DC/DC Boost Converter–Inverter as Driver for a DC Motor: Modeling and Experimental Verification

Abstract

In this paper, the modeling and the experimental verification of the “bidirectional DC/DC boost converter–DC motor” system are presented. By using circuit theory along with the model of a DC motor, the mathematical model of the system is derived. This model was experimentally tested under time-varying duty cycles obtained via the system differential flatness property. The experimental verification was carried out using Matlab-Simulink and a DS1104 board in a built prototype of the system.

1. Introduction

Nowadays, DC/DC power electronic converters are used in several applications [1,2,3,4,5,6]. Such applications include, among others, renewable energies, aircraft, electric vehicles, robots, and telecommunications. If some of these applications require the generation of movement, then electric motors are generally used. In this direction, driving DC motors is achieved by means of PWM. However, PWM switching induces abrupt variations in DC motor voltage and current [7]. These variations can be reduced when the DC motor is fed via DC/DC converters [7,8]. DC/DC converters change an input DC voltage level to a higher or lower output voltage level with a small current ripple. This latter is an intrinsic feature of a power converter because of the capacitor and the inductor composing it. In this paper, a cascaded connection between a DC/DC boost converter, an inverter, and a DC motor is proposed, achieving a bidirectional rotation of the motor shaft.

The state of the art related to works dealing with a boost converter coupled with a DC motor for driving unidirectional angular velocity is as follows. Lyshevski [8] developed the mathematical models of the buck, boost, and Ćuk converters that feed a DC motor. Linares-Flores et al. [9] reported the design of a passivity-based tracking control that is robust to torque variations. Alexandridis and Konstantopoulos [10] presented a nonlinear PI control that does not require current sensors for solving the regulation task, and the same authors developed a regulation nonlinear control that provides a closed-loop passive system that is robust to load variations [11]. Additional works in which different topologies of DC/DC power converters have been used in combination with a DC motor are the following: [12,13,14,15,16,17,18,19,20,21,22,23,24,25] for the buck converter, [26,27] for the buck–boost converter, and [28] for the SEPIC and Ćuk converters. Because of the operation principle of the DC/DC converters, works reported in [7,8,9,10,11,12,13,14,15,16,17,18,19,20,26,27,28] were only focused on driving the motor shaft in an unidirectional fashion for solving both the regulation and tracking problems.

On the other hand, works in which the bidirectional rotation of the motor shaft was addressed, when a DC/DC converter and an inverter were used, are the following. For the “DC/DC buck converter–inverter–DC motor” system, Silva-Ortigoza et al. [29,30] experimentally validated the mathematical model of the system and designed a passivity-based tracking control. Hernández-Márquez et al. proposed and experimentally validated two robust tracking controls in [31]. García-Rodríguez et al. [32] deduced the mathematical model of the “DC/DC boost converter–inverter–DC motor” system, and a passivity-based regulation control was proposed in [33]. It is worth mentioning that in [32,33] only numerical simulations for constant trajectories were reported. Moreover, Hernández-Márquez et al. [34] experimentally validated a mathematical model for the “DC/DC buck–boost converter–inverter–DC motor” system, and a passivity-based tracking control for such a system was designed in [35]. Lastly, Linares-Flores et al. in [36] solved the regulation control problem for the “DC/DC SEPIC converter–inverter–DC motor” system.

On the basis of the aforementioned literature, the contribution of this paper, unlike [32], is to experimentally validate the mathematical model of the DC/DC boost converter–inverter–DC motor system when time-varying duty cycles are considered. Such an aim is achieved by exploiting the differential flatness property of the system, generalizing the work reported in [32].

The rest of this paper is organized as follows. In Section 2, the mathematical model of the DC/DC boost converter–inverter–DC motor system is developed. The built prototype is described in Section 3. The experimental results and conclusions are given in Section 4 and Section 5, respectively.

2. “DC/DC Boost Converter–Inverter–DC Motor” System

Here, the mathematical model of the DC/DC boost converter–inverter–DC motor system is developed. Additionally, the differential parametrization of the model previously deduced is presented. Such a parametrization allows the reference trajectories for the system to be generated.

2.1. Mathematical Model of the System

Figure 1 shows the electronic circuit of the DC/DC boost converter–inverter–DC motor system. This system is composed by the following:

• DC/DC boost converter. This is composed of a power supply E, a switching input $u_1$ that turns on/off transistor $Q_1$, a current i that flows through the inductor L, and a diode D. The output voltage, associated with capacitor C and load R, is denoted by $\upsilon$.
• Inverter. Here, $u_2$ and ${\overline{u}}_2$ are the inputs that turn on/off transistors $Q_2$ and ${\overline{Q}}_2$, respectively, thus achieving the bidirectional rotation of the motor shaft.
• DC motor. Parameters $i_a$, $R_a$, and $L_a$ are the armature current, armature resistance, and armature inductance; $\omega$ denotes the angular velocity of the motor shaft. Additional parameters for the DC motor are J, $k_e$, $k_m$, and b, which correspond to the moment of inertia of the rotor and load, the counter-electromotive force constant, the torque constant, and the viscous friction coefficient, respectively.

The ideal circuit depicted in Figure 2, which is associated with Figure 1, was used to develop the mathematical model of the system. The model was obtained by applying Kirchhoff’s current law and Kirchhoff’s voltage law to node A and mesh I (see Figure 2) and, also, by considering the mathematical model of a DC motor [37].

The operation modes of the electronic circuit shown in Figure 2 are described below. These operation modes appear when transistors are on (i.e., $u_1=1$ or $u_2=1$) or off (i.e., $u_1=0$ or $u_2=-1$) generating the following four operation modes.

2.1.1. Operation Mode 1

Figure 3 shows the ideal electronic circuit when transistor $Q_1$ is off and transistor $Q_2$ is on, that is, $u_1=0$ and $u_2=1$.

After applying Kirchhoff’s laws and using the mathematical model of a DC motor [37], the first operation mode is represented by the following differential equations:

$$\begin{array}{cc}\hfill L\frac{di}{dt}& =E-\upsilon ,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill C\frac{d\upsilon }{dt}& =i-\frac{\upsilon }{R}-i_a,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill L_a\frac{d{i}_a}{dt}& =\upsilon -R_a{i}_a-k_e\omega ,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill J\frac{d\omega }{dt}& =k_m{i}_a-b\omega .\hfill \end{array}$$

(4)

2.1.2. Operation Mode 2

This operation mode is generated when transistors $Q_1$ and $Q_2$ are off, that is, $u_1=0$ and $u_2=-1$. The equivalent circuit is depicted in Figure 4.

When Kirchhoff’s laws are applied to the circuit of Figure 4, the corresponding operation mode is modeled by

$$\begin{array}{cc}\hfill L\frac{di}{dt}& =E-\upsilon ,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill C\frac{d\upsilon }{dt}& =i-\frac{\upsilon }{R}+i_a,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill L_a\frac{d{i}_a}{dt}& =-\upsilon -R_a{i}_a-k_e\omega ,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill J\frac{d\omega }{dt}& =k_m{i}_a-b\omega .\hfill \end{array}$$

(8)

2.1.3. Operation Mode 3

When transistors $Q_1$ and $Q_2$ are on, that is, $u_1=1$ and $u_2=1$, the third operation mode is generated (see Figure 5).

The mathematical model associated with this operation mode, after applying Kirchhoff’s laws, is given by

$$\begin{array}{cc}\hfill L\frac{di}{dt}& =E,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill C\frac{d\upsilon }{dt}& =-\frac{\upsilon }{R}-i_a,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill L_a\frac{d{i}_a}{dt}& =\upsilon -R_a{i}_a-k_e\omega ,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill J\frac{d\omega }{dt}& =k_m{i}_a-b\omega .\hfill \end{array}$$

(12)

2.1.4. Operation Mode 4

When transistor $Q_1$ is on and transistor $Q_2$ is off, that is, $u_1=1$ and $u_2=-1$, the fourth operation mode is achieved (see Figure 6).

In this case, the operation mode is modeled by the following differential equations:

$$\begin{array}{cc}\hfill L\frac{di}{dt}& =E,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill C\frac{d\upsilon }{dt}& =-\frac{\upsilon }{R}+i_a,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill L_a\frac{d{i}_a}{dt}& =-\upsilon -R_a{i}_a-k_e\omega ,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill J\frac{d\omega }{dt}& =k_m{i}_a-b\omega .\hfill \end{array}$$

(16)

After combining the four operation modes, given by Equations (1)–(16), the mathematical model of the DC/DC boost converter–inverter–DC motor system is determined by

$$\begin{array}{cc}\hfill L\frac{di}{dt}& =-(1-u_1)\upsilon +E,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill C\frac{d\upsilon }{dt}& =(1-u_1)i-\frac{\upsilon }{R}-i_a{u}_2,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill L_a\frac{d{i}_a}{dt}& =\upsilon u_2-R_a{i}_a-k_e\omega ,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill J\frac{d\omega }{dt}& =k_m{i}_a-b\omega ,\hfill \end{array}$$

(20)

where $u_1$ and $u_2$ represent the switches’ positions. The mathematical model given by Equations (17)–(20) is known as the switched model because of its binary-valued nature. The average model associated with Equations (17)–(20) is as follows:

$$\begin{array}{cc}\hfill L\frac{di}{dt}& =-(1-u_{1av})\upsilon +E,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill C\frac{d\upsilon }{dt}& =(1-u_{1av})i-\frac{\upsilon }{R}-i_a{u}_{2av},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill L_a\frac{d{i}_a}{dt}& =\upsilon u_{2av}-R_a{i}_a-k_e\omega ,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill J\frac{d\omega }{dt}& =k_m{i}_a-b\omega ,\hfill \end{array}$$

(24)

where $u_{1av}\in [0,1)$ and $u_{2av}\in [-1,1]$.

2.2. Generation of Reference Trajectories

The differential parameterization of the average system given by Equations (21)–(24) is presented below.

The flat output of the boost converter, according to [38,39], is determined by its energy; that is,

$$\mathcal{E}=\frac{1}{2}(L{i}^2+C{\upsilon }^2).$$

(25)

The flat output of a DC motor, according to [15], is given by the angular velocity of the motor shaft $\omega$. After proposing the flat outputs of the DC/DC boost converter–inverter–DC motor average system as $F_1=\mathcal{E}$ and $F_2=\omega$, the following differential parameterization associated to Equations (21)–(24) is found:

$$\begin{array}{cc}\hfill i& =-{\displaystyle \frac{RCE}{2L}}+\alpha ,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \upsilon & ={\left[R\left(-{\displaystyle \frac{RC{E}^2}{2L}}+\alpha E-\beta i_a-{\dot{F}}_1\right)\right]}^{1/2},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill i_a& =\frac{1}{k_m}\left(J{\dot{F}}_2+b{F}_2\right),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \omega & =F_2,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill u_{1av}& ={\displaystyle \frac{1}{\gamma }}\left[{\ddot{F}}_1-{\displaystyle \frac{E^2}{L}}-{\displaystyle \frac{2}{R^{2}C}}{\upsilon }^2-{\displaystyle \frac{3}{RC}}{i}_a\upsilon u_{2av}-{\displaystyle \frac{1}{C}}{i}_a^2{u}_{2av}^2\right]+1,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill u_{2av}& =\frac{\beta }{\upsilon },\hfill \end{array}$$

(31)

with

$$\begin{array}{cc}\hfill \alpha & ={\left\{{\left({\displaystyle \frac{RCE}{2L}}\right)}^2+{\displaystyle \frac{1}{L}}\left[CR\left(\beta i_a+{\dot{F}}_1\right)+2F_1\right]\right\}}^{1/2},\hfill \\ \hfill \beta & =L_a{\displaystyle \frac{d{i}_a}{dt}}+R_a{i}_a+k_e{F}_2,\hfill \\ \hfill \gamma & ={\displaystyle \frac{E}{L}}\upsilon +{\displaystyle \frac{2}{RC}}i\upsilon +{\displaystyle \frac{1}{C}}i{i}_a{u}_{2av}.\hfill \end{array}$$

As can be observed, the average model given by Equations (21)–(24) is now represented by the differential parameterization given by Equations (26)–(31) expressed in terms of variables $F_1$ and $F_2$ and their corresponding derivatives with respect to time. Such a representation, compared to Equations (21)–(24), allows the reference trajectories associated with states i, $\upsilon$, and $i_a$ and inputs $u_{1av}$ and $u_{2av}$ to be found offline [38]. Thus, when $F_1$ and $F_2$ are replaced by ${\mathcal{E}}^{*}$ and ${\omega }^{*}$ in Equations (26)–(31), the reference trajectories are obtained; this is, $i^{*}$, ${\upsilon }^{*}$, $i_a^{*}$, $u_{1av}^{*}$, and $u_{2av}^{*}$. Here, ${\mathcal{E}}^{*}$ is the desired boost converter energy and ${\omega }^{*}$ represents the desired angular velocity of the motor shaft.

3. Built Experimental Prototype

In this section, the experimental platform used to validate the mathematical model of the boost converter–inverter–DC motor system is described.

A schematic diagram of the built platform is presented in Figure 7 and is composed of the following blocks:

• DC/DC boost converter–inverter–DC motor system. The following three subsystems are distinguished within this block: boost converter, inverter, and DC motor. In this direction, the nominal values associated with the converter parameters are

  $$\begin{array}{c}L=4.94\mathrm{mH},R=64\mathsf{\Omega },C=114.4\mu \mathrm{F},E=12\mathrm{V}.\end{array}$$
  The inverter is composed of four IRF840 MOSFET transistors driven by two IR2113 IC’s. The DC motor is an ENGEL GNM5440E-G3.1, whose nominal parameters are

  $$\begin{array}{c}{L}_a=2.22\mathrm{mH},k_e=120.1\times {10}^{-3}\frac{\mathrm{V}·\mathrm{s}}{\mathrm{rad}},\\ R_a=0.965\mathsf{\Omega },k_m=120.1\times {10}^{-3}\frac{\mathrm{N}·\mathrm{m}}{\mathrm{A}},\\ J=118.2\times {10}^{-3}\mathrm{kg}·{\mathrm{m}}^2,b=129.6\times {10}^{-3}\frac{\mathrm{N}·\mathrm{m}·\mathrm{s}}{\mathrm{rad}}.\end{array}$$
  Similarly, variables i, $i_a$, $\upsilon$, and $\omega$ are measured via two A622 Tektronix current probes, a P5200A Tektronix voltage probe, and an E6B2-CWZ6C incremental encoder, respectively.
• Signal conditioning and DSP. Here, the DS1104 board is electrically isolated from the power stage through the NTE3087 and TLP250 optoisolators. Additionally, this block drives the boost converter and the inverter properly by generating the switched signals $u_1$ and $u_2$ via PWM.
• Generation of trajectories. The reference trajectories $i^{*}$, ${\upsilon }^{*}$, $i_a^{*}$, $u_{1av}^{*}$, and $u_{2av}^{*}$ and the desired trajectories ${\mathcal{E}}^{*}$ and ${\omega }^{*}$ are programmed in this block.

Figure 8 shows a picture of the built DC/DC boost converter–inverter–DC motor system. In the picture, the connections between the sensors, the DS1104 board, and the prototype can be observed.

4. Experimental Results

This section presents the experimental results associated with the built prototype. Additionally, comments related to the results are included.

4.1. Experiments Performed

With the aim of verifying the mathematical model of the DC/DC boost converter–inverter–DC motor system developed in Section 2.1, five experiments are presented below.

4.1.1. Experiment 1

Here, the desired trajectories ${\mathcal{E}}^{*}$ and ${\omega }^{*}$ were determined by the following Bézier polynomials [40]:

$$\begin{array}{cc}\hfill {\mathcal{E}}^{*}\left(t\right)& ={\overline{\mathcal{E}}}_i\left(t_i\right)+[{\overline{\mathcal{E}}}_f\left(t_f\right)-{\overline{\mathcal{E}}}_i\left(t_i\right)]\psi \left(t,t_i,t_f\right)\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\omega }^{*}\left(t\right)& ={\overline{\omega }}_i\left(t_i\right)+[{\overline{\omega }}_f\left(t_f\right)-{\overline{\omega }}_i\left(t_i\right)]\psi \left(t,t_i,t_f\right),\hfill \end{array}$$

(33)

where $t_i$ and $t_f$ are the initial and final times of the given trajectories. Pairs $\left[{\overline{\mathcal{E}}}_i\left(t_i\right),{\overline{\mathcal{E}}}_f\left(t_f\right)\right]$ and $\left[{\overline{\omega }}_i\left(t_i\right),{\overline{\omega }}_f\left(t_f\right)\right]$ are the constant energies and angular velocities, which are joined smoothly through ${\mathcal{E}}^{*}$ and ${\omega }^{*}$, respectively, over the time interval $\left[t_i,t_f\right]$. Function $\psi \left(t,t_i,t_f\right)$ is a polynomial defined by

$$\begin{array}{c}\hfill \psi \left(t,t_i,t_f\right)=\left\{\begin{array}{c}0\mathrm{for}t\le t_i,\hfill \\ {\left(\frac{t-t_i}{t_f-t_i}\right)}^5\times \left[252-1050\left(\frac{t-t_i}{t_f-t_i}\right)\right.+1800{\left(\frac{t-t_i}{t_f-t_i}\right)}^2\hfill \\ -1575{\left(\frac{t-t_i}{t_f-t_i}\right)}^3+700{\left(\frac{t-t_i}{t_f-t_i}\right)}^4\left.-126{\left(\frac{t-t_i}{t_f-t_i}\right)}^5\right]\mathrm{for}t\in (t_i,t_f),\hfill \\ 1\mathrm{for}t\ge t_f.\hfill \end{array}\right.\end{array}$$

In order to find the constant energies ${\overline{\mathcal{E}}}_i\left(t_i\right)$ and ${\overline{\mathcal{E}}}_f\left(t_f\right)$, the system given by Equations (21)–(24) was considered in steady state so that $\overline{i}$ could be expressed in terms of $\overline{\upsilon }$ and $\overline{\omega }$ as follows:

$$\begin{array}{cc}\hfill \overline{i}& ={\displaystyle \frac{1}{E}}\left[{\displaystyle \frac{b}{k_m}}\left({\displaystyle \frac{R_{a}b}{k_m}}+k_e\right){\overline{\omega }}^2+{\displaystyle \frac{{\overline{\upsilon }}^2}{R}}\right].\hfill \end{array}$$

(34)

When Equation (34) was replaced in Equation (25), the constant energies ${\overline{\mathcal{E}}}_i\left(t_i\right)$ and ${\overline{\mathcal{E}}}_f\left(t_f\right)$ were obtained and were given by

$$\begin{array}{cc}\hfill {\overline{\mathcal{E}}}_i\left(t_i\right)& ={\displaystyle \frac{1}{2}}{\displaystyle \frac{L}{E^2}}{\left[{\displaystyle \frac{b}{k_m}}\left({\displaystyle \frac{R_{a}b}{k_m}}+k_e\right){\overline{\omega }}_i^2+{\displaystyle \frac{{\overline{\upsilon }}_i^2}{R}}\right]}^2+{\displaystyle \frac{1}{2}}C{\overline{\upsilon }}_i^2,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\overline{\mathcal{E}}}_f\left(t_f\right)& ={\displaystyle \frac{1}{2}}{\displaystyle \frac{L}{E^2}}{\left[{\displaystyle \frac{b}{k_m}}\left({\displaystyle \frac{R_{a}b}{k_m}}+k_e\right){\overline{\omega }}_f^2+{\displaystyle \frac{{\overline{\upsilon }}_f^2}{R}}\right]}^2+{\displaystyle \frac{1}{2}}C{\overline{\upsilon }}_f^2.\hfill \end{array}$$

(36)

In this experiment, the constant voltages and angular velocities, $\left[{\overline{\upsilon }}_i,{\overline{\upsilon }}_f\right]$ and $\left[{\overline{\omega }}_i,{\overline{\omega }}_f\right]$, associated to ${\mathcal{E}}^{*}$ and ${\omega }^{*}$, were chosen as

$$\begin{array}{cc}\hfill {\overline{\upsilon }}_i& =27\mathrm{V},{\overline{\upsilon }}_f=32\mathrm{V},\hfill \\ \hfill {\overline{\omega }}_i& =10\frac{\mathrm{rad}}{\mathrm{s}},{\overline{\omega }}_f=-10\frac{\mathrm{rad}}{\mathrm{s}},\hfill \end{array}$$

(37)

over the time interval $\left[t_i,t_f\right]=\left[4\mathrm{s},6\mathrm{s}\right]$. The experimental results are depicted in Figure 9, while the tracking errors related to variables $\upsilon$, i, $\omega$, and $i_a$ are shown in Figure 10. These tracking errors were defined as $e_{\upsilon }={\upsilon }^{*}-\upsilon$, $e_i=i^{*}-i$, $e_{\omega }={\omega }^{*}-\omega$, and $e_{i_a}=i_a^{*}-i_a$.

4.1.2. Experiment 2

Reference trajectories ${\mathcal{E}}^{*}$ and ${\omega }^{*}$, corresponding to the energy and the angular velocity, were defined again by Equations (32)–(33). Parameters ${\overline{\upsilon }}_i$, ${\overline{\omega }}_i$, ${\overline{\upsilon }}_f$, and ${\overline{\omega }}_f$ were then selected as

$$\begin{array}{cc}\hfill {\overline{\upsilon }}_i& =27\mathrm{V},{\overline{\upsilon }}_f=32\mathrm{V},\hfill \\ \hfill {\overline{\omega }}_i& =-10\frac{\mathrm{rad}}{\mathrm{s}},{\overline{\omega }}_f=10\frac{\mathrm{rad}}{\mathrm{s}},\hfill \end{array}$$

(38)

over the same time interval $\left[t_i,t_f\right]$ as associated with experiment 1. The corresponding results are presented in Figure 11.

4.1.3. Experiment 3

In this experiment, the desired trajectory given by Equation (32) was proposed via the following values for the constant energies ${\overline{\mathcal{E}}}_i\left(t_i\right)$ and ${\overline{\mathcal{E}}}_f\left(t_f\right)$:

$$\begin{array}{cc}\hfill {\overline{\mathcal{E}}}_i\left(t_i\right)& =0.05\mathrm{J},{\overline{\mathcal{E}}}_f\left(t_f\right)=0.08\mathrm{J},\hfill \end{array}$$

(39)

over the same time interval $\left[t_i,t_f\right]$ as associated with experiment 1; ${\omega }^{*}$ was defined as

$$\begin{array}{cc}\hfill {\omega }^{*}\left(t\right)& =A\sin \left(\frac{2\pi }{P_1}t\right)(1-e^{-2t^3}),\hfill \end{array}$$

(40)

with A being the amplitude and $P_1$ being the period, the values of which were $10\frac{\mathrm{rad}}{\mathrm{s}}$ and $3.03\mathrm{s}$, respectively. The experimental results are presented in Figure 12.

4.1.4. Experiment 4

The desired trajectory ${\mathcal{E}}^{*}$ was defined as

$$\begin{array}{cc}\hfill {\mathcal{E}}^{*}\left(t\right)& =0.065+0.015\sin \left(\frac{2\pi }{P_2}t\right),\hfill \end{array}$$

(41)

with $P_2=3.03\mathrm{s}$, while ${\omega }^{*}$ was defined as in Equation (40) and parameters A and $P_1$ were chosen as in the previous experiment. The corresponding experimental results are plotted in Figure 13. The tracking errors related to variables $\upsilon$, i, $\omega$, and $i_a$ are depicted in Figure 14. These tracking errors were defined previously in experiment 1.

4.1.5. Experiment 5

To evaluate the dynamic behavior of the system when abrupt variations appeared in some parameters, the following change in load R was proposed:

$$R_m=\left\{\begin{array}{cc}R\hfill & \mathrm{for}0\le t<7.5\mathrm{s},\\ 30\%R\hfill & \mathrm{for}7.5\le t\le 10\mathrm{s}.\end{array}\right.$$

(42)

The trajectories ${\mathcal{E}}^{*}$ and ${\omega }^{*}$ considered in this experiment were given by Equations (32) and (33), and their parameters were defined by Equation (37). The experimental results are depicted in Figure 15.

4.2. Comments on the Experimental Results

Experiments 1 and 2 (Figure 9 and Figure 11). With the aim of studying the dynamic behavior of the system when constant and time-varying duty cycles are proposed (i.e., $u_{1av}$ and $u_{2av}$), experiments 1 and 2 considered ${\mathcal{E}}^{*}$ and ${\omega }^{*}$ as Bézier polynomials. Regarding $\upsilon$ and ${\upsilon }^{*}$, a small tracking error was observed in both experiments. Although currents i and $i_a$ were similar in shape to $i^{*}$ and $i_a^{*}$, respectively, their magnitudes were clearly different. With the intent of quantifying the tracking errors related to $\upsilon$, i, $\omega$, and $i_a$ of experiment 1, Figure 10 is presented. As can be observed in this figure, the errors do not converge to zero due to parasitic resistances, and energy losses were not considered in the mathematical model. Concerning experiment 2, the corresponding errors are not plotted, as the proposed trajectories were similar to those of experiment 1.

Experiments 3 and 4 (Figure 12 and Figure 13). Here, appropriate trajectories were considered for ${\mathcal{E}}^{*}$ and ${\omega }^{*}$ with the aim of generating the time-varying inputs $u_{1av}$ and $u_{2av}$. It was also observed, on the one hand, that $\omega \to {\omega }^{*}$. On the other hand, the tracking error related to voltages $\upsilon$ and ${\upsilon }^{*}$ was very small, while currents i and $i_a$ were similar in shape to their reference trajectories, although their magnitudes were clearly different. The tracking errors associated with the state of experiment 4, plotted in Figure 14, again did not converge to zero for the same reason as stated in the previous paragraph. The corresponding tracking errors related to experiment 3 are not presented, as the dynamic behavior of the system was similar to that of experiment 4.

Experiment 5 (Figure 15). In this experiment, an abrupt variation was considered for load R. As a consequence, after such variation was applied to the system, signals $\upsilon$ and $\omega$ no longer converged to their reference signals ${\upsilon }^{*}$ and ${\omega }^{*}$, as no control law had been designed for such a purpose.

In this manner, according to the obtained results, the experimental validation of the mathematical model presented here was satisfactory.

5. Conclusions

In this paper, the mathematical model of the DC/DC boost converter–DC motor system was developed and successfully validated. This system allows for the driving of bidirectional angular velocities. The mathematical model was obtained using circuit theory, and the experimental validation was performed on a built prototype of the system using Matlab-Simulink and a DS1104 board. The experimental tests considered time-varying duty cycles for the boost converter and the inverter by exploiting the differential flatness property of the system. Motivated by the experimental results, particularly when abrupt changes were considered in the system parameters (see Figure 15), future research will be focused on designing control techniques for solving the regulation and tracking problems related to $\upsilon$ and $\omega$.