Page 113 - Mechatronics with Experiments

CLOSED LOOP CONTROL 99

PID controler
K p

r + e K I + u
- s +

Ks
D

y
FIGURE 2.42: Block diagram of the standard PID controller.

which shows that the control signal is function of the error between the commanded and
measured output signals, e(t) at time t, as well as the derivative of the error signal ̇ e(t) and
t
the integral of the error signal since the control loop is enabled (t = 0), ∫ e( )d .
0
The discrete time approximation of the PID control algorithm can be implemented by
finite difference approximation to the derivative and integral functions. In digital implemen-
tation, the control signal can be updated at periodic intervals, T, also called the sampling
period. The control is updated at integer multiples of the sampling period. The value of the
signal is kept constant between each update period. At any update instant k, time is t = kT,
and the previous update instant is t − T = kT − T, the next update instant is t = kT + T and
so on, the control signal can be expressed as u(t) = u(kT),
(e(kT) − e(kT − T))
u(kT) = K ⋅ e(kT) + K ⋅ u (kT) + K D ) (2.131)
I
I
p
T
where
u (kT) = u (kT − T) + e(kT) ⋅ T (2.132)
I
I
u (0) = 0.0; at the initialization (2.133)
I
Let us take the Laplace transform of the continuous time domain (analog) version of
the PID control and analyze its effect on a controlled system. Basically, the same results
apply for the discrete time version (digital implementation) provided the sampling period is
short enough (high sampling frequency) relative to the bandwidth of the closed loop system.
( 1 )
u(s) = k + K I + K s e(s) (2.134)
p
D
s
1
D(s) = K + K I + K s (2.135)
D
p
s
( )
1
= K 1 + + T s (2.136)
p D
T s
I
K p
K = , K = K T
I D p D
T
I
Consider a second order mass-force system to study its behavior under various forms of
PID control (Figure 2.43).
m̈ x(t) = f(t) − f (t)
d
1 1
̈ x(t) = f(t) − f (t)
d
m m