# Evaluating the Energy-Saving Potential of Solar Dedicated Ventilation Systems Based on ADRC
## 3 Accurate Control and Analysis Based on ADRC
### Design of ADRC for This Research  
A second-order ADRC is taken as an example to design a temperature control system, whose structure is shown in Figure 2.  
Figure 2: The structure of the ADRC. TD, tracking differentiator; NSEF, nonlinear state error feedback; ESO, extended state observer. For the proposed temperature control system, the input signal is \(v_{0}\), the control quantity of the system is \(u(t)\), the output of the system is \(y(t)\), and the external disturbance is \(w\). The algorithms of the ADRC are shown as Equations (1)-(13). TD equation:  
\[\left\{ \begin{array}{l}v_{1}(k+1)=v_{1}(k)+hv_{2}(k)\\ v_{2}(k+1)=v_{2}(k)+hfst(v_{1}(k)-v_{0},v_{2}(k),r,h_{0})\end{array} \right. \tag{1}\]  
where  
\(v_{1}(k+1)\) is the tracking signal at time \(k+1\);  
\(v_{1}(k)\) is the tracking signal at time \(k\);  
\(h\) is the simulation step;  
\(v_{2}(k)\) is the differential signal of \(v_{1}(k)\) at time \(k\);  
\(v_{2}(k+1)\) is the differential signal of \(v_{1}(k+1)\) at time \(k+1\);  
\(fst()\) is the fastest controlled synthesis function;  
\(v_{0}\) is the input signal;  
\(r\) is the speed factor; and  
\(h_{0}\) is the filtering factor. \[fst(v_{1}(k)-v_{0}(k),v_{2}(k),r,h_{1})=-rsat(g(k),\delta), \tag{2}\]  
\[\delta= h_{1}r,\;\delta_{1}= h_{1}\delta, \tag{3}\]  
\[e(k)=v_{1}(k)-v_{0}(k),y(k)=e(k)-h_{1}v_{2}(k), \tag{4}\]  
\[sat(x,\delta)=\left\{ \begin{array}{c}sign(x),|x|\geq\delta\\ \
icefrac{{x}}{{\delta}}|x|<\delta\end{array} ,\right. \tag{5}\]  
\[g(k)=\left\{ \begin{array}{ll}v_{2}(k)-sign(y(k))(\delta-\sqrt{8r|y(k)|}+ \delta^{2})&|y(k)|\geq\delta_{1}\\ v_{2}(k)+\
icefrac{{v_{1}(k)}}{{h}}&|y(k)|<\delta_{1}\end{array} \right. \tag{6}\]  
where  
\(v_{0}(k)\) is the input signal at time \(k\);  
\(h_{1}\) is an adjustable parameter;  
\(sat()\) is a linear saturation function;  
\(\delta\) and \(\delta_{1}\) are interval length;  
\(e(k)\) is the tracking signal (error signal) at time \(k\);  
\(y(k)\) is the output of the system at time \(k\);  
\(x\) is the independent variable; and  
\(sign(x)\) is the symbolic function. ESO equation:  
\[\varepsilon_{1}=z_{1}(k)-y(k), \tag{7}\]  
\[\left\{ \begin{array}{ll}z_{1}(k+1)=z_{1}(k)+h(z_{2}(k)-k_{1}\varepsilon_{1 })\\ z_{2}(k+1)=z_{2}(k)+h(z_{3}(k)-k_{2}fal(\varepsilon_{1},\alpha_{1},\delta)+ bu(k))\\ z_{3}(k+1)=z_{3}(k)-hk_{3}fal(\varepsilon_{1},\alpha_{2},\delta)\end{array} \right. \tag{8}\]  
where  
\(\varepsilon_{1}\) is the error;  
\(z_{1}(k)\) is the estimation of the output at time \(k\);  
\(z_{1}(k+1)\) is the estimation of the output at time \(k+1\);  
\(z_{2}(k)\) is the estimation of the differential of the output at time \(k\);\(z_{2}(k+1)\) is the estimation of the differential of the output at time \(k+1\);  
\(z_{3}(k)\) is the estimation of total disturbance of the system at time \(k\);  
\(z_{3}(k+1)\) is the estimation of total disturbance of the system at time \(k+1\);  
\(k_{1}\),\(k_{2}\),\(k_{3}\) are the parameters of the observer;  
\(fal()\) is a nonlinear function;  
\(\alpha_{1}\) is the adjustable parameter from 0 to 1;  
\(b\) is a special parameter;  
\(\alpha_{2}\) is the adjustable parameter from 0 to 1; and  
\(u(k)\) is the control amount of the system at time \(k\). NSEF equation:  
\[e_{1}=v_{1}(k)-z_{1}(k), \tag{9}\]  
\[e_{2}=v_{2}(k)-z_{2}(k), \tag{10}\]  
\[u_{0}=\beta_{01}fal(e_{1},\alpha_{1},\delta)+\beta_{02}fal(e_{1},\alpha_{2}, \delta), \tag{11}\]  
\[u(k)=u_{0}-z_{3}(k)/b, \tag{12}\]  
\[fal(\varepsilon,\alpha,\delta)=\left\{ \begin{array}{c}|\varepsilon|^{a}sgn( \varepsilon)|\varepsilon|>\delta\\ \varepsilon/\beta^{-\varepsilon}|\varepsilon|\leq\delta\end{array} \delta>0\right. \tag{13}\]  
where  
\(e_{1}\) is the estimated error between ranged transition process and system output;  
\(e_{2}\) is the differential of errors between ranged transition process and system output;  
\(u_{0}\) is the control component;  
\(\beta_{01}\) is the proportional gain;  
\(\beta_{02}\) is the differential gain;  
\(\alpha\) is the adjustable parameter from 0 to 1; and  
\(\varepsilon\) is the error.