Page 121 - Mechatronics with Experiments

CLOSED LOOP CONTROL 107
Since PD control gains determine the natural frequency and the damping ratio of the closed
loop system, PD control can be efficient in shaping transient response.
Figures 2.46d and 2.46e show the closed loop system pole locations as K varies
p
from zero to infinity. The poles of the closed loop transfer function can be expressed as
2
Δ (s) = s + K s + K = 0 (2.152)
cls D p
(K ∕K ) s + 1
p
D
= 1 + K = 0 (2.153)
p 2
s
1
= 1 + K = 0 (2.154)
p
s(s + K )
D
If we sketch the root locus, it is clear that the closed loop poles will always be stable for any
positive values of K , K (Figure 2.46d). If we plot the root locus for a constant value of
p
D
K ∕K as K varies from zero to infinity, we have the root locus as shown in Figure 2.46d.
D
p
p
If we plot the root locus of the last form of the equation for a given constant value of K D
as K varies from zero to infinity, we have the root locus shown in Figure 2.46e. Although
p
the shape of the root locuses are different for the same closed loop system in question, the
Figure 2.46d is a root locus as both K and K vary such that K ∕K is constant, whereas
p
p
D
D
Figure 2.46e is a root locus as K varies for a constant K . For a given value of K , K ,
D
D
p
p
both root locuses predict the same closed loop root locations, as they should.
1
(ii) Zero command, constant step disturbance case - x = 0; w ≠ 0; w (s) = .The
d
d
d
s
response due to step disturbance is given by
1 1
x(s) =−
2
s + K s + K s
p
D
1 1
x(s) = e(s) =−
2
s + K s + K s
D p
Any non-zero response due to the disturbance is indeed an error. The magnitude of the error
under PD control is
lim e(t) = lim s(s)
t→∞ s→0
1 1
= lim(−s)
2
s→0 s + K s + K s
D p
1
=
K p
Therefore, the PD control alone cannot provide zero steady-state error in the presence of
constant disturbance.
2.12.6 PID Control
PID control is basically a PD control plus PI control. It combines the capabilities of PD
and PI control. PD control is primarily used to shape transient response and stabilize the
system. The D (derivative) action introduces damping into the closed loop system. If the
steady-state error is constant, hence its derivative is zero, the derivative action has no
influence on the steady-state response. PI control is used to reduce the steady state error
and improve disturbance rejection capability. Almost all practical controllers exhibit the
features of PID control. They have control action components which deal with the present
error (proportional – P control), past error using the integral of error (integral – I control),
and the future error using the anticipatory nature of derivative (D-control). There are many
different implementations of PID control. One possible implementation of PID control is