimation results of EKF.
The organization of this paper is as follows: Section II presents the mathematical model of IM. After reviewing EKF and Iterated EKF (IEKF), IIEKF is introduced in section III. Section IV introduces the estimation method to estimate the rotational speed, rotor, and stator resistances of IM. In section V, the experimental results and discussion are presented. The conclusion is presented in section VI.

II. MATHEMATICAL MODEL OF INDUCTION MOTOR
The discrete-time dynamic equations of three-phase IM in a stationary frame, based on the stator currents and rotor fluxes can be written as [22]: where, i ds (k) and i qs (k) are the stator current elements, v ds (k) and v qs (k) are the stator voltage elements, λ ′ dr (k) and λ ′ qr (k) are rotor flux linkage elements in dq reference frame. ω r (k) is the rotor rotational speed. L m , L r , and L s are the mutual inductance, rotor and stator self-inductances, respectively. R r and R s are the rotor and stator resistances, respectively. n p is the number of poles of IM. J is the total inertia of the IM, T l is the load torque and σ is the total leakage coefficient that it can be defined as follows [45]:

III. IMPROVED ITERATED EXTENDED KALMAN FILTER
EKF is an extension of KF for nonlinear dynamics systems. EKF approximates the nonlinearities using linearization around the last estimated value of the state variables. The general framework for EKF was introduced in [46]. IEKF and UKF, as modified versions of EKF, are two alternative filters for linearization first-order approximation errors of the EKF [47]. The estimation performances of UKF and EKF are similar by using the same covariance matrices [44] and greatly degraded in the presence of observation outliers due to their lack of robustness [48], similar to [49] employing a version of IEKF (IIEKF) has been suggested in this study.
Although IEKF requires relatively more computational time compared to EKF, implementation of this filter results in the desired estimation by decreasing estimation error [49]. IEKF equations are presented as follows.
Consider the nonlinear state-space model of a system in the discrete-time domain as (3).
In these equations, w(k) and v(k) are denoted as process noise and measurement noise with covariance matrix Q(k) and R(k), respectively. f(.) and h(.) are nonlinear continuous functions. To linearize nonlinear functions, f(.) and h(.), equations (4) to (6) are represented as follows, where F(.), (.) and H(.) are the Jacobian matrices.
Generally, by determining the initial values as (7), IEKF equations are presented as (8), which can be separated into two parts (measurement update and time update).
Measurement update: Time update: where, P − (k) andx − (k) are the a priori estimation of P(k) and x(k) using Z − = z(1) . . . z(k − 1) , respectively. Since the linearized system may become unobservable in some operating points, it is suggested to check observability before the time update part. If the states are unobservable, the states do not update. In other words, in IIEKF, (8h) is substituted by (18).
In (9), σ min (ϕ) and σ max (ϕ) are the minimum and maximum singular values of ϕ, respectively. ε, as a threshold, can be determined using (9b) when the system is completely observable. It is suggested that ε = 10 −5 .
The flow chart of the introduced IIEKF is shown in Figure 1. In Figure 1 εx i (k) ; i = 1, . . . , n are determined using the convergence of states.

IV. ESTIMATION OF ω r (k), R r , AND R s
In order to estimate ω r (k), R r , and R s using IIEKF a discretetime state-space model should be presented to introduce the dynamic behavior of ω r (k), R r , and R s .
A. STATE-SPACE MODEL FOR ESTIMATING R r AND ω r (k) In order to estimate ω r (k) and R r , the state vector, x r (k), and input vector, u(k), are defined as (11).
It should be noted that in (15) and (23) the variations of R r and R s with time are assumed to be really too small which can be considered constant parameters.
By considering the state-space models of the IM for estimating ω r (k), R r , and R s in (12) and (20), two IIEKFs, which are presented in section III, Figure 1, are used simultaneously to estimate the state variables. In the first IIEKF, R r and ω r (k) will be estimated, where x(k) = x r (k),x(k) =x r (k), f(.) = f r (.), and h(.) = h r (.). Therefore, according to (11), the estimations of R r and ω r (k) are obtained as follows : In the second IIEKF, by defining x(k) = x s (k),x(k) = x s (k), f(.) = f s (.), and h(.) = h s (.), R s and ω r (k) will be estimated as follows:R Since ω r (k) is the state variable in both (11) and (19) state vectors, this variable is estimated twice. To achieve optimal state estimation between (26b) and (27b), we can use the data fusion method as follows [50]: In (28), p 55 r (k) and p 55 s (k) are the 5 th diagonal element of P r (k) and P s (k), respectively. The mathematical definition of P r (k) and P s (k) are as follows: = P(k)in the 1 st IIEKF using to estimate x r (k) P s (k) = p ij s (k) 6×6 ; i, j = 1, . . . , 6 = P(k)in the 2 st IIEKF using to estimatex s (k) (29) It should be noted that P r (k) and P s (k) are the error covariance matrices.

V. EXPERIMENTAL RESULTS AND DISCUSSION
In order to show the effectiveness of the introduced methodology to estimate ω r (k), R r , and R s , the real input-output data of a 1.5 Kw squirrel cage IM were used. The technical specification of the IM has been given in Table 2. The experimental setup and data acquisition system are shown in Figure 2. The load of the IM was a permanent magnet synchronous generator (PMSG) and the load of the PMSG was the resistive load. The PMSG, as a mechanical load for IM, is used to change the load torque. Two experiments were performed under different load torques in the nominal rotor rotational speed  ω r (k) = [0 ∼ 3000](RPM ), and lower than the nominal rotor rotational speed ω r (k) = [0 ∼ 1000](RPM ). By considering stator voltages as inputs, (11), and stator currents as outputs variables, (12) and (20), stator voltages and currents are measured and recorded by NI USB6009. The stator voltages and VOLUME 11, 2023   currents in the nominal speed in the dq stationary reference frame are shown in Figure 3.
According to section IV, the estimation process has been done. A computer with the following characteristics was used, CPU: Intel Core i7-7700HQ CPU @2.80GHz;16GB RAM. OS: Windows 10, 64. The results of the estimation of ω r (k), R r , and R s using two experiments are presented as follows. As described above, two experiments have been performed under a diverse range of operational loads and speeds. Then, the validation of the estimation process based on section IV, estimation of ω r (k), R r , and R s , has been investigated for two ranges of speed as follows: The results of the estimation of ω r (k), R r , R s and the estimation errors,ω r (k),R r , andR s , for the 1 st experiment are shown in Figures 4 to 6. In order to analyze theω r (k),R r , andR s , the root mean square errors and standard deviations ofω r (k),R r , andR s for the 1 st experiment are shown in Table 3. Figures 4 to 6, and the related rows of 1 st experiments in Table 3 show the whiteness of the estimation error, which indicate that the estimated parameters have acceptable accuracy.
In this sub-section, in order to compare IIEKF and EKF, the estimation results of these two methods by using the dataset of 2 nd experiment are shown in Figures 7 to 9. According to Figures 7 to 9, the estimation errors,ω r (k),R r andR s , using IIEKF are less than EKF.
In order to analyze theω r (k),R r , andR s , the root mean square errors and standard deviations ofω r (k),R r , andR s for the 2 nd experiment are shown in Table 3. Similar to the 1 st experiment, analyzing the estimation errors using IIEKF in Figures 7 to 9, and the related rows of the 2 nd experiments in Table 3 (4 th to 6 th rows) show its whiteness and indicate that the estimated parameters have acceptable accuracy. By comparing the estimation errors using IIEKF with the estimation errors using EKF in Table 3 (4 th to 9 th rows), the estimation  errors using IIEKF are less and closer to whiteness than the estimation errors using EKF.
Also, to compare the execution time of both methods, Table 4 is presented. According to Table 4, the execution  time for running IIEKF is more than the execution times for running EKF. Obviously, more execution time in IIEKF VOLUME 11, 2023 because of the delay in obtaining the output signal owing to iterations have occurred. By considering execution times, IIEKF can be employed in the controllers where the execution times for estimating processes are less than the acceptable delay.
Therefore, according to Tables 3, 4, and Figures 4 to 9, the advantage of IIEKF in comparison with using EKF is its fewer estimation errors, and the disadvantage of IIEKF in comparison with using EKF is its execution estimation time.
According to the above results for both scenarios, the IIEKF works properly by using the extracted state-space models of the IM in (12) and (20) for estimating ω r (k), R r , and R s .

VI. CONCLUSION
This paper introduced Improved Iterated Extended Kalman Filter (IIEKF) to estimate the rotational speed, stator, and rotor resistances. The performance of the IIEKF has been verified under real conditions and using experimental data. The results of estimation in the experimental results show that the applied method gives a reliable and accurate estimation for a three-phase IM under different operating conditions. Therefore, the effectiveness of the introduced approach has been shown.
Further research can be done by introducing a novel methodology to estimate rotational speed, rotor, and stator resistances in the presence of uncertainty and fault.