# Operator-Based Adaptive Tracking Capacity Control in Complex Manufacturing Processes
## 3 Operator-Based Adaptive Capacity Control
### Local Adaptive Capacity Control
 \tag{21}\]  
By calculating the Laplace transformation of (21), it can be represented with  
\[\hat{Q}:Y_{i}(s)=\frac{C_{i}^{0}s+\beta C_{i}^{1}}{s^{2}+C_{i}^{0}s+\beta C_{ i}^{1}}\cdot(R_{i}(s)+D_{i}(s)).\]  
Figure 6: Nonlinear feedback tracking control of the MIMO system. Let \(\text{S}_{i}(\cdot)=\text{R}_{i}(\cdot)+\text{D}_{i}(\cdot)\); then, the Lipschitz semi-norm is obtained via  
\[ \begin{split}\|\hat{Q}\|:&=\sup_{F\in(0,\infty)}\sup_{ S\
eq\delta}\frac{\|[\hat{Q}(\text{S})]_{F}-[\hat{Q}(\hat{S})]_{F}\|}{\|[\text{S}]_{F}-[ \hat{\text{S}}]_{F}\|}\\ &=\sup_{F\in(0,\infty)}\mid\frac{C_{i}^{0}\text{s}+\beta C_{i}^{ 1}}{\text{s}^{2}+C_{i}^{0}\text{s}+\beta C_{i}^{1}}\mid\quad\quad\quad\leq 1. \end{split}  \tag{22}\]  
Hence, the above feedback tracking control system is stable. To improve the tracking control performance in complex dynamic environments, the parameters of the PI-type control are usually adaptively tuned by metaheuristic algorithms [33]. In this paper, an artificial neural network (ANN) method was considered, and a single neuron was utilized to adaptively adjust the parameters of the tracking controllers [34] (i.e., \(C_{i}^{0}\), \(C_{i}^{1}\) in (19)). In this case, the inputs of the single neuron were defined as \(x_{i0}=e_{i}^{c}-e_{i}^{cp}\) and \(x_{i1}=e_{i}^{c}=r_{i}-y_{i}\), \(i=1,\cdots,n\), where \(e_{i}^{c}\) is the error between the desired and current feedback of the system, and \(e_{i}^{cp}\) represents the previous error. By using the connection weights of \(x_{ij}\) in the single neuron, the relationship between the input \(x_{ij}\) and the \(i_{th}\) output \(v_{i}\) in the context of incremental PI control is given as  
\[v_{i}=v_{i}^{p}+K_{i}\sum_{j=0}^{1}\hat{w}_{ij}x_{ij}^{T}\quad\quad\quad\hat{ w}_{ij}=w_{ij}/\left|\sum_{j=0}^{1}w_{ij}\right| \tag{23}\]  
where \(w_{ij}\) is a weighting coefficient, and \(K_{i}>0\) is the proportional coefficient of the neuron. Concerning the self-adaptive learning rules, the supervised Hebb learning algorithm was adopted; this included the learning rate \(\tilde{\xi}_{ij}>0\) and the correlative functions of input \(x_{ij}\), output \(v_{i}\), and output error \(e_{i}^{c}\) of the neuron, i.e.,  
\[w_{ij}:=w_{ij}+\tilde{\xi}_{ij}e_{i}^{c}v_{i}x_{ij}^{T} \tag{24}\]  
With the update of the \(w_{ij}\) iteration, the parameters of the tracking controllers \(C_{ij},j=0,1\) for all workstations would be changed, and satisfactory tracking performance could be guaranteed in the presence of disturbances and delays. Overall, the adaptive integral separation PI combined with the RRCF (ISPI-RRCF) method for WIP tracking control via RMTs is outlined in Algorithm 1. ```
0: Given the MIMO system with disturbances \(\mathbf{Q}=(\mathbf{N}+\triangle\mathbf{N})\circ\mathbf{D}^{-1}\);
1: Transfer the MIMO system into a set number of SISO systems via decoupling controllers Z and G;
2: Design RRCF controllers R and S for each decoupled SISO system (workstation) to guarantee stability in the presence of uncertainty and delays. 3: Adopt ISPI controllers C for WIP tracking while decreasing overshoot;
4: Optimize the parameters of the C-based single neuron to improve the transient and robustness performance;
5: Converged WIP level (output) for each workstation within a given prescribed threshold through the assignment of RMTs (input). ```  
**Algorithm 1** Framework of the adaptive ISPI-RRCF