*2.1. Buck–Boost Converter Model*

The DC–DC buck–boost converter was derived from the combination of elementary converters buck and boost. The resulting configuration can provide an output voltage of inverse polarity, either greater or smaller than the input voltage, with the same amount of elements.

Figure 1 shows the electrical diagram of the buck–boost converter. Assuming ideal components and continuous conduction mode (CCM), the averaged model of the buck– boost converter is obtained as follows [20],

$$\begin{array}{rcl} L\frac{d\bar{i}\_L}{dt} & = & V\_{\mathcal{S}}\bar{D} + (1 - \bar{D})v\_{\mathcal{C}'}\\ \mathcal{C}\frac{d\bar{v}\_{\mathcal{C}}}{dt} & = & -(1 - \bar{D})\dot{i}\_L - \frac{v\_{\mathcal{C}}}{\bar{R}} \end{array} \tag{1}$$

where *<sup>D</sup>*¯ <sup>∈</sup> [0, 1], *<sup>V</sup><sup>g</sup>* is the DC power supply, *<sup>L</sup>*, *<sup>C</sup>* and *<sup>R</sup>* are the inductance, capacitance and the resistance, respectively.

**Figure 1.** Electrical diagram of the DC–DC (direct current) buck–boost converter.

Linearization of (1) is performed through the small-signal technique, which consists of perturbing the original model signals to generate its DC and alternating current (AC) components [21]. The resulting AC component will be the linearized small-signal average model of the buck–boost converter, whose state space representation *x*˙ = *Ax* + *Bu* and *y* = *Cx* around the equilibrium point [*iL*, *vC*] = - *<sup>V</sup>gD*¯ /(*R*(<sup>1</sup> <sup>−</sup> *<sup>D</sup>*¯ ) 2 ), *<sup>V</sup>gD*¯ /(<sup>1</sup> <sup>−</sup> *<sup>D</sup>*¯ ) is given by [21],

$$
\begin{bmatrix} \hat{i}\_L \\ \hat{\boldsymbol{\upbeta}}\_\mathbb{C} \end{bmatrix} = \begin{bmatrix} \mathbf{0} & \frac{(1-\bar{D})}{L} \\ \frac{-(1-\bar{D})}{\mathbf{C}} & \frac{-1}{\overline{\mathcal{R}}\mathbf{C}} \end{bmatrix} \begin{bmatrix} \hat{i}\_L \\ \hat{\boldsymbol{\upbeta}}\_\mathbb{C} \end{bmatrix} + \begin{bmatrix} \frac{V\_\mathcal{S}}{L(1-D)} & \frac{D}{L} \\ \frac{V\_\mathcal{S}D}{\mathcal{R}C(1-D)^2} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \hat{d} \\ \boldsymbol{\upbeta}\_\mathcal{S} \end{bmatrix} \tag{2}
$$

and

$$y = \begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} \hat{i}\_L \\ \mathcal{O}\_C \end{bmatrix} \tag{3}$$

where ˆ*iL*, *v*ˆ*C*, ˆ¯*<sup>d</sup>* and *<sup>v</sup>*ˆ*<sup>g</sup>* are the perturbation terms of *<sup>i</sup>L*, *<sup>v</sup>C*, *<sup>D</sup>*¯ and *<sup>V</sup>g*, respectively.

The transfer function of buck–boost converter *Gp*(*s*) will be given by the relation ˆ¯*d*-to-*v*ˆ*<sup>c</sup>* as follows,

$$G\_p(\mathbf{s}) = \mathcal{C}(\mathbf{s}I - A)^{-1}B\_{1\prime} \tag{4}$$

where *B*<sup>1</sup> = h *Vg <sup>L</sup>*(1−*D*¯ ) *VgD*¯ *RC*(1−*D*¯ ) 2 i*T* . Thus, the system transfer function will be given by,

$$\mathbf{G}\_p(\mathbf{s}) = \frac{\left(\frac{V\_\mathcal{S}\bar{D}}{RC(1-D)^2}\right)\mathbf{s} - \left(\frac{V\_\mathcal{S}}{CL}\right)}{\mathbf{s}^2 + \left(\frac{1}{RC}\right)\mathbf{s} + \frac{(1-D)^2}{CL}},\tag{5}$$

which has a right-half plane (RHP) zero, thus it is a non-minimum phase transfer function. The buck–boost converter transfer function (5) can be divided into factors as follows,

$$\mathbf{G}\_p(\mathbf{s}) = \mathbf{G}\_{pm}(\mathbf{s})\mathbf{G}\_{nm}(\mathbf{s}),\tag{6}$$

where

$$G\_{pm}(s) = \frac{\left(\frac{V\_{\bar{\mathcal{K}}}\bar{D}}{\bar{RC}(1-D)^2}\right)\left(s + \frac{R(1-\bar{D})^2}{LD}\right)}{s^2 + \left(\frac{1}{\bar{RC}}\right)s + \frac{(1-D)^2}{CL}},\tag{7}$$

$$G\_{nm}(s) = \frac{s - \frac{R(1-\bar{D})^2}{LD}}{s + \frac{R(1-D)^2}{LD}} \,\prime \tag{8}$$

are the minimum phase and normalized non-minimum phase parts of *Gp*(*s*), respectively. Please note that *Gnm*(*s*) is an all-pass system, i.e., |*Gnm*(*jω*)| = 1, thus, the converter dynamic is given by the minimum phase part *Gpm*(*s*), which will be considered to be the uncontrolled plant. The non-minimum phase part commonly introduces a delay, but it is also responsible for the output polarity inversion.

In the following, the methodology to approximate the non-integer PID controller is described.
