Page 537 - Mechatronics with Experiments

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JWST499-c07
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ELECTROHYDRAULIC MOTION CONTROL SYSTEMS 523
7.8.1 Position Controlled Electrohydraulic Motion Axes
The controlled output, position (x), is determined by the commanded position (x cmd ) signal
and load force (F ). Using basic block diagram algebra, the transfer function from x cmd and
d
F to x can be calculated using the linear models for each component,
d
( ( ) )
1 K pq
X(s) = ⋅ K ⋅ i(s) − ⋅ F (s) (7.313)
q
d
A ⋅ s A
c c
(
1
X(s) = K ⋅ K ⋅ (X cmd (s) − K ⋅ X(s)) (7.314)
sa
q
fx
A ⋅ s
c
( ) )
K pq
− ⋅ F (s) (7.315)
d
A
c
( )
K pq
(A ⋅ s + K ⋅ K ⋅ K ) ⋅ X(s) = K ⋅ K ⋅ X cmd (s) − ⋅ F (s) (7.316)
c
fx
q
sa
q
sa
d
A c
(K ⋅ K ∕A )
sa
c
q
X(s) =+ ⋅ X cmd (s) (7.317)
s + (K ⋅ K ⋅ K ∕A )
sa
fx
c
q
K ∕A 2 c
pq
− ⋅ F (s) (7.318)
d
s + (K ⋅ K ⋅ K ∕A )
fx
q
c
sa
Using the last expression for the transfer function and the final value theorem for Laplace
transforms, we can determine the following key characteristics of the closed loop position
control system.
Steady-state position following error in response to a step position command and no

disturbance force condition, X cmd = X ∕s and F (s) = 0
d
o
lim x(t) = lim s ⋅ X(s) (7.319)
t→∞ s→0
(K ⋅ K ∕A )
c
q
sa
= lim s ⋅ ⋅ X ∕s (7.320)
o
s→0 (s + (K ⋅ K ⋅ K ∕A )
fx
sa
q
c
= (1∕K )X o (7.321)
fx
When a constant displacement is commanded and there is no disturbance force, the
actual position will be proportional to the commanded signal by the feedback sensor
gain. By scaling the commanded signal, the effective ratio between the desired and
actual position can be made to be unity. In other words, the output position is exactly
equal to the commanded position in steady-state when the commanded position is a
constant value.
What is the maximum steady-state position error in response to a ramp command

signal
X cmd (s) = V ∕s 2 (7.322)
0
F (s) = 0 (7.323)
d
The tracking error transfer function is
X (s) = X cmd (s) − K X(s) (7.324)
fx
e
K ⋅ K ⋅ K ∕A c
sa
q
fx
= X (s) − ⋅ X (s) (7.325)
cmd cmd
s + (K ⋅ K ⋅ K ∕A )
sa q fx c
s 2
= ⋅ V ∕s (7.326)
o
(s + K ⋅ K ⋅ K ∕A )
fx
c
q
sa