## 4 Findings--An Initial Model for Zero Defect Manufacturing
### Realization of a Model Enabling ZDM  
On the basis of the above reasoning, a model enabling ZDM is now derived. Firstly, a generic state space representation for a nonlinear dynamic system is used to model \(P\) and \(K\) as follows:  
\[P:\left\{\begin{array}{rl}\dot{x}(t)&=f_{P}(x(t),u_{e}(t),u_{K}(t),d(t), \theta_{f,P}(t))\\ \left[\begin{array}{c}y_{e}(t)\\ y_{K}(t)\end{array}\right]&=h_{P}(x(t),u_{e}(t),u_{K}(t),d(t),\theta_{h,P}(t)) \end{array}\right. \tag{1}\]  
\[K:\left\{\begin{array}{rl}\dot{x}_{K}(t)&=f_{K}(x_{K}(t),r(t),x(t),y_{K}(t), \theta_{f,K}(t))\\ u_{K}(t)&=h_{K}(x_{K}(t),r(t),x(t),y_{K}(t),\theta_{h,K}(t))\end{array}\right. \tag{2}\]  
For the sake of simplicity and with no loss of generality, the time dependency variable will be dropped. The representations for \(P\) and \(K\) are complex networks and have dimensions in accordance with the vector signals in the loop, as \(r\in R^{n_{r}}\), \(u\in R^{n_{u}}\), \(d\in R^{n_{d}}\), \(y\in R^{n_{y}}\) and \(x\in R^{n_{x}}\). There, \(n_{r}\), \(n_{u}\), \(n_{d}\), \(n_{y}\) and \(n_{x}\) are the respective dimensions of the signals. \(x_{K}\) denotes the internal states of the control system, and \(\theta_{\bullet}\) represents the parameter vectors for the respective function in the system equation. It is also assumed that the production process can be approximated by a non-linear differential equation. The presence of \(u_{e}\) and \(u_{K}\) in \(h_{P}\) constitutes the presence of a direct terms in the real-life system, which is quite uncommon, and will be neglected in the sequel.  
It is important to note that the state vector \(x\) and output \(y\) can reflect both operational and condition-based maintenance aspects of the production process \(P\) or \(P_{CL}\). If both aspects are considered, then the state vector and output can be segmented into \(x=[x_{O}\ x_{C}]^{T}\) and \(y=[y_{O}\ y_{C}]^{T}\) by rearranging (1) and (2), respectively. There, \(x_{C}\) and \(y_{C}\) reflect the production process condition as a result of wear and tear in the process. The feedback signal \(y_{K}\) is traditionally only a subset of \(y_{O}\). Condition-based maintenance approaches can then be implemented on the basis of the knowledge of \(x_{C}\) and \(y_{C}\), but require that the model does consider wear and tear aspects.  
Clearly, (1) and (2) are connected differential equations through the feedback that is depicted in Figure 5 and can be restated as a differential equation where \(r\) and \(d\) are the exogenous inputs. In that way the closed-loop system would follow the same structure as given in Figure 3 with an integrated-control system and would constitute a CPS representation. By defining the augmented state  
Figure 5: Block diagram representing a generic production process with a connected control system.  
vector \(x_{A}=[x\ x_{K}]^{T}\) and assuming \(P\) does not have a direct term, the resulting system equations for the closed loop render  
\[P_{CL}:\left\{\begin{array}{ll}\dot{x}_{A}&=\left[\begin{array}{ll}f_{P}(x, h_{K}(x_{K},r,x,h_{P}(x,d,\theta_{h,P}),\theta_{h,K}),u_{e},d,\theta_{f,P})\\ f_{K}(x_{K},r,x,h_{P}(x,d,\theta_{h,P}),\theta_{f,K})\\ y&=h_{P}(x,d,\theta_{h,P})\end{array}\right]\end{array}\right. \tag{3}\]  
The nonlinear functions \(f_{P}\) and \(f_{K}\) are of appropriate size depending on the size of the signal vectors and the state vectors, and we introduce the nonlinear function \(f=[f_{P}(\bullet),f_{K}(\bullet)]^{T}\). In (3) it can also be seen that \(r\) and \(d\) are the new exogenous inputs to the system and that \(r\) is not directly reflected in the output equation for \(y\).  
Moreover, the structure of \(f_{P}\), \(h_{P}\) and \(h_{K}\) determines to what degree different actions of the control system will have an effect on the condition of production process. When it comes to the performance specifications, those are usually reflected through the design of the control system and the operating conditions given by the set points \(r\).