Page 115 - Mechatronics with Experiments

CLOSED LOOP CONTROL 101
If the control action u(t) is decided upon by a proportional control based on the error
between the desired position, x (t) and the actual measured position, x(t),
d
u(t) = K (x (t) − x(t))
d
p
u(s) = K (x (s) − x(s))
p
d
CLS transfer function from the commanded position to the actual position under the
proportional control is
K p
x(s) = x (s)
d
2
s + K p
The proportional control alone on a mass-force system is equivalent to adding a spring
to the system where the spring constant is equal to the proportional feedback gain, K p
(Figure 2.44). The response of this system to a commanded step change in position is
shown in Figure 2.44. Figure 2.46a shows the CLS root locus as K varies from zero to
p
infinity. The steady-state error due to constant disturbance is
1 1
X(s) =− ⋅ F (s) (2.144)
d
2
m s + K
p
1 1
lim x(t) = lim sX(s) =− (2.145)
t→∞ s→0 m K
p
2.12.2 Derivative Control
Let us consider only the derivative control on the same mass-force system. Assume that
the control is proportional to the derivative of position which means proportional to the
velocity,

u(t) =−K ̇ x
D
u(s) =−K sx(s)
D
2
s x(s) =−K sx(s)
D
s(s + K )x(s) = 0
D
If we consider disturbance in the model, the transfer function from the disturbance (i.e.,
wind force) to the position of the mass can be determined as
m̈ x = f(t) − f (t)
d
̈ x = u(t) − w (t)
d
s(s + K )x(s) =−w (s)
D
d
1
x(s) = (−w (s))
d
s(s + K )
D
Let us consider the case that the disturbance is a constant step function, w = 1 and the
d s
resultant response is
1 1
x(s) =−
s s(s + K )
D
a 1 a 2 a 3
= + +
s 2 s (s + K )
D