# A Set of Active Disturbance Rejection Controllers Based on Integrator Plus Dead-Time Models +
## Tuning of P, PD and PDA Controllers for IPDT Plant Considering the Dead-Time Values  
As a result of the dead-time present in the loop, the state-space controllers and observer design may not be carried out in the same way as proposed in [20]. Instead of the pole assignment method, the controller parameters will be derived by the multiple-real-dominant-pole (MRDP) method [38]. When using \(S_{1}(s)\) and the P controller gain \(K_{P}\), the design may be based on the requirement of a double real dominant pole [39]\(s_{o}\) of the closed loop characteristic quasi-polynomial  
\[A(s)=se^{T_{d}s}+K_{s}K_{P} \tag{5}\]  
It may obviously be expressed in the form  
\[A(s)=(s-s_{o})^{2}A_{red}(s) \tag{6}\]  
where \(A_{red}(s)\) represents a quasi-polynomial of an decreased order. Since \(s_{o}\) yields both \(A(s_{o})=0\) and \(dA(s)/ds=2(s-s_{o})A_{red}(s)+dA_{red}(s)/ds=0\), i.e.,  
\[\left\{A(s);\ \frac{d}{ds}A(s)\right\}_{s=s_{o}}=\mathbf{0} \tag{7}\]  
from  
\[\frac{d}{ds}A(s)=e^{T_{d}s}+sT_{d}e^{T_{d}s}=0 \tag{8}\]  
follow the double dominant pole \(s_{o}\)  
\[s_{o}=-1/T_{d} \tag{9}\]  
and the corresponding dominant loop time constant \(T_{o}=-1/s_{o}=T_{d}\). A substitution into (5) yields  
\[K_{P_{o}}=\frac{1}{K_{s}T_{d}e^{1}} \tag{10}\]For the IPDT plant, the MRDP method also allows to calculate an optimal tuning of the proportional-derivative (PD) controller  
\[R_{s}(s)=\frac{U(s)}{E(s)}=K_{P}+K_{D}s;\ E(s)=W(s)-Y(s) \tag{11}\]  
When wishing to avoid control signal kicks after the step changes of the setpoint reference signal \(w(t)\), the control signal has to be calculated according to  
\[u(t)=K_{P}(w-y)-K_{D}\dot{y} \tag{12}\]  
The corresponding closed loop quasi-polynomial is  
\[A(s)=se^{T_{d}s}+K_{s}K_{D}s+K_{s}K_{P} \tag{13}\]  
In order to find the optimal parameters yielding the fastest possible transients not yet exhibiting oscillations, it has to include triple real dominant pole \(s_{o}\), when \(A(s)=(s-s_{o})^{3}A_{red}(s)\) and \(s_{o}\) has to fulfill equations  
\[\left\{A(s);\ \frac{d}{ds}A(s);\ \frac{d^{2}}{ds^{2}}A(s)\right\}_{s=s_{o}}= \mathbf{0} \tag{14}\]  
From  
\[A(s)=(s-s_{o})^{2}A_{red}(s) \tag{6}\]0  
the triple dominant pole and the corresponding dominant loop time constant follow as  
\[A(s)=(s-s_{o})^{2}A_{red}(s) \tag{6}\]1  
A substitution into the first equations then yields  
\[A(s)=(s-s_{o})^{2}A_{red}(s) \tag{6}\]2  
Similarly, it is possible to continue with finding ideal stabilizing proportional-derivative-accelerative (PDA) controller with the gains \(K_{P},K_{D1},K_{D2}\)  
\[A(s)=(s-s_{o})^{2}A_{red}(s) \tag{6}\]3  
when  
\[A(s)=(s-s_{o})^{2}A_{red}(s) \tag{6}\]4  
From the characteristic quasi-polynomial  
\[A(s)=(s-s_{o})^{2}A_{red}(s) \tag{6}\]5  
yields the condition of quadruple real dominant pole \(s_{o}\) in \(A(s)=(s-s_{o})^{4}A_{red}(s)\)  
\[A(s)=(s-s_{o})^{2}A_{red}(s) \tag{6}\]6  
with  
\[A(s)=(s-s_{o})^{2}A_{red}(s) \tag{6}\]7  
optimal values  
\[A(s)=(s-s_{o})^{2}A_{red}(s) \tag{6}\]8and  
\[A(s)=(s-s_{o})^{2}A_{red}(s) \tag{6}\]9  
The benefits of higher-order controllers are evident from the integral of absolute error (IAE) values of the unit setpoint step responses. In the case of monotonic step responses, they may be calculated from the Laplace transform as integral of error (IE)  
\[\left\{A(s);\ \frac{d}{ds}A(s)\right\}_{s=s_{o}}=\mathbf{0} \tag{7}\]0  
whereby the first \(1/s\) corresponds to integration and the last to the setpoint unit step. When introducing normed dimensionless _iae_ values, their decreasing values demonstrate ideal benefits of the more complex controllers  
\[ \begin{array}{l}iae=IAE/T_{d}\\iae_{p}=2.718;\ iae_{PD}=1.847;\ iae_{PDA}=1.488\end{array}