**1. Introduction**

Electronic power converters as drivers for DC motors have been recently studied [1–30]. According to the literature on power converters, the Buck [1–23], the Boost [24–26], and the Buck-Boost [27–30] topologies are the most used. The Buck topology received the most attention. This is, in part, due to the fact that the mathematical model of the Buck topology is linear and, compared with the Boost and the Buck-Boost topologies, the Buck topology does not have a nonminimum phase

output variable [31]. As the present paper focuses on the Buck power converter as a driver for a DC motor, a review of state-of-the-art of this topic is presented below.

#### *1.1. Related Work*

The literature reviewed has been divided into two approaches: (1) DC/DC Buck converter–DC motor systems [1–20] and (2) DC/DC Buck converter–inverter–DC motor systems [21–23].

Regarding a DC motor when it is fed by a DC/DC Buck converter, the most relevant literature is the following; Lyshevski, in [1], designed a nonlinear PI control that regulates the velocity of the motor shaft. Ahmad et al., in [2], presented a performance assessment of the PI, fuzzy PI, and LQR controls for the tracking problem. Bingöl and Paçaci reported, in [3], a virtual laboratory based on neural networks to control the angular velocity. Sira-Ramírez and Oliver-Salazar [4] used the concepts of active disturbance rejection and differential flatness to design a tracking control for two configurations of a DC/DC Buck converter connected to a DC motor. In [5], Silva-Ortigoza et al. introduced a two-stage sensorless tracking control based on flatness, whose implementation was executed via a Σ–Δ-modulator. In [6], Hoyos et al. designed a robust adaptive quasi-sliding mode regulation control, which is generated through the zero average dynamics (ZAD) technique and a fixed point inducting control (FPIC). Later, Silva-Ortigoza et al. proposed a robust hierarchical control approach based on differential flatness in [7]. Wei et al. in [8] reported a robust adaptive controller based on dynamic surface and sliding mode. A two-stage controller based on sliding mode plus PI control and flatness was reported by Silva-Ortigoza et al. in [9]. Hernández-Guzmán et al., in [10], proposed a simple control scheme by using sliding mode, to regulate the converter current, and three PI controls. These latter to regulate the converter voltage, the motor current, and the angular velocity. Moreover, via sensorless load torque estimation schemes, a passive tracking control based on the exact tracking error dynamics was proposed by Kumar and Thilagar in [11]. Khubalkar et al., in [12], presented the design and realization of standalone digital fractional order PID control for the Buck converter–DC motor system, where the dynamic particle swarm optimization (dPSO) technique is used to tune the gains and the order of the control. By using the concept of differential flatness and a derivative-free nonlinear Kalman filter, Rigatos et al., in [13], designed a control to solve the trajectory tracking problem. Another solution was proposed by Nizami et al. in [14], where a neuroadaptive backstepping tracking control was developed for the system. The dynamic analysis of the Buck converter that uses the combined ZAD-FPIC technique to control the speed of the DC motor, when different reference values are considered, was developed in [15] by Hoyos et al. Khubalkar et al., in [16], presented for the DC/DC Buck converter driving a DC motor a digital implementation of a fractional order PID control, whose parameters are tuned through the improved inertia weight dPSO technique. A flatness-based tracking control implemented in successive loops was presented by Rigatos et al. in [17]. More recently, the speed regulation problem was addressed by Yang et al. in [18], by using a robust predictive control via a discrete-time reduced-order GPI observer. Additionally, other important contributions related to the connection of a DC/DC Buck converter and a DC motor have been reported in [19,20].

On the other hand, regarding a DC motor when it is fed by a DC/DC Buck converter–inverter, the literature is as follows. In [21], Silva-Ortigoza et al. developed and experimentally validated a mathematical model associated with the DC/DC Buck converter–inverter–DC motor system. Silva-Ortigoza et al. reported, in [22], a passive tracking control based on the exact tracking error dynamics. Robust tracking controls were proposed by Hernández-Márquez et al. in [23].

#### *1.2. Discussion of Related Work and Contribution*

In accordance with the aforementioned, different approaches have been proposed for a DC motor fed by a DC/DC Buck converter when the unidirectional rotation of the motor shaft is considered [1–19]. This unidirectional rotation emerges because the Buck converter only delivers unipolar voltages. In this regard, when an inverter is integrated between the converter and the DC motor, bidirectional rotation of the motor shaft is achieved, giving rise to the "DC/DC Buck converter–inverter–DC motor" system [21]. Related to this system, the trajectory tracking problem has been addressed in [22,23]. Note that as such a system includes an inverter connected to the DC motor, an abrupt behavior of the voltages and currents is generated because of the hard switching of the transistors composing the inverter; consequently, the useful life of the DC motor could be reduced. One manner to attenuate the abruptness of the voltages and currents and at the same time to drive a bipolar voltage to the DC motor is through the full-bridge Buck inverter, giving rise to the new "full-bridge Buck inverter–DC motor" system [32]. Thus, compared with [32], the contribution of the present paper is fourfold:


The remainder of this paper is organized as follows. In Section 2, the generalities of the full-bridge inverter DC–motor system and some static/dynamic properties and the generation of the reference trajectories, via the flatness concept, are presented. To validate the proposed mathematical model, in Section 3, circuit simulation and experimental results are shown. Later, a discussion of the obtained results is introduced in Section 4. Finally, concluding remarks are given in Section 5.

#### **2. Materials and Methods**

This section describes the key concepts of the full-bridge Buck inverter–DC motor system. First, the mathematical model of the system is obtained by using the circuit theory and the mathematical model of a DC motor. Second, some static and dynamic properties related to the deduced model are listed. Last, the generation of the reference trajectories is introduced.

#### *2.1. Model of the "Full-Bridge Buck Inverter–DC Motor" System*

In the following, the generalities of the full-bridge Buck inverter–DC motor system and the deduction of its corresponding mathematical model are presented.

#### 2.1.1. Generalities of the Full-Bridge Buck Inverter–DC Motor System

The electronic diagram of the full-bridge Buck inverter–DC motor topology is drawn in Figure 1a. Such a circuit can be divided into two parts: (1) the full-bridge Buck inverter and (2) the DC motor. The full-bridge Buck inverter modulates and supplies the bipolar voltage *υ* to the DC motor via the input signal *u*. This part also contains the following; a power supply *E*; an array of four MOSFET transistors— *Q*1, *Q*1, *Q*2, and *Q*2; and a low-pass filter composed of *R*, *C*, and *L* (where the current *i* flows through). In addition, the DC motor is the actuator of the system and is made up of the elements *La*, *Ra* and a variable *ia*, corresponding to the inductance, resistance, and armature current. Here, *ω* is the angular velocity of the shaft. Other important values of the DC motor are *b*, *km*, *J*, and *ke*, which correspond to the viscous friction coefficient of the motor, the motor torque constant, the moment of inertia of the rotor, and the counterelectromotive force constant, respectively. Additionally, Figure 1b depicts the ideal configuration of the system, which will be described in the following section.

**Figure 1.** The full-bridge Buck inverter–DC motor topology: (**a**) proposed structure and (**b**) ideal configuration.

2.1.2. Mathematical Model of the Proposed "Full-Bridge Buck Inverter–DC Motor" Topology

In Figure 1b, the ideal structure of the proposed "full-bridge Buck inverter–DC motor" topology is depicted. In this figure, the transistors *Q*1, *Q*2, *Q*1, and *Q*2 are replaced by switches *S*1, *S*2, *S*1, and *S*2, respectively. When switches *S*1 or *S*2 are on, the possible values of the input signal *u* are 1 or −1. These values depend on the voltage polarity, or operating cycle, that is desired to be generated in load *R*. That is, for a positive voltage, or positive cycle, the switch *S*1 will be on and the input signal *u* will be 1. For a negative voltage, or negative cycle, the switch *S*2 will be on and the output signal *u* will be −1. On the other hand, when switches *S*1 and *S*2 are off, the input signal *u* is 0. During this commutation process, the switches *S*1 and *S*2 are activated complementarily to *S*1 and *S*2. With the aim of easing the deduction of the mathematical model of the full-bridge Buck inverter–DC motor topology, different structures (associated with the positions of the inverter switches) will be analyzed. Thus, in accordance with the bipolarity of the voltage *υ*, the deduction of the mathematical model will be divided into positive and negative cycles.
