Cleveland State University
MCE441: Intr. Linear Control Systems
Lecture 7:Time Response
Pole-Zero Maps
Influence of Poles and Zeros
Higher Order Systems and Pole Dominance Criterion
Prof. Richter
1 / 26
Test Inputs
⊲ Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
Real inputs contain randomness (for example, noise):
unpredictable.
Test inputs help to predict response quality under other
inputs.
–
–
–
–
step inputs are useful to simulate sudden changes
(startup, jump loading, etc)
sinusoidal inputs are used to test frequency response
(more later)
impulses are used to simulate shock conditions
ramp inputs are used to simulate transitions between
setpoints
2 / 26
Pole-Zero Maps
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
Im
⊲
G(s) =
(s−z1 )(s−z̄1 )(s−z2 )
(s−p5 )(s−p̄5 )(s−p2 )2 (s−p1 )(s−p4 )(s−p̄4 )(s−p5 )(s−p̄5 )
p5
p4
z1
z2
p2
z¯1
p1
Re
p¯4
p¯5
3 / 26
First-Order Systems
Test Inputs
Pole-Zero Maps
First-Order
Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
First-order systems are described by the I/O differential equation
⊲
τ ċ + c = r(t)
or in TF form
C(s)
1
=
R(s)
τs + 1
The number τ is called the time constant of the system. The TF
has one real pole at s = − τ1 .
Step Response
Applying the unit step input R(s) =
1
s
gives the response
− τt
c(t) = 1 − e
4 / 26
First-Order Step Response
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
(Stable) first-order systems always track the step command (the error approaches zero asymptotically).
The time it takes for the output to reach approx. 98% of its steady value is called settling time, also referred to
as 2 % settling time. It equals 4 time constants.
5 / 26
First-Order Ramp Response
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
First-order systems cannot track a ramp command. The tracking error is always equal to the time constant.
6 / 26
Second-Order Systems
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
The standard form of a second-order system is:
C(s)
wn2
= 2
R(s)
s + 2ζwn s + wn2
where wn and ζ are called the natural frequency (in rad/s) and
damping ratio (dimensionless), respectively. The response will
depend on the nature of the poles:
Underdamped: 0 < ζ < 1
Marginally Stable: ζ = 0
Critically Damped: ζ = 1
Overdamped: ζ > 1
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Case 1: Underdamped (0 < ζ < 1)
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
The poles are complex conjugates. The response to a unit step input is
given by
!
p
−ζwn t
e
1 − ζ2
−1
c(t) = 1 − p
sin wd t + tan
2
ζ
1−ζ
where wd is the damped natural frequency:
p
wd = wn 1 − ζ 2
The damped natural frequency is the one obtained by counting the
cycles per unit time in an experimental (and underdamped) response
and converting the result to radians per second.
8 / 26
Underdamped Case
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
Im
p
wd = wn 1 − ζ 2
Poles: −ζwn ± wd j
cos β = ζ
wn
β
−ζwn
0
Re
−wd
9 / 26
Marginally Stable and Overdamped Cases
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
Marginally Stable Case (ζ = 0)
When ζ = 0 the poles lie on the imaginary axis and the response to a
step input is
c(t) = 1 − cos wn t
that is, the oscillation does not die out. The system oscillates with its
natural frequency wn .
Critically Damped Case (ζ = 1)
When ζ = 1 we get a real pole of multiplicity two at s = −wn . The
response is given by c(t) = 1 − e−wn t (1 + wn t). No oscillations occur
for a step input.
10 / 26
Overdamped Case (ζ > 1)
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
When ζ > 1 the poles lie are real and unequal. The response
features two decaying exponential terms (no oscillations):
−s1 t
−s
t
2
wn
e
e
c(t) = 1 + p
−
2
s1
s2
2 ζ −1
where the poles are
s1,2 = (ζ ±
p
ζ 2 − 1)wn
11 / 26
Normalized Second-Order Response
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
12 / 26
Time-Domain Transient Specifications
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
Many practical design cases call for time-domain
specifications
Step response specs are often demanded (worst-case, drastic
situation)
Common specifications
1.
2.
3.
4.
Rise time (Tr )
Peak time (Tp )
Percent overshoot (P.O.)
Settling time (Ts )
13 / 26
Transient Specs: Step Response
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
Second−Order Step Response Characteristics
Overshoot
Final
Value
1.02*F.V
FV
0.98*F.V
Setpoint
Setpoint
0
0
Tr
Tp
Steady
Error
Tset 2%
Time (sec)
14 / 26
Design Aids: 2nd-Order Systems
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order
Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
2 % settling time: ts =
Rise time: tr =
π−β
wd
Peak time: tp =
π
wd
Peak value: Mpt = (e
4
ζwn
−( √
ζ
1−ζ 2
)π
Percent overshoot: P.O.= 100e
+ 1)y(∞)
−( √
ζ
1−ζ 2
)π
⊲
15 / 26
Design Chart: 2nd-Order Systems
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order
Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
16 / 26
Pole Locations and Response
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations
and Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
17 / 26
Effect of Zeros
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
The above design formulas are valid when no zeros are
present.
In general, zeros spoil the response by increasing the
overshoot.
If zeros are found on the r.h.p., the system is said to be
nonminimum phase, and the response is worsened.
Zeros may be neglected in some cases (more about this later
in this lecture)
⊲
18 / 26
Higher-Order Systems: Pole Dominance
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
When more than two poles are present, or when two distinct
real poles are present, it can be difficult to predict the
response without computer simulation
Frequently, one or two poles are dominant. Remember that
a
each single real pole gives rise to a term of the form s(s+a)
when a step input is applied. If we invert the term, the
contribution of the pole is 1 − e−at . So if the time constant
of the pole (1/a) is very small, the transient is very short and
does not contribute much to the overall response. On the
other hand, poles with large time constants are the ones
dictating the response. We say that such poles are dominant.
The same applies to complex conjugate pairs, where the time
constant taken into consideration is ζw1 n .
19 / 26
Pole Dominance Criterion
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
Begin by finding all poles of the TF and finding their time
constants. Remember: For a real pole at s = −a, the time
constant is 1/a, and for a complex conjugate pair with
0 < ζ < 1, it is ζw1 n .
Sort out the time constants: (high) τ1 , τ2 ..., τn (low).
Attempt to find a gap of a factor of eight or more between
two consecutive time constants, starting with the highest. If
such a gap is found, neglect all poles on the fast side of the
gap (small time constants).
Careful! when reassembling the TF. You may be
inadvertently changing the TF gain (a.k.a. DC gain): Make
sure that the responses to a step input have the same final
value. If not, correct the reduced TF with an appropriate
constant.
20 / 26
Pole Dominance: Example
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
G(s) =
3
s4 +6.4s3 +14.4s2 +27.2s+32
Poles: s1 = −2, s2 = −4, s3,4 = −0.2 ± 1.989i(ζ = 0.1, wn = 2). Time constants for real poles:
τ1 = 0.5 and τ2 = 0.25. For the complex poles : τ = 5.
3
The factored TF is: G(s) =
.
2
(s+2)(s+4)(s +0.4s+4)
Time constants: 5, 0.5, 0.25. A gap of a factor of 10 is found, therefore we neglect the fast poles at
s2 = −4 and s1 = −2.
If we simply eliminate the factors (s + 2) and (s + 4), we have a gain mismatch. In order to have the same
final value, we correct the gain to obtain:
Gr (s) =
3/8
s2 + 0.4s + 4
Check with the final value theorem: lims→0 sGr (s) = lims→0 sG(s).
⊲
21 / 26
Effect of a Zero
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
For second-order systems, we can predict the effect of a single real zero on the overshoot. Consider
G(s) =
2
(wn
/a)(s + a)
2
s2 + 2ζwn s + wn
If a/ζwn > 8, neglect the zero due to dominance: Gr (s) =
2
wn
2
s2 +2ζwn s+wn
. If not, use the chart:
⊲
22 / 26
Example
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
Sketch the response of the following TF when the input is a step
of size 2:
3(s + 5)
G(s) =
(s + 23)(s2 + 5s + 16)
⊲
23 / 26
Solution
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
24 / 26
Solution
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
⊲
25 / 26
Matlab Simulation
⊲
0.1
c(t )=0.0919
p
0.09
0.08
0.07
c(∞)=0.0815
Overshoot=12.76%
0.06
c(t)
Test Inputs
Pole-Zero Maps
First-Order Systems
First-Order Step
Response
First-Order Ramp
Response
Second-Order
Systems
Time-Domain
Transient
Specifications
Transient Specs:
Step Response
Design Aids:
2nd-Order Systems
Design Chart:
2nd-Order Systems
Pole Locations and
Response
Effect of Zeros
Higher-Order
Systems: Pole
Dominance
Effect of a Zero
0.05
0.04
0.03
0.02
t =1.3 s
s
0.01
0
0
0.5
1
1.5
2
2.5
t
3
3.5
4
4.5
5
26 / 26