# Modeling a Hydraulically Powered Flight Control Actuation System
## 5 HPFCAS Model Development and Simulations
### Simulations of Model in a Transient State  
Simulations in a transient state were completed with fault injection conditions, hence the equation of the steady-state flow changed to Equation (4) to give the force balance for the piston movement of the integrated actuator [35]. \[mv_{a}=(P_{a}A_{a}-P_{b}A_{b})-(\beta_{\gamma}+F_{fr})-F_{L} \tag{4}\]  
where \(mv_{a}=\) mass of the piston and the double derivative of the piston displacement, respectively. \(P_{a}A_{a}-P_{b}A_{b}=\) Pressure derivative equations from two chambers,  
\(\beta_{\gamma}+F_{fr}=\) Viscous and frictional forces in the actuator cylinder,  
\(F_{L}=\) External load. This represents a dynamic model of the actuator, which contains pressure derivative equations from two chambers expressed by a continuity mass equation and a double derivative equation for the motion of the actuator piston [36]. From the developed HPFCAS model, a significant quantity of data was generated from the various conditions of operations considered for all the principal components. For instance, the pump speed was adjusted for five different conditions of speed from 200 rpm to 600 rpm, and the valves were adjusted from 10 to 100% operations. The model itself was set to run in two ways. These were either \"Run to flow\", where you set a flow rate and run all simulations, or \"Run to rpm\", in which case, a specific rpm of the pump is set at a constant value. In this work, the run-to-flow option was used because, in practical terms, the engine-driven pump or electric-driven pump expects the fluid supply to be at a preset flow rate, to maintain a constant flow [37]. Thereafter, the Simulink model was transformed into Matlab/Simscape to observe the responses of the actuator as follows:  
The first step was to test the frequency response characteristics of the actuator system model using a control approach without tuning and with tuning to establish how the actuator vibration would be managed on the Simulink model, as shown in Figure 17a,b. Figure 17: The responses of the actuator in Simulink without tuning and with tuning in (**a,b**), respectively. In Figure 17a without tuning of the actuator, the magnitude of the actuator frequency responses, when simulated for 10 s oscillates, take a longer time to settle and show the nature of the vibrations in the actuator before settling down, while the phase margin does not change. These are established faults associated with HPFCAS. However, with the right tuning parameters after a series of iterative processes in Figure 17b, the actuator responses dampened in a short period of just 2s, which shows that the actuator overcomes vibrations within a short time. This makes its parameters suitable for integration in the Simscape model. Meanwhile, the phase margin was still unaffected. Secondly, the actuator was tested for its saturation limit and its high gain despite the result obtained in the first instance before it was used to develop the Simscape model. Thus, in the Simscape model simulation, the result of the actuator response was obtained by different sensor measurements for the piston position, velocity, acceleration, and force. This was required to verify and observe the behavior of the actuator as its parameters were adjusted. Figure 18 shows the Simscape integration of the actuator. The result was a video clip animation of the dynamic movement of the actuator piston, as it extends and retracts which was captured on the simulation of the Simscape integration model, shown in Figure 19.  
In Figure 19, the horizontal linear response of the actuator is captured during the simulation of an extension. The green color is the cylinder, the blue-colored block is the piston, and the violet color represents the loading due to forces. The black and white cycles  
Figure 19: The responses of the actuator in the Simscape model showing extension and retraction. Figure 18: The Simscape integration of the actuator to verify its behavior. are the center of gravities of the cylinder, piston, and loads, respectively.