# Operator-Based Adaptive Tracking Capacity Control in Complex Manufacturing Processes
## 3 Operator-Based Adaptive Capacity Control
### Mathematical Preliminaries  
In operator theory, a job shop manufacturing system can be represented by an operator \(Q\), which is a mapping from the input space \(\mathbb{U}\) to the output space \(\mathbb{Y}\). The domain and range of \(Q\) are defined with \(\mathcal{D}(Q)\) and \(\mathcal{R}(Q)\), respectively. \(\mathcal{N}(\mathbb{U},\mathbb{Y})\) donates the set of all nonlinear operators, and the Lipschitz semi-norm of \(Q\) on \(\mathcal{N}(D_{s},\mathbb{Y})\) is defined via  
\[\|Q\|:=\sup_{x,\xi\in D,\delta x\
eq\bar{x}}\frac{\|Q(x)-Q(\hat{x})\|_{ \mathbb{Y}}}{\|x-\bar{x}\|_{\mathbb{U}}}.\]  
Figure 1: The general model structure of the \(n\)-workstation job shop system [31].  
For such complex job shop systems, the purpose is to design an adaptive capacity controller to guarantee that the job shop works at a predefined level. When the semi-norm is finite, the system is called finite-gain input-output stable (simply called'stable'). The operator-based definitions and theorems are given as follows; cf. [5] for details.  
Consider a causal and stabilizable operator \(Q:\mathcal{D}(Q)\rightarrow\mathcal{R}(Q)\). It has the right factorization \(Q=N\circ D^{-1}\), where \(N\) is causal and stable, while \(D\) is causal, stable, and invertible. Then, the right coprime factorization (RCF) of \(Q\), which is shown in Figure 2, is defined based on the Bezout identity:  
\[R\circ N+S\circ D=M, \tag{7}\]  
where \(R:\mathcal{R}(N)\rightarrow\mathcal{D}(Q)\) and \(S:\mathcal{R}(D)\rightarrow\mathcal{D}(Q)\) are causal and stable, and \(M:\mathcal{D}(Q)\rightarrow\mathcal{D}(Q)\) is a unimodular operator.  
As shown in Figure 2, except for the input signal \(u\) and the output signal \(y\), there exists a quasi-state signal \(w\) from the right factorization. \(v\) and \(l\) are the referred and feedback signals, and \(e\) is the error. With this definition, the dynamics of this control system can be described with the following theorem [5]. Consider a causal and stabilizable operator \(Q:\mathcal{D}(Q)\rightarrow\mathcal{R}(Q)\). If it has RCF, then this system is stable and can be equivalent to \(y=N\circ M^{-1}(v)\).  
When considering bounded uncertainties or disturbances, if the above system \(Q\) still has the right factorization with  
\[Q=(N+\triangle N)\circ D^{-1} \tag{8}\]  
where \(\triangle N\) is an unknown bounded operator, then the following definition is given. Consider a bounded disturbed system \(Q\) (8). If there exist two operators \(R\) and \(S\) satisfying the Bezout identity \(R\circ(N+\triangle N)+S\circ D=\tilde{M}\), then, the system has robust right coprime factorization (RRCF), as shown in Figure 3, where \(\tilde{M}\) is also a unimodular operator.  
Similarly to the Theorem 3.1, the RRCF of the disturbed system can be equivalent to \(y=(N+\triangle N)\circ\tilde{M}^{-1}(v)\). These definitions provide an effective tool for the control and analysis of a class of nonlinear systems, especially for complex dynamic job shops. The details of the controller design process are introduced in the following; these include the decoupling controller design for MIMO systems and the local adaptive capacity control of SISO systems.  
Figure 2: The right coprime factorization for an SISO system.