# Operator-Based Adaptive Tracking Capacity Control in Complex Manufacturing Processes
## 3 Operator-Based Adaptive Capacity Control
### Local Adaptive Capacity Control  
The above decoupling controller transformed the MIMO system into \(n\) independent SISO systems. Then, while considering each decoupled SISO system, the local adaptive capacity controller is designed in two parts, namely, the RRCF controller and adaptive tracking controller for the robustness and tracking performance, respectively. For the decoupled SISO system, the RRCF operators \(R_{i}\) and \(S_{i}\) are designed with the Bezout identity: \(R_{i}\circ(N_{i}+\triangle N_{i})+S_{i}\circ L_{i}=\bar{M}_{i}\). Then, the following theorem is presented. **Theorem 3**. : _Consider the MIMO system (4) with the decoupling controller in Proposition 1. If the ith workstation has its local RRCF controller,_  
\[R_{i}(s(\cdot))=(1-K_{i})\cdot s(\cdot)^{\prime} \tag{14}\]  
\[S_{i}^{-1}(s(\cdot))=\frac{(h_{i}v_{i}^{R}-1)s(\cdot)}{v_{i}^{R}K_{i}}-\frac{ v_{i}^{D}n_{i}^{D}}{v_{i}^{R}}. \tag{15}\]  
_for \(i=1,2,\cdots,n\), and the RRCF control parameter \(K_{i}\in(0,1)\), then the feedback control system shown in Figure 5 is stable._  
Proof. : The coupling system (4) can be represented with \(y_{i}(t)=N_{i}L_{i}^{-1}\) for \(i=1,2,\cdots,n\) by using Proposition 1. According to the definition of RRCF 2, let  
\[L_{i}^{-1}(s(\cdot))=\frac{s(\cdot)v_{i}^{R}+v_{i}^{D}n_{i}^{D}}{ h_{i}v_{i}^{R}-1} \tag{16}\] \[(N_{i}+\triangle N_{i})(s(\cdot))=\int(s(\cdot))dt. \tag{17}\]  
Additionally, according to Theorem 1, the control system can be equivalent, and its Lipschitz semi-norm is  
\[\|N_{i}+\triangle N_{i}\|: =\sup_{s,s\in D_{i},s\
eq s}\frac{\|(N_{i}+\triangle N_{i})(s)-( N_{i}+\triangle N_{i})(\tilde{s})\|_{Y_{s}}}{\|s-s\|_{U_{s}}}<1+|X^{D}(t)|. \tag{18}\]  
Figure 5: RRCF control of the decoupled MIMO systems. Because the uncertainty \(X_{i}^{D}(t)\) is bounded, the Lipschitz semi-norm \(\|(N_{i}+\triangle N_{i})\|\) is finite. Finally, this can prove that the above local RRCF control system is stable. Based on the RRCF controller designed for robust stability, a PI-type tracking controller (PI-RRCF), as proposed in [16], was investigated. Though integration could eliminate static differences and improve control accuracy, it may cause a large deviation in the system output during the start-up or in face of significant changes with respect to the set value, which would lead to the accumulation of integrations with the resulting overshoot or oscillation [32]. To solve these problems, therefore, an integral-separation PI tracking controller is proposed, which is called ISPI-RRCF algorithm; cf. Figure 6 for a sketch. Then, the tracking controller \(C_{i}\) in the \(j_{th}\) workstation is designed as  
\[C_{i}(s(\cdot))=C_{i}^{0}\cdot s(\cdot)+\beta\cdot C_{i}^{1}\cdot\int s(\cdot) \tag{19}\]  
where \(C_{i}^{0}\) and \(C_{i}^{1}\) are tracking parameters, \(\beta\) is the integral term selector switch coefficient, and \(\epsilon\) is the threshold, i.e.,  
\[\beta= \begin{cases}0&|s(\cdot)|>\epsilon\\ 1&|s(\cdot)|\leq\epsilon\end{cases}  \tag{20}\]  
From (19) and (20), it can be seen that the integration does not work in the case of overshoot when the deviation exceeds the given threshold, and conversely. the integration would be (re-)introduced to minimize the static error once the deviation falls into the threshold. Now, the ISPI-RRCF feedback tracking control system is discussed with the following theorem. Given the bounded and disturbed MIMO system (4) and the tracking operator \(C_{i}\) in (19), the closed-loop system in Figure 6 is stable. Proof. : When the system is at the initial zero state, as shown in Figure 6, it obtains \(y_{i}=(N_{i}+\triangle N_{i})C_{i}(e_{i}^{\epsilon}(\cdot))\) with  
\[ \begin{split} y_{i}(t)&=\int C_{i}^{0}(e_{i}^{ \epsilon})(t)+\beta C_{i}^{1}\int(e_{i}^{\epsilon})(\tau)d\tau dt+\int C_{i}^{ 0}(X_{i}^{D})(t)+\beta C_{i}^{1}\int(X_{i}^{D})(\tau)d\tau dt\\ &=\int C_{i}^{0}(r_{i}-y_{i})(t)+\beta C_{i}^{1}\int(r_{i}-y_{i}) (\tau)d\tau dt+\int C_{i}^{0}(X_{i}^{D})(t)+\beta C_{i}^{1}\int(X_{i}^{D})( \tau)d\tau dt.\end{split}