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ble resistance to flow.
11. Friction forces on the ram are viscous (no dry friction).
12. External load force is small enough to make assumption (5) possible.
Most of these assumptions are usually valid for small displacements of
the load. A notable exception is 11) which proved wrong in Shearer’s
work. Several components of the complete system as shown in Fig. 18.1
can therefore be regarded as ideal and do not have to be modelled.
18.2 Model of Control Valves
Proportional directional control valves are described and modelled in detail
in Chap. 16. These models describe the large signal behaviour. They in-
clude a number of non-linearities and are very well suited for digital
simulation. For the design of control systems simpler models are needed
because most analysis and design methods are based on the assumption of
linear and time-invariant behaviour of the plant and actuator. The follow-
ing linear model is a “violent approximation of the real process” (Wi-
kander 1988) but can be useful to describe the behaviour for small valve
openings, e.g. up to 20 % of the maximum input signal. It can be used for
1 The earliest paper on control of a pneumatic piston may be from Wintergerst
(1950) who gives a very simple linear third order model and some stability
analysis. A similar system was studied by Brann (1966).an initial selection of controller structure and parameters. More detailed
studies have to use either a non-linear model or the actual drive.
In Chap. 16 the model of the control valve is divided into a mechanical
and a pneumatic part. The same approach can be chosen for a controller
design oriented model. The mechanical part consists of the spool, a spring
and a force generating part, today mostly a coil, but torque motors or flap-
per-nozzle pilot valves have also been used in the past. The mass of the
spool and the compliance of the spring lead to a second order system. The
force generation can be described by a first order system that either models
the inductance of the coil or the pressure build-up in the pilot stage. Figure
18.2 gives a block diagram of the model where the limitations due to hard
stops and saturation are explicitly shown. Proportional control valves usu-
ally have an internal control loop for the spool displacement which can
modify the dynamic behaviour considerably. Depending on the design, it
may be sufficient to model the dynamics with a first or second order sys-
tem.
The pneumatic part of the control valve describes the air flow as a func-
tion of the spool displacement and pressure. The nozzle model from Chap.
5 can be used to describe the flow through a metering nozzle when the
sonic conductance C is known as a function of the spool displacement
sSpool. The critical pressure ratio b is often modelled as independent of the
spool displacement, i.e. a constant value.where m & mass flow rate in kg/s,
p1 pressure upstream in Pa,
C(sspool) sonic conductance as a function of spool displacement
in m3
/(s⋅Pa),
ρ0 density of air at reference conditions in kg/m³,
T0 temperature of air at reference conditions in K,
T1 upstream temperature of air in K,
p2 pressure downstream in Pa,
b(sspool) critical pressure ratio as a function of spool displacement,
sspool displacement of spool in m.
To describe all flow paths in a 5/3-way valve, four metering nozzles are
required.
Equation (18.1) is non-linear and not suited for controller design. In a
first step towards a simple model, Schwenzer (1983:29–42) replaces it by a
linear relationship between mass flow rate and pressure:
1 K valve coefficient.
He gives comparisons between model and measurements of a sudden
pressure build-up or pressure release in a pneumatic circuit and shows that