## 2 Materials and Methods
### Speed Control of a DC Motor  
As a speed control is applied to the DC motor \(W_{m}\), the reference speed \(W_{mref}\) must be followed in the rotary load. Thus, in the sampling period \(kT\), the tracking error is defined in Equation (10):  
\[e(kT)=W_{m}(kT)-W_{mref}(kT) \tag{10}\]  
Furthermore, by considering that the system is of fourth order, the sliding surface \(s(x)\)[29] describes the third order dynamic behavior of the error (\(e(kT)\)), given by Equation (11):  
\[s(kT)=e(kT)+k_{s1}\frac{de(kT)}{d(kT)}+k_{s2}\frac{d^{2}e(kT)}{d(kT)^{2}}+k_{s 3}\frac{d^{3}e(kT)}{d(kT)^{3}} \tag{11}\]  
The coefficients \(k_{s1},k_{s2},\) and \(k_{s3}\) are parameterized as a function of the constant of the inductive-capacitive (LC) filter multiplied by the factors \(K_{S1},K_{S2},\) and \(K_{S3},\) as shown in Equation (12). The factors \(K_{S1},K_{S2},\) and \(K_{S3}\) are the parameters of the ZAD controller and can be adjusted to impose a desired dynamic in the closed loop system. Note that, such parameters can be considered to construct dimensional bifurcation diagrams.  
\[\begin{array}{c}k_{s1}=K_{S1}\sqrt{LC}\\ k_{s2}=K_{S2}LC\\ k_{s3}=K_{S3}LC\sqrt{LC}\end{array} \tag{12}\]  
When the reference signal \(W_{mref}\) is constant, Equation (11) can be written as shown in Equation (13), and the derivate is equal to Equation (14):  
\[s(kT)=W_{m}(kT)-W_{mref}(kT)+k_{s1}\frac{dW_{m}(kT)}{d(kT)}+k_{s2}\frac{d^{2}W _{m}(kT)}{d(kT)^{2}}+k_{s3}\frac{d^{3}W_{m}(kT)}{d(kT)^{3}} \tag{13}\]  
\[\dot{s}(kT)=\frac{dW_{m}(kT)}{d(kT)}+k_{s1}\frac{d^{2}W_{m}(kT)}{d(kT)^{2}}+k _{s2}\frac{d^{3}W_{m}(kT)}{d(kT)^{3}}+k_{s3}\frac{d^{4}W_{m}(kT)}{d(kT)^{4}} \tag{14}\]  
The duty cycle can be calculated as expressed in Equation (15):  
\[d_{k}(kT)=\frac{2s(kT)+T\dot{s}_{-}(kT)}{T\big{(}\dot{s}_{-}(kT)-\dot{s}_{+}( kT)\big{)}} \tag{15}\]  
where \(s(kT)\) is calculated with Equation (11) at the beginning of each switching period for the system given in Equation (3). Thus, \(s(kT)=s(kT)|\,\dot{s}_{=ON}\); \(\dot{s}_{+}(kT)\) is calculated with Equation (12) for the system given in Equation (3); that is, \(\dot{s}_{+}(kT)=\dot{s}(kT)|\,_{S=ON}\). Additionally, the parameter \(\dot{s}_{-}(kT)\) is calculated with Equation (14) for the system given by Equation (5); that is, \(\dot{s}_{-}(kT)=\dot{s}(kT)|\,_{S=OFF}\).  
Given the period of delay in the control action, the new duty cycle is calculated, as in Equation (16).  
\[d_{k}(kT)=\frac{2s((k-1)T)+T\dot{s}_{-}((k-1)T)}{T\big{(}\dot{s}_{-}((k-1)T)- \dot{s}_{+}((k-1)T)\big{)}} \tag{16}\]By considering the ZAD-FPIC technique, the new duty cycle is calculated to ensure that the load and the motor rotate at the desired speed (\(W_{mref}\)), leading to the expression for the duty cycle given in Equation (17).  
\[d_{ZADFPIC}(kT)=\frac{d_{k}(kT)+Nd^{*}}{N+1} \tag{17}\]  
Combining Equations (16) and (17), the following control law of the ZAD-FPIC is obtained, as shown in Equation (18):  
\[d_{ZADFPIC}(kT)=\left(\frac{2s((k-1)T)+\dot{Ts}_{-}((k-1)T)}{T\big{(}\dot{s}_{ -}((k-1)T)-\dot{s}_{+}((k-1)T)\big{)}}+Nd^{*}\right)\left(N+1\right)^{-1} \tag{18}\]  
where \(d^{*}\) is calculated at the beginning of each period, as shown in Equation (19):  
\[d^{*}=d_{k}(kT)|_{\text{estado~{}stable}} \tag{19}\]  
Finally, a saturation function must be applied to consider the actual physical limits of the duty cycle between 0 and 1, leading to the saturation function described in Equation (20):  
\[d=\left\{\begin{array}{ccc}d_{ZADFPIC}(kT)&\text{if}&0<d_{ZADFPIC}(kT)<1\\ 1&\text{if}&1\leq d_{ZADFPIC}(kT)\\ 0&\text{if}&d_{ZADFPIC}(kT)\leq 0\end{array}\right. \tag{20}\]