Page 622 - The Mechatronics Handbook

0066_Frame_C20 Page 92 Wednesday, January 9, 2002 5:49 PM

Hypothesizing a bipolar reference signal, it can be assumed that the range V ref > 0 corresponds to the
supply–user connection, while the field V ref < 0 corresponds to the user–discharge connection. The
appropriate equation of flow above must be rewritten in the same way.
Calculation of the conductance of the flow proportional valve is made considering the static and
dynamic link between the reference voltage V ref and the opening of the passage aperture A V in accordance
with modeling of the second order of the type:

2
----------- + 2zs n --------- + s n A V = K s s n V ref (20.65)
d A V
2
2
dA V
dt 2 dt
where ζ is the damping factor, σ n is the valve’s natural frequency, and K s is its area static gain.
Assuming a static relation of the linear type between the opening A V and the conductance C, as an
initial approximation, we get
C = K c A V = K c K s V ref = K V V ref (20.66)

where K V is the flow static gain of the valve, function of the maximum conductance C max , and of the
maximum value of the reference voltage V ref max :

K V = --------------- (20.67)
C max
V ref max

The complete dynamic relation between reference voltage and conductance is, therefore,

2
d C dC 2 2
--------- + 2zs n ------- + s n C = K c s n V ref (20.68)
dt 2 dt

The nonlinear model of the pneumatic servosystem with the position reference x set and the force
disturbance F e as inputs, is made up of a nonlinear system of nine equations, of order eight overall, of
the type:

a) C 1 = C 1 (V ref 1 , t) order 2 conductance of valve V 1 (see (20.68))
b) C 2 = C 2 (V ref 2 , t) order 2 flow rate of valve V 1 (see (20.68))
c) G 1 = G 1 (C 1 , P 1 ) order 0 flow rate of valve V 1 (see (20.61)–(20.64))
d) G 2 = G 2 (C 2 , P 2 ) order 0 flow rate of valve V 2 (see (20.61)–(20.64))
e) G 1 = G 1 (P 1 , P ˙ 1 , x, ) x ˙ order 1 continuity chamber 1 (see (20.58))
f) G 2 = G 2 (P 2 , P ˙ 2 , x, ) x ˙ order 1 continuity chamber 2 (see (20.59))
g) x ˙˙ = (F e , P 1 , P 2 , ) x ˙˙ x ˙ order 2 piston equilibrium (see (20.60))
h) V ref 1 = V ref 1 (x set , x ret ) order 0 V 1 valve control
i) V ref 2 = V ref 2 (x set , x ret ) order 0 V 2 valve control
If we want to carry out a linear analysis, it can be assumed that the equations a), b), g), h), i) are already
written in linear form.
As far as the flow rates of valves c) and d) are concerned, it is hypothesized that the flow rate for each
of them is subsonic in feed, with V ref > 0, and sonic in discharge, with V ref < 0. This means that for valve
V 1 , for example, the pressure P 1 must be within the range bP s < P 1 ≤ P s in feed and in the range P 1 ≥ P amb /b

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