Page 540 - Mechatronics with Experiments

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However, the amplifier gain cannot be made arbitrarily large due to the closed loop band-
width limitations set by the natural frequency of the open loop system,
f (i , e ) ≤ K ≤ f (w ) (7.349)
1 db max sa 2 wb
K ⋅ K ⋅ K fx 1
q
sa
w = K = ≤ ⋅ w (7.350)
bw vx n
A 3
c
which indicates that the lower limit of the amplifier gain is set by the deadband and
desired positioning accuracy, and the upper limit is set by the open loop bandwidth of the
hydraulic axis.
7.8.2 Load Pressure Controlled Electrohydraulic
Motion Axes

Let us also consider the same EH motion system, except that this time the closed loop
control objective is to control the load pressure. The load pressure is measured as the
differential pressure between the two output ports of the valve. We will assume that the
pressure dynamics between the valve output ports and the actuator ports is negligable
(Figure 7.91). The commanded signal represents the desired load pressure, ΔP cmd .Using
basic block diagram algebra, the transfer function from the commanded load pressure and
the external load speed to the load pressure can be obtained as
K Q 1
c
t
ΔP(s) = ⋅ (K ⋅ i(s) − K pq ⋅ ΔP (s) − A ⋅ V (s)) (7.351)
l
q
L
c
A A s
c
c
Notice that the internal leakage term due K pq ⋅ ΔP(s) is relatively small for servo valves
and can be neglected in the analysis. Assuming the leakage term is neglected, the closed
loop transfer function can be expressed as
( 2 )
K ⋅ Q ∕A c ⋅ K K )
sa q
c
t
L
ΔP (s) =+ ( 2 ) ⋅ ΔP cmd (s) (7.352)
s + K K K K Q ∕A c
c
fp sa q t
(K Q ∕A )
t
c
c
l
− ( ) ⋅ V (s) (7.353)
s + K K K K Q ∕A 2 c
fp sa q t
c
In terms of command signal and output signal relationship, the transfer function behavior
of the force servo is similar to the position servo. In the above linear models, the only
dynamics included is the integrator behavior of the actuator. The transient response of the
valve from current to flow is not modeled. Experimental studies indicate that the transient
response of the proportional or servo valve from current to flow can be approximated by
a first-order or a second-order filter depending on the accuracy of approximation needed.
The transient response model of a valve is then,
Q (s) = K q (7.354)
o
i(s) ( s + 1)( s + 1)
v1 v2
where v1 and v2 are the two time constants for the second-order model. One of them is
set to zero for a first order model.
Remark The flow rate through an orifice, that is valve, is expressed in two different
common equation forms,
√
Q(t) = C ⋅ A(x ) ⋅ Δp(t) (7.355)
v
d
√
A(x ) Δp(t)
v
Q(t) = Q ⋅ ⋅ (7.356)
r
A max p r