Page 141 - Mechatronics with Experiments

CLOSED LOOP CONTROL 127
of difference equations and inverse Z-transform of z-domain transfer functions to
obtain difference equations. For real-time algorithmic implementation, we need the
difference equation form of the controller.
u(z) aTz −1
=
e(z) 1 − (1 − aT)z −1
a.T
=
z − (1 − aT)
a
= (2.216)
((z − 1)∕T) + a
Notice the substitution relationship between s and z using this approximation
a a
⟶ (2.217)
s + a z−1 + a
T
z − 1
s ⟶ ; z = sT + 1 (2.218)
T
(ii) Backward difference approximation:
Another possible approximation is to use the backward difference rule

u(kT) = u(kT − T) + T [−au(kT) + ae(kT)] (2.219)
Again, the above equation is in a form suitable for real-time implementation in
software. In order to obtain a more generic relationship for this type of approximation,
let us take the Z-transform of the above equation,
−1
u(z) = z u(z) − Ta u(z) + Ta e(z) (2.220)
−1
(1 + Ta − z )u(z) = Ta e(z) (2.221)
u(z) Ta zTa
= =
e(z) 1 + Ta − z −1 z − 1 + Taz
a
= (2.222)
z−1 + a
zT
The backward approximation is equivalent to the following substitution between s
and z,
a a
⟶ (2.223)
s + a z−1 + a
Tz
z − 1
s ⟶ (2.224)
Tz
(iii) Trapezoidal approximation (Tustin’s method, bilinear transformation)
Finally, we will consider the trapezoidal rule approximation among the finite differ-
ence approximations to the integration,
T
u(kT) = u(kT − T) + [−a[u(kT − T) + u(kT)] + [a(e(kT − T) + e(kT)]] (2.225)
2
Similarly, we take the Z-transform of the above equation,
T
zu(z) = u(z) + [−a(1 + z)u(z) + a(1 + z)e(z)] (2.226)
2
[ Ta ] T ⋅ a
z + (1 + z) − 1 u(z) = (1 + z)e(z) (2.227)
2 2