On Modelling and State Estimation of DC Motors

Abstract

Direct current motors are widely used in a plethora of applications, ranging from industrial to modern electric (and intelligent) vehicle applications. Most recent operation methods of these motors involve drives that are designed based on an adequate knowledge of the motor dynamics and circulating currents. However, in spite of its simplicity, accurate discrete-time models are not always attainable when utilising the Euler method. Moreover, these inaccuracies may not be reduced when estimating the currents and rotor speed in sensorless direct current motors. In this paper, we analyse three discretisation methods, namely the Euler, second-order Taylor method and second-order Runge–Kutta method, applied to three common types of direct current motor: separately excited, series, and shunt. We also analyse the performance of two of the most simple Bayesian filtering methods, namely the Kalman filter and the extended Kalman filter. For the comparison of the models and the state estimation techniques, we performed several Monte Carlo simulations. Our simulations show that, in general, the Taylor and Runge–Kutta methods exhibit similar behaviours, whilst the Euler method results in less accurate models.

1. Introduction

Direct current (DC) motors are among the oldest technologies for converting electrical energy into mechanical energy. DC motors have been used for marine propulsion for over a century. Until the 1980s, conventional brushed DC machines were the primary choice for high-performance industrial applications requiring precise speed and torque control. This made them a natural fit for naval propulsion systems. However, brushed DC drives have largely been replaced by more efficient alternate current (AC) drive systems, such as vector control. Despite this, DC commutator motors are still utilised in specialised vessels such as icebreakers, submarines, tugboats, minesweepers, dredges, and research vessels. A notable modern application of DC machines is the hybrid diesel–electric and gas propulsion system of the Royal Navy’s Type 23 frigates, introduced in the late 1980s. This system consists of two 3-MW GEC DC separately excited motors, powered by six-pulse thyristor rectifiers for the armature and antiparallel rectifiers for the field circuit, enabling regenerative braking. Moreover, Siemens continues to produce DC propulsion systems, such as the SINAVY DC-Prop, used in the Type 209 submarines commissioned in 2019. This system includes two DC excitation motors mounted on a shared shaft, operating in parallel or series configurations depending on the speed requirements. Armature and field circuit DC choppers provide control, while auxiliary windings are employed in parallel configurations to balance the load. Moreover, four diesel generators support battery charging [1].

On the other hand, the simplicity, reliability, and broad speed range of DC motors [2,3,4] make them ideal for modern applications, such as industrial robotics, electric vehicles, electric cranes, and precision speed control systems [3,4,5,6,7]. Moreover, DC machine control schemes constitute the basis of AC machine control schemes.

In this context, accurate modelling of DC motors has become a critical issue for many applications, since control and drive system designs depend heavily on model precision. For instance, advanced control techniques like model predictive control are significantly influenced by the model’s accuracy.

The modern development and construction of electric motors and drives includes the compliance with electromagnetic compatibility standards; see, e.g., [8] and the references therein. This is due to the fact that control signals are generated using switching electronic components that emit electromagnetic radiation at a number of frequency ranges, see, e.g., [8,9,10,11]. These high frequency components can be modelled as additive non-Gaussian noise in the state equations of electric motors. However, the resulting stochastic model would have to be identified for different discretisation techniques, which is outside the scope of the paper.

In the literature, and to the best of our knowledge, the explicit discrete-time model of DC motors that arises from discretisation techniques is seldom provided. For instance, in [12,13,14] the discrete-time model is not provided. In [15,16,17,18,19], Runge–Kutta methods are used, but the explicit model is not provided. In [20], the Taylor method is used to obtain a discrete-time model of a DC motor. However, the model is simplified, and the resulting discrete-time model is not defined. Since discrete-time models are needed for driver and controller development and implementation on microcontrollers, field programmable gate arrays (FPGAs), or dedicated signal processing processors, our contribution is as follows:

(i)

We provide detailed description of discrete-time state–space models arising from the Euler, Taylor, and Runge–Kutta methods for the separately excited, shunt, and series DC motors, as a function of the electromechanical parameters of the motor and the sampling period.

(ii)

We analyse, via a numerical example, the accuracy of each discretisation technique as a function of the sampling period.

(iii)

We analyse, via a numerical example, the effect of each discretisation technique on the accuracy of Kalman-based (i.e., Kalman filter/smoother and extended Kalman filter/smoother) state estimators for real-world stochastic models.

2. Direct Current Motor Model [21,22]

The DC machine consists of a field winding, placed on the stator, which is responsible for generating the excitation flux, and an armature winding that produces a fixed field while the armature winding rotates, due to the introduction of a commutator circuit. Torque is developed through the interaction of both field and armature fluxes.

Depending on the connection and distribution of armature and field windings, DC machines can be categorised into shunt, series, and compound types. Each type exhibits distinct torque-speed characteristics. Initially, each type of machine was employed for specific applications, but this approach has evolved with the advancement in power electronics.

The shunt DC machine, particularly the separately excited type, has become widely used in applications requiring the precise control of both the field and armature currents. By independently controlling these currents, various torque–speed characteristics can be achieved, making the machine versatile for different applications. In contrast, for series and compound DC machines, the armature current also contributes to the excitation flux, along with the field winding. In a series machine, the field and armature currents are the same, thus becoming a very suitable machine for application where high torque ratings are required.

On the other hand, compound DC machines exhibit a combination of both the separately excited and the series machine main characteristics: in a cumulative compound machine, the flux generated by the armature current aligns with the flux from the field winding, providing high torque operation and speed regulation capability. In a differential compound machine, the flux produced by the armature current opposes the flux from the field winding, thus improving the compensation of the voltage drop due to the induced armature voltage (back-emf); for this reason, they have a better performance as DC generators when providing high current ratings. Because of the complexity in their winding design and manufacturing and the development of highly efficient torque and speed control schemes, compound DC machines are rarely used nowadays, compared to shunt, separately excited, and series machines. So this work will focus on these three DC machine configurations. For completeness of the presentation, we next include the continuous-time model of the DC shunt, separately excited, and series machines motors. Moreover, for simplicity of presentation, we consider that the motor load torque is proportional to the angular velocity, given by $K_L\omega$, where $K_L$ is a proportionality constant.

2.1. Separately Excited Motor

In this type of DC motor, both the field and armature windings are independent circuits, fed from independent DC sources $V_f$ and $V_a$, respectively, as shown in Figure 1. In the same equivalent circuit, the armature parameters are given by the armature resistance $R_a$ and the armature inductance $L_a$, while $i_a$ is the state variable, corresponding to the armature current. For the field circuit, $R_f$ and $L_f$ are the field resistance and inductance, respectively (field circuit parameters), and $i_f$ is the state variable corresponding to the field current. The induced armature voltage (back-emf) is given as $E_a=L_{af}{i}_f\omega$, where $L_{af}$ is the mutual inductance between the armature and the field coils, and $\omega$ is the angular velocity.

In a separately excited (SE) motor, before supplying power to the armature circuit, the field circuit should be providing nominal flux, e.g., the nominal field current has to be sustained. This is to avoid starting the DC motor with no load at field weakening, which can overspend the motor, damaging the armature winding and commutator brushes. Therefore, the field current is considered as constant when operating in the constant torque region which yields the following linear model:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{i}_a}{dt}}& ={\displaystyle \frac{1}{L_a}}\left(V_a-R_a{i}_a-L_{af}{i}_f\omega \right),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\omega }{dt}}& ={\displaystyle \frac{1}{J}}\left(L_{af}{i}_a{i}_f-K_L\omega \right),\hfill \end{array}$$

(2)

where J is the inertia of the motor. The developed SE motor torque, given by $L_{af}{i}_a{i}_f$, appears in the rotor velocity dynamics Equation (2). Notice that we have left the dependency of time implicitly.

2.2. Shunt DC Motor

In the shunt DC motor, the armature and the field circuit are connected in parallel and fed from the same DC source $V_L$, as shown in Figure 2, where $i_f$ and $i_a$ correspond to the state variables (field and armature currents, respectively), and it is possible to identify the line current (input current) $i_L=i_a+i_f$. This implies that, unlike in the separately excited motor, the field current is not constant, because it depends on the input voltage $V_L$. Therefore, the nonlinear continuous-time (CT) dynamic equations that characterise the shunt motor are the following:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{i}_a}{dt}}& ={\displaystyle \frac{1}{L_a}}\left(V_L-R_a{i}_a-L_{af}{i}_f\omega \right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{i}_f}{dt}}& ={\displaystyle \frac{1}{L_f}}\left(V_L-R_f{i}_f\right),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\omega }{dt}}& ={\displaystyle \frac{1}{J}}\left(L_{af}{i}_a{i}_f-K_L\omega \right).\hfill \end{array}$$

(5)

Notice that the nonlinear system arises from the product of the field current and the velocity in (3). Additionally, the developed torque in a series DC machine is given by $L_{af}{i}_a{i}_f$, which introduces a quadratic nonlinearity in (7) due to the product of the armature and the field currents in (5).

Figure 2. Equivalent circuit of a shunt DC motor.

Figure 2. Equivalent circuit of a shunt DC motor.

2.3. Series DC Motor

Series DC motors are characterised by the series connection of the field and armature circuits and fed from a single DC source, as shown in Figure 3. This implies that the armature current is equal to the field current, as $i_L=i_a=i_f$. Thus, the series motor is modelled by the following CT dynamic equations:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{i}_L}{dt}}& ={\displaystyle \frac{1}{L_L}}\left(V_L-R_L{i}_L-L_{af}{i}_L\omega \right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\omega }{dt}}& ={\displaystyle \frac{1}{J}}\left(L_{af}{i}_L^2-K_L\omega \right),\hfill \end{array}$$

(7)

where the series resistance is given by $R_L=R_a+R_f$, and the series inductance: $L_L=L_a+L_f$. Notice that this is also a nonlinear system, as it involves the product of the line current and the velocity in (6). Additionally, the developed torque in a series DC machine is given by $L_{af}{i}_L^2$, which introduces a quadratic nonlinearity in (7) due to the squared current term.

3. A Brief Review of the Euler, Second-Order Taylor, and Second-Order Runge–Kutta Discretisation Methods

Modern drive and control systems are based on the utilisation of microcontrollers or dedicated digital signal processing devices, such as microprocessors or field programmable gate arrays; see, e.g., [23,24,25,26,27,28]. This clearly implies that we will only have a set of samples of the currents available. Therefore, it is important to consider the effect of different discretisation methods on the correct representation of the motor.

3.1. Forward Euler Method Applied to DC Motors

The forward Euler method is based on the idea that, based on a regular sampling process with sampling period $T_s$, we can approximate the derivative of a function utilising the difference of the value of two consecutive samples (i.e., computing the slope of the function of interest using two values) [29,30]. That is, for a function $f\left(t\right)$ and two samples of it obtained at a sampling rate of $T_s$ samples per second at time instants $k{T}_s$ and $(k+1)T_s$, the derivative ${\displaystyle \frac{d}{dt}f\left(t\right)}$ can be computed as

$${\displaystyle \frac{d}{dt}}f\left(t\right)={\displaystyle \frac{f_{k+1}-f_k}{T_s}},$$

(8)

where $f_k$ and $f_{k+1}$ are the samples of $f\left(t\right)$ at time instants $k{T}_s$ and $(k+1)T_s$, respectively.

Remark 1. 

Notice that if $T_s\to 0$, we obtain the definition of the derivative of $f\left(t\right)$ at $t=k{T}_s$:

$${\displaystyle \frac{d}{dt}}f\left(t\right)=\underset{T_s\to 0}{lim}{\displaystyle \frac{f(t+T_s)+f\left(t\right)}{T_s}}.$$

3.1.1. Euler Method for the SE Motor

If we apply the Euler method in (8) to the SE motor equations in (1) and (2), we obtain the folowing discrete-time (DT) model:

$$\begin{array}{cc}\hfill i_{k+1}^{\left(a\right)}& =a_{11}^{\mathrm{E},\mathrm{SE}}{i}_k^{\left(a\right)}+a_{12}^{\mathrm{E},\mathrm{SE}}{\omega }_k+B^{\mathrm{E},\mathrm{SE}}Va,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\omega }_{k+1}& =a_{21}^{\mathrm{E},\mathrm{SE}}{i}_k^{\left(a\right)}+a_{22}^{\mathrm{E},\mathrm{SE}}{\omega }_k,\hfill \end{array}$$

(10)

where

$$\begin{array}{cc}\hfill a_{11}^{\mathrm{E},\mathrm{SE}}& =1-T_s{\displaystyle \frac{R_a}{L_a}},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill a_{12}^{\mathrm{E},\mathrm{SE}}& =-T_s{\displaystyle \frac{L_{af}{i}_f}{L_a}},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill a_{21}^{\mathrm{E},\mathrm{SE}}& =T_s{\displaystyle \frac{L_{af}{i}_f}{J}},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill a_{22}^{\mathrm{E},\mathrm{SE}}& =1-T_s{\displaystyle \frac{K_L}{J}},\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill B^{\mathrm{E},\mathrm{SE}}& ={\displaystyle \frac{T_s}{L_a}}.\hfill \end{array}$$

(15)

Since the armature voltage and the field current are constant, they do not depend on the time index k in (9), (12), and (13).

3.1.2. Euler Method for the Shunt Motor

The CT equations of the shunt motor in (3)–(5) exhibit a nonlinear behaviour that is preserved in the DT equation. This is because, when applying the Euler method, the products of the variables are maintained, resulting in

$$\begin{array}{cc}\hfill i_{k+1}^{\left(a\right)}& =a_{11}^{\mathrm{E},\mathrm{Sh}}{i}_k^{\left(a\right)}+a_{12}^{\mathrm{E},\mathrm{Sh}}{i}_k^{\left(f\right)}{\omega }_k+b_1^{\mathrm{E},\mathrm{Sh}}{V}_L,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill i_{k+1}^{\left(f\right)}& =a_{22}^{\mathrm{E},\mathrm{Sh}}{i}_k^{\left(f\right)}+b_2^{\mathrm{E},\mathrm{Sh}}{V}_L,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\omega }_{k+1}& =a_{31}^{\mathrm{E},\mathrm{Sh}}{i}_k^{\left(a\right)}{i}_k^{\left(f\right)}+a_{32}^{\mathrm{E},\mathrm{Sh}}{\omega }_k,\hfill \end{array}$$

(18)

where

$$\begin{array}{cc}\hfill a_{11}^{\mathrm{E},\mathrm{Sh}}& =1-{\displaystyle \frac{T_s{R}_a}{L_a}},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill a_{12}^{\mathrm{E},\mathrm{Sh}}& =-{\displaystyle \frac{T_s{L}_{af}}{L_a}},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill a_{21}^{\mathrm{E},\mathrm{Sh}}& =1-{\displaystyle \frac{T_s{R}_f}{L_f}},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill a_{31}^{\mathrm{E},\mathrm{Sh}}& ={\displaystyle \frac{T_s{L}_{af}}{J}},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill a_{32}^{\mathrm{E},\mathrm{Sh}}& =1-{\displaystyle \frac{T_s{K}_L}{J}},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill b_1^{\mathrm{E},\mathrm{Sh}}& ={\displaystyle \frac{T_s}{L_a}},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill b_2^{\mathrm{E},\mathrm{Sh}}& ={\displaystyle \frac{T_s}{L_f}}.\hfill \end{array}$$

(25)

3.1.3. Euler Method for the Series Motor

In this model, the CT nonlinearities are also preserved in the DT model when applying the Euler method, yielding

$$\begin{array}{cc}\hfill i_{k+1}^{\left(a\right)}& =a_{11}^{\mathrm{E},\mathrm{s}}{i}_k^{\left(a\right)}+a_{12}^{\mathrm{E},\mathrm{s}}{i}_k^{\left(a\right)}{\omega }_k+b^{\mathrm{E},\mathrm{s}}{V}_L,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\omega }_{k+1}& =a_{21}^{\mathrm{E},\mathrm{s}}{\left(i_k^{\left(a\right)}\right)}^2+a_{22}^{\mathrm{E},\mathrm{s}}{\omega }_k,\hfill \end{array}$$

(27)

where

$$\begin{array}{cc}\hfill a_{11}^{\mathrm{E},\mathrm{s}}& =1-T_s{\displaystyle \frac{R_L}{L_L}},\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill a_{12}^{\mathrm{E},\mathrm{s}}& =-T_s{\displaystyle \frac{L_{af}}{L_L}},\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill a_{21}^{\mathrm{E},\mathrm{s}}& =T_s{\displaystyle \frac{L_{af}}{J}},\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill a_{22}^{\mathrm{E},\mathrm{s}}& =1-T_s{\displaystyle \frac{K_L}{J}},\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill b^{\mathrm{E},\mathrm{s}}& ={\displaystyle \frac{T_s}{L_L}}.\hfill \end{array}$$

(32)

3.2. Second-Order Taylor Method Applied to DC Motors

It is well known that any infinitely differentiable function $f\left(t\right)$ can be expressed in terms of its Taylor series expansion about the point $t=t_0$ as

$$f\left(t\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{f^{\left(n\right)}\left(t_0\right)}{n!}}{(t-t_0)}^n,$$

(33)

where $f^{\left(n\right)}\left(t_0\right)$ is the $n$-th derivative of $f\left(t\right)$ evaluated at $t=t_0$.

If we only consider two terms of the series about the point $t=k{T}_s$ and we evaluate it at $t=(k+1)T_s$, we obtain the forward Euler method in (8). However, if we now consider three terms (up to the second derivative of $f\left(t\right)$), we obtain the second-order Taylor method as follows:

$$f_{k+1}=f_k+T_s{f}^{\left(1\right)}\left(k{T}_s\right)+{\displaystyle \frac{T_s^2}{2}}{f}^{\left(2\right)}\left(k{T}_s\right).$$

(34)

Thus, if we want to obtain the DT model of a system described by ${\displaystyle \frac{d}{dt}f\left(t\right)=g\left(t\right)}$, the first derivative is given by the system itself, whilst $f^{\left(2\right)}\left(t\right)$ is obtained as ${\displaystyle f^{\left(2\right)}\left(t\right)=\frac{d}{dt}g\left(t\right)}$.

3.2.1. Second-Order Taylor Method for the SE Motor

The SE motor model in (1) and (2) corresponds to a second-order CT system, and the order of the system is preserved when applying the second-order Taylor method. The main difference from (9) and (10) are the coefficients of the model:

$$\begin{array}{cc}\hfill i_{k+1}^{\left(a\right)}& =a_{11}^{\mathrm{T},\mathrm{SE}}{i}_k^{\left(a\right)}+a_{12}^{\mathrm{T},\mathrm{SE}}{\omega }_k+b_1^{\mathrm{T},\mathrm{SE}}{V}_a,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\omega }_{k+1}& =a_{21}^{\mathrm{T},\mathrm{SE}}{i}_k^{\left(a\right)}+a_{22}^{\mathrm{T},\mathrm{SE}}{\omega }_k+b_2^{\mathrm{T},\mathrm{SE}}{V}_a,\hfill \end{array}$$

(36)

where

$$\begin{array}{cc}\hfill a_{11}^{\mathrm{T},\mathrm{SE}}& =1+{\displaystyle \frac{T_s}{2}}\left(-2{\displaystyle \frac{R_a}{L_a}}+T_s\left({\displaystyle \frac{R_a^2}{L_a^2}}-{\displaystyle \frac{L_{af}^2{i}_f^2}{L_{a}J}}\right)\right),\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill a_{12}^{\mathrm{T},\mathrm{SE}}& ={\displaystyle \frac{T_s}{2}}\left(-{\displaystyle \frac{2L_{af}{i}_f}{L_a}}+T_s\left({\displaystyle \frac{R_a{L}_{af}{i}_f}{L_a^2}}+{\displaystyle \frac{L_{af}{i}_f{K}_L}{L_{a}J}}\right)\right),\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill a_{21}^{\mathrm{T},\mathrm{SE}}& ={\displaystyle \frac{T_s}{2}}\left({\displaystyle \frac{2L_{af}{i}_f}{J}}-T_s\left({\displaystyle \frac{R_a{L}_{af}{i}_f}{L_{a}J}}+{\displaystyle \frac{L_{af}{i}_f{K}_L}{J^2}}\right)\right),\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill a_{22}^{\mathrm{T},\mathrm{SE}}& =1+{\displaystyle \frac{T_s}{2}}\left(-{\displaystyle \frac{2K_L}{J}}+T_s\left({\displaystyle \frac{K_L^2}{J^2}}-{\displaystyle \frac{L_{af}^2{i}_f^2}{L_{a}J}}\right)\right),\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill b_1^{\mathrm{T},\mathrm{SE}}& ={\displaystyle \frac{T_s}{L_a}}-{\displaystyle \frac{T_s^2{R}_a}{2L_a^2}},\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill b_2^{\mathrm{T},\mathrm{SE}}& ={\displaystyle \frac{T_s^2{L}_{af}{i}_f}{2L_{a}J}}.\hfill \end{array}$$

(42)

3.2.2. Second-Order Taylor Method for the Shunt Motor

For this motor, the DT model we obtain from applying (34) to (3)–(5) includes new nonlinearities when compared to (3)–(5), as can be seen in the following equations:

$$\begin{array}{cc}\hfill i_{k+1}^{\left(a\right)}& =a_{11}^{\mathrm{T},\mathrm{Sh}}{i}_k^{\left(a\right)}+a_{12}^{\mathrm{T},\mathrm{Sh}}{i}_k^{\left(f\right)}{\omega }_k+a_{13}^{\mathrm{T},\mathrm{Sh}}{i}_k^{\left(a\right)}{\left(i_k^{\left(f\right)}\right)}^2+a_{14}^{\mathrm{T},\mathrm{Sh}}{V}_L{\omega }_k+b_1^{\mathrm{T},\mathrm{Sh}}{V}_L,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill i_{k+1}^{\left(f\right)}& =a_{22}^{\mathrm{T},\mathrm{Sh}}{i}_k^{\left(f\right)}+b_2^{\mathrm{T},\mathrm{Sh}}{V}_L,\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill {\omega }_{k+1}& =a_{31}^{\mathrm{T},\mathrm{Sh}}{i}_k^{\left(a\right)}{i}_k^{\left(f\right)}+a_{32}^{\mathrm{T},\mathrm{Sh}}{\left(i_k^{\left(f\right)}\right)}^2{\omega }_k+a_{33}^{\mathrm{T},\mathrm{Sh}}{\omega }_k+a_{34}^{\mathrm{T},\mathrm{Sh}}{i}_k^{\left(a\right)}{V}_L+a_{35}^{\mathrm{T},\mathrm{Sh}}{i}_k^{\left(f\right)}{V}_L,\hfill \end{array}$$

(45)

where

$$\begin{array}{cc}\hfill a_{11}^{\mathrm{T},\mathrm{Sh}}& =1-{\displaystyle \frac{T_s{R}_a}{L_a}}+{\displaystyle \frac{T_s^2{R}_a^2}{2L_a^2}},\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill a_{12}^{\mathrm{T},\mathrm{Sh}}& =-{\displaystyle \frac{T_s{L}_{af}}{L_a}}+{\displaystyle \frac{T_s^2}{2}}\left({\displaystyle \frac{R_a{L}_{af}}{L_a^2}}+{\displaystyle \frac{R_f{L}_{af}}{L_a{L}_f}}+{\displaystyle \frac{L_{af}{K}_L}{L_{a}J}}\right),\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill a_{13}^{\mathrm{T},\mathrm{Sh}}& =-{\displaystyle \frac{T_s^2{L}_{af}^2}{2L_{a}J}},\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill a_{14}^{\mathrm{T},\mathrm{Sh}}& =-{\displaystyle \frac{T_s^2{L}_{af}}{2L_a{L}_f}},\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill a_{22}^{\mathrm{T},\mathrm{Sh}}& =1-{\displaystyle \frac{T_s{R}_f}{L_f}}+{\displaystyle \frac{T_s^2{R}_f^2}{2L_f^2}},\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill a_{31}^{\mathrm{T},\mathrm{Sh}}& ={\displaystyle \frac{T_s{L}_{af}}{J}}-{\displaystyle \frac{T_s^2}{2}}\left({\displaystyle \frac{R_a{L}_{af}}{L_{a}J}}+{\displaystyle \frac{R_f{L}_{af}}{L_{f}J}}+{\displaystyle \frac{L_{af}{K}_L}{J^2}}\right),\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill a_{32}^{\mathrm{T},\mathrm{Sh}}& =-{\displaystyle \frac{T_s^2{L}_{af}^2}{2L_{a}J}},\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill a_{33}^{\mathrm{T},\mathrm{Sh}}& =1-{\displaystyle \frac{T_s{K}_L}{J}}+{\displaystyle \frac{T_s^2{K}_L^2}{2J^2}},\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill a_{34}^{\mathrm{T},\mathrm{Sh}}& ={\displaystyle \frac{T_s^2{L}_{af}}{2L_{f}J}},\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill a_{35}^{\mathrm{T},\mathrm{Sh}}& ={\displaystyle \frac{T_s^2{L}_{af}}{2L_{a}J}},\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill b_1^{\mathrm{T},\mathrm{Sh}}& ={\displaystyle \frac{T_s}{L_a}}-{\displaystyle \frac{T_s^2{R}_a}{2L_a^2}},\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill b_2^{\mathrm{T},\mathrm{Sh}}& ={\displaystyle \frac{T_s}{L_f}}-{\displaystyle \frac{T_s^2{R}_f}{2L_f^2}}.\hfill \end{array}$$

(57)

3.2.3. Second-Order Taylor Method for the Series Motor

Similar to the shunt motor, applying the second-order Taylor method results in additional nonlinearities for the series motor when compared against the CT model as shown in the following equations:

$$\begin{array}{cc}\hfill i_{k+1}^{\left(a\right)}& =a_{11}^{\mathrm{T},\mathrm{s}}{i}_k^{\left(a\right)}+a_{12}^{\mathrm{T},\mathrm{s}}{i}_k^{\left(a\right)}{\omega }_k+a_{13}^{\mathrm{T},\mathrm{s}}{i}_k^{\left(a\right)}{\omega }_k^2+a_{14}^{\mathrm{T},\mathrm{s}}{\left(i_k^{\left(a\right)}\right)}^3+a_{15}^{\mathrm{T},\mathrm{s}}{\omega }_k{V}_L+b^{\mathrm{T},\mathrm{s}}{V}_L,\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill {\omega }_{k+1}& =a_{21}^{\mathrm{T},\mathrm{s}}{\left(i_k^{\left(a\right)}\right)}^2+a_{22}^{\mathrm{T},\mathrm{s}}{\omega }_k+a_{23}^{\mathrm{T},\mathrm{s}}{\left(i_k^{\left(a\right)}\right)}^2{\omega }_k+a_{24}^{\mathrm{T},\mathrm{s}}{i}_k^{\left(a\right)}{V}_L,\hfill \end{array}$$

(59)

where

$$\begin{array}{cc}\hfill a_{11}^{\mathrm{T},\mathrm{s}}& =1-T_s{\displaystyle \frac{R_L}{L_L}}+{\displaystyle \frac{T_s^2{R}_L^2}{2L_L^2}},\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill a_{12}^{\mathrm{T},\mathrm{s}}& =-T_s{\displaystyle \frac{L_{af}}{L_L}}+{\displaystyle \frac{T_s^2}{2}}\left({\displaystyle \frac{2R_L{L}_{af}}{L_L^2}}+{\displaystyle \frac{L_{af}{K}_L}{L_{L}J}}\right),\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill a_{13}^{\mathrm{T},\mathrm{s}}& ={\displaystyle \frac{T_s^2{L}_{af}^2}{2L_L^2}},\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill a_{14}^{\mathrm{T},\mathrm{s}}& =-{\displaystyle \frac{T_s^2{L}_{af}^2}{2L_{L}J}},\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill a_{15}^{\mathrm{T},\mathrm{s}}& =-{\displaystyle \frac{T_s^2{L}_{af}}{2L_L^2}},\hfill \end{array}$$

(64)

$$\begin{array}{cc}\hfill a_{21}^{\mathrm{T},\mathrm{s}}& =T_s{\displaystyle \frac{L_{af}}{J}}-{\displaystyle \frac{T_s^2}{2}}\left({\displaystyle \frac{2R_L{L}_{af}}{L_{L}J}}+{\displaystyle \frac{L_{af}{K}_L}{J^2}}\right),\hfill \end{array}$$

(65)

$$\begin{array}{cc}\hfill a_{22}^{\mathrm{T},\mathrm{s}}& =1-T_s{\displaystyle \frac{K_L}{J}}+{\displaystyle \frac{T_s^2{K}_L^2}{2J^2}},\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill a_{23}^{\mathrm{T},\mathrm{s}}& =-{\displaystyle \frac{T_s^2{L}_{af}^2}{L_{L}J}},\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill a_{24}^{\mathrm{T},\mathrm{s}}& ={\displaystyle \frac{T_s^2{L}_{af}}{L_{L}J}},\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill b^{\mathrm{T},\mathrm{s}}& ={\displaystyle \frac{T_s}{L_L}}-{\displaystyle \frac{T_s^2{R}_L}{2L_L^2}}.\hfill \end{array}$$

(69)

3.3. Second-Order Runge–Kutta Method Applied to DC Motors

The idea behind the second-order Runge–Kutta (RK) method is to approximate the second derivative in (34) using sampling subintervals instead of computing the derivative. This usually produces a similar accuracy to the Taylor method.

For the system defined by ${\displaystyle \frac{d}{dt}f\left(t\right)=g\left(t\right)}$, the RK method can be defined by the following expression:

$$f_{k+1}=f_k+T_s\varphi (f_k,k{T}_s),$$

(70)

where $\varphi (f_k,k{T}_s)$ is an increment function that extrapolates $f_{k+1}$ from $f_k$. When $g\left(t\right)$ does not depend explicitly on t, the increment function for the second-order RK method is given by

$$\varphi (f_k,k{T}_s)={\alpha }_1{\gamma }_1+{\alpha }_2{\gamma }_2,$$

(71)

where ${\gamma }_1=g\left(f_k\right)$, ${\gamma }_2=g(f_k+T_s{\gamma }_1)$, ${\alpha }_1=0.5$, and ${\alpha }_2=0.5$.

3.3.1. Second-Order Runge–Kutta Method for the SE Motor

When applying the second-order RK method, the DT model of the SE motor preserves the order, just as the second-order Taylor method. Hence, we obtain

$$\begin{array}{cc}\hfill i_{k+1}^{\left(a\right)}& =a_{11}^{\mathrm{RK},\mathrm{SE}}{i}_k^{\left(a\right)}+a_{12}^{\mathrm{RK},\mathrm{SE}}{\omega }_k+b_1^{\mathrm{RK},\mathrm{SE}}Va,\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill {\omega }_{k+1}& =a_{21}^{\mathrm{RK},\mathrm{SE}}{i}_k^{\left(a\right)}+a_{22}^{\mathrm{RK},\mathrm{SE}}{\omega }_k+b_2^{\mathrm{RK},\mathrm{SE}}{V}_a,\hfill \end{array}$$

(73)

where

$$\begin{array}{cc}\hfill a_{11}^{\mathrm{RK},\mathrm{SE}}& =1+{\displaystyle \frac{T_s}{2}}\left(-2{\displaystyle \frac{R_a}{L_a}}+T_s\left({\displaystyle \frac{R_a^2}{L_a^2}}-{\displaystyle \frac{L_{af}^2{i}_f^2}{L_{a}J}}\right)\right),\hfill \end{array}$$

(74)

$$\begin{array}{cc}\hfill a_{12}^{\mathrm{RK},\mathrm{SE}}& ={\displaystyle \frac{T_s}{2}}\left(-{\displaystyle \frac{2L_{af}{i}_f}{L_a}}+T_s\left({\displaystyle \frac{R_a{L}_{af}{i}_f}{L_a^2}}+{\displaystyle \frac{L_{af}{i}_f{K}_L}{L_{a}J}}\right)\right),\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill a_{21}^{\mathrm{RK},\mathrm{SE}}& ={\displaystyle \frac{T_s}{2}}\left({\displaystyle \frac{2L_{af}{i}_f}{J}}-T_s\left({\displaystyle \frac{R_a{L}_{af}{i}_f}{L_{a}J}}+{\displaystyle \frac{L_{af}{i}_f{K}_L}{J^2}}\right)\right),\hfill \end{array}$$

(76)

$$\begin{array}{cc}\hfill a_{22}^{\mathrm{RK},\mathrm{SE}}& =1+{\displaystyle \frac{T_s}{2}}\left(-{\displaystyle \frac{2K_L}{J}}+T_s\left({\displaystyle \frac{K_L^2}{J^2}}-{\displaystyle \frac{L_{af}^2{i}_f^2}{L_{a}J}}\right)\right),\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill b_1^{\mathrm{RK},\mathrm{SE}}& ={\displaystyle \frac{T_s}{L_a}}-{\displaystyle \frac{T_s^2{R}_a}{2L_a^2}},\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill b_2^{\mathrm{RK},\mathrm{SE}}& ={\displaystyle \frac{T_s^2{L}_{af}{i}_f}{2L_{a}J}}.\hfill \end{array}$$

(79)

Notice that in this case, the second-order RK method and second-order Taylor method yield the same model. That is, the system equations in (72) and (73) and the parameters in (74)–(79) are exactly the same as (35) and (36) and (37)–(42), respectively.

3.3.2. Second-Order Runge–Kutta Method for the Shunt Motor

In this case, the second-order RK method results in the following DT model:

$$\begin{gathered} \begin{array}{cc}\hfill i_{k+1}^{\left(a\right)}=& a_{11}^{\mathrm{RK},\mathrm{Sh}}{i}_k^{\left(a\right)}+a_{12}^{\mathrm{RK},\mathrm{Sh}}{i}_k^{\left(f\right)}{\omega }_k+a_{13}^{\mathrm{RK},\mathrm{Sh}}{i}_k^{\left(a\right)}{\left(i_k^{\left(f\right)}\right)}^2+a_{14}^{\mathrm{RK},\mathrm{Sh}}{i}_k^{\left(a\right)}{i}_k^{\left(f\right)}{V}_L+a_{15}^{\mathrm{RK},\mathrm{Sh}}{V}_L{\omega }_k\hfill \end{array} \\ \begin{array}{cc}\hfill & +b_1^{\mathrm{RK},\mathrm{Sh}}{V}_L,\hfill \end{array} \end{gathered}$$

(80)

$$\begin{array}{cc}\hfill i_{k+1}^{\left(f\right)}=& a_{22}^{\mathrm{RK},\mathrm{Sh}}{i}_k^{\left(f\right)}+b_2^{\mathrm{RK},\mathrm{Sh}}{V}_L,\hfill \end{array}$$

(81)

$$\begin{gathered} \begin{array}{cc}{\omega }_{k+1}=\hfill & a_{31}^{\mathrm{RK},\mathrm{Sh}}{i}_k^{\left(a\right)}{i}_k^{\left(f\right)}+a_{32}^{\mathrm{RK},\mathrm{Sh}}{\left(i_k^{\left(f\right)}\right)}^2{\omega }_k+a_{33}^{\mathrm{RK},\mathrm{Sh}}{\omega }_k+a_{34}^{\mathrm{RK},\mathrm{Sh}}{i}_k^{\left(f\right)}{\omega }_k{V}_L+a_{35}^{\mathrm{RK},\mathrm{Sh}}{i}_k^{\left(a\right)}{V}_L\hfill \end{array} \\ \begin{array}{cc}\hfill & \hfill +a_{36}^{\mathrm{RK},\mathrm{Sh}}{i}_k^{\left(f\right)}{V}_L+b_3^{\mathrm{RK},\mathrm{Sh}}{V}_L^2,\end{array} \end{gathered}$$

(82)

where

$$\begin{array}{cc}\hfill a_{11}^{\mathrm{RK},\mathrm{Sh}}& =1+{\displaystyle \frac{T_s}{2}}\left({\displaystyle \frac{T_s{R}_a^2}{L_a^2}}-{\displaystyle \frac{2R_a}{L_a}}\right),\hfill \end{array}$$

(83)

$$\begin{array}{cc}\hfill a_{12}^{\mathrm{RK},\mathrm{Sh}}& ={\displaystyle \frac{T_s}{2}}\left({\displaystyle \frac{T_s{L}_{af}{R}_a}{L_a^2}}+{\displaystyle \frac{T_s{L}_{af}{K}_L}{L_{a}J}}+{\displaystyle \frac{T_s{L}_{af}{R}_f}{L_a{L}_f}}-{\displaystyle \frac{T_s^2{L}_{af}{R}_f{K}_L}{L_a{L}_{f}J}}-{\displaystyle \frac{2L_{af}}{L_a}}\right),\hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill a_{13}^{\mathrm{RK},\mathrm{Sh}}& ={\displaystyle \frac{T_s}{2}}\left({\displaystyle \frac{T_s^2{L}_{af}^2{R}_f}{L_a{L}_{f}J}}-{\displaystyle \frac{T_s{L}_{af}^2}{L_{a}J}}\right),\hfill \end{array}$$

(85)

$$\begin{array}{cc}\hfill a_{14}^{\mathrm{RK},\mathrm{Sh}}& =-{\displaystyle \frac{T_s^3{L}_{af}^2}{2L_a{L}_{f}J}},\hfill \end{array}$$

(86)

$$\begin{array}{cc}\hfill a_{15}^{\mathrm{RK},\mathrm{Sh}}& ={\displaystyle \frac{T_s}{2}}\left({\displaystyle \frac{T_s^2{L}_{af}{K}_L}{L_a{L}_{f}J}}-{\displaystyle \frac{T_s{L}_{af}}{L_a{L}_f}}\right),\hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill a_{22}^{\mathrm{RK},\mathrm{Sh}}& =1+{\displaystyle \frac{T_s}{2}}\left({\displaystyle \frac{T_s{R}_f^2}{L_f^2}}-{\displaystyle \frac{2R_f}{L_f}}\right),\hfill \end{array}$$

(88)

$$\begin{array}{cc}\hfill a_{31}^{\mathrm{RK},\mathrm{Sh}}& ={\displaystyle \frac{T_s{L}_{af}}{J}}\left(1-{\displaystyle \frac{T_s{R}_a}{2L_a}}-{\displaystyle \frac{T_s{R}_f}{2L_f}}+{\displaystyle \frac{T_s^2{R}_a{R}_f}{2L_a{L}_f}}-{\displaystyle \frac{T_s{K}_L}{2J}}\right),\hfill \end{array}$$

(89)

$$\begin{array}{cc}\hfill a_{32}^{\mathrm{RK},\mathrm{Sh}}& ={\displaystyle \frac{T_s{L}_{af}}{2J}}\left({\displaystyle \frac{T_s^2{L}_{af}{R}_f}{L_a{L}_f}}-{\displaystyle \frac{T_s{L}_{af}}{L_a}}\right),\hfill \end{array}$$

(90)

$$\begin{array}{cc}\hfill a_{33}^{\mathrm{RK},\mathrm{Sh}}& =1+{\displaystyle \frac{T_s}{2}}\left({\displaystyle \frac{T_s{K}_L^2}{J^2}}-{\displaystyle \frac{2K_L}{J}}\right),\hfill \end{array}$$

(91)

$$\begin{array}{cc}\hfill a_{34}^{\mathrm{RK},\mathrm{Sh}}& =-{\displaystyle \frac{T_s^3{L}_{af}^2}{2L_a{L}_{f}J}},\hfill \end{array}$$

(92)

$$\begin{array}{cc}\hfill a_{35}^{\mathrm{RK},\mathrm{Sh}}& ={\displaystyle \frac{T_s^2{L}_{af}}{2L_{f}J}}\left(1-{\displaystyle \frac{T_s{R}_a}{L_a}}\right),\hfill \end{array}$$

(93)

$$\begin{array}{cc}\hfill a_{36}^{\mathrm{RK},\mathrm{Sh}}& ={\displaystyle \frac{T_s^2{L}_{af}}{2L_{a}J}}\left(1-{\displaystyle \frac{T_s{R}_f}{L_f}}\right),\hfill \end{array}$$

(94)

$$\begin{array}{cc}\hfill b_1^{\mathrm{RK},\mathrm{Sh}}& ={\displaystyle \frac{T_s}{L_a}}\left(1-{\displaystyle \frac{T_s{R}_a}{2L_a}}\right),\hfill \end{array}$$

(95)

$$\begin{array}{cc}\hfill b_2^{\mathrm{RK},\mathrm{Sh}}& ={\displaystyle \frac{T_s}{L_f}}\left(1-{\displaystyle \frac{T_s{R}_f}{2L_f}}\right),\hfill \end{array}$$

(96)

$$\begin{array}{cc}\hfill b_3^{\mathrm{RK},\mathrm{Sh}}& ={\displaystyle \frac{T_s^3{L}_{af}}{2L_a{L}_{f}J}}.\hfill \end{array}$$

(97)

3.3.3. Second-Order Runge–Kutta Method for the Series Motor

The application of the second-order RK method results in a DT model with nonlinearities involving the second and third powers of the armature current, similar to the model in (58) and (59). In addition, the model also includes the square of the applied voltage:

$$\begin{gathered} \begin{array}{cc}\hfill i_{k+1}^{\left(a\right)}=& a_{11}^{\mathrm{RK},\mathrm{s}}{i}_k^{\left(a\right)}+a_{12}^{\mathrm{RK},\mathrm{s}}{i}_k^{\left(a\right)}{\omega }_k+a_{13}^{\mathrm{RK},\mathrm{s}}{i}_k^{\left(a\right)}{\omega }_k^2+a_{14}^{\mathrm{RK},\mathrm{s}}{\left(i_k^{\left(a\right)}\right)}^3{\omega }_k+a_{15}^{\mathrm{RK},\mathrm{s}}{\left(i_k^{\left(a\right)}\right)}^3\hfill \end{array} \\ \begin{array}{cc}\hfill & \hfill +a_{16}^{\mathrm{RK},\mathrm{s}}{\left(i_k^{\left(a\right)}\right)}^2{V}_L+a_{17}^{\mathrm{RK},\mathrm{s}}{\omega }_k{V}_L+b_1^{\mathrm{RK},\mathrm{s}}{V}_L,\end{array} \end{gathered}$$

(98)

$$\begin{gathered} \begin{array}{cc}\hfill {\omega }_{k+1}=& a_{21}^{\mathrm{RK},\mathrm{s}}{\left(i_k^{\left(a\right)}\right)}^2+a_{22}^{\mathrm{RK},\mathrm{s}}{\omega }_k+a_{23}^{\mathrm{RK},\mathrm{s}}{\left(i_k^{\left(a\right)}\right)}^2{\omega }_k+a_{24}^{\mathrm{RK},\mathrm{s}}{\left(i_k^{\left(a\right)}\right)}^2{\omega }_k^2+a_{25}^{\mathrm{RK},\mathrm{s}}{i}_k^{\left(a\right)}{V}_L\hfill \end{array} \\ \begin{array}{cc}\hfill & \hfill +a_{26}^{\mathrm{RK},\mathrm{s}}{i}_k^{\left(a\right)}{\omega }_k{V}_L+b_2^{\mathrm{RK},\mathrm{s}}{V}_L^2,\end{array} \end{gathered}$$

(99)

where

$$\begin{array}{cc}\hfill a_{11}^{\mathrm{RK},\mathrm{s}}& =1-T_s{\displaystyle \frac{R_L}{L_L}}+{\displaystyle \frac{T_s^2{R}_L^2}{2L_L^2}},\hfill \end{array}$$

(100)

$$\begin{array}{cc}\hfill a_{12}^{\mathrm{RK},\mathrm{s}}& =-{\displaystyle \frac{T_s{L}_{af}}{L_L}}+{\displaystyle \frac{T_s^2{L}_{af}}{2L_L}}\left({\displaystyle \frac{2R_L}{L_L}}+{\displaystyle \frac{K_L}{J}}-{\displaystyle \frac{T_s{R}_L{K}_L}{L_{L}J}}\right),\hfill \end{array}$$

(101)

$$\begin{array}{cc}\hfill a_{13}^{\mathrm{RK},\mathrm{s}}& ={\displaystyle \frac{T_s^2{L}_{af}^2}{2L_L^2}}\left(1-{\displaystyle \frac{T_s{K}_L}{J}}\right),\hfill \end{array}$$

(102)

$$\begin{array}{cc}\hfill a_{14}^{\mathrm{RK},\mathrm{s}}& ={\displaystyle \frac{T_s^3{L}_{af}^3}{2L_L^{2}J}},\hfill \end{array}$$

(103)

$$\begin{array}{cc}\hfill a_{15}^{\mathrm{RK},\mathrm{s}}& ={\displaystyle \frac{T_s^2{L}_{af}^2}{2L_{L}J}}\left({\displaystyle \frac{T_s{R}_L}{L_L}}-1\right),\hfill \end{array}$$

(104)

$$\begin{array}{cc}\hfill a_{16}^{\mathrm{RK},\mathrm{s}}& =-{\displaystyle \frac{T_s^3{L}_{af}^2}{2L_L^{2}J}},\hfill \end{array}$$

(105)

$$\begin{array}{cc}\hfill a_{17}^{\mathrm{RK},\mathrm{s}}& ={\displaystyle \frac{T_s^2{L}_{af}}{2L_L^2}}\left({\displaystyle \frac{T_s{K}_L}{J}}-1\right),\hfill \end{array}$$

(106)

$$\begin{array}{cc}\hfill a_{21}^{\mathrm{RK},\mathrm{s}}& ={\displaystyle \frac{T_s{L}_{af}}{J}}\left(1+{\displaystyle \frac{T_s}{2}}\left({\displaystyle \frac{T_s{R}_L^2}{L_L^2}}-{\displaystyle \frac{2R_L}{L_L}}-{\displaystyle \frac{K_L}{J}}\right)\right),\hfill \end{array}$$

(107)

$$\begin{array}{cc}\hfill a_{22}^{\mathrm{RK},\mathrm{s}}& =1-T_s{\displaystyle \frac{K_L}{J}}+{\displaystyle \frac{T_s^2{K}_L^2}{2J^2}},\hfill \end{array}$$

(108)

$$\begin{array}{cc}\hfill a_{23}^{\mathrm{RK},\mathrm{s}}& ={\displaystyle \frac{T_s^2{L}_{af}^2}{L_{L}J}}\left({\displaystyle \frac{T_s{R}_L}{L_L}}-1\right),\hfill \end{array}$$

(109)

$$\begin{array}{cc}\hfill a_{24}^{\mathrm{RK},\mathrm{s}}& ={\displaystyle \frac{T_s^3{L}_{af}^3}{2L_L^{2}J}},\hfill \end{array}$$

(110)

$$\begin{array}{cc}\hfill a_{25}^{\mathrm{RK},\mathrm{s}}& ={\displaystyle \frac{T_s^2{L}_{af}}{L_{L}J}}\left(1-{\displaystyle \frac{T_s{R}_L}{L_L}}\right),\hfill \end{array}$$

(111)

$$\begin{array}{cc}\hfill a_{26}^{\mathrm{RK},\mathrm{s}}& =-{\displaystyle \frac{T_s^3{L}_{af}^2}{L_L^{2}J}},\hfill \end{array}$$

(112)

$$\begin{array}{cc}\hfill b_1^{\mathrm{RK},\mathrm{s}}& ={\displaystyle \frac{T_s}{L_L}}\left(1-{\displaystyle \frac{T_s{R}_L}{2L_L}}\right),\hfill \end{array}$$

(113)

$$\begin{array}{cc}\hfill b_2^{\mathrm{RK},\mathrm{s}}& ={\displaystyle \frac{T_s^3{L}_{af}}{2L_L^{2}J}}.\hfill \end{array}$$

(114)

3.4. Performance Comparison

In this section, we show a simulation analysis in order to understand the behaviour of the three discretisation techniques applied to the three types of DC motors. To this end, we considered three specifications with the parameters shown in

Table 1. Simulation parameters for the three types of DC motors considered in the analysis.

	Separately Excited	Shunt	Series
Armature voltage ($V_a$)	170 V	–	–
Field voltage ($V_f$)	210 V	–	–
Line voltage ($V_L$)	–	170 V	230 V
Nominal armature current	$12.4$ A	17 A	$12.5$ A
Nominal field current	$1.1$ A	$0.9$ A	$12.5$ A
Nominal velocity	1000 rpm	1100 rpm	2000 rpm
Nominal electric torque	$15.5$ N m	$17.8$ N m	$10.7$ N m
Armature inductance ($L_a$)	$17.8$ mH	$17.8$ mH	$0.12$ mH
Field inductance ($L_f$)	1 µH	10 H	30 mH
Mutual inductance ($L_{af}$)	$1.136$ H	$1.1634$ H	$68.5$ mH
Armature resistance ($R_a$)	$3.1533$$\Omega$	$2.9051$$\Omega$	$3.3576$$\Omega$
Field resistance ($R_f$)	$190.909$$\Omega$	$188.889$$\Omega$	$0.7$$\Omega$
Inertia (J)	$0.0142$ kg m2	$0.0142$ kg m2	$0.015$ kg m2
($K_L$)	$0.148$ N m s	$0.1545$ N m s	$0.0511$ N m s

. In addition, we considered a sampling period $T_s=2$ ms for the three discretisation methods considered in the paper. In order to fully understand the behaviour of these methods when applied to the three types of DC motors in this paper, we considered as the CT model (i.e., ground truth) the one obtained using an eighth-order RK method [31].

The parameters for the separately excited and shunt connected motors in Table 1 were obtained from the brochure in [32], whilst the parameters from the series motor were obtained from [33] (Ch. 2).

3.4.1. Separately Excited Motor

For the SE motor, we obtained the CT behaviour shown in Figure 4. Notice that the motor is started from rest.

When applying the three discretisation methods considered in this paper, we obtained similar results in terms of the instantaneous values of the armature current and the rotational speed. However, a closer inspection shows that the Euler method performed slightly worse than the Taylor and RK methods. This is shown in Figure 5, where the difference between the CT signals and the DT signals (at every sample) are shown. For the Euler method, the error in the armature current has an undershoot and an overshoot (notice that when the discrete-time signals are smaller than the continuous-time signals at a given time, the error signals will be positive. Conversely, if the discrete-time signals are larger than the continuous-time signals at a given time, the error signals will be negative), tending to zero after $0.1$ s (which is equivalent to 50 samples), whilst the error in the angular velocity exhibits first an overshoot and then an undershoot, tending to zero after $0.12$ s (which is equivalent to 60 samples). In contrast, the error in the armature current for both the Taylor and RK methods (recall that in this case the Taylor and RK methods yield the same DT model) exhibits an overshoot that is much smaller than the undershoot we obtain with the Euler method. In addition, the error tends to zero at around $t=0.04$ s (which is equivalent to 20 samples). With the Taylor and RK methods, the error in the angular velocity is also much smaller than the error with the Euler method, and it also tends to zero much quicker (at around $t=0.05$ s).

To summarise the error for the three methods, we present

Table 2. Mean square error for the three discretisation methods applied to the separately excited motor.

	Euler	Taylor	Runge–Kutta
Armature current MSE	1.7583	$0.0249$	$0.0249$
Angular velocity MSE	1.6776	$8.269\times {10}^{-3}$	$8.269\times {10}^{-3}$

, where the mean square error (MSE) with respect to the true CT armature current is computed as

$${\mathrm{MSE}}_{i_a}={\displaystyle \frac{1}{N}}\sum _{k=0}^{N-1}{(i_a\left(k{T}_s\right)-i_k^{a,\mathrm{DT}})}^2,$$

(115)

where $i_k^{a,\mathrm{DT}}$ corresponds to the armature current obtained with the Euler, Taylor, and RK methods, and $N=101$ samples. Similarly, the MSE with respect to the true CT angular velocity is computed as

$${\mathrm{MSE}}_{\omega }={\displaystyle \frac{1}{N}}\sum _{k=0}^{N-1}{(\omega \left(k{T}_s\right)-{\omega }_k^{\mathrm{DT}})}^2,$$

(116)

where ${\omega }_k^{\mathrm{DT}}$ corresponds to the angular velocity obtained with the Euler, Taylor, and RK methods. From Table 2, we observe that the Taylor and RK2 methods yield more accurate models. This behaviour is expected since higher-order discretisation methods are able to better capture the dynamics of the system. However, the improvement from the Euler method is within two to three orders of magnitude, which suggests that it is better to model the SE motor using second-order discretisation methods.

3.4.2. Shunt Motor

The CT armature current, field current, and angular velocity for the shunt motor are shown in Figure 6, starting the motor from rest. The three discretisation methods yield similar DT signals, in spite of resulting in different DT models. However, the Euler method yields less accurate signals. This can be seen in Figure 7, where the Taylor and RK methods result in almost identical armature and field currents, whilst the error in the angular velocity is about the same order of magnitude. In

Table 3. Mean square error for the three discretisation methods applied to the shunt motor.

	Euler	Taylor	Runge–Kutta
Armature current MSE	$0.643$	$8.9525\times {10}^{-3}$	$9.1522\times {10}^{-3}$
Field current MSE	$12.9414\times {10}^{-6}$	$2.1223\times {10}^{-6}$	$2.1223\times {10}^{-6}$
Angular velocity MSE	$0.1574$	$1.3874\times {10}^{-3}$	$0.3689\times {10}^{-3}$

, we summarise the MSE for the three discretisation techniques with $N=251$ samples (from $t=0$ to $t=500$ ms). The MSE for the armature current and the angular velocity are computed using (115) and (116), respectively. The MSE for the field current is computed using

$${\mathrm{MSE}}_{i_f}={\displaystyle \frac{1}{N}}\sum _{k=0}^{N-1}{(i_f\left(k{T}_s\right)-i_k^{f,\mathrm{DT}})}^2,$$

(117)

where $i_k^{f,\mathrm{DT}}$ corresponds to the armature current obtained with the Euler, Taylor and RK methods.

Again, the performance of the Euler method is worse than both the Taylor and RK methods. In this nonlinear system, we observe a similar behaviour in terms of the MSE and the orders of magnitude between the MSE performance of the Euler method and the Taylor and RK methods for the field current. In addition, the Euler method yields an MSE two orders of magnitude larger than the MSE obtained with the Taylor and RK for the armature current and the angular velocity. Notice that in spite of the nonlinearities, the Euler method does not perform worse than it does for the SE motor.

3.4.3. Series Motor

The CT signals (i.e., armature current and angular velocity) for the series motor are shown in Figure 8. Notice that the armature current reaches a peak after a few miliseconds and a steady-state behaviour after around $0.3$ s. On the other hand, the discretisation of the series motor yields the same behaviour observed for the shunt and the SE motor: the Euler method results in a worse performance than the Taylor and RK methods. This can be seen in Figure 9, where the armature current error and the angular velocity error obtained with the Euler method are much larger than the ones obtained with the Taylor and RK methods. This can also be seen in

Table 4. Mean square error for the three discretisation methods applied to the series motor.

	Euler	Taylor	Runge–Kutta
Armature current MSE	$0.6574$	$1.2613\times {10}^{-3}$	$13.8958\times {10}^{-3}$
Angular velocity MSE	$4.7641$	$15.4825\times {10}^{-3}$	$25.2639\times {10}^{-3}$

, where the MSE of the armature current and the angular velocity were computed using (115) and (116), respectively.

For the series motor, the performance of the three discretisation methods is similar to the ones obtained for the other two DC motors studied in this paper. This strongly suggests that, for deterministic modelling, it is better to utilise higher-order discretisation techniques than the Euler method.

3.5. Discrete-Time Model Accuracy as a Function of the Sampling Period

It is well known that for continuous-time linear systems, accurate discrete-time representations can be obtained if the inverse of the sampling period (i.e., the sampling frequency) is at least twice the maximum frequency of interest of the system (which is usually greater than the frequency of the faster pole). For nonlinear systems, the sampling period can have an important effect on the corresponding discrete-time system.

In order to understand the effect of the sampling period on the accuracy of the discrete-time models of each DC motor, we considered several values ranging from $T_s=0.1$ ms to $T_s=3.5$ ms for the three discretisation methods and the three DC motors considered in this paper. To illustrate the accuracy of each method, we considered the MSE for the armature current, angular velocity, and the field current for the shunt motor. The results of varying the sampling period are summarised in Figure 10, Figure 11 and Figure 12, where we present the MSE of the SE, shunt, and series DC motors, respectively, for different sampling periods.

Across all DC motors, the discrete models arising from the Taylor and RK methods perform similarly, since both are second-order approximations. For the shunt DC motor, the RK performs slightly better than the Taylor in terms of the angular velocity MSE when the sampling time exceeds $2.5$ ms, as shown in Figure 11. Conversely, for the series DC motor, the Taylor performs slightly better than the RK under the same condition, as shown in Figure 12. For sampling times smaller than $2.5$ ms, the Taylor and RK perform similarly for the three DC motors.

Regarding Euler discretisation, as the sampling time increases, the armature current MSE in the SE DC motor grows about three and six times compared to the shunt and series DC motors, respectively. Conversely, in the series DC motor, the angular velocity MSE increases about three and fifty times compared to the SE and shunt DC motors, respectively.

On the other hand, the difference between the second-order approximations and the Euler method is notable. As the sampling time increases, the Euler MSE grows dramatically, resulting in an MSE of more than two orders of magnitude greater than the the MSE with the Taylor or RK methods.

With respect to the selection of the sampling period and discretisation method, our simulations suggest that there is a great variability in the errors. Therefore, the sampling period and discretisation method should be chosen depending on the overall system tolerance to those errors. In particular, applications that require high precision (such as remote surgery), the Taylor or Runge–Kutta methods with a small sampling period would be adequate. In contrast, applications that do not require high precision (such as electro-mobility or propulsion systems) could cope with the intrinsic errors from the Euler method.

4. Bayesian Filtering

4.1. Stochastic Dynamic Model of a Motor

In control systems and signal processing, stochastic models are usually utilised to provide a model that incorporates discrepancies between the observed signals and the deterministic model; see, e.g., [34,35] for detailed modelling and analysis of stochastic dynamic systems. These discrepancies may arise from the discretisation method that has been utilised, perturbations (such as vibrations), changes in the model (such as an increase in the value of a resistance or a physical change of a component due to an increase in temperature), to name a few. These perturbations are typically incorporated in the model as additive independent random variables, which result in stochastic state variables, and an output equation with additive random noise that accounts for the fact that the measurement equipment is comprised of electronic (i.e., solid-state) devices that are subject to electronic thermal noise; see, e.g., [36]. Moreover, in sensorless systems, there is no direct measurement of the angular velocity. Thus, a general sensorless stochastic state–space model for a DC motor is as follows:

$$\left[\begin{array}{c}{i}_{k+1}^{\left(a\right)}\\ i_{k+1}^{\left(f\right)}\\ {\omega }_{k+1}\end{array}\right]=f\left(\left[\begin{array}{c}{i}_k^{\left(a\right)}\\ i_k^{\left(f\right)}\\ {\omega }_k\end{array}\right],\left[\begin{array}{c}{V}_a\\ V_L\end{array}\right]\right)+\left[\begin{array}{c}{e}_k^{\left(a\right)}\\ e_k^{\left(f\right)}\\ e_k^{\left(\omega \right)}\end{array}\right],$$

(118)

$$\left[\begin{array}{c}{y}_k^{\left(a\right)}\\ y_k^{\left(f\right)}\end{array}\right]=\mathbf{C}\left[\begin{array}{c}{i}_k^{\left(a\right)}\\ i_k^{\left(f\right)}\\ {\omega }_k\end{array}\right]+\left[\begin{array}{c}{\eta }_k^{\left(a\right)}\\ {\eta }_k^{\left(f\right)}\end{array}\right],$$

(119)

where $f(·)$ is the function obtained using a discretisation method, ${\mathit{e}}_k={\left[\begin{array}{c}{e}_k^{\left(a\right)},e_k^{\left(f\right)},e_k^{\left(\omega \right)}\end{array}\right]}^T\sim \mathcal{N}(\mathbf{0},{\mathbf{\Sigma }}_e)$, where ${\mathbf{\Sigma }}_e$ is a diagonal matrix, $y_k^{\left(a\right)}$ is the measurement of the armature current, $y_k^{\left(f\right)}$ the measurement of the field current, $\mathit{C}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]$, and ${\mathit{\eta }}_k={\left[\begin{array}{c}{\eta }_k^{\left(a\right)},{\eta }_k^{\left(f\right)}\end{array}\right]}^T\sim \mathcal{N}(\mathbf{0},{\mathbf{\Sigma }}_{\eta })$, where ${\mathbf{\Sigma }}_{\eta }$ is a diagonal matrix.

4.2. Kalman Filter

The stochastic model for the SE motor is linear for the three discretisation techniques utilised in this paper.

The stochastic model for the SE motor using Euler method is as follows:

$$\begin{array}{cc}\hfill i_{k+1}^{\left(a\right)}& = a_{11}^{\mathrm{E},\mathrm{SE}}{i}_k^{\left(a\right)}+a_{12}^{\mathrm{E},\mathrm{SE}}{\omega }_k+B^{\mathrm{E},\mathrm{SE}}Va+e_k^{\left(a\right)},\hfill \end{array}$$

(120)

$$\begin{array}{cc}\hfill {\omega }_{k+1}& = a_{21}^{\mathrm{E},\mathrm{SE}}{i}_k^{\left(a\right)}+a_{22}^{\mathrm{E},\mathrm{SE}}{\omega }_k+e_k^{\left(\omega \right)},\hfill \end{array}$$

(121)

$$\begin{array}{cc}\hfill y_k& = i_k^{\left(a\right)}+{\eta }_k^{\left(a\right)},\hfill \end{array}$$

(122)

where $e_k^{\left(a\right)}\sim \mathcal{N}(0,Q_a^{\mathrm{SE}})$, $e_k^{\left(\omega \right)}\sim \mathcal{N}(0,Q_{\omega }^{\mathrm{SE}})$, ${\eta }_k^{\left(a\right)}\sim \mathcal{N}(0,{\mathrm{Y}}_a^{\mathrm{SE}})$, and the remaining parameters are defined in (11)–(15).

The stochastic model for the SE motor using Taylor method is as follows:

$$\begin{array}{cc}\hfill i_{k+1}^{\left(a\right)}& = a_{11}^{\mathrm{T},\mathrm{SE}}{i}_k^{\left(a\right)}+a_{12}^{\mathrm{T},\mathrm{SE}}{\omega }_k+b_1^{\mathrm{T},\mathrm{SE}}{V}_a+e_k^{\left(a\right)},\hfill \end{array}$$

(123)

$$\begin{array}{cc}\hfill {\omega }_{k+1}& = a_{21}^{\mathrm{T},\mathrm{SE}}{i}_k^{\left(a\right)}+a_{22}^{\mathrm{T},\mathrm{SE}}{\omega }_k+b_2^{\mathrm{T},\mathrm{SE}}{V}_a+e_k^{\left(\omega \right)},\hfill \end{array}$$

(124)

$$\begin{array}{cc}\hfill y_k^{\left(a\right)}& = i_k^{\left(a\right)}+{\eta }_k^{\left(a\right)},\hfill \end{array}$$

(125)

where $e_k^{\left(a\right)}\sim \mathcal{N}(0,Q_a^{\mathrm{T}})$, $e_k^{\left(\omega \right)}\sim \mathcal{N}(0,Q_{\omega }^{\mathrm{T}})$, ${\eta }_k^{\left(a\right)}\sim \mathcal{N}(0,{\mathrm{Y}}_a^{\mathrm{T}})$, and the remaining parameters are defined in (37)–(42).

The stochastic model for the SE motor using RK method is as follows:

$$\begin{array}{cc}\hfill i_{k+1}^{\left(a\right)}& = a_{11}^{\mathrm{RK},\mathrm{SE}}{i}_k^{\left(a\right)}+a_{12}^{\mathrm{RK},\mathrm{SE}}{\omega }_k+b_1^{\mathrm{RK},\mathrm{SE}}Va+e_k^{\left(a\right)},\hfill \end{array}$$

(126)

$$\begin{array}{cc}\hfill {\omega }_{k+1}& = a_{21}^{\mathrm{RK},\mathrm{SE}}{i}_k^{\left(a\right)}+a_{22}^{\mathrm{RK},\mathrm{SE}}{\omega }_k+b_2^{\mathrm{RK},\mathrm{SE}}{V}_a+e_k^{\left(\omega \right)},\hfill \end{array}$$

(127)

$$\begin{array}{cc}\hfill y_k^{\left(a\right)}& = i_k^{\left(a\right)}+{\eta }_k^{\left(a\right)},\hfill \end{array}$$

(128)

where $e_k^{\left(a\right)}\sim \mathcal{N}(0,Q_a^{\mathrm{RK}})$, $e_k^{\left(\omega \right)}\sim \mathcal{N}(0,Q_{\omega }^{\mathrm{RK}})$, ${\eta }_k^{\left(a\right)}\sim \mathcal{N}(0,{\mathrm{Y}}_a^{\mathrm{RK}})$, and the remaining parameters are defined in (74)–(79).

In all three models for the SE motor, the angular velocity is not measured, and the armature current is measured with additive white Gaussian noise. Hence, since the models are linear, the estimation of the state variables can be carried out using the Kalman filter (KF) [37]. This is due to the fact that the KF is the optimal (minimum variance) linear state estimator [38].

The KF can be obtained from the theory of Bayesian filtering, using the fact that the state–space model of the DC motor is defined as a first-order Markov model; see, e.g., [35,37,38].

In Bayesian filtering, the goal is to obtain the marginal posterior probability density function (pdf) of the states given the measurements. This is typically achieved via filtering and smoothing algorithms that operate recursively; see, e.g., [39]. The recursivity is based on the first-order Markovian property of state–space models and the Chapman–Kolmogorov equation [38,39]. The Markovian property establishes that for a system, with known input $v_k$, described as

$$\begin{array}{cc}\hfill x_{k+1}& = f(x_k,v_k)+e_k,\hfill \end{array}$$

(129)

$$\begin{array}{cc}\hfill y_k& = h\left(x_k\right)+{\eta }_k,\hfill \end{array}$$

(130)

the conditional pdf of the state at time k, given the states up to time $k-1$, satisfies $p\left(x_k\right|x_{k-1},x_{k-2},\dots ,x_1,)=p\left(x_k\right|x_{k-1})$. This expression results in the following Chapman–Kolmogorov equation (for mode details, see, e.g., [39]):

$$p\left(x_k\right|y_{1:k-1})=\int p\left(x_k\right|x_{k-1})p\left(x_{k-1}\right|y_{1:k-1})d{x}_{k-1},$$

(131)

where $y_{1:k-1}=\{y_1,y_2,...,y_{k-1}\}$. On the other hand, the output equation defines the pdf of the measurement at time k, given the state at time k, $p\left(y_k\right|x_k)$, which in turn defines the measurement-update equation

$$p\left(x_k\right|y_{1:k})={\displaystyle \frac{p\left(y_k\right|x_k)p\left(x_k\right|y_{1:k-1})}{p\left(y_k\right|y_{1:k-1})}}.$$

(132)

Notice that both the Chapman–Kolmogorov equation and the measurement-update equation can be obtained recursively from the pdf of the initial condition of the state, $x_0$.

Finally, the posterior pdf of the state at time k, given all the measurements (up to time $k=N$), is computed using the Bayesian smoothing equation [39]:

$$p\left(x_k\right|y_{1:N})=p\left(x_k\right|y_{1:k})\int {\displaystyle \frac{p\left(x_{k+1}\right|y_{1:N})p\left(x_{k+1}\right|x_k)}{p\left(x_{k+1}\right|y_{1:k})}}d{x}_{k+1}.$$

(133)

In the case when the system is linear, and the process noise and measurement noise are Gaussian, the Chapman–Kolmogorov equation and the measurement-update equation result in the KF, which yields a Gaussian pdf of the state at time k, given the measurements up to time k. For time-invariant systems, the KF is summarised in

Table 5. Kalman filter for linear systems with Gaussian process and measurement noise.

System model
	$\begin{array}{ccc}\hfill {\mathbf{x}}_{k+1}& ={\mathbf{Ax}}_k+{\mathbf{Bv}}_k+{\mathit{e}}_k,\hfill & \\ \hfill {\mathbf{y}}_k& ={\mathbf{Cx}}_k+{\eta }_k,\hfill & \\ & {\mathit{e}}_k\sim \mathcal{N}(\mathbf{0},\mathit{Q}),\hfill & {\eta }_k\sim \mathcal{N}(\mathbf{0},\mathbf{Y}).\hfill \end{array}$
Time update
	$\begin{array}{ccc}\hfill {\widehat{\mathbf{x}}}_{k|k-1}& =\mathbf{A}{\widehat{\mathbf{x}}}_{k-1|k-1}+{\mathbf{Bv}}_k\hfill & \\ \hfill {\mathbf{P}}_{k|k-1}& ={\mathbf{AP}}_{k-1|k-1}{\mathbf{A}}^T+\mathit{Q},\hfill \end{array}$
Measurement update
	$\begin{array}{ccc}\hfill {\epsilon }_k& ={\mathbf{y}}_k-{\mathbf{Cx}}_{k-1|k-1},\hfill & \\ \hfill {\mathbf{S}}_k& ={\mathbf{CP}}_{k|k-1}{\mathbf{C}}^T+\mathbf{Y},\hfill & \\ \hfill {\mathbf{K}}_k& ={\mathbf{P}}_{k|k-1}{\mathbf{C}}^T{\mathbf{S}}_k^{-1},\hfill & \\ \hfill {\widehat{\mathbf{x}}}_{k|k}& ={\widehat{\mathbf{x}}}_{k|k-1}+{\mathbf{K}}_k{\epsilon }_k,\hfill & \\ \hfill {\mathbf{P}}_{k|k}& ={\mathbf{P}}_{k|k-1}-{\mathbf{K}}_k{\mathbf{S}}_k{\mathbf{K}}_k^T.\hfill \end{array}$

and the Kalman smoother (KS) in

Table 6. Kalman smoother for linear systems with Gaussian process and measurement noise.

RTS smoother

$\begin{array}{ccc}\hfill {\mathbf{G}}_k& ={\mathbf{P}}_{k|k}{\mathbf{A}}^T{\mathbf{P}}_{k+1|k}^{-1},\hfill & \\ \hfill {\widehat{\mathbf{x}}}_{k|N}& ={\widehat{\mathbf{x}}}_{k|k}+{\mathbf{G}}_k({\widehat{\mathbf{x}}}_{k+1|N}-{\widehat{\mathbf{x}}}_{k+1|k}),\hfill & \\ \hfill {\mathbf{P}}_{k|N}& ={\mathbf{P}}_{k|k}-{\mathbf{G}}_k({\mathbf{P}}_{k+1|N}-{\mathbf{P}}_{k+1|k}){\mathbf{G}}_k^T.\hfill \end{array}$

. In Table 5, the subscript $k|k-1$ refers to the estimation at time k given the measurements up to time $k-1$. Similarly, the subscript $k|k$ refers to the estimation at time k, given the measurements up to time k. The estimation of the state (which effectively corresponds to the estimation of the mean of the posterior distribution) is represented by $\widehat{\mathbf{x}}$. Therefore, ${\mathbf{x}}_k|{\mathbf{y}}_k\sim \mathcal{N}({\widehat{\mathbf{x}}}_{k|k},{\mathbf{P}}_{k|k})$. In Table 6, the subscript $k|N$ refers to the estimation at time k, given all the measurements (up to time N). Therefore, ${\mathbf{x}}_k|{\mathbf{y}}_N\sim \mathcal{N}({\widehat{\mathbf{x}}}_{k|N},{\mathbf{P}}_{k|N})$. Notice that the Kalman smoothers in Table 6 are recursive backward equations that correspond to the Rauch–Tung–Striebel (RTS) smoother [39].

In order to understand each stochastic model and their respective filtering/smoothing for the SE motor, we considered the following simulation setup:

• The true stochastic system was modelled utilising the eighth-order RK discretisation method, with additive Gaussian process noise with variance equal to $0.2$ for the armature current and $0.06$ for the angular velocity.
• The process noise variances for the armature currents were $Q_a^E=6.667\times {10}^{-3}$, $Q_a^{\mathrm{T}}=0.367\times {10}^{-3}$, and $Q_a^{RK}=0.367\times {10}^{-3}$.
• The process noise variances for the angular velocities were $Q_{\omega }^E=2\times {10}^{-3}$, $Q_{\omega }^T=0.1$, and $Q_{\omega }^{RK}=0.11$.
• The measurement noise variance was ${\mathrm{Y}}_a^E={\mathrm{Y}}_a^{\mathrm{T}}={\mathrm{Y}}_a^{RK}=0.4$.
• The motor was started from rest.
• The number of Monte Carlo simulations is $M=1000$.

The results of all the Monte Carlo simulations are summarised in

Table 7. Mean square error of the estimation utilising the Kalman smoother for the three discretisation methods applied to the SE motor.

	Euler	Taylor	Runge–Kutta
Armature current MSE (${\mathrm{MSE}}_{i_a}$)	$960.5\times {10}^{-3}$	$68.83\times {10}^{-3}$	$68.83\times {10}^{-3}$
Angular velocity MSE (${\mathrm{MSE}}_{\omega }$)	$0.7185$	$0.7021$	$0.7021$

, where the MSE is computed as

$$\begin{array}{cc}\hfill {\mathrm{MSE}}_{i_a}& ={\displaystyle \frac{1}{NM}}\sum _{m=1}^M\sum _{k=1}^N(i_k^{(a,m),\mathit{True}}-{\widehat{i}}_{k|N}^{(a,m)}{)}^2,\hfill \end{array}$$

(134)

$$\begin{array}{cc}\hfill {\mathrm{MSE}}_{\omega }& ={\displaystyle \frac{1}{NM}}\sum _{m=1}^M\sum _{k=1}^N{({\omega }_k^{\left(m\right),\mathit{True}}-{\widehat{\omega }}_{k|N}^{\left(m\right)})}^2.\hfill \end{array}$$

(135)

${(·)}^{\left(m\right),\mathit{True}}$ represents the true value of the state for the mth Monte Carlo simulation, and ${\widehat{i}}_{k|N}^{(a,m)}$ and ${\widehat{\omega }}_{k|N}^{\left(m\right)}$ are the estimates of the armature current and the angular velocity, respectively, obtained from the Kalman smoother for the mth Monte Carlo simulation and each discretisation method.

The performance of the KS for the stochastic models follows a similar trend as the deterministic models. However, the difference in the MSE is of just one order of magnitude for the armature current, and the MSEs for the angular velocity are very similar.

4.3. Extended Kalman Filter

In nonlinear systems, the posterior pdfs $p\left(x_k\right|y_{1:k-1})$, $p\left(x_k\right|y_{1:k})$, and $p\left(x_k\right|y_{1:N})$ are usually difficult to obtain in closed-form expressions. Their computation is usually subject to approximations that make them computationally tractable; see, e.g., [37,39]. One of the most popular approximations is based on the linearisation of the nonlinear system around the current estimates and the Gaussian approximation of the posterior pdfs. The result is the extended Kalman filter (EKF) and the extended Kalman smoother (EKS).

In the EKF, we approximate the nonlinear state in (129) by its first order Taylor series at time k around ${\widehat{x}}_{k|k}$. The output equation in (130) is approximated by its first order Taylor series at time k around ${\widehat{x}}_{k|k-1}$. These two approximations yield the following time-variant model:

$$\begin{array}{cc}\hfill x_{k+1}& =F_k{x}_k+(f\left({\widehat{x}}_{k|k}\right)-F_k{\widehat{x}}_{k|k})+e_k,\hfill \end{array}$$

(136)

$$\begin{array}{cc}\hfill y_k& =H_k{x}_k+(h\left({\widehat{x}}_{k|k-1}\right)-H_k{\widehat{x}}_{k|k-1})+{\eta }_k,\hfill \end{array}$$

(137)

where $F_k={\left.{\displaystyle \frac{\partial f\left(x_k\right)}{\partial x_k}}\right|}_{x_k={\widehat{x}}_{k|k},v_k},$ and $H_k={\left.{\displaystyle \frac{\partial h\left(x_k\right)}{\partial x_k}}\right|}_{x_k={\widehat{x}}_{k|k-1}}$. The recursive equations that define the KF as a solution to the Chapman–Kolmogorov equation in (131) and the measurement update equation in (132) are shown in

Table 8. Extended Kalman filter for nonlinear systems with additive Gaussian process and measurement noise.

System model
	$\begin{array}{ccc}\hfill {\mathbf{x}}_{k+1}& =f({\mathbf{x}}_k,{\mathbf{v}}_k)+{\mathit{e}}_k,\hfill & \\ \hfill {\mathbf{y}}_k& =h\left({\mathbf{x}}_k\right)+{\eta }_k,\hfill & \\ & {\mathit{e}}_k\sim \mathcal{N}(\mathbf{0},\mathit{Q}),\hfill & {\eta }_k\sim \mathcal{N}(\mathbf{0},\mathbf{Y}).\hfill \end{array}$
Time update
	$\begin{array}{ccc}\hfill {\widehat{\mathbf{x}}}_{k|k-1}& =f({\widehat{\mathbf{x}}}_{k-1|k-1},{\mathbf{v}}_k)\hfill & \\ \hfill {\mathbf{P}}_{k|k-1}& ={\mathbf{F}}_{k-1}{\mathbf{P}}_{k-1|k-1}{\mathbf{F}}_{k-1}^T+\mathit{Q},\hfill \end{array}$
Measurement update
	$\begin{array}{ccc}\hfill {\epsilon }_k& ={\mathbf{y}}_k-h\left({\mathbf{x}}_{k-1|k-1}\right),\hfill & \\ \hfill {\mathbf{S}}_k& ={\mathbf{H}}_k{\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^T+\mathbf{Y},\hfill & \\ \hfill {\mathbf{K}}_k& ={\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^T{\mathbf{S}}_k^{-1},\hfill & \\ \hfill {\widehat{\mathbf{x}}}_{k|k}& ={\widehat{\mathbf{x}}}_{k|k-1}+{\mathbf{K}}_k{\epsilon }_k,\hfill & \\ \hfill {\mathbf{P}}_{k|k}& ={\mathbf{P}}_{k|k-1}-{\mathbf{K}}_k{\mathbf{S}}_k{\mathbf{K}}_k^T.\hfill \end{array}$

, whilst the recursive equations that define the EKS as solution to the smoothing equation in (133) are shown in

Table 9. Extended Kalman smoother for nonlinear systems with additive Gaussian process and measurement noise.

RTS smoother

$\begin{array}{ccc}\hfill {\mathbf{G}}_k& ={\mathbf{P}}_{k|k}{\mathbf{F}}_k^T{\mathbf{P}}_{k+1|k}^{-1},\hfill & \\ \hfill {\widehat{\mathbf{x}}}_{k|N}& ={\widehat{\mathbf{x}}}_{k|k}+{\mathbf{G}}_k({\widehat{\mathbf{x}}}_{k+1|N}-{\widehat{\mathbf{x}}}_{k+1|k}),\hfill & \\ \hfill {\mathbf{P}}_{k|N}& ={\mathbf{P}}_{k|k}-{\mathbf{G}}_k({\mathbf{P}}_{k+1|N}-{\mathbf{P}}_{k+1|k}){\mathbf{G}}_k^T.\hfill \end{array}$

.

Remark 2. 

Notice that, in our problem of interest, the armature current (for the SE motor and the series motor) is directly measured, which results in $g(i_k^{\left(a\right)},{\omega }_k)=i_k^{\left(a\right)}$. Therefore, $G_k=1$. Similarly, for the shunt motor, we have that $g(i_k^{\left(a\right)},i_k^{\left(f\right)},{\omega }_k)=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]\left[\begin{array}{c}{i}_k^{\left(a\right)}\\ i_k^{\left(f\right)}\\ {\omega }_k\end{array}\right]$. Therefore, $G_k=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]$.

4.3.1. EKF/EKS Applied to the Shunt Motor

The filtering and smoothing equations in Table 8 and Table 9, respectively, were applied to the shunt motor with the following simulation parameters:

• The true stochastic system was modelled utilising the eighth-order RK discretisation method, with additive Gaussian process noise with variance equal to $0.3$ for the armature current, ${10}^{-4}$ for the field current, and $0.02$ for the angular velocity.
• The process noise variances for the armature currents were $Q_a^E=0.06$, $Q_a^{\mathrm{T}}=0.175$, and $Q_a^{RK}=0.13$.
• The process noise variances for the field currents were $Q_f^E=2\times {10}^{-6}$, $Q_f^{\mathrm{T}}=5.833\times {10}^{-6}$, and $Q_f^{RK}=4.33\times {10}^{-6}$.
• The process noise variances for the angular velocities were $Q_{\omega }^E=4\times {10}^{-4}$, $Q_{\omega }^T=11.667\times {10}^{-3}$, and $Q_{\omega }^{RK}=8.667\times {10}^{-3}$.
• The measurement noise variances were ${\mathrm{Y}}_a^E={\mathrm{Y}}_a^{\mathrm{T}}={\mathrm{Y}}_a^{RK}=0.1$, and ${\mathrm{Y}}_{\omega }^E={\mathrm{Y}}_{\omega }^{\mathrm{T}}={\mathrm{Y}}_{\omega }^{RK}=5\times {10}^{-5}$.
• The motor was started from rest.
• The number of Monte Carlo simulations was $M=1000$.

The results of all the Monte Carlo simulations are summarised in

Table 10. Mean square error of the estimation utilising the extended Kalman smoother for the three discretisation methods applied to the shunt motor.

	Euler	Taylor	RK
Armature current MSE (${\mathrm{MSE}}_{i_a}$)	$74.173\times {10}^{-3}$	$56.3795\times {10}^{-3}$	$56.1337\times {10}^{-3}$
Field current MSE (${\mathrm{MSE}}_{i_f}$)	$34.0361\times {10}^{-6}$	$28.9177\times {10}^{-6}$	$28.9177\times {10}^{-6}$
Angular velocity MSE (${\mathrm{MSE}}_{\omega }$)	$0.23$	$0.2279$	$0.2142$

, where the MSE of the estimation of the field current is computed as

$${\mathrm{MSE}}_{i_f}={\displaystyle \frac{1}{NM}}\sum _{m=1}^M\sum _{k=1}^N(i_k^{(f,m),\mathit{True}}-{\widehat{i}}_{k|N}^{(f,m)}{)}^2.$$

(138)

${(·)}^{\left(f\right),\mathit{True}}$ represents the true value of the state for the mth Monte Carlo simulation, and ${\widehat{i}}_{k|N}^{(f,m)}$ are the estimates of the field current obtained from the Kalman smoother for the mth Monte Carlo simulation and each discretisation method.

The performance of the EKS is similar for the three nonlinear models. However, the Euler method results in the worst performance (albeit very similar to the performance obtained using the other two discretisation methods). Again, the Taylor and RK methods exhibit an almost identical performance.

4.3.2. EKF/EKS Applied to the Series Motor

For the series motor, we considered the following simulation setup:

• The true stochastic system was modelled utilising the eighth order RK discretisation method, with additive Gaussian process noise, with a variance equal to $0.4$ for the armature current and $0.1$ for the angular velocity.
• The process noise variances for the armature currents were $Q_a^E=0.14$, $Q_a^{\mathrm{T}}=6.667\times {10}^{-3}$, and $Q_a^{RK}=0.02$.
• The process noise variances for the angular velocities were $Q_{\omega }^E=0.35$, $Q_{\omega }^T=1.667\times {10}^{-3}$, and $Q_{\omega }^{RK}=5\times {10}^{-3}$.
• The measurement noise variance was ${\mathrm{Y}}_a^E={\mathrm{Y}}_a^{\mathrm{T}}={\mathrm{Y}}_a^{RK}=0.1$.
• The motor was started from rest.
• The number of Monte Carlo simulations was $M=1000$.

The results of all the Monte Carlo simulations are summarised in

Table 11. Mean square error of the estimation utilising the extended Kalman smoother for the three discretisation methods applied to the series motor.

	Euler	Taylor	Runge–Kutta
Armature current MSE (${\mathrm{MSE}}_{i_a}$)	$137.6553\times {10}^{-3}$	$66.656\times {10}^{-3}$	$72.0259\times {10}^{-3}$
Angular velocity MSE (${\mathrm{MSE}}_{\omega }$)	$2.5498$	$1.3078$	$1.5499$

.

Simlar to the shunt motor, the performance of the EKS with the Euler discretisation method was very similar to the performance of the EKS with the Taylor and RK methods. In fact, the MSE with the Euler method was almost twice the MSE with the other methods.

5. Conclusions

In this paper, we have provided detailed expressions for modelling DC motors in discrete-time based on the Euler method, second-order Taylor method, and second-order Runge-Kutta method. Modelling DC motors in discrete-time requires the utilisation of adequate discretisation techniques. Despite its popularity and ubiquity, the Euler method resulted in the worst performance compared to the second-order Taylor and RK methods. This is due to the fact that second-order methods are able to better capture the dynamics of the system.

However, the difference in performance (in terms of the MSE) was not large for the stochastic models of the DC motor. In particular, the MSE of the armature current in the stochastic shunt motor was within an order of magnitude (in the order of ${10}^{-3}$) for the Euler, Taylor, and RK methods, whilst it was also within an order of magnitude (in the order of ${10}^{-6}$) for the field current and (in the order of ${10}^{-1}$) for the angular velocity. The MSE of the armature current in the stochastic series motor with the Euler method was roughly twice the MSE obtained with both the Taylor and RK methods. The MSE of the angular velocity in the stochastic series motor was also within an order of magnitude for the three discretisation methods. This implies that the selection of a particular discretisation technique (and therefore stochastic model) greatly depends on the type of application. In contrast, for deterministic systems, the Taylor and RK discretisation methods will provide better performance.