*2.3. Synthesis of Fractional-Order PID Approximation*

The standard representation of a PID controller *Gc*(*s*) = *kp*(1 + 1/(*T<sup>i</sup> s*) + *T<sup>d</sup> s*) was modified to consider the non-integer approach in its integral/derivative modes. The fractionalorder PI*α*D*<sup>µ</sup>* structure is a special case of PID controller with additional degrees of freedom which is described as follows [24]

$$\mathbf{G}\_{\mathcal{L}}(\mathbf{s}) = k\_p \left( 1 + \frac{1}{T\_l s^a} + T\_d s^\mu \right) \tag{13}$$

where 0 < *α*, *µ* < 1, *k<sup>p</sup>* is the proportional gain, *T<sup>i</sup>* and *T<sup>d</sup>* are the integral and derivative time constants, respectively.

Different optimization strategies have suggested that *T<sup>i</sup>* and *T<sup>d</sup>* are related through *T<sup>i</sup>* = *ηT<sup>d</sup>* , where *η* is a constant. The first suggestion derived in *η* = 4 aiming to achieve a compromise between controller performance and its viability to be implemented [25]. Motivated by the need to ensure unique solutions of (13), some other results showed that smaller values of *η* produced significant improvements [25,26]. By slightly modifying (13) and by setting *α* = *µ*, *η* = 1, (13) can be expressed as follows,

$$G\_{\mathfrak{c}}(\mathbf{s}) = \frac{k\_{\mathfrak{c}}(T\_i \mathbf{s}^{\mathfrak{a}} + 1)^2}{\mathbf{s}^{\mathfrak{a}}},\tag{14}$$

which directly simplifies the PID structure through a perfect square trinomial [27], where *k<sup>c</sup>* = *kp*/*T<sup>i</sup>* .

The phase of the plant to be controlled *φpm*, the controller phase *φ<sup>c</sup>* and phase margin *φ<sup>m</sup>* are related through *φ<sup>c</sup>* + *φpm* = −*π* + *φ<sup>m</sup>* at the phase crossover frequency *ωpc*, which implies that *φ<sup>c</sup>* = *φ<sup>m</sup>* − *π* − *φpm*, thus,

$$\mathfrak{a} = \frac{(-\pi - \phi\_{pm} + \phi\_{m})}{(\pi/2)}.\tag{15}$$

In the next section, the fractional-order PID controller approximation is validated numerically and experimentally. A generalization of its structure for buck and boost modes of the converter is derived as well.
