# Design of an IMCPID Optimized Neural Network for Stepless Flow Control of Reciprocating Machinery
## 4 Algorithm Simulation and Results
### Anti-Interference and Robustness Test  
To illustrate the stability of the proposed IMCPIDNN controller, some anti-interference and robustness tests had been done. We want to know how the control effect changes when the values of the transfer function time constant and gain are mismatched with the actual system and how these controllers perform when the control process encounters external disturbance. Figures 8 and 9 present the step response with +40% variation on time constant and gains. We can see that the proposed IMCNNPID controller is very stable to the changes of model parameters. Figure 8 shows that the overshoot of PID, IMCPID, FLSM and IMCPIDNN in the face of model mismatch is 3.60%, 16.59%, 0% and 1.79%, respectively. Additionally, it can be seen that the IMCPIDNN controller has a faster adjustment speed than the other three controllers. Figure 9 shows the process outputs when the model mismatch increases to +40%. The most interesting aspect of these two figures is that the overshoot obtained by the IMCPIDNN controller is slightly larger than Figures 10 and 11, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Algorithm** & \(M_{p}\) **(\%)** & \(t_{\text{s}}\) **(s)** & **IAE(\(\times\)103)** & **ISE(\(\times\)105)** & **ITAE(\(\times\)105)** & **ITSE(\(\times\)106)** \\ \hline \multirow{4}{*}{Output1} & PID & 2.01 & 85.37 & 1.01 & 0.12 & 7.86 & 8.80 \\  & IMCPID & 0.93 & 113.68 & 1.28 & 0.19 & 9.62 & 13.79 \\  & FLSMC & 0 & 53.85 & 1.02 & 0.17 & 7.38 & 12.52 \\  & IMCPIDNN & 0 & 42.32 & 0.65 & 0.12 & 4.72 & 8.89 \\ \hline \multirow{4}{*}{Output2} & PID & 0.11 & 182.10 & 3.20 & 0.56 & 47.59 & 81.71 \\  & IMCPID & 0.76 & 99.97 & 2.06 & 0.37 & 30.18 & 53.67 \\ \cline{1-1}  & FLSMC & 0 & 83.90 & 1.56 & 0.27 & 22.05 & 39.57 \\ \cline{1-1}  & IMCPIDNN & 0 & 81.43 & 1.49 & 0.27 & 21.44 & 38.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance comparison of the four algorithms used for sequential step responses. Figure 7: Comparison of step responses with sequential changes in set-points: (**a**) Output1; (**b**) Output2. Next, 20 kPa of external disturbance was added to output 1 and output 2 at \(t=100\) s and \(t=600\) s, respectively. The simulation results are shown in Figures 12-14. Compared with the PID, IMCPID and FLSMC controllers, the settle time and overshoot of the proposed IMCPIDNN controller performed better, obviously. Under the action of the IMCPIDNN controller, the output can return to the set-point faster and more steadily, and the other loop is not interfered. This is meaningful for a steless flow control system as it can ensure that the system runs more safely in the presence of external interference and reduce the failure rate of the equipment. constructed. In terms of efficiently adjusting control parameters, the BP neural network algorithm with dynamically adjustable parameters has played a key role. Numerical experiments show that the proposed IMCPIDNN controller has good setpoint tracking performance and adaptability to a system model mismatch. Compared with PID, IMCPID and FLSMC controllers under the same conditions, the proposed IMCPIDNN controller performs best in terms of settle time, overshoot, robustness and anti-interference ability. More importantly, the process of the proposed IMCPIDNN controller to adjust the control parameters is the simplest because of the fewest adjustable parameters, which will provide great convenience for actual industrial applications. Finally, the proposed controller can be applied to any linear or non-linear MIMO system. We will continue to study process modeling, stability analysis, system convergence and robustness analysis to achieve better control effects. Conceptualization, H.H. and Z.J. ; methodology, H.H., W.M., W.X. and J.Z. ; software, H.H., W.M. and W.L. ; validation, H.H. and Y.W. ; formal analysis, H.H. and W.X. ; investigation, Y.W. ; resources, W.L. ; data curation, H.H. ; writing--original draft preparation, H.H. ; writing--review and editing, H.H. and Z.J. ; visualization, J.Z. ; supervision, Z.J.