## System Modeling

Valves are complex devices that may include elements in the mechanical, electrical, hydraulic, and pneumatic domains. To illustrate in detail our modeling methodology, we take a pneumatic valve as an example. A diagram of a normally-closed pneumatic valve with a linear cylinder actuator is shown below. The valve is closed by filling the chamber above the piston with gas, and evacuating the chamber below the piston. The valve is opened by filling the chamber below the piston and evacuating the chamber above the piston. The spring ensures that when pressure is lost, the valve will close, i.e., the valve shown below is a normally-closed valve.

Pneumatic Valve

The system state includes the position of the valve, the velocity of the valve, and the masses of gas in the piston chambers. The b and t subscripts refer to quantities for the gas chambers above and below the piston, respectively.

The sum of forces includes the forces from the pneumatic gas, the forces from the fluid flowing through the valve, the weight of the plug, stem, and piston, the spring force, friction forces, and the contact forces at the boundaries of the valve motion:

where Ap is the area of the piston that contacts the pneumatic gas, pfl and pfr are the fluid pressures on either side of the valve (considered to be inputs to this model), Av is the area of the valve contacting the fluid, k is the spring constant and xo is the amount of spring compression when the valve is closed, kc is the contact force constant, and Ls is the stroke length.

The pressures pt and pb are calculated as:

where we assume an isothermal process in which the gas temperature is constant at T, Rg is the gas constant for the pneumatic gas, and Vt0 and Vb0 are the minimum gas volumes for the gas chambers above and below the piston, respectively.

The gas flows are given by

where fg defines gas flow through an orifice for choked and nonchoked flow conditions:

where γ is the ratio of specific heats, and Z is the gas compressibility factor. Pressures ut(t) and ub(t) are considered inputs to this model and represent the input pressures to the gas chambers above and below the piston. They will alternately be set to the supply pressure and the exhaust pressure depending on the commanded valve position. We select our measurement vector as

where fv is the fluid flow through the valve:

where Cv is the (dimensionless) flow coefficient of the valve, and ρ is the liquid density. The open(t) and closed(t) signals are from discrete sensors which output 1 if the valve is in the fully opened or fully closed state:

The following figure shows a typical valve cycle. The valve is commanded to open at 0 s and to close at 15 s. It takes the valve about 8 s to open and about 5 s to close. The valve closes faster than it opens due to the action of the return spring. Most of the delay in the valve movement is due to the time it takes the pneumatic gas to flow into and out of the valve actuator.

Nominal Valve Operation