## 3 Particle Filter with Weight Adjustment Design for the Compensation of Sensor Noise and Time Delay
### Operator Based Control System Design
Other components are defined in accordance with Figure 1.  
In the component GPF/WAPF, controlled variables \(T_{ci}(t)\) and \(T_{ho}(t)\) are assumed as state variables \(x_{T_{ci},t}\) and \(x_{T_{ho},t}\), which will be estimated as the sum of weighted particles (corresponding to \(\hat{x}\) in Figure 2), manipulated variable \(U_{h(t)}\) is \(U_{h,t}\), and controlled variable \(T_{co,t}\) is measurement variable \(y_{T_{co,t}}\) which is predicted using Equation (6) (corresponding to \(\hat{y}\) in Figure 2). To formulate a paradigm for the particle filter, the process model is established as Equation (13), \begin{table}
\begin{tabular}{l l l} \hline \(r\) & Target temperature value & 35 \({}^{\circ}\)C \\ \(T_{hi}\) & Hot fluid outlet temperature & 40 \({}^{\circ}\)C \\ \(T_{ci}(0)\) & Initial cold fluid inlet temperature & 26 \({}^{\circ}\)C \\ \(T_{co}(0)\) & Initial cold fluid temperature & 26 \({}^{\circ}\)C \\ \(U_{hmax}\) & Max of hot fluid flow rate & 5.4 L/min \\ \(U_{c}\) & Cold fluid flow rate & 4.3 L/min \\ \(a\) & Archimedes’ spiral equation constant & \(8\times 10^{-7}\) m/rad \\ \(\lambda\) & Thermal conductivity of SUS304 & 16.7 W/(m\({}^{\circ}\)C) \\ \(R_{c}\) & Reynolds number & 22,000 \\ \(P_{r}\) & Prandtl number & 7 \\ \(A\) & Cross-section area of flow path & \(5.5\times 10^{-4}\) m\({}^{2}\) \\ \(c_{\rho}\) & Specific heat of water & 4.2 kJ/(kg\({}^{\circ}\)C) \\ \(\rho\) & Density of water & 1000 kg/m\({}^{3}\) \\ \(\delta\) & Thickness of heat exchanger’s wall & \(1.83\times 10^{-3}\) m \\ \(dr\) & Width of flow path & \(5\times 10^{-3}\) m \\ \(m\) & Mass of cold fluid flow rate & 0.0717 kg \\ \(\delta m\) & Uncertainty of \(m\) & 0.015 kg \\ \(M\) & Mass of cold fluid in TANK2 & 31.8 kg \\ \(\alpha\) & \(T_{ci}\)-\(U_{h}\) Design parameter & 0.3 L/min \\ \(\beta\) & \(T_{ho}\)-\(U_{h}\) Design parameter & 0.03 \\ \(\gamma\) & Design parameter for valve of hot fluid & 1.25 \\ \(\kappa\) & Design parameter for flow change of hot fluid & 0.026 \\ \(K\) & Design parameter of \(S,R\) & 0.7 \\ \(K_{\kappa}\) & Proportional gain of C & 2000 \\ \(K_{g}\) & Integral gain of C & 97 \\ \(\Delta t\) & Sampling time & 1 s \\ \(T_{end}\) & Simulation time & 2401 s \\ \(\sigma\) & Standard deviation of likelihood function & 0.01 \({}^{\circ}\)C \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters. Figure 2: Overall control system incorporated with particle filter considering noise and time delay. \[T_{ci}(t) =\frac{m+\Delta m}{m+\Delta m+M}T_{co}(t-\Delta t)+\frac{M}{m+\Delta m +M}T_{ci}(t-\Delta t) \tag{13}\] \[X(t) =\frac{U_{h}(t)+\alpha}{U_{h}(t)+U_{c}}T_{hi}+\frac{U_{c}-\alpha}{ U_{h}(t)+U_{c}}T_{ci}(t)\] \[T_{ho}(t) =(1-\beta)T_{ho}(t-\Delta t)+\beta X(t),\]  
and the measurement model is established as Equation (14),  
\[Y(t) =\frac{\gamma U_{h}(t)}{U_{hmax}} \tag{14}\] \[Z(t) =(1-\kappa)Z(t-\Delta t)+\kappa\dot{Y}(t)\] \[T_{co}(t) =T_{ci}(t)+[T_{hi}-T_{ho}(t)]Z(t),\]  
where the process noise and measurement noise comply with Gaussian distribution. For the physical meaning and values in the equations, please refer to Appendix A and Table 1.  
It can be noticed that in the first equation of the state transition model shown in Equation (13), the calculation of \(x_{T_{ci},t}\) consists of the information of measurement value \(y_{T_{co},t-1}\), which seems not to meet the Markovian assumption in the particle filter. However, in a generic particle filter, the prior distribution is by far the most popular importance density in the literature [25], and such importance density is also adopted in this paper, as illustrated in Section 2.2. In such a way, the posterior probability modification effect on particle sampling is ignored.