2.2.3. Controllability

The controllability property of a dynamic system is crucial in control theory. This property states that if an input to a system can be found such that it takes the vector state from a desired initial state to a desired final state, the system is controllable; in other case, the system is uncontrollable. With the aim of determining whether a system is controllable or not, a controllability matrix C can be constructed. If matrix C is of rank *n*, being *n* the dimension of the vector state, then the system is completely controllable.

Regarding the full-bridge Buck inverter–DC motor system, represented by Equation (33), the associated controllability matrix is given by

$$\begin{array}{l} \mathcal{C} = \begin{bmatrix} \mathbf{B} & \mathbf{AB} & \mathbf{A}^2 \mathbf{B} & \mathbf{A}^3 \mathbf{B} \end{bmatrix} \\\\ \begin{bmatrix} \frac{E}{L} & 0 & -\frac{E}{L^2 \mathcal{C}} & \frac{E}{RL^2} \\\\ 0 & \frac{E}{L\mathcal{C}} & -\frac{E}{RL} & -\frac{E(R^2 L \mathcal{C} + R^2 L\_d \mathcal{C} - L L\_d \mathcal{C}^4)}{R^2 L^2 L\_d \mathcal{C}^3} \\\\ 0 & 0 & \frac{E}{L L\_d \mathcal{C}} & -\frac{E(R R\_d + L\_d \mathcal{C})}{R L L\_d^2 \mathcal{C}} \\\\ 0 & 0 & 0 & \frac{E k\_m}{L L\_d \mathcal{C}} \end{bmatrix} . \tag{37}$$

with matrices **A** and **B** defined in (34). In this way, after calculating the determinant of matrix C, one obtains

$$\det \mathcal{C} = \frac{E^4 k\_m}{f L^4 L\_a^2 \mathcal{C}^3} \neq 0,\tag{38}$$

meaning that the system is controllable.

On the other hand, an important property directly linked with controllability is that of differential flatness. This latter states that if a system is differentially flat [33], then it is controllable. This, in turn, means that the vector state and the input can be differentially parameterized in terms of the flat output and a finite number of its derivatives with respect to time. Moreover, there is a relation between the differential parametrization and the steady-state behavior, as the latter can be also obtained when the time derivatives of the flat output are equating to zero.

Note that the differential flatness property has been exploited during the past few years in DC/DC power converters-DC motor and DC/DC power converters-inverter-DC motor systems for different purposes. The most common ones are: (a) as a generator of time-varying reference trajectories to be used in validating mathematical models [21,25] and in passive controls [22,24,28] and (b) for control design purposes [4,5,7,9,23,29]. This paper exploits the flatness property with the intention of generating the reference trajectories for validating the obtained mathematical model, as will be presented in the following section.

#### *2.3. Generation of Reference Trajectories via Differential Flatness*

After finding that det C = 0, i.e., the full-bridge Buck inverter–DC motor system is differentially flat, the flat output of the overall system is found through the following mathematical statement:

$$[0\ 0\ 0\ 1] \mathcal{C}^{-1} \mathbf{x} = \frac{JLL\_{\text{u}} \mathcal{C}}{Ek\_{\text{m}}} \omega\_{\text{\textdegree}} \tag{39}$$

and, without loss of generality, the flat output of the system described by Equations (21)–(24) can be taken as

$$\mathcal{S} = \omega\_{\prime} \tag{40}$$

which corresponds to the angular velocity of the full-bridge Buck inverter–DC motor system. Therefore, the variables *ia*, *υ*, *i*, and the input *uav* of the system can be expressed in terms of S and its successive derivatives with respect to time as follows,

$$
\omega = \mathbb{S},
\tag{41}
$$

$$
\dot{q}\_a = \frac{J}{k\_m} \mathcal{S} + \frac{b}{k\_m} \mathcal{S},
\tag{42}
$$

*Electronics* **2019**, *8*, 1216

$$\begin{aligned} \boldsymbol{\nu} &= \frac{\boldsymbol{I} \mathbf{L}\_{\text{d}}}{k\_{\text{m}}} \boldsymbol{\mathcal{S}} + \left( \frac{\boldsymbol{b} \mathbf{L}\_{\text{d}} + \boldsymbol{f} \mathbf{R}\_{\text{d}}}{k\_{\text{m}}} \right) \boldsymbol{\mathcal{S}} + \left( \frac{\boldsymbol{b} \mathbf{R}\_{\text{d}}}{k\_{\text{m}}} + k\_{\text{c}} \right) \boldsymbol{\mathcal{S}}, \\\ \boldsymbol{\mathcal{S}} &= \left( \boldsymbol{f} \boldsymbol{L}\_{\text{d}} \mathbf{C} \right) \boldsymbol{\mathcal{S}}^{(3)} \boldsymbol{\mathcal{S}}^{(3)} + \left( \boldsymbol{b} \mathbf{R} \boldsymbol{L}\_{\text{d}} \mathbf{C} + \boldsymbol{f} \mathbf{R} \mathbf{R}\_{\text{d}} \mathbf{C} + \boldsymbol{f} \mathbf{L}\_{\text{d}} \right) \boldsymbol{\mathcal{S}} \end{aligned} \tag{43}$$

$$\begin{split} i &= \left(\frac{fL\_{\rm a}\mathbb{C}}{k\_{\rm m}}\right)\mathcal{S}^{(3)} + \left(\frac{bRL\_{\rm a}\mathbb{C} + fRR\_{\rm a}\mathbb{C} + fL\_{\rm a}}{Rk\_{\rm m}}\right)\ddot{\mathcal{S}} \\ &+ \left(\frac{bL\_{\rm a} + fR\_{\rm a} + fR + bRR\_{\rm a}\mathbb{C} + RK\_{\rm c}K\_{\rm m}\mathbb{C}}{k\_{\rm m}R}\right)\dot{\mathcal{S}} + \left(\frac{bR\_{\rm a} + k\_{\rm c}k\_{\rm m} + bR}{k\_{\rm m}R}\right)\mathcal{S}, \end{split} \tag{44}$$

$$\begin{split} \mu\_{\text{av}} &= \left(\frac{\|I\|\_{\text{d}}\mathrm{L}\mathbb{C}}{\mathrm{E}k\_{\text{m}}}\right) \mathcal{S}^{(4)} + \left(\frac{b\mathrm{R}\mathrm{L}\mathrm{L}\_{\text{d}}\mathrm{C} + \mathrm{J}\mathrm{R}\mathrm{R}\_{\text{d}}\mathrm{L}\mathrm{C} + \mathrm{J}\mathrm{L}\mathrm{L}\_{\text{d}}}{\mathrm{E}k\_{\text{m}}\mathrm{R}}\right) \mathcal{S}^{(3)} \\ &+ \left(\frac{b\mathrm{L}\mathrm{L}\_{\text{d}} + \mathrm{J}\mathrm{R}\_{\text{d}}\mathrm{L} + \mathrm{J}\mathrm{R}\mathrm{L} + b\mathrm{R}\mathrm{R}\_{\text{d}}\mathrm{L}\mathrm{C} + \mathrm{K}\_{\text{c}}\mathrm{K}\_{\text{m}}\mathrm{R}\mathrm{L}\mathrm{C} + \mathrm{J}\mathrm{R}\mathrm{L}\_{\text{d}}}{\mathrm{E}k\_{\text{m}}\mathrm{R}}\right) \mathcal{S} \\ &+ \left(\frac{b\mathrm{R}\_{\text{d}}\mathrm{L} + k\_{\text{c}}k\_{\text{m}}\mathrm{L} + b\mathrm{R}\mathrm{L} + b\mathrm{R}\mathrm{L}\_{\text{d}} + \mathrm{J}\mathrm{R}\mathrm{R}\_{\text{d}}}{\mathrm{E}k\_{\text{m}}\mathrm{R}}\right) \mathcal{S} + \left(\frac{b\mathrm{R}\_{\text{d}} + k\_{\text{c}}k\_{\text{m}}}{\mathrm{E}k\_{\text{m}}}\right) \mathcal{S}. \end{split}$$

From the previous results, if a desired trajectory S∗ is proposed, i.e., *<sup>ω</sup>*<sup>∗</sup>, then from Equation (45), the input to be introduced into the full-bridge Buck inverter–DC motor system is

$$\begin{split} u\_{dv}^{\*} &= \left(\frac{I\mathcal{L}\_{d}\mathcal{L}\mathcal{C}}{\mathcal{E}k\_{m}}\right) \mathcal{S}^{\*(4)} + \left(\frac{b\mathcal{R}\mathcal{L}\mathcal{L}\_{d}\mathcal{C} + I\mathcal{R}\mathcal{R}\_{d}\mathcal{L}\mathcal{C} + I\mathcal{L}\mathcal{L}\_{d}}{\mathcal{E}k\_{m}\mathcal{R}}\right) \mathcal{S}^{\*(3)} \\ &+ \left(\frac{b\mathcal{L}\mathcal{L}\_{d} + I\mathcal{R}\_{d}\mathcal{L} + I\mathcal{R}\mathcal{L} + b\mathcal{R}\mathcal{R}\_{d}\mathcal{L}\mathcal{C} + \mathcal{K}\_{c}\mathcal{K}\_{m}\mathcal{R}\mathcal{L}\mathcal{C} + I\mathcal{R}\mathcal{L}\_{d}}{\mathcal{E}k\_{m}\mathcal{R}}\right) \mathcal{S}^{\*} \\ &+ \left(\frac{b\mathcal{R}\_{d}\mathcal{L} + k\_{c}k\_{m}\mathcal{L} + b\mathcal{R}\mathcal{L} + b\mathcal{R}\mathcal{L}\_{d} + I\mathcal{R}\mathcal{R}\_{d}}{\mathcal{E}k\_{m}\mathcal{R}}\right) \mathcal{S}^{\*} + \left(\frac{b\mathcal{R}\_{d} + k\_{c}k\_{m}}{\mathcal{E}k\_{m}}\right) \mathcal{S}^{\*}, \end{split}$$

achieving that *ω* be similar to *<sup>ω</sup>*<sup>∗</sup>, meaning that the mathematical model is sufficiently accurate. In addition, when S∗ is replaced in Equations (42)–(44), the reference trajectories of the system, i.e., *i*∗*a* , *<sup>υ</sup>*<sup>∗</sup>, and *i*<sup>∗</sup>, are obtained offline.
