*Positive Cycle*

The generation of a positive voltage, i.e., the clockwise movement of the motor shaft, is executed when the ideal circuit of Figure 1b is simplified to the circuit shown in Figure 2a. It is noteworthy that switches *S*2 and *S*2 are in a fixed position, whereas switches *S*1 and *S*1 work complementarily, switching their position according to the input signal *u*. Thus, similar to a Buck converter, the energy charging in the *LC* filter occurs when the input signal *u* = 1, whereas energy discharging occurs when *u* = 0. This behavior is summarized in Figure 2b.

**Figure 2.** The full-bridge Buck inverter–DC motor system: (**a**) ideal circuit for positive cycle; (**b**) in positive cycle *Q*1 and *Q*1 of Figure 1 operate as a transistor and as a diode, respectively; (**c**) energy charging mode of operation; and (**d**) energy discharging mode of operation.

**(i) Energy charging.** In this mode of operation, a part of the energy supplied by the power supply is stored in the *LC* filter. Figure 2c depicts the structure of this operating mode. By using the mathematical model of a DC motor [34,35] and applying Kirchhoff's laws to the electric circuit of Figure 2c, the following system of differential equations is obtained:

$$L\frac{di}{dt} = -\nu + E,\tag{1}$$

$$C\frac{dv}{dt} = i - \frac{v}{R} - i\_{a\prime} \tag{2}$$

$$L\_a \frac{di\_a}{dt} = \upsilon - R\_a i\_a - k\_c \omega\_\prime \tag{3}$$

$$J\frac{d\omega}{dt} = k\_m i\_a - b\omega.\tag{4}$$

**(ii) Energy discharging.** Here, as the *LC* filter is no longer connected to the power supply, the energy stored in the filter is released directly to the resistance *R* and to the DC motor. Figure 2d shows the connection of this operating mode. By using the mathematical model of a DC motor and applying Kirchhoff's laws, the following system associated with the circuit of Figure 2d is obtained:

$$L\frac{di}{dt} = -\upsilon\_{\prime} \tag{5}$$

$$C\frac{d\upsilon}{dt} = i - \frac{\upsilon}{R} - i\_{a\prime} \tag{6}$$

$$L\_a \frac{di\_a}{dt} = \upsilon - R\_a i\_a - k\_c \omega\_\prime \tag{7}$$

$$J\frac{d\omega}{dt} = k\_m i\_a - b\omega.\tag{8}$$

*Negative Cycle*

In generating the negative voltage, the ideal circuit of Figure 1b reduces to the one shown in Figure 3a. Here, switches *S*1 and *S*1 are in a fixed position, whereas switches *S*2 and *S*2 work complementarily, switching their position according to the input *u*. Similarly to the positive cycle, there is energy charging or discharging in the *LC* filter. This behavior is summarized in Figure 3b.

**Figure 3.** The full-bridge Buck inverter–DC motor system: (**a**) ideal circuit for negative cycle; (**b**) in negative cycle *Q*2 and *Q*2 of Figure 1 operate as a transistor and as a diode, respectively; (**c**) energy charging mode of operation; and (**d**) energy discharging mode of operation.

**(i) Energy charging.** The circuit allowing energy charging for negative cycles is shown in Figure 3c. The model related to the circuit of Figure 3c, after applying Kirchhoff's laws and considering the mathematical model of a DC motor, is given by

$$L\frac{di}{dt} = -\upsilon - E,\tag{9}$$

$$C\frac{d\upsilon}{dt} = i - \frac{\upsilon}{R} - i\_{a\prime} \tag{10}$$

$$L\_a \frac{di\_a}{dt} = \upsilon - R\_a i\_a - k\_a \omega\_\prime \tag{11}$$

$$J\frac{d\omega}{dt} = k\_m i\_a - b\omega.\tag{12}$$

**(ii) Energy discharging.** Lastly, the energy discharging in this mode of operation is presented in Figure 3d. For this mode (see Figure 3d), the mathematical model is determined by the following system of differential equations:

$$L\frac{di}{dt} = -\upsilon\_{\prime} \tag{13}$$

$$C\frac{dv}{dt} = i - \frac{v}{R} - i\_{a\prime} \tag{14}$$

$$L\_d \frac{di\_d}{dt} = \upsilon - R\_d i\_d - k\_d \omega\_\prime \tag{15}$$

$$J\frac{d\omega}{dt} = k\_m i\_d - b\omega.\tag{16}$$

By unifying the four modes (see Figures 2c,d, and 3c,d), represented by Equations (1)–(16), the model of the full-bridge Buck inverter–DC motor topology is given by

$$L\frac{di}{dt} = -\upsilon + Eu\_\prime \tag{17}$$

$$\mathcal{C}\frac{d\upsilon}{dt} = i - \frac{\upsilon}{R} - i\_{a\prime} \tag{18}$$

$$L\_{\mathfrak{a}} \frac{d\dot{a}\_{\mathfrak{a}}}{dt} = \upsilon - R\_{\mathfrak{a}} \dot{\mathfrak{a}}\_{\mathfrak{a}} - k\_{\mathfrak{a}} \omega\_{\mathfrak{a}} \tag{19}$$

$$J\frac{d\omega}{dt} = k\_m i\_a - b\omega,\tag{20}$$

where *u* ∈ {−1, 0, 1} are the positions of the switches. Due to the discrete nature of the system modeled by Equations (17)–(20), it is usual to call it a "switched model". In contrast, the continuous model or "average model" associated with the full-bridge Buck inverter–DC motor system is described by

$$L\frac{di}{dt} = -\nu + Eu\_{av\prime} \tag{21}$$

$$C\frac{d\upsilon}{dt} = i - \frac{\upsilon}{R} - i\_{a\prime} \tag{22}$$

$$L\_d \frac{di\_d}{dt} = \upsilon - R\_d i\_d - k\_d \omega\_\prime \tag{23}$$

$$J\frac{d\omega}{dt} = k\_m i\_a - b\omega\_\prime \tag{24}$$

with *uav* ∈ [−1, 1] the average input.

#### *2.2. Properties of the "Full-Bridge Buck Inverter–DC Motor" System*

This section presents the most relevant static and dynamic properties of the full-bridge Buck inverter–DC motor system. These properties bring qualitative information about the behavior of such a system. Particularly, the steady-state, stability, controllability, and flatness properties of the system are described.
