## 4 Findings--An Initial Model for Zero Defect Manufacturing
### Reflecting Interconnected Large-Scale Systems  
The generic model \(P_{CL}\) can be used to approximate the behavior of physically entangled CPSs. When several such systems are interconnected in a production chain at a producer site or multiple producers in a supply chain, individual \(P_{CL}\) can be interconnected into a larger system description. To this end, assume there are \(m\) individual production processes \(P_{CL}\) denoted as \(P_{CL}^{(i)}\) with \(i\in[1,m]\), and that the system \(P_{CL}\) can be approximated by a partial linear system in accordance with [38] or an input affine non-linear system [39]. Further, it will be assumed that the input \(u_{e}\) can be split into two components \([u_{e,l},u_{e,g}]^{T}\), where \(u_{e,l}\) are known local exogenous inputs and \(u_{e,g}\) are known global exogenous inputs. The global exogenous inputs can thus be outputs from other processes. An approximation of (3) as a input-affine non-linear system is given as  
\[P_{CL}^{\prime}:\left\{ \begin{array}{ll}\dot{x}_{A}&=f(x_{A},\theta)+g_{l}( x_{A},\theta)[u_{e,l},r,d]^{T}+g_{g}(x_{A},\theta)u_{e,g}\\ y&=h(x_{A},\theta)+h_{d}(x_{A},\theta)d\end{array} \right. \tag{4}\]  
where \(\theta=[\theta_{h,P},\theta_{h,K},\theta_{f,K}]^{T}\), \(g_{l}\) is the local input function, \(g_{g}\) is the global input function, and \(h_{d}\) is the local disturbance output function. Here, \(h_{d}\) is assumed to be zero. Using (4) the combined Large-Scale (LS) system description can be stated as  
\[p_{CL}^{LS}:\left\{ \begin{array}{ll}\left[\begin{array}{ll}\dot{x}_{A}^{(1 )}\\ \dot{x}_{A}^{(2)}\\ \vdots\\ \dot{x}_{A}^{(m)}\end{array}\right]&=\left[\begin{array}{ll}f^{(1)}&g_{g}^{(1,2)}h^{(2)}&\cdots&g_{g}^{(1,m)}h^{(m)}\\ g_{g}^{(2,1)}h^{(1)}&f^{(2)}&\cdots&g_{g}^{(2,m)}h^{(m)}\\ \vdots&\vdots&\ddots&\vdots\\ g_{g}^{(m,1)}h^{(1)}&g_{g}^{(m,2)}h^{(2)}&\cdots&f^{(m)}\end{array}\right]+ \left[\begin{array}{ll}g_{l}^{(1)}[u_{e,l}^{(1)},r^{(1)},d^{(1)}]^{T}\\ g_{l}^{(2)}[u_{e,l}^{(2)},d^{(2)}]^{T}\\ \vdots\\ g_{l}^{(m)}[u_{e,l}^{(m)},r^{(m)},d^{(m)}]^{T}\end{array}\right]\end{array} \right. \tag{5}\]  
where arguments are omitted for the sake of clarity and the interconnections are represented by the nonlinear functions \(g_{g}^{(ij)}\). In cases where all outputs \(y^{(j)}\) are forwarded unaltered from system \(P_{CL}^{(j)}\) to \(P_{CL}^{(i)}\), the function \(g_{g}^{(ij)}\) does not select specific outputs. Otherwise, the function \(g_{g}^{(ij)}\) might be applied to \(h_{P}^{(j)}\), which selects and alters certain measurable outputs. In (5), material and information flows can be modeled by appropriately defining the interconnection functions \(g\). It is also possible to realize sequential and circular dependencies between production processes. When the interconnections contain a dynamic behavior themselves, then an intermediate production process needs to be created that contains the states of the interconnection. A typical example for such an interconnection is storage facilities that are used in the interconnection, or when estimators are used to derive information on the connected production processes states. In Figure 6, an example for a large production system with five unit processes as subsystems is considered. The five subsystems are part of a sequential production chain, where the forward flow and interaction between the unit processes is indicated by the outputs that are forwarded to the individual unit processes. It can be seen that the outputs of \(P_{CL}^{(3)}\) and \(P_{CL}^{(5)}\) are split into two subsets \(A\) and \(B\). The subset \(A\) is always forwarded to the next process, while subset \(B\) is recycled or returned to a previous process in the chain. The resulting model \(P_{CL}^{LS}\) can then be stated as follows in (6):  
\[P_{CL}^{LS}:\left\{