# Gain-Scheduled Disturbance Observer-Based Saturated Controllers for Non-Linear First-Order System +
## 3 Illustrative Example
### Performance Evaluation for Unconstrained Control  
In a detailed comparison of different controller implementations, the input and output signals of the process are evaluated. The speed of the setpoint responses is reciprocal to the integral of the absolute error (\(IAE\)), defined as  
\[IAE=\int_{0}^{\infty}\lvert e(t)\rvert dt\ ;\ \ e=w-y\, \tag{68}\]  
Figure 7: Parameters of the linearized system model of process (63) calculated with \(Y=1\), \(w\in[0,1]\) and the tuning parameters (67) according to (65) and (66) for all gain-scheduled controllers considered in Table 2 and \(T_{c}=0.1\) (left) and \(T_{c}=0.5\) (right).  
Since a setpoint response corresponding to a first-order exponential \(1-e^{-t/T_{c}}\) gives the value \(IAE=T_{c}\), the \(IAE\) values obtained by simulation are normalized by the time constant \(T_{c}\) (67) and the value \(\Delta w=0.1\) according to the relation  
\[iae=\frac{IAE}{\Delta wT_{c}} \tag{69}\]  
For an ideal match with the first-order exponential response given by \(T_{c}\
eq 0\), the performance measure (69) should give the value 1. Although the use of \(IAE\) values may be more useful than other alternative measures [72], the minimum of \(IAE\) corresponds to a moderate overshoot. Therefore, it is necessary to avoid such a case by obtaining additional information about the transients of the process output by also considering shape-related measures related to the deviation from monotonicity of the signal.  
A response \(y\) is monotonic if its total variation (\(TV(y)\)) represented by the sum of the absolute values of all increments  
\[TV(y)=\sum_{i=0}^{\infty}\lvert y_{i+1}-y_{i}\rvert \tag{70}\]  
corresponds to the total change from an initial value \(y_{0}\) (e.g., \(y_{0}=0\)) to a final value \(y_{\infty}\to w\). To evaluate the deviation from the monotonicity of the output after a change in the setpoint, the performance measure \(TV_{0}(y)\) can therefore be applied [73], which is defined as  
\[TV_{0}(y)=\sum_{i=0}^{\infty}\lvert y_{i+1}-y_{i}\rvert-\lvert y_{\infty}-y_{ 0}\rvert. \tag{71}\]  
Using the values of \(TV_{0}(y)\) in relation to the step value \(\Delta w\), it can be determined that the overshoot percentage (\(\Delta y\)) will be less than half of \(100*TV_{0}(y)/\Delta w\).  
The analysis of the process output must be supplemented by the qualitative and quantitative analysis of the process input (controller output). Again, the analysis will be based on modification of the total sum of increments \(TV(u)\) according to  
\[TV_{1}(u)=\sum_{i}^{\infty}\lvert u_{i+1}-u_{i}\rvert-\lvert 2u_{m}-u_{ \infty}-u_{0}\rvert. \tag{72}\]  
It enumerates summary deviations over two monotonic input intervals, which form a pulse with an extreme point. The first monotonic interval corresponds to the transition from an initial value \(u_{0}\) to the extreme value \(u_{m}\), which forces the output to follow the new setpoint. The next monotonic segment corresponds to the reduction in the controller output from the extreme value \(u_{m}\) to a stationary value \(u_{\infty}\). All increments contributing to \(TV_{1}(u)>0\) can be considered as redundant increments.