*3.1. Control Design and Numerical Results*

The suggested algorithm to synthesize the approximation of the fractional-order PID (FOPID) controller for the buck–boost converter comprises the following steps:


Since buck–boost converter operates in both elementary conversion modes and considering that its transfer function is one of varying parameters, controller design is developed under the following conditions: buck–boost converter transfer function (5) and (6), whose parameter values are shown in Table 1, desired stability margins of *g<sup>m</sup>* ≥ 10 dB, <sup>30</sup>◦ <sup>≤</sup> *<sup>φ</sup>dm* <sup>≤</sup> <sup>60</sup>◦ and equilibrium points generated with the average duty cycle *<sup>D</sup>*¯ <sup>=</sup> 0.375 and *D*¯ = 0.583 for buck and boost conversion modes to produce an output voltage of *V<sup>o</sup>* = 15 V and *V<sup>o</sup>* = 35 V, respectively.

**Element Notation Value** DC voltage source *V<sup>g</sup>* 25 V Capacitor *C* 30 µF Inductor *L* 10 mH Resistance *R* 10 Ω Switching frequency *f<sup>s</sup>* 20 kHz

**Table 1.** Parameter values of buck–boost converter in Figure 1.

By considering the described conditions, the minimum phase transfer function *Gpm*(*s*) of buck–boost converter is linearized around equilibrium points [*iL*, *vC*] = [2.4, −15] for buck conversion mode and [*iL*, *vC*] = [8.4, −35] for boost conversion mode. In Table 2, computation of phase margin *φm*, uncontrolled plant phase *φpm*, and fractional-order *α* for buck and boost conversion modes are provided.

**Table 2.** Values of *φm*, *φpm* and *α* for buck/boost conversion modes.


Therefore, the controller phase contributions are −60.7◦ and −60.5◦ for buck and boost conversion modes, respectively. Please note that the algorithm for controller design provides a very similar result for both conversion modes up to this point. However, it should be kept in mind that the transfer function is of varying parameters; thus, its frequency response changes depending on the equilibrium point considered in the linearization. In Figure 3, the fractional-order approximations that are generated with the computed values of *α* are shown. One can see their similarity in shape and phase contribution, but they differ in their frequency band.

**Figure 3.** Frequency response of fractional-order approximation for buck/boost conversion modes.

The controller *Gc*(*s*) can be obtained by manipulating (9) as *s <sup>α</sup>* <sup>≈</sup> *<sup>T</sup>*(*s*) <sup>≡</sup> *<sup>N</sup>*(*s*)/*D*(*s*) and then substituting in (14), thus the controller structure will be given by,

$$G\_{\mathcal{L}}(\mathbf{s}) = k\_{\mathfrak{c}} \frac{(T\_i N(\mathbf{s}) + D(\mathbf{s}))^2}{N(\mathbf{s}) D(\mathbf{s})},\tag{16}$$

from which one concludes that controller effect can be varied between integral and derivative depending on the value of *T<sup>i</sup>* . In Figure 4 the transition between both effects are shown. As can be seen, integral (derivative) effect is achieved as *T<sup>i</sup>* → 0 (*T<sup>i</sup>* → ∞).

**Figure 4.** Transition between integral and derivative effect as function of *T<sup>i</sup>* .

By setting *T<sup>i</sup>* = 0.001 and *k<sup>c</sup>* = 3, the controller *Gc*(*s*) produces the necessary effect to fulfill the required stability margins in both conversion modes. The resulting controller structure for converter in Figure 1 is given by,

$$G\_c(s) = k \frac{s^4 + \rho\_1 s^3 + \rho\_2 s^2 + \rho\_3 s + \rho\_4}{s^4 + \psi\_1 s^3 + \psi\_2 s^2 + \psi\_3 s + \psi\_4},\tag{17}$$

whose parameters are listed in Table 3.

**Table 3.** Coefficients for approximation of the fractional-order PID controller (17).


The step response of the closed-loop system will allow us to determine the effectiveness of the synthesized controller. In Figure 5, the regulation capacity of the proposed controller can be confirmed. One can see that the response exhibits a fast and stable tracking characteristic for both conversion modes, which can be corroborated quantitatively through performance parameters in Table 4, column 2.

**Figure 5.** Closed-loop step response for both (**a**) buck and (**b**) boost conversion modes of converter in Figure 1.

**Table 4.** Closed-loop response performance parameters for buck/boost conversion modes, where *ess*, *τ*, *tr*, *tp*, *t<sup>s</sup>* and %*M* stand for steady-state error, time constant, rising time, peak time, settling time and overshoot, respectively.


Although regulation velocity of the response is acceptable, the proposed controller performs different depending on the selected conversion mode. These differences are attributed to the operating frequency band of the approximation, being the one for boost conversion mode of a higher frequency range, as shown in Figure 3.

A comparison of the fractional-order PID approximation with a typical PID controller allows us to determine that the former outperforms the latter. By using MatLab algorithm to tune PID controller, targeting 60◦ phase margin and a compromise between robustness and performance, typical PID controllers were tuned, resulting [*kp*, *T<sup>i</sup>* , *T<sup>d</sup>* ]buck = [0.068, 3.85 <sup>×</sup> <sup>10</sup>−<sup>5</sup> , 9.92 <sup>×</sup> <sup>10</sup>−<sup>9</sup> ] and [*kp*, *T<sup>i</sup>* , *T<sup>d</sup>* ]boost = [0.021, 2.59 <sup>×</sup> <sup>10</sup>−<sup>5</sup> , 8.04 <sup>×</sup> <sup>10</sup>−<sup>9</sup> ] for both conversion modes. Performance of inter-order PID controller is quantified in Table 4 column 3. By comparing columns 2 and 3, the proposed approach superiority can be determined through time constants since the ones obtained with the approximation of fractional-order PID controller are much smaller. Although typical PID achieves the output regulation with zero steady-state error, it took longer in both conversion modes to reach the reference value.

To determine if a typical PID structure can equalize performance of the proposed approach and considering that an increase in performance reduces robustness and vice versa, a second option of PID controller was tuned with a less restrictive phase margin aiming to obtain performance improvement, resulting [*kp*, *T<sup>i</sup>* , *T<sup>d</sup>* ]buck = [0.034, 1.51 <sup>×</sup> <sup>10</sup>−<sup>5</sup> , 3.78 × 10−<sup>6</sup> ] and [*kp*, *T<sup>i</sup>* , *T<sup>d</sup>* ]boost = [0.123, 1.37 <sup>×</sup> <sup>10</sup>−<sup>6</sup> , 3.43 <sup>×</sup> <sup>10</sup>−<sup>7</sup> ] for both conversion modes. In Table 4, column 4 performance parameters produced by the second PID controller are provided. Please note that despite the increase in performance, the second option of integer-order PID controller produces time constants that are not competitive with those of the proposed approach.

The comparison is moved to the frequency domain to corroborate the stability margins and determine the effect of controller on the magnitude/phase curves of the closed-loop response. In Figure 6, frequency response of closed-loop system with fractional-order PID and typical PID controllers, operating in buck and boost conversion modes, are shown. One can see that both controllers were able to achieve the desired phase margin. Note the shape similarity of the magnitude curves. They both have their peaks around the same value, which corroborates that overshoot is similar for both controllers. The operating frequency band is wider for the system controlled with the fractional-order PID approximation. The latter is consistent with the response velocity measured for system controlled with the proposed controller, since the wider the bandwidth the shorter the rising time, due to the higher-frequency signals pass through the system more easily.

**Figure 6.** Frequency response of closed-loop system for both (**a**) buck and (**b**) boost conversion modes.

Lastly, frequency response of the sensitivity and complementary sensitivity functions allows us to determine the robustness of the controller through its disturbance and noise rejection characteristics. Recalling that sensitivity function *S* = 1/(1 + *GpGc*), and complementary sensitivity function *T* = *GpGc*/(1 + *GpGc*) determine how disturbances/perturbations and noise affect respectively the output, it is expected the controller *G<sup>c</sup>* to produce a curve with attenuation in low frequencies for *S* and a curve with attenuation in high ones for *T* ([28], Chap. 4). In Figure 7, frequency response of sensitivity *S* and complementary sensitivity *T* functions is shown for both conversion modes, where *L* = *GpG<sup>c</sup>* is the loop gain, when using fractional-order PID and typical PID controllers.

**Figure 7.** Frequency response of sensitivity *S*, complementary sensitivity *T* functions and loop gain *L* for both (**a**) buck and (**b**) boost conversion modes.

Note the magnitude flatness of sensitivity function *S* produced by the fractional-order PID controller, which implies that it attenuates better disturbances/perturbations for a wider frequency band. On the other side, complementary sensitivity function *T* successfully attenuates high-frequency noise. Thus, a better disturbance/perturbation and noise rejection characteristic is obtained with the fractional-order PID controller approximation and therefore, the closed-loop system will exhibit a robust performance.

In the following section, a generation of the fractional-order PID approximation to facilitate controller implementation is derived. Experimental results are also provided and described in that section.
