## 4 Simulations
### Numerical Simulations
#### 4.3.1 Case 1: Both Fluid Flow Rates \(\rm F_{p},F_{u}\) Are Constant  
The objective of this simulation was to prove the convergence of the observers in a common situation where both fluid flow rates remained constant over a long time. The computed inlet flow rate of the utility fluid \(\rm F_{u}\) was \(4.22\times 10^{-5}\) m\({}^{3}\)s\({}^{-1}\), and the inlet flow rate of the process fluid \(\rm F_{p}\) was a constant \(4.17\times 10^{-6}  
The main contribution of the proposed method is the capacity that, besides states, it can estimate the unknown connection of an interconnected system by the final measured outputs, where the unknown connection represents the inlet fluid flow rate \(\mathrm{F_{P}},\mathrm{F_{u}}\) in the IHEX system. Figures 5 and 6 give the encouraging results. Figure 5 shows the computation and estimation of the process fluid flow rate \(\mathrm{F_{P}}\), using a UIO and Observer 2. Figure 6 is the results for the utility fluid flow rate \(\mathrm{F_{u}}\). As expected, \(\mathrm{\hat{F}_{P}andF_{u}}\) follow different trajectories before they converge toward the true value (computed value) in a relatively short transient period, whether or not noise exists. However, the convergence speed of the proposed method is obviously faster than that of the UIO. In addition, the proposed observer was more robust with noise. For \(\mathrm{F_{P}}\), as shown in Figure 5, if no noise was present, after less than 5 s, the three curves overlapped. Compared with the curve of the UIO on the dashed, dotted line, the curve of the proposed Observer 2 in dashed, dotted line only needed less than 1 s to track the simulated curve of the solid line. Moreover, from Figure 5, we can see that the curve of the UIO in the dashed line was significantly more affected by noise than that of Observer 2 in the dashed, dotted line. The similar results are shown in Figure 6 with respect to \(\mathrm{F_{u}}\). According to Figure 6, it took about 1.5 s for the three curves to be overlapped, while for the curve of Observer 2 on the dashed line, less than 0.5 s was taken. The impact of noise is relatively obvious on the curve of the UIO, while for the curve of Observer 2, the influence was less significant.  
Figure 4: Output of the utility fluid temperature \(\mathrm{T_{u}}\). The solid line is the measured value, and the dashed line is estimated by Observer 1. (**a**) represents Noise-free case while (**b**) represents Noise-corrupted case.  
Figure 5: The computation and estimation of the process fluid flow rate \(\mathrm{F_{P}}\). The solid line is the computed value; the dashed line is estimated by the unknown input observer (UIO); and the dashed, dotted line is estimated by Observer 2. (**a**) represents Noise-free case while (**b**) represents Noise-corrupted case.  
In both fluids, these differences were caused by their varied initial values. However, it is proven that if adequate values of the tuning parameters k and \(\theta\) are selected, no matter the degree of deviation of the initial value of \(\hat{\mathrm{F}}_{\mathrm{p}}^{0}\) and \(\hat{\mathrm{F}}_{\mathrm{u}}^{0}\) from the simulated values in the system model, convergences are guaranteed. Larger values of these tuning parameters ensure a smaller convergence time, while smaller values have the opposite effect. However, large tuning values should be avoided, since the observer may become too sensitive to measurement noise in real-time applications. According to the above analysis, we can readily conclude that the proposed interconnected observer works effectively and robustly for the designed purposes with proper tuning parameters.