Improving a pipeline hybrid dynamic model using 2DOF PID

2.1. Construction of the hybrid model without 2DOF PID

In the experimental setup shown in Fig. 1 accelerometer (a) is located at the right end of the pipe section, accelerometer (b) is located in the middle of the section and accelerometer (c) is located at the left of the section. An instrumented hammer applies the excitation force at the right end of the pipe. DSpace and its data acquisition system measures the hammer force as well as the accelerometer responses.

The experimental data is utilized in a hardware-in-the-loop format as shown in Fig. 2. to produce an optimal estimate of the pipe vibration at any desired location. It should be noted that only one accelerometer measurement is used to update the model (i.e., feedback signal), the remaining accelerometers are used for comparison with estimates at corresponding locations.

The system identification toolbox in MATLAB is utilized to obtain the initial model from raw experimental data [6, 7]. Moreover, the initial model is used to construct the hybrid model which uses the LQE and LQR techniques to generate optimal estimates of the system’s states such that the associated cost functions are minimized.

Fig. 1Experimental setup

Fig. 2Schematic of the hybrid model without 2DOF PID

2.2. Improving the model with 2DOF PID

A 2DOF PI controller is used to improve the performance of the hybrid model, the general block diagram of the PID control system [8] is shown in Fig. 3.

Fig. 3Block diagram of 2DOF control system

In the process,$G_c\left(\mathrm{s}\right)$, $G_{ff}\left(\mathrm{s}\right)$ and ${\mathrm{G}}_p\left(\mathrm{s}\right)$ are transfer functions which are feedback and feedforward controllers as shown in Eqs. (1), (2) and (3):

Eq. (1) and (2) represent the PI controllers, in Eq. (2) the parameter $b$ has no influence on disturbance rejection, while it has a significant influence on the set point response. In Eq. (3) $k_p$, $T$ and $\tau$ are the process gain, time constant and dead time, respectively.

Fig. 4Parameter setting panel of 2DOF PID

The parameter setting panel of 2DOF PID, in Simulink, is shown in Fig. 4. The parameter “Derivative (D)” is set to 0, to avoid adding any fictitious damping to the original model. Other parameters are properly adjusted to achieve proper tracking of the actual response of the pipeline at measurements location. The 2DOF controller is incorporated into the model as shown in Fig. 5. It is clear that the model uses feedback from the actual system to generate better estimates of the plant (actual pipe) states. The model produced can estimate, with good accuracy, the vibration at any location along the span of the pipeline without the need for any physical measurements at that location. This makes the proposed approach unique, as it uses minimal feedback measurements to yield good estimates at any desired location on the structure. This method can be used for any dynamic structure and it is not limited to pipelines only.

Fig. 5Schematic of improving the Hybrid Model with 2DOF PID