Page 108 - Mechatronics with Experiments

94 MECHATRONICS
impose conditions that CLS poles must have a real part smaller than a negative real value

Re(p ) < 0 or − a (2.107)
i
or in the frequency domain it can be specified in terms of gain and phase margins,
GM > GM min , PM > PM min (2.108)

The response quality is generally divided into two groups: transient response and
steady-state response. The transient response is generally specified as the step response.
The step response is quantified using percent overshoot, settling time, and rise time spec-
ifications: (PO%, t , t ). For a second-order closed loop system, the PO% and t uniquely
s r s
determine the closed loop system poles with damping ratio ( ) and natural frequency (w ):
n
√
2
p 1,2 =− w ± j 1 − w n (2.109)
n
4.0
t = ; (2.110)
s
w n
√
− ∕ 1− 2
PO = e ; for 0 ≤ < 1.0 (2.111)
In the frequency domain, the cross-over frequency (w ) of the loop transfer function and
cr
bandwidth (w ) of the closed loop transfer function, along with the phase margin of the
bw
loop transfer function, correlate well with the transient response. Cross-over frequency is
defined for the loop transfer function where the magnitude of the loop transfer function
is 1 (or 0 dB). Bandwidth frequency is defined for the closed loop transfer function where
the magnitude is 0.707 or (−3 dB). Bandwidth closely relates to the speed of response (t ),
s
and phase margin (PM) is closely related to the damping ratio. For a second-order closed
loop system as shown in Figure 2.38, it can be shown [6] that
2
−1
PM =tan √ (2.112)
√
1 + 4 − 2 2
2
≈ 100 for ≤ 0.6 (2.113)
√
√
2
2
w = −2 + 4 + 1 ⋅ w ; w = w at |G(jw)| = 1.0 (2.114)
cr
n
cr
;20 log |G(jw)| = 0.0 dB (2.115)
10
√
√
2
2
2
w bw = w ⋅ (1 − 2 ) + (2 − 1) + 1 (2.116)
n
; w = w bw at |G(jw)∕(1 + G(jw))| = 0.707 (2.117)
;20 log |G(jw)∕(1 + G(jw))| =−3 dB (2.118)
10
√
2
w = w 1 − 2 ⋅ ; where |G(jw)∕(1 + G(jw))| is maximum (2.119)
p n
d
; w = w at (|G(jw)∕(1 + G(jw))|) = 0.0 (2.120)
p
dw
; for 0 ≤ ≤ 0.707 (2.121)
(2.122)
M =max |G(jw)∕(1 + G(jw))|
p
1
= √ ; for 0 ≤ ≤ 0.707 (2.123)
2 1 − 2