# Gain-Scheduled Disturbance Observer-Based Saturated Controllers for Non-Linear First-Order System +
### GS-DOB0 Controller Based on Ultralocal Models  
For the \"nominal\" ultralocal linearized model \(K_{s}/s\), the use of the disturbance reconstruction according to  
\[d_{f}=\frac{1}{K_{s}}\frac{s}{T_{f01}s+1}\Delta y-\frac{\Delta u}{T_{f01}s+1} \tag{45}\]is equivalent to a stabilizing controller  
\[R^{01}(s)=K_{P}+\frac{1}{K_{s}}\frac{s}{T_{f01}s+1}. \tag{46}\]  
By substituting \(K_{P}=1/(K_{s}T_{01})\), the disturbance-compensating controller takes the form of the automatic reset (series PI) controller  
\[C^{01}(s)=K_{P}\frac{1+T_{f01}s}{T_{f01}s}+\frac{1}{K_{s}T_{f01}}=\frac{1}{K_{s }}\Bigg{(}\frac{1}{T_{01}}+\frac{1}{T_{f01}}\Bigg{)}+\frac{1}{K_{s}}\frac{1}{T_ {01}T_{f01}s} \tag{47}\]  
The closed-loop transfer function becomes  
\[\begin{split}& F^{01}_{xy}(s)=\frac{C^{01}(s)S(s)}{1+C^{01}(s)S(s)}= \frac{(1+T_{f01}s)K_{P}K_{s}}{T_{f00}s(s+a)+K_{P}K_{s}(1+T_{f01}s)}\\ &=\frac{1+(T_{f01}+T_{01})s}{T_{f01}T_{01}s^{2}+(T_{f01}+T_{01}+T _{f01}T_{01}a)s+1}.\end{split} \tag{48}\]  
The closed-loop stability is guaranteed for  
\[T_{f01}+T_{01}+T_{f01}T_{01}a>0. \tag{49}\]  
For stable processes with \(a>0\), the loop remains stable for any positive \(T_{f01}>0\) and \(T_{01}>0\). For unstable processes with \(a<0\), the system reaches the stability limit for  
\[a=-\Bigg{(}\frac{1}{T_{f01}}+\frac{1}{T_{01}}\Bigg{)}. \tag{50}\]  
Using the more complex DOB (45) and the P controller with \(K_{P}=1/\left(K_{s}T_{01}\right)\), the stability region can be extended compared to (43) (the simplified DOB (38) with \(T_{01}=T_{00}\)). Since the numerator of \(F^{01}_{xy}\) is equal to \(1+(T_{f01}+T_{01})s\), an overshoot of the setpoint step responses is expected. The easiest way to eliminate such an overshoot is to use a pre-filter  
\[F^{01}_{p}(s)=1/(1+T_{p}s);\ T_{p}=T_{f01}+T_{01}. \tag{51}\]  
An important practical issue for the GS-DOB0 controller is the effect of measurement noise \(\delta\) due to the derivative term in (45). It can be adjusted by changing the gain \(K_{P01}=1/(K_{s}T_{01})\) of the stabilizing controller and the AR filter time constant \(T_{f01}\).  
**Remark 1** (Attenuation of the measurement noise).: _The amplification of the noise \(\delta\) can be estimated by calculating the instantaneous change in the controller output (signals \(\Delta u_{\delta 00}\) and \(\Delta u_{\delta 01}\)) caused by the noise increment \(\delta\). From the expressions (33), (45), and (47), the changes in the controller outputs can be calculated as follows:_  
\[\begin{split}\Delta u_{\delta 00}&=K_{P00}\delta=\frac{1}{K_{ s}T_{00}}\delta,\text{or}\\ \Delta u_{\delta 01}&=\frac{1}{K_{s}}\Bigg{[}\frac{1}{T_ {01}}+\frac{1}{T_{f01}}\Bigg{]}\delta.\end{split} \tag{52}\]  
_Obviously, for sufficiently long time constants \(T_{f01}\) and \(T_{00}=T_{01}\), the difference between \(\Delta u_{\delta 00}\) and \(\Delta u_{\delta 01}\) can be neglected. However, the use of a more complex DOB may no longer make sense. The relationships (52) also mean that the noise attenuation for ARC0 depends on \(T_{00}\) and for DOB0 on \(T_{01}\) and \(T_{f01}\)._