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2.004 Dynamics and Control II
Spring 2008
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Massachusetts Institute of Technology
Department of Mechanical Engineering
2.004 Dynamics and Control II
Spring Term 2008
Lecture 201
Reading:
• Nise: Secs. 4.1 – 4.6 (pp. 153 - 177)
1
Standard Forms for First- and Second-Order Systems
These are (a) all pole system (with no zeros), and (b) have unity gain (limt→∞ ystep (t) = 1).
1.1
First-Order System:
We define the first-order standard form as
G(s) =
1
,
τs + 1
where the single parameter τ is the time constant. As a differential equation
τ
dy
+ y = u(t).
dt
and the system has a single real pole at s = −1/τ .
jw
s - p la n e
a s in g le r e a l p o le
-
1
x
s
1
t
c D.Rowell 2008
copyright �
20–1
The step response is
�
ystep = L
0 .9 8
0 .9
1
y
s te p
−1
1 1/τ
s s + 1/τ
�
= 1 − e−t/τ
(t)
in itia l s lo p e is
1
t
0 .1
0
1.1.1
t
0
T
T s
R
»
4
t
t
Common Step Response Descriptors:
(a) Settling Time:
The time taken for the response to reach 98% of its final value. Since
ystep (t) = 1 − e−t/τ
and e−4 = 0.0183 ≈ 0.02, we take
Ts = 4τ
as the definition of Ts .
(b) Rise Time: Commonly taken as the time taken for the step response to rise from
10% to 90% of the steady-state response to a step input. It is found from the step
response as follows
0.1 = 1 − e−t0.1 /τ ⇒ t0.1 = τ ln(0.9)
0.9 = 1 − e−t0.9 /τ ⇒ t0.9 = τ ln(0.1)
TR = t0.9 − t0.1 = (ln(0.1) − ln(0.9))τ = 2.2τ
1.2
Second-Order Systems
The standard unity gain second-order system has a transfer function
G(s) =
ωn2
s2 + 2ζωn s + ωn2
with two parameters (1) ωn – the undamped natural frequency, and (ii) ζ – the damping
ratio (ζ ≥ 0). The system poles are the roots of s2 + 2ζωn s + ωn2 = 0, that is
�
p1 , p2 = −ζωn ± ωn ζ 2 − 1,
leading to four cases
20–2
i) ζ > 1 – the poles are real
� and distinct
p1 , p2 = −ζωn ± ωn ζ 2 − 1,
ii) ζ = 1 – the poles are real and coincident
p1 , p2 = −ζωn ,
iii) 0 < ζ < 1 – the poles �
are complex conjugates
p1 , p2 = −ζωn ± jωn 1 − ζ 2 , or
(iv) ζ = 0 – the poles are purely imaginary
p1 , p2 = ±jωn .
jw
s - p la n e
c o n ju g a te p o le s
x
x
c o in c id e n t p o le s
im a g in a r y p o le s
xx
x
s
r e a l p o le
x
x
1.2.1
Pole Positions For an Underdamped Second-Order System
p1 , p2 = −ζωn ± jωn
�
1 − ζ2
and when plotted on the s-plane
jw
x
jw
w
-w
-z w
n
s - p la n e
n
n
1 - z
2
n
w
q
s
n
w
x
jw
n
1 - z
2
w
n
q = c o s
z w n
n
- jw
n
we note that
20–3
-1
z
(a) The poles lie at a distance ωn from the origin, and
(b) The poles lie on radial lines at an angle
θ = cos−1 (ζ)
as shown above.
The influence of ζ and ωn on the pole locations may therefore be summarized:
jw
w
n
in c r e a s in g
jw
s - p la n e
s - p la n e
z = 0
z = 1
s
s
z = 1
lin e s o f c o n s ta n t w
1.2.2
z = 0
n
lin e s o f c o n s ta n t z
Step Responses
(a) The over damped case (ζ > 1)
ystep (t) = 1 − C1 e−p1 t − C2 e−p2 t
where the constants C1 and C2 are determined from p1 and p2 .
1
ys
te p
(t)
n o o v e rs h o o t
z > 1
0
t
(b) The critically damped case (ζ = 1)
step response takes a special form
With two coincident poles p1 = p2 = p, the
ystep (t) = 1 − C1 ept − C2 te−pt
20–4
ys
te p
(t)
1
n o o v e rs h o o t
z = 1
0
t
(c) The under damped case (0 < ζ < 1)
With a pair of complex conjugate poles
�
p1 , p2 = −ζωn ± jωn 1 − ζ 2
the step response becomes oscillatory
��
e−ζωn t � � �
ystep (t) = 1 − �
cos ωn 1 − ζ 2 t − φ
1 − ζ2
where
�
−1
φ = tan
�
ζ
�
1 − ζ2
and if we define the damped natural frequency ωd as
�
ωd = ωn 1 − ζ 2
we can write the step response as
e−ζωn t
ystep (t) = 1 − �
(cos (ωd t − φ))
1 − ζ2
ys
te p
(t)
o v e rs h o o t
1
d a m p e d o s c illa to r y r e s p o n s e
z < 1
0
t
20–5
(d) The undamped case (ζ = 0)
In this case
G(s) =
s2
ωn
+ ωn2
and the poles are p1 , p2 = ±jωn . Then
ystep (t) = 1 − cos (ωn t)
ys
te p
(t)
p u r e o s c illa to r y r e s p o n s e
z = 0
2
1
0
t
Note: For any second-order system, the initial slope of the step response is zero,
since by definition the system is at rest at time t = 0, that is ystep (0) = 0, and
ẏstep (0) = 0.
1.2.3
Step Response Based Second-Order System Specifications
(a) Rise Time (TR ):
Applies to over- and under-damped systems. As in the case of
first-order systems, the usual definition is the time taken for the step response to rise
from 10% to 90% of the final value:
ys
0 .9
0 .1
0
ys
(t)
te p
(t)
te p
1
n o o v e rs h o o t
z > 1
T
t0
.1
R
t
0 .9
1
0 .9
t
0 .1
0
T
t0
.1
d a m p e d o s c illa to r y r e s p o n s e
z < 1
R
t
0 .9
t
For a second-order system there is no simple (general) expression for TR . The following
figure - from Nise, Fig. 4.16, (p. 172) is derived empirically:
20–6
2 .8
R is e tim e x n o r m a liz e d fr e q u e n c y
2 .6
2 .4
2 .2
2
1 .8
1 .6
1 .4
1 .2
1
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
d a m p in g r a tio
0 .7
0 .8
z
0 .9
(b) Peak Time (Tp ): Applies only to under-damped systems, and is defined as the time
to reach the first peak of the oscillatory step response.
y
p e a k
ys
te p
(t)
1
d a m p e d o s c illa to r y r e s p o n s e
z < 1
0
T
t
P
Tp is found by differentiating the step response ystep (t), and equating to zero. (See Nise
p. 170 for details.)
π
π
Tp = �
=
2
ωd
ωn 1 − ζ
** Transient response specifications continued in Lecture 21. **
20–7