## 4 Simulations
### Numerical Simulations
#### 4.3.2 Case 2: The Fluid Flow Rates Vary Due to Parameter Changes  
The present computations were executed to get an accurate screening of the variation of the observer estimate by corroborating if they were in agreement with the simulated fluid flow rates which underwent either an abrupt change or a gradual variation due to, for example, aging or erosion. The parameter effects were taken into account in the following way. Initial values of \(\mathrm{F_{u}}\ =\ 4.22\times 10^{-5}\ \mathrm{m^{3}s^{-1}}\) and \(\mathrm{F_{p}}\ =\ 4.17\times 10^{-6}\ \mathrm{m^{3}s^{-1}}\) are considered, followed by an abrupt change of \(\mathrm{F_{u}}\) at 80 s. The reason for this change is the variation of parameter \(\Delta\mathrm{P}\), with the value changing from 0.6 MPa to 0.4 MPa. Several factors can be attributed to this kind of variation, such as valve clogging or an unexpected pressure drop across the control valves. After that, at t = 150 s, \(\mathrm{F_{p}}\) began to deteriorate due to an increase of the spring compliance \(\mathrm{k_{2}}\) in the process fluid actuator. One main reason contributing to this change is erosion. Because of erosion, the gland packing of the valve may loosen, which leads to stem vibration. In the simulation, a value of 1000 nm\({}^{-1}\) was added to the spring compliance \(\mathrm{k_{2}}\). This simulation was carried out using the same constants used in the previous simulation and the same values of \(\mathrm{T_{pi}}\) and \(\mathrm{T_{ui}}\), as well as \(\mathrm{T_{p}^{0}}\) and \(\mathrm{T_{u}^{0}}\), were used. The initial conditions of both observers, as well as the observer parameters (\(\mathrm{k_{1}},\mathrm{k_{2}},\mathrm{k_{3}},\mathrm{k_{4}}\), and \(\theta\)) were the same as the previous ones. These variations are illustrated in Figures 7-10. Figure 6: The computation and estimation of the utility fluid flow rate \(\mathrm{F_{u}}\). The solid line is the computed value; the dashed line is estimated by the UIO; and the dashed, dotted line is estimated by Observer 2. (**a**) represents Noise-free case while (**b**) represents Noise-corrupted case. Figure 8: Outlet temperature of the utility fluid \(\mathrm{T_{u}}\). 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 7: Outlet temperature of the process fluid \(\mathrm{T_{p}}\). 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. another drop happened, and a new stable level was expected with a 0.9 \({}^{\circ}\)C reduction. These decreases imply the influences of parameter changes in the fluid actuators, and no further variations illustrate the occurrence of additional changes. A similar result was obtained in the estimated \(\rm\hat{T}_{u}\) of the utility fluid in Figure 8. It is shown that, due to changes of \(\rm\Delta P_{1}\) and \(\rm k_{2}\), the measured \(\rm T_{u}\) dropped 0.2 \({}^{\circ}\)C and 0.5 \({}^{\circ}\)C at 80 s and 150 s, respectively. The estimated \(\rm\hat{T}_{u}\) on the dashed line tracks \(\rm T_{u}\) after the observer converges. The simulation curves indicate that the proposed observer was effective at tracking system performance. However, it is shown that, because of noise, the drops caused by local parameter changes at 80 s could not be visually observed on the global output directly. In order to analyze the influence of these changes, we have to detect it through the local dynamics supervision. Therefore, it is very meaningful to monitor both the local and global dynamics of a control system. It can be seen from Figure 9 that, in a noise-free situation, two estimate curves on the dashed line and the dashed, dotted line converge to the simulated value \(\rm F_{P}\) of the solid line quickly after a transient response.