# Design of an IMCPID Optimized Neural Network for Stepless Flow Control of Reciprocating Machinery
## 4 Algorithm Simulation and Results
### Step Response Test and Validation  
To demonstrate the characteristics of the proposed IMCPIDNN controller, step response tests are made for the reciprocating compressor numerical model. The PID controller, IMCPID controller and FLSMC controller are constructed to make comparisons under the same condition. The simulations are carried out with reference pressure profile under the simulated condition to verify the control performance of the proposed controller. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Parameter** & **Value** \\ \hline \(\eta\) & Learning coefficient & 0.5 \\ \(\alpha\) & Inertia coefficient & 0.2 \\ \(l\) & Number of input layer nodes & 5 \\ \(j\) & Number of hidden layer nodes & 6 \\ \(i\) & Number of output layer nodes & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the IMCPIDNN controller. In order to evaluate the performance of the proposed controllers, we use the integral of absolute value error criterion (IAE), integral of time multiplied by the absolute value of error criterion (ITAE), integral of square error criterion (ISE) and integral of time multiplied by squared error criterion (ITSE) as the evaluation function [33]. Two-system step response performance indicators have also been applied and they are \(t_{s}\) and \(M_{p}\), where \(t_{s}\) means the shortest time that reaches the step response steady-state value of 2% error and \(M_{p}\) means the overshoot, respectively. Figure 6 shows that the dynamic output of each loop meets the desired set point. The dynamic tracking performances obtained by PID, IMCPID, FLSMC and IMCPIDNN controllers for the two-input and two-output nonlinear system were compared. The process of starting the system from the shutdown state is imitated, where the set-points of the first output and the second output change from 0 kPa to 120 kPa and 220 kPa at moment t = 0 synchronously. Obviously, IMCPIDNN has the shortest settle time and the smallest overshoot, and it performs the best among the four algorithms while PID performs worst. Table 3 presents the best control performance indices obtained by each controller. What stands out in the table is all the performance indices of the IMCPIDNN controller are optimal in the closed-loop step response. The value of the settle time \(t_{s}\) of the step responses for the two loops are: 61.91 s and 146.78 s, respectively. They are smaller than those obtained by the other three algorithms. The IAE, ITAE, ISE and ITSE values obtained with the proposed algorithm for loop 1 are: 2.62, 1.98, 0.42 and 1.96, and for loop 2: 10.60, 14.05, 4.01 and 31.14. Compared with the PID controller, IMCPID controller and FLSMC controller, the proposed IMCPIDNN controller performs better in terms of set-point tracking. At the moments t = 100 and t = 800, the set-point value is changed sequentially in the two loops. The simulation result in Figure 7 shows the step response and control effect obtained by the proposed IMCPIDNN controller. It is relevant to note that the performance of the PID and IMDPID controllers are not satisfied, and this is expected because of the strong interactions. The common problem of the two controllers is that they cannot eliminate the system coupling problem. For example, the step change at moment