We use an air tank shown in Figure S10d to characterize the open degree of solenoid valves, which includes four steps: i) Inflate the air tank to 60 kPa (other air pressures are also feasible). ii) Open the valve and start recording the pressure in the tank simultaneously. iii) After an interval of \(\Delta t(\Delta t \le 1s)\) , close the valve. iv) Continue recording pressure until \(t=1\)  s.
As is shown in Figure S10e, both the opening and closing time between fully close and open states are about 0.2 s. The experimental results overall agree with the work of Xu et al. [65]. Notably, the response time of different valves varies due to inconsistent manufacturing quality. However, a more complicated situation is when the system generates high-frequency actions, the solenoid valve usually does not change between the fully close and open states. The state change may occur at any point between these two extreme states. We change \(\Delta t\)  from 0.05 s to 0.25 s with an interval of 0.05 s, and the results are shown in Figure S10f. Here, we propose a time-based piecewise linear model to describe the open degree function.
ยทIf the valve command changes from ON to OFF