# On Invertibility of an Interconnected System Composed of Two Dynamic Subsystems
## 5 Numerical Simulations
### Inverse System Representing
#### 5.4.1 Case 1: Both Pneumatic Pressure \(p_{c1},p_{c2}\) Are Fixed  
Both local inputs of pneumatic pressures \(p_{c1}\), \(p_{c2}\) were considered to be fixed in this case. The simulation purpose is to confirm if the local inputs pneumatic pressure \(p_{c1}\). \(p_{c2}\) can be uniquely recovered by the global measured temperatures \(T_{H}\), \(T_{C}\), under given initial conditions. Figures 4-9 confirm the reconstructability and robustness of the interconnected system. Figures 4 and 5 show the measured temperatures of both fluids under noise-free and noise-corrupted situations. From Figures 4a and 5a, it can be seen that, after a relatively short transient time, the temperatures of both fluids stabilized at a new level, and from Figures 4b and 5b, it can be seen that even measurements were corrupted by the colored noise, convergence can also be ensured. These measurements were then employed to reconstruct inputs of high-level subsystems. As shown in Figure 6, the expected computed value of process fluid flow rates \(Q_{H}\) are plotted in the black solid lines, and red dash lines represent the reconstructed values via the inverse of the high-level subsystem. It is illustrated in Figure 6a that, after a short transient period, if the measurement is not corrupted by noise, the reconstructed value in the red dash curve tracks the computed value in the black solid curve correctly. And from Figure 6b, when measurement suffers noise, the reconstructed value can also track the computed one with acceptable accuracy. It is obvious that the inverse of the high-level subsystem is capable of recovering its inputs with acceptable accuracy. Figure 4: (**a**) Measured \(T_{H}\) (noise-free situation) in case 1; (**b**) measured \(T_{H}\) (noise-corrupted situation) in case 1. Figure 5: (**a**) Measured \(T_{\rm C}\) (noise-free situation) in case 1; (**b**) measured \(T_{\rm C}\) (noise-corrupted situation) in case 1. Figure 6: (**a**) Reconstructed and computed \(Q_{H}\) (noise-free situation) in case 1; (**b**) reconstructed and computed \(Q_{H}\) (noise-corrupted situation) in case 1. When it comes to the utility fluid case in Figure 7a,b, the computed value in black solid lines overlapped the reconstructed values in red dash lines after a short transient time, no matter with or without measurement noise. Figure 7a illustrates the noise-free case while Figure 7b represents the noise-corrupted situation. From Figure 7b, compared with measured temperature, it can be seen that noise value impacts reconstruct value more significantly. Whatever the case, the input reconfiguration capability of the high-level subsystem is confirmed. Thereafter, these reconstructed values were used as inputs of the inverse of the low-level subsystem, with the aim of checking the reconfiguration capacities. It can be seen from Figure 8a,b that the input reconfiguration capability of the low-level subsystem is confirmed. It is obvious in Figure 8a,b that the recovered pneumatic pressure of the process fluid \(p_{c1}\) in red dash lines converge to the measurement value in black solid lines correctly, and the short transient time is less than 2 s. From Figure 8a, during the first two seconds, oscillation is observed. It can be seen from Figure 8b that the reconstructed value is influenced clearly by measurement noise. Since recovered pneumatic pressure \(p_{c1}\) is also the original input of the interconnected system, thus reconfiguration capability of the global interconnected system is also confirmed. For utility fluid flowrate, similar results to process fluid are obtained. From Figure 9a,b, it can be seen that the recovered pneumatic pressure \(p_{c2}\) in red dash lines converge to the measured pneumatic pressure in black solid curves after a short transient time, which are also the outputs of the inverse interconnected system. Figure 7: (**a**) Reconstructed and computed \(Q_{C}\) (noise-free situation) in case 1; (**b**) reconstructed and computed \(Q_{C}\) (noise-corrupted situation) in case 1.