## 4 Control System Synthesis  
The cascade structure for regulating the output voltage of the FIBC is illustrated in Figure 2. The inner current control loop ensures the inductor current exactly tracks the current reference that is generated by the outer voltage control loop in finite time despite the variation of input voltage. To control the ON-OFF state of the power switches, two PWM signals are constituted based on the control actions constructed by the inner loop. Meanwhile, the output voltage of the FIBC is guaranteed to closely follow the voltage reference in the presence of load change, which always exists in fuel cell applications.  
Figure 1: The schematic of the considered DC–DC boost converter.  
From (41), the equilibrium point \((dv_{o}/dt=0;dI_{Lref}/dt=0)\) is determined as  
\[\begin{cases}v_{o}=V_{ref}\\ i_{L}=I_{Lref}=\left(\frac{v_{o}+v_{in}}{v_{in}}\right)\frac{v_{o}}{2R}\end{cases} \tag{42}\]  
By linearizing (41) about the equilibrium point, we have [39]  
\[\frac{dv_{o}}{dt}=-\frac{2}{RC}\frac{v_{in}+2v_{o}}{v_{in}+v_{o}}v_{o}+\frac{4 }{C}\frac{v_{in}}{v_{in}+v_{o}}I_{Lref}-\frac{2L}{RC}\frac{V_{ref}}{v_{in}} \frac{dI_{Lref}}{dt} \tag{43}\]  
Hence, (42) can be rewritten in the following form  
\[\hat{v}_{o}=\alpha v_{o}+\beta u+\gamma\dot{u} \tag{44}\]  
where \(u=I_{Lref}\) is the fictitious control input and coefficients \(\alpha\), \(\beta\), and \(\gamma\) are given by  
\[\alpha=-\frac{2}{RC}\frac{v_{in}+2v_{o}}{v_{in}+v_{o}};\beta=\frac{4}{C}\frac {v_{in}}{v_{in}+v_{o}};\gamma=-\frac{2L}{RC}\frac{V_{ref}}{v_{in}} \tag{45}\]  
**Remark 2**.: _From (44), it can be recognized that the output voltage of the FIBC is disturbed by not only the variation in input voltage but also the load change. In addition, the control gain \(\beta\) is unknown since it depends on the unmeasurable input voltage. Therefore, an active disturbance rejection control framework should be employed to compensate for the adverse effects of the above disturbances and achieve a high voltage regulation performance._  
To address the unknown control gain, the dynamic of the output voltage is transformed into  
\[\dot{v}_{o}=\beta_{0}u+f(v_{o},u,\dot{u},w) \tag{46}\]  
where \(\beta_{0}\) is a positive constant in (46), which is determined based on nominal parameters of the FIBC and \(f(v_{o},u,\dot{u},w)=\gamma\dot{u}+\alpha v_{o}+(\beta-\beta_{0})u\) is the total disturbance term with \(w\) characterized as the input variation and load change effects.  
**Assumption 1**.: _The total disturbance \(f(v_{o},u,\dot{u},w)\) is differential and the upper bounded of its first-order derivative is restricted by an unknown constant, i.e., \(\left|\dot{f}(v_{o},u,\dot{u},w)\right|\leq H\)._  
Defining the state vector \(x=[x_{1},x_{2}]^{T}=[v_{o},f]^{T}\), (46) can be converted into  
\[\begin{split}\dot{x}_{1}&=\beta_{0}u+x_{2}\\ \dot{x}_{2}&=h(x_{1},u,\dot{u},w(t))\end{split} \tag{47}\]  
where \(h(x_{1},u,\dot{u},w(t))=\dot{f}(x_{1},u,\dot{u  
where \(\eta_{1}\) and \(\eta_{2}\) are positive constants.  
**Remark 3**.: _Considering the system dynamics (47) and appropriately choosing the parameter \(\omega\) of the observer (48), the estimated values \(\hat{x}_{1}\) and \(\hat{x}_{2}\) reach their actual values \(x_{1}\) and \(x_{2}\) with arbitrarily small estimation errors. Compared to the conventional ESO, the observer (48) can observe the lumped disturbance more exactly by using the fractional order of the error \(\tilde{x}_{1}\)._  
The voltage regulation error is given by  
\[e=V_{ref}-v_{o} \tag{50}\]  
Taking the derivative of it and combining with (46) yields  
\[\dot{e}=\dot{V}_{ref}-\beta_{0}u-f(v_{o},u,\dot{u},w(t)) \tag{51}\]  
It is assumed that the total disturbance is well estimated to compensate for the effect of the lumped disturbance on the system performance, and since \(\dot{V}_{ref}=0\), the control law is developed as  
\[u=\frac{1}{\beta_{0}}\big{(}-\hat{x}_{2}+k_{p}e\big{)} \tag{52}\]  
where \(k_{p}>0\) is the controller gain.