# A Set of Active Disturbance Rejection Controllers Based on Integrator Plus Dead-Time Models +
## 4 Real-Time Experiments and Analysis
### Overall Loop Optimization  
In the present considerations, the individual controllers were characterized by three quantitative measures: \(IAE\) (representing the speed of transients), \(TV_{1}(u)\) (characterizing excessive control effort) and \(TV_{1}(y)\) (excessive output wobbling), which could be related both to the setpoint and disturbance steps. When choosing the optimal solution, however, it would be far easier to use instead of these individual indicators a combined speed-effort (SE) cost function determined, for example, as  
\[J_{u}=\alpha_{s}IAE_{s}^{k_{s}}TV_{1}(u_{s})+\alpha_{d}IAE_{d}^{k_{d}}TV_{1}(u_ {d}) \tag{45}\]  
where \(\alpha_{s}\), \(k_{s}\), \(\alpha_{d}\) and \(k_{d}\) represent appropriate weighting coefficients. Another such a combined cost function denoted as a speed-output wobbling (SW) cost function  
\[J_{y}=\alpha_{s}IAE_{s}^{k_{s}}TV_{0}(y_{s})+\alpha_{d}IAE_{d}^{k_{d}}TV_{1}(y_ {d}) \tag{46}\]  
could focus on output evaluation. Requirements from practical applications appear to create an infinite number of different combinations of individual weighting coefficients. To illustrate their use, we will consider the simplest situations, when focussing just on the setpoint, or disturbance responses with the choice  
\[J^{k}(u_{s}),J^{k}(y_{s}):\ \ \alpha_{s}=1;\ \alpha_{d}=0;\ k_{s}=k \tag{47}\]  
or  
\[J^{k}(u_{d}),J^{k}(y_{d}):\ \ \alpha_{s}=0;\ \alpha_{d}=1;\ k_{d}=k \tag{48}\]  
Wight of the speed of the transients expressed in terms of the IAE values will be denoted by the parameter \(k\). Next, we will compare these four parameters for two different values \(k=1\) and \(k=3\). For ADRC\({}_{1}\) controllers (Figure 12), at \(k=1\) we get conclusions similar to the conclusions from the analysis of individual performance measures: higher dependence of structures with \(\overline{T}_{d}=0\) on the value of \(T_{c}\), but the possibility of reducing the cost functions by increasing \(T_{c}\). With the tuning \(T_{c}=2T_{d}\) and \(k=1\) we get the globally minimal values \(J^{1}(u_{s})\) and \(J^{3}(u_{s})\). Figure 12: _Cont._With \(\overline{T}_{d}=T_{d}\), the performance dependence on \(T_{c}\) is weaker. A low dependence of \(\mathfrak{f}^{k}(u_{s})\) values on tuning parameter \(T_{1}\) is given by ADRC\({}_{2}\) with \(T_{c}=T_{d}\) (Figure 13). For \(k=1\) and \(T_{c}=T_{d}\) it holds also for \(\mathfrak{f}^{k}(u_{d})\) values. However, for \(k=3\) and disturbance steps, lower \(\mathfrak{f}^{k}(u_{d})\) and \(\mathfrak{f}^{k}(y_{d})\) values correspond to the faster ESO with \(T_{c}=T_{d}/2\). This controller yields the globally optimal values \(\mathfrak{f}^{1}(y_{s})\) (for \(T_{1}=T_{d}/4\)), \(\mathfrak{f}^{3}(y_{s})\) (for \(T_{1}=T_{d}/2\)) and \(\mathfrak{f}^{3}(u_{d})\) (for \(T_{1}=T_{d}/4\)). Compared to ADRC\({}_{1}\), we can state an improvement here mainly with regard to the lower dependence on the optional parameter. Because we evaluated up to 9 settings with ADRC\({}_{3}\) with significantly different levels of corresponding cost functions, we will omit the evaluation using bar-graphs here. The globally lowest \(\mathfrak{f}^{1}(u_{d})\) value is given by \(T_{c}=T_{d},T_{2}=3T_{d}/8\). The best \(\mathfrak{f}^{1}(y_{d})\) value corresponds to \(T_{c}=T_{d}/2,T_{2}=3T_{d}/8\) and the best \(\mathfrak{f}^{3}(u_{d})\) value to \(T_{c}=T_{d}/2,T_{2}=3T_{d}/8\)  
Figure 12: ADRC\({}_{1}\) controllers for \(T_{c}=\{0.15,0.3,0.6\}\) with \(T_{dm}=\overline{T}_{d}=0\) and \(T_{dm}=\overline{T}_{d}=0.3\): Holistic cost functions (45) and (46) characterizing performance at the plant input and output for both setpoint and disturbance steps and \(k=1\) and \(k=3\). **Remark 3** (No globally optimal solution). : _We did not carry out the above analysis of the achieved solutions in order to find some specific optimal control for the given system. Rather, it was a question of whether solutions considering higher-order controllers were of any relevance to practice.