Page 538 - Mechatronics with Experiments

JWST499-Cetinkunt
JWST499-c07
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Using the final value theorem, it can be determined that the steady-state position
following error while tracking a ramp position command (constant speed command) is
lim X (t) = lim s ⋅ X (s) (7.327)
e
e
t→∞ s→0
1
= ⋅ V o (7.328)
(K ⋅ K ⋅ K ∕A )
q
fx
sa
c
Another important characteristic is the closed loop stiffness of the positioning system.
What is the steady-state change in position if there is a constant disturbance force
acting on the system, F (s) = F ∕s, X (s) = 0.0, K = F ∕X? Using the final
d do cmd cl do
value theorem,
2
(K ∕A )
pq
c
X(s) =− ⋅ F (s) (7.329)
d
s + (K ⋅ K ⋅ K ∕A )
c
sa
fx
q
2
(K ∕A )
pq
lim s ⋅ X(s) =−s ⋅ c ⋅ F ∕s (7.330)
d0
s→0 s + (K ⋅ K ⋅ K ∕A )
sa q fx c
2
x(∞) = (K ∕A )∕K ⋅ F (7.331)
pq c vx do
F do
= K cl (7.332)
x(∞)
The closed loop steady-state stiffness is
K vx ⋅ A 2 c
K = (7.333)
cl
K pq
K sa ⋅K q ⋅K fx 2
⋅ A
A c c
= (7.334)
K pq
K ⋅ K ⋅ K ⋅ A c
q
fx
sa
= (7.335)
K
pq
The leakage coefficient for servo valves is approximately
Q r
K pq = 0.02 ⋅ (7.336)
P S
Hence, the stiffness of the closed position control servo with such a valve is approxi-
mately,
2
A ⋅ P S
c
K = 50 ⋅ K vx (7.337)
cl
Q r
which clearly indicates the effect of the leakage coefficient on the closed loop stiffness.
It is important to recall that the natural frequency of the electrohydraulic system and
its load imposes an upper limit on the closed loop system bandwidth. The position servo
control system described above has approximately the closed loop bandwidth (closed loop
transfer function pole) equal to K vx and the value of it is limited by the natural frequency
of the system due to stability considerations,
1
w = K ≤ ⋅ w (7.338)
bw vx n
3
The characteristics of the valve around its null position significantly affects the
steady-state positioning accuracy of the closed loop system.