Page 120 - Mechatronics with Experiments

106 MECHATRONICS
and three poles at the roots of the following equation,
3
s + K s + K = 0
I
p
K p ( K I )
1 + s + = 0
s 3 K p
s + K I
K p
1 + K p = 0
s 3
Let us study the locus of the roots of this equation for various values of K , K
p I
(Figure 2.46c). Closed loop system dominant poles are such that one of them is on the
negative real axis, but the other two have positive real parts, limited to 1 K I , and are
2 K p
unstable. If we want to stabilize the system, we must introduce more damping using
derivative (D) control, which is considered next. The pure mass-force system is unstable
under a PI controller for any positive values of the PI gains (K , K ). Many real systems
I
p
have some inherent damping in open loop. In other words, the loop force-position transfer
1 1
function is instead of . If the open loop damping is large enough, the closed loop
s(s+c) s 2
system under PI control would be stable for a finite range of K , K gains. That is the reason
p
I
why many physical second-order systems are stable and well controlled by a PI controller
alone, without the derivative control action.
2.12.5 PD Control
Now, we will consider the characteristics of the mass-force of a system under proportional
plus derivative (PD) control. The PD control algorithm is given by,
u(t) = K (x (t) − x(t)) − K ̇ x(t)
D
d
p
u(s) = K (x (s) − x(s)) − K ̇ x(s)
d
D
p
Substituting this into the mass-force model,
2
s x(s) = u(s) − w (s)
d
2
(s + K s + K )x(s) = K x (s) − w (s)
p d
D
p
d
We will consider the dominant response to a step command in the desired position, and the
steady-state response to a constant disturbance.
(i) Transient response in the s-domain
K p
x(s) = (1∕s)
2
s + K s + K
D p
and the time domain response is found by taking the inverse Laplace transform,
1 √
− n t
2
x(t) = 1 − e √ sin( 1 − t + )
n
1 − 2
where
K = 2 n
p
K = 2 n
D
(√ )
1 − 2
=tan −1