# Operator-Based Adaptive Tracking Capacity Control in Complex Manufacturing Processes
###### Abstract
To improve the tracking performance, a feedback-linearization-based PID method was proposed for the path tracking of micro-actuators to reduce the tracking error in [22]. [PERSON] et al. designed a non-singular fast terminal sliding-mode control for the path tracking of nonlinear second-order systems with compound disturbances in [23]. For adaptive control, a neural network (NN) is an effective method. [PERSON] et al. recommended an adaptive robust position control technique by integrating an radial basis function NN (RBF-NN) and NN-based disturbance observer for disturbed electro-hydraulic servo systems [24]. [PERSON] et al. also proposed an RBF-NN adaptive sliding-mode controller for nonlinear electromechanical actuator systems with uncertainty and disturbances [25]. Considering the key performance of job shops, WIP is essential, as it greatly influences many key performance indicators, e.g., energy efficiency, throughput, and delivery date reliability [26; 27]. Hence, the purpose of control is to guarantee that the manufacturing process works on an expected WIP level via flexible capacity control with RMTs and to ensure the closed-loop stability in which the performance indicators (e.g., WIP) remain bounded while converging toward desired values or an acceptable stability region. In [28], the authors investigated the problem of WIP regulation via RMT assignment in a job shop with a constant flow probability. Desirable control performance and the stability of the closed-loop system were achieved by using MPC with terminal endpoint constraints. However, the preferred assignment of RMTs for capacity adjustment belonged to a continuous optimization case. To solve this problem, the authors further extended their work and employed MPC in association with a deterministic method (e.g., branch and bound) and stochastic optimization techniques (e.g., genetic algorithm) for the integer assignment of RMTs [29]. However, transportation delays were not taken into consideration, and the computational complexity of this method was very high due to the iterative online optimization. In [30], the complexity of workstations was described as a nonlinear operator, and the WIP was controlled by considering the productivity and customer requirements. This also indicated the applicability of operator theory in the capacity control process. To measure the control performance, transient and robustness were two key factors facing stochastic customer demands. As a key transient performance indicator, overshoot reflects the bottleneck level of workstations and the capacity distribution; therefore, it is expected to have a low value. The above survey revealed that the integration of RMTs and RRCF was an effective means of capacity control. However, there is still much work to be done to improve the performance in capacity control systems while considering practical applications. This research is concerned with quickly solving the serious bottleneck represented by the high overshoot and unbalanced capacity reflected by the robustness in job shops facing stochastic demands. To solve this problem, an improved adaptive tracking capacity control algorithm is proposed by integrating the RRCF and ISPI with a single-neuron algorithm. The main contribution can be summarized as follows:  
* To decrease the complexity of a high-coupling job shop manufacturing process and autonomously realize rapid responsiveness among workstations, a decoupling-controller-based operator was designed to decompose the complex MIMO system into multiple SISO systems. * To ensure the steady-state performance and keep the WIP level for each workstation in the vicinity of the planned values while considering disturbances and delays, robust controllers were theoretically designed by using the RRCF method based on the Bezout identity.