# Operator-Based Adaptive Tracking Capacity Control in Complex Manufacturing Processes
## 5 Conclusions  
This paper focused on the adaptive tracking capacity control of complex job shops. Considering stochastic customer demands, an improved ISPI-RRCF adaptive capacity control method was proposed to enhance the transient and robustness performance. Considering the customized flexibility of RMTs and the complexity of job shops, including stochastic customer demands, reconfiguration delays of RMTs, and transportation delays between workstations, a funnel-based mathematical model was introduced for the capacity controller design. Based on this model, an operator-based capacity controller was designed, which included a decoupling controller for the MIMO system and, later, a local ISPI-RRCF capacity controller for decoupled SISO systems. In particular, the local ISPI-RRCF controller was designed by including an RRCF controller and ISPI tracking controller, where the parameters were dynamically optimized by a single neuron based on the supervised Hebb learning algorithm. The stability of the closed-loop system was theoretically proven based on the Lipschitz norm. Finally, a case study for the capacity control of a job shop system was implemented in a simulation. The transient and robustness performance of the  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Number of the Workstation** & **1** & **2** & **3** & **4** \\ \hline MRMT in PI-RRCF & 2.22 & 0.99 & 0.87 & 1.65 \\ MRMT in ISPI-RRCF & 2.23 & 1.01 & 0.87 & 1.66 \\ SDRMT in PI-RRCF & 1.22 & 0.41 & 0.69 & 1.52 \\ SDRMT in ISPI-RRCF & 1.17 & 0.52 & 0.49 & 1.11 \\ MAE in PI-RRCF & 18.85 & 34.62 & 51.97 & 60.90 \\ MAE in ISPI-RRCF & 17.24 & 13.25 & 26.65 & 21.55 \\ SDAE in PI-RRCF & 16.28 & 16.37 & 49.79 & 65.72 \\ SDAE in ISPI-RRCF & 10.43 & 10.46 & 28.89 & 34.74 \\ Overshoots in PI-RRCF & 0.72 & 0.15 & 0.41 & 0.96 \\ Overshoots in ISPI-RRCF & 0.60 & 0.14 & 0.02 & 0.12 \\ Setting time in PI-RRCF & 21 & 5 & 32 & 48 \\ Setting time in ISPI-RRCF & 6 & 5 & 13 & 18 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Statistics of the capacity control systems.  
proposed ISPI-RRCF control system was analyzed and compared with that of a previous PI-RRCF control system. The results showed that when considering the system performance indexes, both the proposed ISPI-RRCF and the previous PI-RRCF capacity control systems could keep the WIP of all workstations in the vicinity of the planned levels and could keep the output rate of the products close to the input rates. However, for the control performance indexes, the proposed ISPI-RRCF system had a better transient performance with smaller overshoots and shorter settling times, as well as better robustness, with a smaller MAE and SDAE. Therefore, the effectiveness of the proposed adaptive ISPI-RRCF capacity control algorithm could be proven due to the avoidance of serious bottlenecks and the balancing of capacity distribution problems. This research also provides new opportunities for manufacturers for higher productivity, lower costs, and greater customer satisfaction when facing challenges from quickly changing markets, new technological revolutions, and sustainable manufacturing.  
Conceptualization, P.L.; methodology, P.L.; software, P.L.; validation, P.L. and Q.Z.; investigation, P.L., A.W. and J.P.; writing--original draft preparation, P.L. and Q.Z.; writing--review and editing, S.W.; visualization, A.W.; supervision, J.P.; project administration, P.L. All authors have read and agreed to the published version of the manuscript.  
This research was partially supported by the National Key Research and Development Program of China (grant number 2020 YFB1712403) and the Key Scientific and Technological Project of Henan Province (grant numbers 212102210380 and 212102210080).  
Not applicable.  
The authors declare no conflicts of interest.