# On Invertibility of an Interconnected System Composed of Two Dynamic Subsystems
## 5 Numerical Simulations
### Inverse System Representing
#### 5.4.2 Case 2: Pneumatic Pressure \(p_{c1}\) Remains Fixed, Pneumatic Pressure \(p_{c2}\)  
Increases Abruptly  
In this case, the aim is to illustrate that the proposed local input recovered algorithm is also available even if the local inputs are time-varying. To achieve this purpose, a sudden increase was supposed to apply to the pneumatic pressure \(p_{c2}\) of the utility fluid, where another 0.5 MPA was assumed at time 120 s. Figures 10-15 report the simulation results. Figures 10 and 11 represent the measured temperature of both fluids, Figure 10a describes the noise-free situation, and Figure 10a,b means that measurement of process temperature \(T_{H}\) is corrupted by the colored noise. It can be seen from Figures 10 and 11 that the temperature of both fluids at the global level varies at 120 s due to variation of pneumatic pressure \(p_{c1}\) at the local level. The temperature of the utility fluid in Figure 10b is not influenced by noise since measured noise was supposed on the process fluid. Thus, from Figure 11b, it is clear that the measured process fluid temperature is fluctuant with a small amplitude due to noise. These measured fluid temperatures were fed to the high-level inverse subsystem to reconstruct inputs of the high-level subsystem. From Figure 12a,b, it can be obtained that the reconstructed flow rates \(Q_{H}\) of the process fluid in red dash curves overlap the computed values in black solid lines after a short transient time. Resulting from the supposed variation of pneumatic pressure \(p_{c2}\) at 120 s, the reconstructed values fluctuate a little. For the noise-free case in Figure 12a, after less than two seconds, the reconstructed flow rate \(Q_{H}\) of the process fluid converges back to the computed values in black solid lines again. For the noise-corrupted situation in Figure 12b, the reconstructed value fluctuates significantly. Compared with the measured value, the influence of noise on the reconstructed value is more obvious. Figure 11: (**a**) Measured \(T_{\text{C}}\) (noise-free situation) in case 2; (**b**) measured \(T_{\text{C}}\) (noise-corrupted situation) in case 2. Figure 12: (**a**) Reconstructed and computed \(Q_{H}\) (noise-free situation) in case 2; (**b**) reconstructed and computed \(Q_{H}\) (noise-corrupted situation) in case 2. For the utility fluid flow rate \(Q_{C}\) in Figure 13a,b, it can be seen that the reconstructed values in red dash lines converge to the computed values in the black solid line after transient time. Because of changes at 120 s, reconstructed fluid flow rates vary accordingly and reach a new stable level. Similar to the process fluid, the reconstructed utility fluid \(Q_{C}\) fluctuates significantly in the noise-corrupted situation in Figure 13b. Since the reconstructed values were retrieved from the global measured temperatures by higher-level subsystems, it can be seen that the variation of variables of the local subsystem has a significant impact on the global measurements, which is consistent with the hypothesis. Moreover, the simulation results show that the influence of noise on the reconstructed value is obviously stronger than the measured value. Next, the output of the inverse of the high-level subsystem was fed back to the inverse of the low-level subsystem to reconstruct the original local input at the low-level subsystem, with the aim to identify the reconfigurability of the global interconnected system. The simulation results are shown in Figures 13-15. As we can see from Figure 14a,b, recovered pneumatic pressure \(p_{c1}\) in red dash lines follow the measured values in black solid lines after a short transient time. At time 120 s, the recovered value in the red dash line varies suddenly, but fortunately, it converges to the measured curve quickly. The reason for this variation is caused by the change of pneumatic pressure \(p_{c2}\).