On Accurate Discrete-Time Dynamic Models of an Induction Machine

Abstract

Induction machines have become the standard for highly demanding industrial applications. This has led to the utilization of modern discrete-time control techniques (such as model predictive control) that require the estimation of internal variables that are not subject to measurement (such as the rotational velocity in sensorless applications). From this point of view, it is fundamental to have accurate discrete-time models of induction machines, particularly given their nonlinear nature, so that control techniques perform according to design requirements. In spite of the above, the modeling of induction machines has not received much attention in the literature, even though more powerful machines and faster microcontrollers are currently being used. To better understand induction machine models for control, in this paper, we develop and compare various discrete-time models of induction machines based on Euler, Taylor, and Runge–Kutta methods. In addition, we compare the Extended Kalman Filter and Unscented Kalman Filter for state estimation in terms of accuracy and computational burden. The models are derived and compared through extensive Monte Carlo simulations and the state estimation techniques are compared in terms of root mean squared error, execution time, and maximum absolute error. Our simulations show that, in general, the Taylor method yields more accurate models than both the Runge–Kutta and Euler methods. In particular, the Taylor method results in a root mean square error that is one order of magnitude smaller than the Euler method for stator current and rotor flux linkages. For rotor angular speed, the Runge–Kutta methods are more accurate than both the Taylor and Euler methods, resulting in a root mean square error that is two orders of magnitude smaller than the Euler method. On the other hand, the Extended Kalman Filter results in smaller execution time than the Unscented Kalman Filter, up to two orders of magnitude. In terms of root mean squared error and maximum absolute error, both filtering algorithms perform similarly.

1. Introduction

Induction machines (IM) are basically an alternating current (AC) polyphase machine connected to an AC power source, either in the stator or in the rotor. The winding arrangement on the stator produces a rotating magnetic field in the machine airgap, which induces circulating currents that result in an induced rotating magnetic field in the rotor [1].

Induction machines have become the standard for highly demanding industrial applications. This is due to the simplicity in their construction and working principle, robustness, and wide speed range operation [2]. On the other hand, control strategies such as Field Oriented Control or Direct Torque Control are today’s industrial standard control schemes, since they allow for high energy efficiency, i.e., they develop high torque per Ampere ratio [1,2]. Among modern control methods, Model Predictive Control (MPC) has been widely studied in the literature [3,4]. Moreover, technological advances have allowed for the implementation of MPC strategies, see, e.g., [5,6]. It is important to note that power converters play a key role in driving the machine, delivering the necessary input voltage according to the control requirements. In general, one of the practical limitations in machine drives is the switching frequency of the power converter, which in turn limits the maximum sampling rate. This limitation arises from the fact that the higher the voltage the power converter handles, the lower the switching frequency it can withstand [7]. This sampling rate restriction may limit the performance of the controller since a low sampling rate could result in an inaccurate discrete-time model.

In variable-speed drives, speed is conventionally measured using a sensor, which is susceptible to electromagnetic noise; it requires maintenance and it is unsuitable for abrasive environments [8,9]. For these reasons, speed sensors are not used today. Instead, the rotor speed is estimated using an observer. In addition, the benefits of speed estimation encompass diminished hardware complexity and costs, reduced physical space requirements for the drive, and decreased maintenance demands, to name a few [8,9]. On the other hand, modern control techniques typically require knowledge of the rotor magnetic flux position. Given the cost implications and challenges associated with integrating electromagnetic flux sensors directly into the machine, the flux position is commonly estimated. Therefore, only the stator currents are measured whilst rotor speed and magnetic fluxes are estimated. This kind of drive is known as sensorless [8].

To estimate the rotor speed and rotor flux linkages, model-based methods are typically used, such as model adaptive systems [10,11], full-order observers [12], Luenberger observers [13], and sliding mode observers [14,15], among others. The methods previously mentioned are based on a deterministic formulation of the system. However, deterministic models do not take into account parameter offset from the nominal value, friction, lubricant ageing, or the effects of increased temperature (under normal operation) on the windings, all of which can be understood as model uncertainties. These uncertainties are typically modeled using random variables [16,17,18,19]. In this framework, the unknown variables are estimated using Bayesian filtering. In linear systems, the Kalman Filter (KF) is used, whereas for addressing nonlinear state estimation, the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF) are typically used [20,21,22,23,24].

The well-known Kalman Filter has been used in state estimation of induction machines in several works under the assumption of constant speed, see, e.g., [25,26,27]. This assumption results in a linear system, and since the Kalman Filter is the optimal linear filter [28,29], it is the best option in this case. However, in our problem of interest, the speed is not constant, particularly in the numerical examples where we have considered direct start and load application. This non-constant speed results in a nonlinear system, which in turn requires the utilization of nonlinear state estimation techniques. To the best of our knowledge, the Extended Kalman Filter is the most widely used technique for state estimation in induction machines, see, e.g., [30,31,32,33,34]. Additionally, the Euler method is typically used for the attainment of discrete-time induction machine models, and whilst at high sampling rates it can yield adequate models, our work suggests that higher-order methods yield more accurate models that result in more accurate state estimation in terms of root mean square error.

On the other hand, the state estimation using UKF for an induction machine is presented in [35,36,37], where it is compared with EKF. In [35,37], it is determined that UKF has a slightly better performance than EKF in state estimation for the induction machine. Conversely, in [36] it is reported that the EKF yields a slightly better performance than UKF. Moreover, UKF has been used in sensorless control [37,38,39,40]. In [38], in addition to estimating the states, the rotor resistance and the load torque are also estimated. In [40], UKF is compared with EKF, and UKF provides slightly better estimation. However, in terms of execution time, the UKF algorithm takes six times longer than the EKF algorithm.

Currently, due to the digital nature of controllers and processors, it has become necessary to discretize continuous-time models for their implementation. Historically, in machine drives, the Euler method was widely adopted in the literature to obtain a discrete-time model of the machine due to its simplicity, see, e.g., [41,42,43]. In fact, in Bayesian filtering applications on IM drives using EKF [27,32,33,34] and UKF [35,37,38,39,40], Euler discretization methods are typically considered. Higher-order methods like the Taylor or Runge–Kutta, which involve greater computational costs and usually yield more complex systems, are not commonly utilized. The implementation of Taylor discretization, for instance, requires explicit calculations of higher-order derivatives. In contrast, Runge–Kutta employs estimated intermediate points between each sample to construct the model’s evolution [44]. Notably, Runge–Kutta methods are chosen to solve the nonlinear differential equations that define the machine behavior, resulting in “approximate” continuous-time currents, magnetic fluxes, and rotor speed. Moreover, Runge–Kutta methods are extensively utilized in simulation software such as Simulink and Plecs [45,46].

The contribution of this article is twofold: (i) a formulation and comparison of discrete-time models for induction machines using Euler, Taylor, and Runge–Kutta methods are presented, and (ii) a thorough comparison of EKF and UKF applied to the induction machine using the different discrete-time models is presented. Our analysis shows that the Euler method accurately represents the machine’s dynamics, but at very high sampling rates. On the other hand, at lower sampling rates, various discrete-time models based on the Taylor and Runge–Kutta methods can accurately represent the machine model. Finally, for the different discrete-time models, EKF and UKF algorithms are compared through extensive Monte Carlo simulations, in terms of root mean squared error (RMSE), execution time, and maximum absolute error.

The reminder of this article is organized as follows. In Section 2, the continuous-time model of the induction machine is described. In Section 3, the Euler, Taylor, and Runge–Kutta discrete-time models of the induction machine are formulated and compared with the continuous-time model in terms of RMSE. In Section 4, EKF and UKF algorithms are presented based on a stochastic model of the machine. In Section 5, EKF and UKF algorithms using different discrete-time models are compared in terms of execution time, RMSE, and maximum absolute error. Finally, we present our conclusions in Section 6.

2. Continuous-Time Induction Machine Model

The main parts of the induction machine are the stator and the rotor. The stator is made up of three symmetrically distributed coils. On the other hand, the rotor consists of aluminum bars shorted at the ends forming a cylinder. The three-phase coils are represented on a complex space, whose axes are named α-β. The stator space vectors are

$$\begin{array}{cc}\hfill {\mathit{v}}_s^{\alpha \beta }=& v_s^{\alpha }+j{v}_s^{\beta }=\frac{2}{3}\left({\mathit{a}}^0{v}_a+\mathit{a}{v}_b+{\mathit{a}}^2{v}_c\right)\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\mathit{i}}_s^{\alpha \beta }=& i_s^{\alpha }+j{i}_s^{\beta }=\frac{2}{3}\left({\mathit{a}}^0{i}_a+\mathit{a}{i}_b+{\mathit{a}}^2{i}_c\right)\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\mathit{\psi }}_s^{\alpha \beta }=& {\psi }_s^{\alpha }+j{\psi }_s^{\beta }=\frac{2}{3}\left({\mathit{a}}^0{\psi }_a+\mathit{a}{\psi }_b+{\mathit{a}}^2{\psi }_c\right)\hfill \end{array}$$

(3)

where ${\mathit{v}}_s^{\alpha \beta }$ is the stator voltage space vector on the stationary reference frame α-β; in the same way, ${\mathit{i}}_s^{\alpha \beta }$ and ${\mathit{\psi }}_s^{\alpha \beta }$ are the stator current and flux linkage, respectively, on the α-β reference frame. The unit space vector $\mathit{a}=e^{j\frac{2\pi }{3}}$ denotes the position of the coil on the space. The subscript $a,b,c$ denotes the three-phase component.

The stator electrical equation on the α-β reference frame is

$${\mathit{v}}_s^{\alpha \beta }=R_s{\mathit{i}}_s^{\alpha \beta }+\frac{d}{dt}{\mathit{\psi }}_s^{\alpha \beta }$$

(4)

where $R_s$ is the stator resistance.

Similarly, the rotor electrical equation is

$$0=R_r{\mathit{i}}_r^{\alpha \beta }+\frac{d}{dt}{\mathit{\psi }}_r^{\alpha \beta }+jp{\omega }_r{\mathit{\psi }}_r^{\alpha \beta }$$

(5)

where ${\mathit{i}}_r^{\alpha \beta }$ and ${\mathit{\psi }}_r^{\alpha \beta }$ are the rotor current and flux linkage, respectively, on the α-β reference frame; $R_r$ is the rotor resistance; p is the number of pole pairs; and ${\omega }_r$ is the rotor shaft angular speed.

The flux linkages of the stator and rotor are

$$\begin{array}{cc}\hfill {\mathit{\psi }}_s^{\alpha \beta }& =L_s{\mathit{i}}_s^{\alpha \beta }+L_m{\mathit{i}}_r^{\alpha \beta }\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\mathit{\psi }}_r^{\alpha \beta }& =L_r{\mathit{i}}_r^{\alpha \beta }+L_m{\mathit{i}}_s^{\alpha \beta }\hfill \end{array}$$

(7)

where $L_s$ and $L_r$ are the stator and rotor self-inductances, respectively, while $L_m$ is the mutual inductance between stator and rotor.

On the other hand, the mechanical equation at the rotor shaft is

$$J\frac{d}{dt}{\omega }_r={\mathcal{T}}_d-{\mathcal{T}}_l$$

(8)

where J is the inertia; ${\mathcal{T}}_d$ is the machine developed torque; and ${\mathcal{T}}_l$ represents the load torque. The machine developed torque is

$${\mathcal{T}}_d=\frac{3}{2}p\mathfrak{Re}\left\{j{\mathit{\psi }}_r^{\alpha \beta }{\mathit{i}}_s^{\left(\alpha \beta \right)\ast }\right\}$$

(9)

where the superscript ${}^{\ast }$ denotes the complex conjugate and $\mathfrak{Re}\left\{·\right\}$ denotes the real part of a complex number.

State-Space Representation

A state-space representation is used to analyze the IM model given in (2)–(9). For convenience, the selected state variables are the stator current ${\mathit{i}}_s^{\alpha \beta }$, the rotor flux linkage ${\mathit{\psi }}_r^{\alpha \beta }$, and the rotor shaft speed ${\omega }_r$. By manipulating these variables, torque and speed control is achievable [2]. Furthermore, in sensorless drives only the stator current is measured.

Typically, the state vector is expanded and the load torque dynamics is modeled as well [32,39,40]. Including the load torque as a state variable, the IM dynamic equations are

$$\begin{array}{cc}\hfill \frac{d}{dt}{i}_s^{\alpha }& =-a_1{i}_s^{\alpha }+a_2{\psi }_r^{\alpha }+a_3{\omega }_r{\psi }_r^{\beta }+b_1{v}_s^{\alpha }\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \frac{d}{dt}{i}_s^{\beta }& =-a_1{i}_s^{\beta }+a_2{\psi }_r^{\beta }-a_3{\omega }_r{\psi }_r^{\alpha }+b_1{v}_s^{\beta }\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \frac{d}{dt}{\psi }_r^{\alpha }& =a_4{i}_s^{\alpha }-a_5{\psi }_r^{\alpha }-a_6{\omega }_r{\psi }_r^{\beta }\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \frac{d}{dt}{\psi }_r^{\beta }& =a_4{i}_s^{\beta }-a_5{\psi }_r^{\beta }+a_6{\omega }_r{\psi }_r^{\alpha }\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill \frac{d}{dt}{\omega }_r& =a_7\left[{\psi }_r^{\alpha }{i}_s^{\beta }-{\psi }_r^{\beta }{i}_s^{\alpha }\right]-a_8{\mathcal{T}}_l\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \frac{d}{dt}{\mathcal{T}}_l& =g({\mathcal{T}}_l,{\omega }_r)\hfill \end{array}$$

(15)

where a and b coefficients are

$$\begin{array}{cccccc}\hfill a_1& =\frac{R_{\sigma }}{\sigma L_s}\hfill & \hfill a_2& =\frac{L_m}{\sigma L_s{\tau }_r{L}_r}\hfill & \hfill a_3& =\frac{L_m}{\sigma L_s{L}_r}p\hfill \\ \hfill a_4& =\frac{L_m}{{\tau }_r}\hfill & \hfill a_5& =\frac{1}{{\tau }_r}\hfill & \hfill a_6& =p\hfill \\ \hfill a_7& =\frac{3}{2}\frac{1}{J}p\frac{L_m}{L_r}\hfill & \hfill a_8& =\frac{1}{J}\hfill & \hfill b_1& =\frac{1}{\sigma L_s}\hfill \end{array}$$

The above coefficients are in terms of already-named machine parameters and additionally by

$$\begin{array}{cccccc}\hfill R_{\sigma }& =R_s+R_r\frac{L_m^2}{L_r^2}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}\sigma & =1-\frac{L_m^2}{L_s{L}_r}\hfill & \hfill \hspace{1em}\hspace{1em}\hspace{1em}{\tau }_r& =\frac{L_r}{R_r}\hfill \end{array}$$

where $R_{\sigma }$ is known as the stator dynamic transient resistance; $\sigma$ as the leakage coefficient; and ${\tau }_r$ as the rotor time constant. The function $g({\mathcal{T}}_l,{\omega }_r)$ represents any load torque function. The load torques that are generally found in electric drives are given in

Table 1. Typical load torque functions [47].

Load Torque	Model
Constant	${\mathcal{T}}_l={\mathcal{T}}_{l0}$
Linear	${\mathcal{T}}_l=K{\omega }_r$
Quadratic	${\mathcal{T}}_l=K{\omega }_r^2$
Inverse	${\mathcal{T}}_l=K{\omega }_r^{-1}$

, where K is a proportional constant.

The dynamic equations are depicted in the block diagrams shown in Figure 1, where the α-β components of the space vectors are jointly represented. The states are obtained by integrating the right-hand side of Equations (10)–(15). Specifically, Figure 1a illustrates the joint behavior of the α-β stator current, while Figure 1b captures the joint dynamic behavior of the α-β rotor flux linkages. Figure 1c represents the dynamic behavior of the rotor shaft angular speed, with ⊗ representing the cross product of the rotor flux linkages and stator currents (${\mathit{\psi }}_r^{\alpha \beta }\times {\mathit{i}}_s^{\alpha \beta }$); finally, Figure 1d illustrates the dynamic behavior of the load torque.

In compact form, the representation of the IM model in state-space is

$$\begin{array}{cc}\hfill \frac{d}{dt}\mathit{x}=& \mathit{f}\left(\mathit{x},\mathit{u}\right)\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \mathit{y}=& \mathit{H}\mathit{x}\hfill \end{array}$$

(17)

Equation (16) is given by (10)–(15); $\mathit{x}$ is the state vector and $\mathit{u}$ is the input vector, in this case ${\mathit{v}}_s^{\alpha \beta }$. The output (17) is related to the state vector by the matrix $\mathit{H}$. Since it is a sensorless drive, only the stator current $i_s^{\alpha \beta }$ is measured, hence

$$\mathit{H}=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\end{array}\right]$$

Note that the state variables depend implicitly on time t (i.e., $\mathit{x}=\mathit{x}\left(t\right)$). Also, note that the IM state Equation (16) exhibits a nonlinear behavior.

3. Discrete-Time Induction Machine Model

Due to the discrete nature of controllers, mostly discrete signals are processed today. In order to achieve a precise and accurate discretization, the discrete-time model must be as close as possible to the continuous-time model; hence, the error between continuous-time and discrete-time models must be close to zero. The discrete-time model is described in the form

$$\begin{array}{cc}\hfill {\mathit{x}}_{k+1}=& {\mathit{f}}_d\left({\mathit{x}}_k,{\mathit{u}}_k\right)\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\mathit{y}}_k=& \mathit{H}{\mathit{x}}_k\hfill \end{array}$$

(19)

where ${\mathit{x}}_k=\mathit{x}\left(k{T}_s\right)$ and ${\mathit{x}}_{k+1}=\mathit{x}(k{T}_s+T_s)$; $T_s$ is the time interval between samples. The input ${\mathit{u}}_k$ is considered as a zero-order hold (ZOH) sinusoidal wave.

The goal is to approximate numerically the IM dynamic model. There are several methods to numerically approximate differential equations. In this section, three methods are discussed: Euler, Taylor, and Runge–Kutta.

3.1. Euler Method

The Euler method is intuitive and straightforward, and is the basis of the two other methods. It approximates the solution to the differential equation as follows

$${\mathit{x}}_{k+1}\approx {\mathit{x}}_k+T_s\frac{d}{dt}\mathit{x}{|}_{{\mathit{x}}_k}$$

(20)

The IM model (16) approximated by the Euler Method is

$$\begin{array}{cc}\hfill i_{s,k+1}^{\alpha }& =i_{s,k}^{\alpha }+T_s\left(-a_1{i}_{s,k}^{\alpha }+a_2{\psi }_{r,k}^{\alpha }+a_3{\omega }_{r,k}{\psi }_{r,k}^{\beta }+b_1{v}_{s,k}^{\alpha }\right)\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill i_{s,k+1}^{\beta }& =i_{s,k}^{\beta }+T_s\left(-a_1{i}_{s,k}^{\beta }+a_2{\psi }_{r,k}^{\beta }-a_3{\omega }_{r,k}{\psi }_{r,k}^{\alpha }+b_1{v}_{s,k}^{\beta }\right)\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\psi }_{r,k+1}^{\alpha }& ={\psi }_{r,k}^{\alpha }+T_s\left(a_4{i}_{s,k}^{\alpha }-a_5{\psi }_{r,k}^{\alpha }-a_6{\omega }_{r,k}{\psi }_{r,k}^{\beta }\right)\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {\psi }_{r,k+1}^{\beta }& ={\psi }_{r,k}^{\beta }+T_s\left(a_4{i}_{s,k}^{\beta }-a_5{\psi }_{r,k}^{\beta }+a_6{\omega }_{r,k}{\psi }_{r,k}^{\alpha }\right)\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\omega }_{r,k+1}& ={\omega }_{r,k}+T_s\left(a_7\left[{\psi }_{r,k}^{\alpha }{i}_{s,k}^{\beta }-{\psi }_{r,k}^{\beta }{i}_{s,k}^{\alpha }\right]-a_8{\mathcal{T}}_{l,k}\right)\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill {\mathcal{T}}_{l,k+1}& ={\mathcal{T}}_{l,k}+T_{s}g\left({\mathcal{T}}_{l,k},{\omega }_{r,k}\right)\hfill \end{array}$$

(26)

For the Euler method, the local truncation error (i.e., the error committed at each time step) is of the order $\mathcal{O}\left(T_s^2\right)$.

3.2. Taylor Method

The derivative of a function is approximated using Taylor series expansions

$${\mathit{x}}_{k+1}\approx {\mathit{x}}_k+T_s\frac{d}{dt}\mathit{x}{|}_{{\mathit{x}}_k}+\frac{T_s^2}{2}\frac{d^2}{d{t}^2}\mathit{x}{|}_{{\mathit{x}}_k}+\frac{T_s^3}{3!}\frac{d^3}{d{t}^3}\mathit{x}{|}_{{\mathit{x}}_k}+\cdots$$

(27)

The more terms in the series considered for the approximation, the more accurate the model. According to [48], in order to capture completely the effect of a ZOH-input at the next sampling instant, the input must appear explicitly in the state equations, that is, a discrete-time relative degree $\ge 1$. As simply as possible, the discretization of relative degree $=1$ is achieved by expanding by second-order Taylor series the rotor flux, speed, and load torque given in (12)–(15). The discrete-time IM model with relative degree $=1$ is given by

$$\begin{array}{cc}\hfill i_{s,k+1}^{\alpha }=& i_{s,k}^{\alpha }+T_s\left(-a_1{i}_{s,k}^{\alpha }+a_2{\psi }_{r,k}^{\alpha }+a_3{\omega }_{r,k}{\psi }_{r,k}^{\beta }+b_1{v}_{s,k}^{\alpha }\right)\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill i_{s,k+1}^{\beta }=& i_{s,k}^{\beta }+T_s\left(-a_1{i}_{s,k}^{\beta }+a_2{\psi }_{r,k}^{\beta }-a_3{\omega }_{r,k}{\psi }_{r,k}^{\alpha }+b_1{v}_{s,k}^{\beta }\right)\hfill \end{array}$$

(29)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\psi }_{r,k+1}^{\alpha }=& {\psi }_{r,k}^{\alpha }+T_s\left(a_4{i}_{s,k}^{\alpha }-a_5{\psi }_{r,k}^{\alpha }-a_6{\omega }_{r,k}{\psi }_{r,k}^{\beta }\right)+\frac{T_s^2}{2}\left[a_4\left(-a_1{i}_{s,k}^{\alpha }+a_2{\psi }_{r,k}^{\alpha }+a_3{\omega }_{r,k}{\psi }_{r,k}^{\beta }+b_1{v}_{s,k}^{\alpha }\right)\right]\hfill \\ \hfill & +\frac{T_s^2}{2}\left\{-a_6\left[{\omega }_{r,k}\left(a_4{i}_{s,k}^{\beta }-a_5{\psi }_{r,k}^{\beta }+a_6{\omega }_{r,k}{\psi }_{r,k}^{\alpha }\right)+{\psi }_{r,k}^{\beta }\left(a_7\left({\psi }_{r,k}^{\alpha }{i}_{s,k}^{\beta }-{\psi }_{r,k}^{\beta }{i}_{s,k}^{\alpha }\right)-a_8{\mathcal{T}}_{l,k}\right)\right]\right\}\hfill \\ \hfill & -\frac{T_s^2}{2}\left[a_5\left(a_4{i}_{s,k}^{\alpha }-a_5{\psi }_{r,k}^{\alpha }-a_6{\omega }_{r,k}{\psi }_{r,k}^{\beta }\right)\right]\hfill \end{array}\end{array}$$

(30)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\psi }_{r,k+1}^{\beta }=& {\psi }_{r,k}^{\beta }+T_s\left(a_4{i}_{s,k}^{\beta }-a_5{\psi }_{r,k}^{\beta }+a_6{\omega }_{r,k}{\psi }_{r,k}^{\alpha }\right)+\frac{T_s^2}{2}\left[a_4\left(-a_1{i}_{s,k}^{\beta }+a_2{\psi }_{r,k}^{\beta }-a_3{\omega }_{r,k}{\psi }_{r,k}^{\alpha }+b_1{v}_{s,k}^{\beta }\right)\right]\hfill \\ \hfill & +\frac{T_s^2}{2}\left\{a_6\left[{\omega }_{r,k}\left(a_4{i}_{s,k}^{\alpha }-a_5{\psi }_{r,k}^{\alpha }-a_6{\omega }_{r,k}{\psi }_{r,k}^{\beta }\right)+{\psi }_{r,k}^{\alpha }\left(a_7\left({\psi }_{r,k}^{\alpha }{i}_{s,k}^{\beta }-{\psi }_{r,k}^{\beta }{i}_{s,k}^{\alpha }\right)-a_8{\mathcal{T}}_{l,k}\right)\right]\right\}\hfill \\ \hfill & -\frac{T_s^2}{2}\left[a_5\left(a_4{i}_{s,k}^{\beta }-a_5{\psi }_{r,k}^{\beta }+a_6{\omega }_{r,k}{\psi }_{r,k}^{\alpha }\right)\right]\hfill \end{array}\end{array}$$

(31)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\omega }_{r,k+1}=& {\omega }_{r,k}+T_s\left(a_7\left[{\psi }_{r,k}^{\alpha }{i}_{,}{sk}^{\beta }-{\psi }_{r,k}^{\beta }{i}_{s,k}^{\alpha }\right]-a_8{\mathcal{T}}_{l,k}\right)+\frac{T_s^2}{2}\left[a_7{i}_{s,k}^{\beta }\left(a_4{i}_{s,k}^{\alpha }-a_5{\psi }_{r,k}^{\alpha }-a_6{\omega }_{r,k}{\psi }_{r,k}^{\beta }\right)\right]\hfill \\ \hfill & +\frac{T_s^2}{2}\left[a_7{\psi }_{r,k}^{\alpha }\left(-a_1{i}_{s,k}^{\beta }+a_2{\psi }_{r,k}^{\beta }-a_3{\omega }_{r,k}{\psi }_{r,k}^{\alpha }+b_1{v}_{s,k}^{\beta }\right)\right]-\frac{T_s^2}{2}\left[a_7{i}_{s,k}^{\alpha }\left(a_4{i}_{s,k}^{\beta }-a_5{\psi }_{r,k}^{\beta }+a_6{\omega }_{r,k}{\psi }_{r,k}^{\alpha }\right)\right]\hfill \\ \hfill & -\frac{T_s^2}{2}\left[a_7{\psi }_{r,k}^{\beta }\left(-a_1{i}_{s,k}^{\alpha }+a_2{\psi }_{r,k}^{\alpha }+a_3{\omega }_{r,k}{\psi }_{r,k}^{\beta }+b_1{v}_{s,k}^{\alpha }\right)\right]-\frac{T_s^2}{2}{a}_{8}g({\mathcal{T}}_{l,k},{\omega }_{r,k})\hfill \end{array}\end{array}$$

(32)

$$\begin{array}{cc}\hfill {\mathcal{T}}_{l,k+1}=& {\mathcal{T}}_{l,k}+T_{s}g({\mathcal{T}}_{l,k},{\omega }_{r,k})+\frac{T_s^2}{2}\dot{g}({\mathcal{T}}_{l,k},{\omega }_{r,k})\hfill \end{array}$$

(33)

For a second-order Taylor series expansion, the local truncation error is $\mathcal{O}\left(T_s^3\right)$.

3.3. Runge–Kutta Method

The Runge–Kutta (RK) method approximates the solution of the differential equation based on estimates of the solution at different points. In this subsection, two RK methods are taken into account: RK 2 and RK 4. The RK 2 approximation considers two estimates, whereas RK 4 considers four. The advantage that Runge–Kutta has over Taylor is that it is not necessary to explicitly calculate higher-order derivatives of $\mathit{f}(\mathit{x},\mathit{u})$.

3.3.1. Runge–Kutta 2

Runge–Kutta 2 is a second-order approximation. This method requires two evaluations of $\mathit{f}(\mathit{x},\mathit{u})$ at each time step. The IM model (16) is approximated by

$${\mathit{x}}_{k+1}={\mathit{x}}_k+\frac{T_s}{2}\left({\mathit{r}}_1+{\mathit{r}}_2\right)$$

(34)

where

$$\begin{array}{cc}\hfill {\mathit{r}}_1& =\mathit{f}({\mathit{x}}_k,{\mathit{u}}_k)\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\mathit{r}}_2& =\mathit{f}\left({\mathit{x}}_k+T_s{\mathit{r}}_1,\mathit{u}(t_k+T_s)\right)\hfill \end{array}$$

(36)

and $t_k=k{T}_s$. Note that ${\mathit{r}}_2$ estimation requires future input values. Since the system is assumed to be causal and future inputs are unknown, ${\mathit{u}}_k$ is used instead. For the IM, ${\mathit{r}}_1$ and ${\mathit{r}}_2$ are found in Appendix A.1.

3.3.2. Runge–Kutta 4

Runge–Kutta 4 is a fourth-order method that requires four evaluations of $\mathit{f}(\mathit{x},\mathit{u})$ in one sample period. The approximation of IM model (16) by RK 4 is

$${\mathit{x}}_{k+1}={\mathit{x}}_k+\frac{T_s}{6}\left({\mathit{r}}_1+2{\mathit{r}}_2+2{\mathit{r}}_3+{\mathit{r}}_4\right)$$

(37)

where

$$\begin{array}{cc}\hfill {\mathit{r}}_1& =\mathit{f}({\mathit{x}}_k,{\mathit{u}}_k)\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {\mathit{r}}_2& =\mathit{f}\left({\mathit{x}}_k+\frac{T_s}{2}{\mathit{r}}_1,\mathit{u}(t_k+\frac{T_s}{2})\right)\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill {\mathit{r}}_3& =\mathit{f}\left({\mathit{x}}_k+\frac{T_s}{2}{\mathit{r}}_2,\mathit{u}(t_k+\frac{T_s}{2})\right)\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\mathit{r}}_4& =\mathit{f}\left({\mathit{x}}_k+T_s{\mathit{r}}_3,\mathit{u}(t_k+T_s)\right)\hfill \end{array}$$

(41)

Since the input is ZOH and no future input values are taken into account, ${\mathit{u}}_k$ is used in ${\mathit{r}}_2$, ${\mathit{r}}_3$, and ${\mathit{r}}_4$. For the IM, the estimate points ${\mathit{r}}_1$, ${\mathit{r}}_2$, ${\mathit{r}}_3$, and ${\mathit{r}}_4$ are found in Appendix A.2.

3.4. Comparison between Models

The previously mentioned discrete-time IM models were contrasted with each other and with a continuous-time model. The simulations were carried out in Matlab. To compare the discrete-time models with the continuous-time one, the latter was obtained using a fifth-order Dormand–Prince (DOPRI 5) method with the same discretization time as the discrete-time models. More information about this method is given in Appendix B. For the simulations, $T_s=200$ μs and a 4 kW IM was used; its nominal parameters are $V=$ 380 V; $I=$ 8.6 A; $f=$ 50 Hz; two pole pairs; $R_s=$ 1.32 $\Omega$; $R_r=$ 2.63 $\Omega$; $L_m=$ 0.1889 H; $L_s=$ 0.1972 H; $L_r=$ 0.2012 H; and $J=$ 0.528 kg·m².

To illustrate the differences between discrete-time and continuous-time models, two scenarios are taken into account. First, the IM is directly started from the grid. Second, a 15 Nm step-type load impact occurs suddenly at 4 s. The total simulation time was 6 s.

Table 2. Models’ root mean squared error.

State	Euler	Taylor	RK 2	RK 4
$i_s^{\alpha }$	2.3288	0.3743	0.5830	0.4188
$i_s^{\beta }$	2.3286	0.3723	0.5985	0.4177
${\psi }_r^{\alpha }$	0.0567	0.0091	0.0245	0.0191
${\psi }_r^{\beta }$	0.0567	0.0089	0.0286	0.0190
${\omega }_r$	21.6914	11.3117	1.9997	0.1401
${\mathcal{T}}_l$	0.0046	0.0046	7.0356 $\times {10}^{-5}$	7.0171 $\times {10}^{-9}$

contrasts the discrete-time models with the continuous-time model by presenting the computed root mean squared error (RMSE) between them, calculated as follows

$$RMSE\left(\mathit{x}\right)=\sqrt{\frac{1}{N}\sum _{i=1}^N{\left({\mathit{x}}_{i,cont}-{\mathit{x}}_{i,disc}\right)}^2}$$

(42)

To begin with, in Table 2, the best approximation for the stator current and rotor flux linkage is obtained by Taylor. The RK 2 and RK 4 models are not as accurate as Taylor, due to the fact that RK 2 and RK 4 do not take into account future input estimation for the approximation. Conversely, for the rotor speed RK 4 and RK 2 provide the best approximation. Finally, Euler results in the worst approximation as expected.

Figure 2 shows the α-β stator currents and rotor flux linkages for the discrete-time and continuous-time models with no load condition at steady state. Figure 2a shows that the Taylor and RK 4 methods yield discrete-time values that are almost identical to the continuous-time values. In the same figure, we can observe that the RK 2 method yields a larger difference since the amplitude of the currents is smaller than the continuous-time currents, whilst the Euler method results in even larger errors since the resulting currents have a much smaller amplitude. In Figure 2b, the rotor flux linkages for each method are shown. The Taylor, RK 2, and RK 4 signals resemble the continuous-time signals. However, the Euler method results in a slightly larger difference from the continuous-time flux than the Taylor and RK methods. On the other hand, Figure 3 illustrates the rotor angular speed and zero load torque for all models described. As shown, both the Taylor and Euler models are less accurate under no-load conditions than the RK methods. In particular, RK 4 provides the most accurate speed approximation, followed by RK 2.

On the other hand, notice that after the sudden step-type load torque impact occurs, stator current increases while rotor speed decreases. Figure 4 shows the stator currents and rotor flux linkages for the discrete-time and continuous-time models after the torque impact occurs. In this case, the Euler method yields a bad approximation, since the resulting currents not only exhibit a smaller amplitude but also a large phase error. In contrast, the Taylor, RK 2, and RK 4 models resemble the continuous-time model. Figure 5 illustrates the rotor angular speed and the load torque impact for all models. We can observe that, once the speed reaches the steady state, RK 4 and Taylor result in a very similar curve to the continuous-time model, followed by RK 2 and finally Euler. In addition, since load torque does not depend on rotor angular speed or any other state variable, the differences between the discrete-time models and the continuous-time model are, in general, small, as shown in Figure 5b.

In general, accurate models are desired since, in modern industrial systems, modern control techniques are applied. If the model is not accurate, model-based observers may fail to represent the actual behavior of the states, providing inaccurate feedback to the controller, hence resulting in poor control performance. Since with the Euler method the model error is reflected not only in the magnitude but also in the phase of the signals of interest, the overall control loop’s performance decreases greatly. This situation is naturally avoided in order to satisfy control system performance and requirements.

Changes in Sample Time

In this section, the behavior of the RMSE between the discrete-time models and the continuous-time model is observed as the discretization time varies. The discretization time is adjusted until the error of the model discretized by Euler is nearly zero. Two scenarios are considered; first, the IM is directly started, and then a sudden load impact occurs at 4 s. This follows the same approach as the previous section.

Figure 6 illustrates how RMSE varies according to $T_s$ for α-β stator currents, α-β rotor flux linkages, and rotor shaft speed. Since the error between the $\alpha$ and $\beta$ components of the stator current and rotor flux linkage components is similar, the average value is used in this figure. As expected, as the sampling time decreases, error decreases as well. In particular, the Taylor discrete-time model is better than RK 2 and RK 4 in terms of stator current and rotor flux linkage RMSE. Conversely, RK 2 and RK 4 are more accurate for speed approximation. Regarding the load torque approximation, RK 2 and RK 4 are better; nevertheless, Euler and Taylor are near zero. On the other hand, Euler RMSE rises dramatically as $T_s$ increases. Thus, using Euler for approximating the IM model is not advisable when the $T_s$ exceeds 50 μs. To conclude, Taylor is the best approximation, prioritizing stator currents and rotor flux linkage approximations.

4. Filtering

The continuous-time deterministic model of the IM was described in (16) and (17). Even so, those equations do not include model uncertainties or unknown disturbances, such as winding temperature rises, mechanical vibrations, switching noise, etc. All these uncertain effects can be represented by adding Gaussian noise in the state Equation (16) and output Equation (17) [16,17,18,19]. Also, considering the discrete nature of the measurement and control, a discrete-time representation of the model is used. Therefore, the discrete-time stochastic model of the IM is as follows

$$\begin{array}{cc}\hfill {\mathit{x}}_{k+1}=& {\mathit{f}}_d\left({\mathit{x}}_k,{\mathit{u}}_k\right)+{\mathbf{w}}_k\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill {\mathit{y}}_k=& \mathit{H}{\mathit{x}}_k+{\mathbf{v}}_k\hfill \end{array}$$

(44)

where w${}_k$ and v${}_k$ are Gaussian noise with zero mean and covariance matrix $\mathit{Q}$ and $\mathit{R}$, respectively. The term w${}_k$ is associated with model uncertainties, while v${}_k$ represents measurement noise.

Under a stochastic point of view, the state estimation is obtained from noisy measurements. In Bayesian inference, the estimation corresponds to calculating the joint probability density function (pdf) of the states given the measurements $\mathbb{P}\left({\mathit{x}}_{0:T}\right|{\mathit{y}}_{1:T})$, which incorporates all the statistical information about the state $\mathbb{P}\left({\mathit{x}}_{0:T}\right)$ that is contained in the available observation $\mathbb{P}\left({\mathit{y}}_{1:T}\right|{\mathit{x}}_{0:T})$ and the initial condition $\mathbb{P}\left({\mathit{x}}_{0:T}\right)$ [19,21]. Using the Bayes theorem, the joint pdf is

$$\mathbb{P}\left({\mathit{x}}_{0:T}\right|{\mathit{y}}_{1:T})={\displaystyle \frac{\mathbb{P}\left({\mathit{y}}_{1:T}\right|{\mathit{x}}_{0:T})\mathbb{P}\left({\mathit{x}}_{0:T}\right)}{\mathbb{P}\left({\mathit{y}}_{1:T}\right)}}$$

(45)

where the denominator $\mathbb{P}\left({\mathit{y}}_{1:T}\right)$ is a normalization constant. It is necessary to mention that ${\mathit{x}}_{0:T}$ and ${\mathit{y}}_{1:T}$ are the time series of the state vector $\{{\mathit{x}}_0,\cdots ,{\mathit{x}}_T\}$ and of the measurements $\{{\mathit{y}}_1,\cdots ,{\mathit{y}}_T\}$ vector, respectively. For real-time applications, it is unnecessary and computationally inefficient to compute the full joint distribution. For this reason, the marginal distribution of the current state given all the measurements $\mathbb{P}\left({\mathit{x}}_k\right|{\mathit{y}}_{1:k})$, also known as posterior, is computed instead. Additionally, assuming the IM model to be Markovian, it is possible to compute the posterior recursively [19,21]. The recursion is based on calculating the posterior distribution in two steps, starting from an initial prior distribution $\mathbb{P}\left({\mathit{x}}_0\right)$.

• First: predict. The prediction of the distribution of the states at the current time k is computed from the dynamic model by means of the Chapman–Kolmogorov equation

  $$\mathbb{P}\left({\mathit{x}}_k\right|{\mathit{y}}_{1:k-1})=\int \mathbb{P}\left({\mathit{x}}_k\right|{\mathit{x}}_{k-1})\mathbb{P}\left({\mathit{x}}_{k-1}\right|{\mathit{y}}_{1:k-1})d{\mathit{x}}_{k-1}$$

  (46)
• Second: update. Given the measurement ${\mathit{y}}_k$ at time k, the posterior distribution is calculated by solving

  $$\mathbb{P}\left({\mathit{x}}_k\right|{\mathit{y}}_{1:k})=\frac{1}{Z_k}\mathbb{P}\left({\mathit{y}}_k\right|{\mathit{x}}_k)\mathbb{P}\left({\mathit{x}}_k\right|{\mathit{y}}_{1:k-1})$$

  (47)

  where $Z_k$ is a normalization constant.

Unfortunately, since the IM model is nonlinear, it is not possible to solve the filtering Equations (46) and (47) in closed form. Nevertheless, there exist state estimation methods for nonlinear models; two of the classic methods are based on the Kalman Filter (KF): the Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF). Both are addressed in this section.

4.1. Extended Kalman Filter

The most well-known nonlinear filtering method is the Extended Kalman Filter, which propagates the mean and covariance through a linearization of the system resulting in a Gaussian posterior pdf. However, EKF does not ensure the convergence of the estimation, since it depends on the linearization of the model around an operating point that is also estimated [20,21,22,23,24]. The implementation steps of EKF for N samples are summarized in Algorithm 1, where $\hat{\mathit{x}}$ is the state estimate and $\mathit{P}$ is its covariance matrix. The subscript k denotes the posterior and $k|k-1$ the prior. $\mathit{K}$ is the Kalman gain; $\mathit{I}$ is the identity matrix; and ${\mathit{F}}_{k|k-1}$ is the Jacobian matrix of ${\mathit{f}}_d\left(\mathit{x},\mathit{u}\right)$ evaluated at $k-1$.

Algorithm 1: EKF Algorithm

4.2. Unscented Kalman Filter

On the other hand, there is the Unscented Kalman Filter, which by means of the Unscented Transform (UT) propagates representative points of the pdf through the nonlinearity of the system to estimate the state. The advantage that UKF has over EKF is that the representative points are adjustable by the user, attaining in general a better estimation [20,21,22,23,24]. These representative points, named Sigma Points (SPs), ensure the propagation of the mean and the covariance [49]. A total of $2n+1$ SP are required, with $n=dim\left\{\mathit{y}\right\}$. The SPs are computed as follows

$$\begin{array}{cccc}\hfill {\mathcal{X}}^{\left(0\right)}& =\mathit{x}\hfill & \hfill & \end{array}$$

(53)

$$\begin{array}{cccc}\hfill {\mathcal{X}}^{\left(i\right)}& =\mathit{x}+\sqrt{(n+\lambda )}{\left(\sqrt{\mathit{P}}\right)}_i\hfill & \hfill i& =1,...,n\hfill \end{array}$$

(54)

$$\begin{array}{cccc}\hfill {\mathcal{X}}^{\left(i\right)}& =\mathit{x}-\sqrt{(n+\lambda )}{\left(\sqrt{\mathit{P}}\right)}_i\hfill & \hfill i& =n+1,...,n\hfill \end{array}$$

(55)

where $\lambda$ is a scale factor that modifies the dispersion of the SP around the mean. The term $\sqrt{{\mathit{P}}_i}$ is computed using Cholesky factorization. Also, weight factors $W_i^{\left(m\right)}$ and $W_i^{\left(c\right)}$ are associated to the mean and covariance, defined as

$$\begin{array}{cccc}\hfill W_0^{\left(m\right)}& =\frac{\lambda }{n+\lambda }\hfill & \hfill W_0^{\left(c\right)}& =\frac{\lambda }{n+\lambda }+(1-{\alpha }^2+\beta )\hfill \\ \hfill W_{i\ne 0}^{\left(m\right)}& =\frac{\lambda }{2(n+\lambda )}\hfill & \hfill W_{i\ne 0}^{\left(c\right)}& =\frac{\lambda }{2(n+\lambda )}\hfill \end{array}$$

where typically, $\beta =2$ for Gaussian distributions; $0\le \alpha \le 1$; $\lambda ={\alpha }^2(n-\kappa )-n$ with $\kappa =3-n$.

The steps to implement the UKF are summarized in Algorithm 2. ${\mathit{S}}_k$ is the covariance matrix of the measurement; ${\mathit{C}}_k$ is the cross-covariance between the state and the measurement; and ${\mathit{K}}_k$ is the filter gain.

Algorithm 2: UKF algorithm

5. Numerical Examples

In this section, EKF and UKF were implemented for each IM discrete-time model discussed above. The simulations were carried out in Matlab. For the simulations, all the initial condition are zero; the discretization time is $T_s=200$ μs; the noise covariance matrices are $\mathit{Q}=diag\left\{2.12\times {10}^{-2},2.12\times {10}^{-2},{10}^{-6},{10}^{-6},{10}^{-3},9.64\times {10}^{-4}\right\}$ and $\mathit{R}=diag\left\{1/9,1/9\right\}$. The SP coefficients were selected empirically as in [40], where $\alpha ={10}^{-1}$, $\beta =2$, and $\kappa =-3$. The same noise covariance matrices were used for EKF and UKF. To test the IM models in filtering, a thousand Monte Carlo simulations were carried out, which is equivalent to measuring the same event a thousand times, in this case a direct start from a 50 Hz grid. All models in both filtering algorithms were compared in terms of execution time RMSE, where the actual system was simulated using DOPRI 5. Additionally, the maximum absolute error for each model using EKF and UKF was acquired to compare the maximum deviation from the actual value for each state, particularly at startup and after starting.

Table 3. EKF average root mean squared error.

State	EKF
	Euler	Taylor	RK 2	RK 4
$i_s^{\alpha }$	0.3612	0.1977	0.2029	0.2026
$i_s^{\beta }$	0.3577	0.1967	0.2017	0.2013
${\psi }_r^{\alpha }$	0.0777	0.0377	0.0433	0.0433
${\psi }_r^{\beta }$	0.0784	0.0379	0.0456	0.0456
${\omega }_r$	28.4063	27.2101	24.2762	24.5003
${\mathcal{T}}_l$	0.1038	0.1038	0.1042	0.1042

and

Table 4. UKF average root mean squared error.

State	UKF
	Euler	Taylor	RK 2	RK 4
$i_s^{\alpha }$	0.3611	0.1978	0.2029	0.2026
$i_s^{\beta }$	0.3575	0.1966	0.2016	0.2012
${\psi }_r^{\alpha }$	0.0777	0.0412	0.0431	0.0429
${\psi }_r^{\beta }$	0.0784	0.0425	0.0441	0.0443
${\omega }_r$	28.7982	28.0307	24.6992	24.8631
${\mathcal{T}}_l$	0.1038	0.1038	0.1042	0.1042

show the average RMSE from the Monte Carlo simulations in filtering using EKF and UKF, respectively. Both tables consider the Euler, Taylor, and Runge–Kutta IM models. From the tables, for the same IM model a considerable similarity between EKF and UKF is seen. In regard to IM models in both algorithms, there was a significant improvement in state estimation with the Taylor and Runge–Kutta models as expected. In currents and flux linkages, Taylor has less error than Runge–Kutta, due to the fact that RK models do not take into account future values of the input since they are unknown. Nevertheless, Runge–Kutta speed estimation is better than Taylor. Finally, load torque estimation is nearly the same in all discrete-time models.

On the other hand,

Table 5. EKF and EKF average execution times per time step.

Method	EKF Execution Time	UKF Execution Time
Euler	6.9456 × ${10}^{-6}$ s	9.9939 × ${10}^{-5}$ s
Taylor	7.4185 × ${10}^{-6}$ s	1.3481 × ${10}^{-4}$ s
RK 2	8.0369 × ${10}^{-6}$ s	1.1440 × ${10}^{-4}$ s
RK 4	9.7486 × ${10}^{-6}$ s	1.4439 × ${10}^{-4}$ s

shows average execution time per sample of the EKF and UKF algorithms for different IM models. This analysis is of particular interest in practical applications since IMs are usually coupled with modern control techniques that rely on state estimation, among other things. Hence, a low execution time allows for carrying out other computing tasks, such as optimization in a Model Predictive Control setup [6]. From Table 5, we can observe that as the complexity of the model increases, so does the execution time when utilizing both EKF and UKF. In particular, when utilizing UKF, apart from Euler, RK 2 has the smallest execution time followed by Taylor and finally RK 4. As for EKF and UKF, clearly the former requires much less execution time than the latter, no matter which discretization technique is used. In fact, our results show that UKF requires more than half of the sampling interval during the execution of the algorithm. This could leave not enough time for other tasks. However, more research is needed in this area in a practical setup since filtering can be implemented in different computing platforms with different architectures, such as a microcontroller or a field programmable gate array.

Table 6. Max absolute error EKF at startup.

State	EKF
	Euler	Taylor	RK 2	RK 4
$i_s^{\alpha }$	3.1673	3.1673	3.1673	3.1673
$i_s^{\beta }$	1.3279	1.3279	1.3280	1.3280
${\psi }_r^{\alpha }$	6.3224	6.3765	6.6773	6.6699
${\psi }_r^{\beta }$	5.8992	5.8999	5.9002	5.9041
${\omega }_r$	103.6	104.44	97.951	98.636
${\mathcal{T}}_l$	0.3806	0.3804	0.3805	0.3805

and

Table 7. Max absolute error UKF at startup.

State	UKF
	Euler	Taylor	RK 2	RK 4
$i_s^{\alpha }$	3.1673	3.1673	3.1673	3.1673
$i_s^{\beta }$	1.3279	1.3279	1.3280	1.3239
${\psi }_r^{\alpha }$	6.3166	6.3211	6.6758	6.4512
${\psi }_r^{\beta }$	5.8992	5.9000	5.9001	5.8696
${\omega }_r$	103.92	105.17	98.554	99.191
${\mathcal{T}}_l$	0.3806	0.3804	0.3805	0.3805

show the maximum absolute error at the startup of each state for each model using EKF and UKF algorithms, respectively, at startup. The maximum error is not symmetric, especially in the $\alpha$-$\beta$ components of stator currents and rotor flux linkages. This asymmetry is primarily due to the high uncertainty of the rotor angular speed. At the start, all initial conditions are zero; therefore, obtaining an accurate approximation is challenging. Notice that the maximum absolute errors are similar for all the discretization methods and both EKF and UKF. By themselves, these values only show what the maximum deviation is but not the overall behavior of each technique. However, they show what to expect in terms of the worst estimation in a single experiment. Moreover, if we also consider the average root mean squared error from Table 3 and Table 4, we can conclude that the Taylor and RK methods exhibit a reduced spread around the mean when compared to the Euler method. On the other hand,

Table 8. Max absolute error EKF after starting.

State	EKF
	Euler	Taylor	RK 2	RK 4
$i_s^{\alpha }$	1.4441	1.0330	1.0395	1.0388
$i_s^{\beta }$	1.4868	1.0503	1.0483	1.0443
${\psi }_r^{\alpha }$	0.1287	0.0413	0.0682	0.0683
${\psi }_r^{\beta }$	0.1297	0.0377	0.0620	0.0621
${\omega }_r$	23.818	11.507	14.465	13.498
${\mathcal{T}}_l$	0.5516	0.5519	0.5518	0.5518

and

Table 9. Max absolute error UKF after starting.

State	UKF
	Euler	Taylor	RK 2	RK 4
$i_s^{\alpha }$	1.4440	1.0356	1.0395	0.7169
$i_s^{\beta }$	1.4866	1.0514	1.0484	0.5494
${\psi }_r^{\alpha }$	0.1286	0.0406	0.0682	0.0344
${\psi }_r^{\beta }$	0.1296	0.0357	0.0620	0.0049
${\omega }_r$	23.812	11.784	14.459	3.116
${\mathcal{T}}_l$	0.5516	0.5519	0.5518	0.0867

illustrate the maximum absolute error after the startup of each state for each model using EKF and UKF algorithms, respectively. For the Euler, Taylor, and RK 2 models, EKF and UKF’s maximum absolute error are similar; in particular, UKF RK 4 is slightly less than EKF RK 4. Once the rotor speed reaches the steady state, the estimation shows better performance compared to the startup.

6. Conclusions

In this article, we formulated and compared different discretization techniques, namely, Euler. The results show that the Taylor method exhibits slightly lower RMSE compared to RK methods for stator currents and rotor flux linkages. This is mainly because, in the RK methods, it was assumed that the future values of the input were equal to the input at the current time instant. Since a direct start from the grid was simulated, the retention of the input does not fully reflect the continuous-time sinusoidal input behavior. However, when implementing these methods with an input provided by an inverter, input retention is fulfilled at every time instant. Therefore, a more accurate representation of the system is expected in practical setups.

Overall, Taylor exhibits the smallest α-β stator current and rotor flux linkage RMSE, followed by RK 4 and then RK 2. Conversely, the RK 4 model demonstrates the smallest rotor shaft speed error, followed by the RK 2 and Taylor models. The Taylor method resulted in a similar rotor shaft speed error to the Euler method under no-load conditions; however, when a load torque was applied, the Taylor method exhibited a performance similar to RK models and signals that resembled the continuous-time signals. Regarding the load torque, all models performed similarly. These behaviors persisted as $T_s$ varied. Also, it was demonstrated that the Euler method can only accurately represent the machine at very high sampling rates.

We also compared state estimation techniques, namely, EKF and UKF, utilizing the discrete-time models from the Euler, Taylor, and RK methods. EKF and UKF show similar RMSE; however, in terms of execution time, EKF required fewer computations. In most cases, as the complexity of the model increases, the execution time increases as well. Nonetheless, for the UKF algorithm, the RK 2 model is faster than the Taylor model. The execution time required for one iteration of the EKF and UKF utilizing the Taylor and RK methods is less than the sampling period. Therefore, these techniques are suitable for implementation in modern control techniques. In contrast, when utilizing the Euler method at high sampling rates, both filtering techniques result in an execution time per iteration that exceeds the sampling period. Finally, in terms of the maximum error, EKF and UKF are similar; therefore, both algorithms are similar in the maximum deviation from the actual state. Particularly, UKF RK 4 is slightly better than EKF RK 4.