# Gain-Scheduled Disturbance Observer-Based Saturated Controllers for Non-Linear First-Order System +
## 3 Illustrative Example
### Comparison with Gain-Scheduled PI Control  
[PERSON] et al. analyzed in [6] the reasons for a loss of performance or even instability of the feedback system linearized about the operating points and as a solution they proposed velocity implementation of controllers [11; 81]. For a fairly general class of solutions that are usually referred to as tracking controllers, differentiating some of the measured outputs before they are fed back to the gain-scheduled controller is applied. Due to the integration of control signal provided at the output of the controllers, despite the use of differentiation, this scheme does not introduce additional noise amplification and allows for relatively easy consideration of control saturation without causing windup. The PI controller with a gain \(K_{p}\) and integral time constant \(T_{i}\) can be defined by the transfer function  
\[C(s)=\frac{U(s)}{E(s)}=K_{p}\bigg{(}1+\frac{1}{T_{i}s}\bigg{)}=\frac{K_{p}}{T_ {i}}\frac{1+T_{i}s}{s}. \tag{75}\]  
With \(K_{i}=K_{p}/T_{i}\) and for a piecewise constant setpoint signal \(w\), the velocity controller implementation can then be described by  
\[U(s)=\frac{1}{s}(K_{i}E(s)+K_{p}sE(s))=\frac{1}{s}(K_{i}E(s)-K_{p}sY(s)). \tag{76}\]  
For the process (8), application of the PI controller yields the close-loop transfer function  
\[F_{wy}(s)=\frac{K_{s}K_{p}s+K_{s}K_{i}}{s(s+a)+K_{s}(K_{p}s+K_{i})}=\frac{K_{s }K_{p}s+K_{s}K_{i}}{s^{2}+s(a+K_{s}K_{p})+K_{s}K_{i}} \tag{77}\]  
Figure 15: Performance measures of the constrained setpoint step responses from Figure 14 for \(T_{c}=0.02\) (left) and \(T_{c}=0.06\) (right); \(T_{f}=T_{c}\), \(T_{f}=5T_{c}\) and \(T_{c}=10T_{f}\): Above—\(TV_{1}(u)\); below—\(TV_{0}(y)\). By requiring a double real closed-loop pole \(\gamma\), or a time constant \(T_{c}=-1/\gamma\), corresponding to the chosen polynomial \((s-\gamma)^{2}\) in (77), the PI parameters can be determined according to  
\[K_{p}=-(2\gamma+a)/K_{s};\ K_{i}=\gamma^{2}/K_{s};\ \gamma=-1/T_{c}. \tag{78}\]  
With the scheduling parameter \(\alpha=w\) (26), this controller can be scheduled according to (66). Let us recall that the velocity implementation of the PI controller (76) requires the availability of the derivative of the output, which can generally be achieved by using a filter with a derivative time constant \(T_{f}\) according to  
\[U(s)=\frac{1}{s}(K_{i}E(s)+K_{p}sE(s))=\frac{1}{s}(K_{i}E(s)-K_{p}\frac{s}{T_{ f}s+1}Y(s)). \tag{79}\]  
However, completely different rules will apply to the optimal setting of the value of \(T_{f}\) than when choosing \(T_{f}\) in controllers with DOB. As shown by the setpoint step responses and the corresponding performance measures in Figure 16, the gain-scheduled PI controller according to [6] denoted as 1 DOF PI1 (red curves) provides good responses with zero shape-related deviations from ideal responses, which are, however, dependent on the magnitude of the setpoint variable \(w\) and slower than the responses with 2 DoF model-based controllers ARCO-1 and DOB0-1. **Remark 6** (Different dynamics requirements of ARC, DOB, and PI controllers). : _A more detailed analysis of the performance achieved would require noting the differences in the first-order closed-loop dynamics corresponding to gain-scheduled ARC0 and DOB0 controllers with disturbance reconstruction and compensation according to Figure 3 and the second-order dynamics corresponding to the 1 DoF PI1 controller with velocity implementation according to (79). While for the ARC0 and DOB0 controllers, the gain \(K_{P}\) is designed from the requirement of the time constant of the first-order exponential (31), for GS-PI according to [6] (1 DOF PI1) the controller setting is calculated based on the requirement of double time constant \(T_{c}\) for setpoint tracking.