Lecture 8B
Frequency Response
Frequency Response
• The most interesting type of system response
• Uses a sine wave forcing function input
• Interest is in the steady-state response instead of
the transient response
• Steady-state response to a step change only
provides information about process gain
• Transient response to sine wave change involves a
fast change (gone within 4 Tp values) to consistent
response oscillating at the input frequency
Attributes of Frequency Response
• Two major characteristics of frequency response:
1. Output amplitude with respect to the input
2. Output phase lag or lead with respect to the input
• Amplitude is attenuated or amplified depending
on input frequency
• Output oscillating out of phase with input also
depends on input frequency
First Order Frequency Response
For a sine wave input to a first order process
the output equals
By partial fraction decomposition, numerators are:
First Order Frequency Response
Now, by inverting back to the time domain:
dies out as t increases
Note: φ is a function of Tp and ω, but not of t.
So for values of t/Tp ≥ 4.0 (e-4 = 0.018), the process
output
, is a sine wave attenuated by
whose value depends on ω being fast or slow.
First Order Frequency Response
Attenuation and time shift between input and output
sine waves (Kp=1). The phase angle φ of the output
signal is calculated from φ = – 180*Δt/P, where Δt is
the time shift and P is the period of oscillation.
Bodé Diagrams
- for a First Order System
Note that ωn = 1/Tp ,
the natural frequency
of the system, so the
X-axis is the relative
frequency ω/ωn
Why is Frequency Response
Analysis so Useful?
• The amplitude ratio and phase lag provide info
about system stability
• These can be determined using the Laplace
Transform directly to generate a Bodé Diagram
• Can be used to study:
• System identification (create a model)
• Controller tuning
• Stability of the control system
• Robustness of the control system
• Designing noise filters
Laplace and Complex Numbers
Consider a complex number W defined as
a + bj
where a represents the real part of W or Re(W)
b represents the imaginary part of W or Im(W).
Define the following terms to characterize this
number in the complex domain (polar coordinates)
The Modulus
The Phase Angle
Laplace and Complex Numbers
These two terms can be used to plot the complex number
in a polar coordinate plane
a = |W|cosθ and b = |W|sinθ
and W = |W|cosθ + j|W|sinθ
Laplace and Complex Numbers
Since cosθ = (e+jθ + e-jθ)/2 and sinθ = (e+jθ – e-jθ)/2j
Then W = |W|[(e+jθ + e-jθ)/2] + j|W|[(e+jθ – e-jθ)/2j]
W = |W|e+jθ
Apply this complex knowledge to a First Order System:
Substitute jω for s:
Laplace and Complex Numbers
In polar coordinate form, this becomes:
So by substituting jω for s and rearranging the transform
into polar coordinate form, the amplitude ratio and phase
angle for the system are obtained directly.
Second Order Frequency Response
The original second order transform:
Substitute jω for s:
Second Order Frequency Response
In polar coordinate form, this becomes:
So, steady state response is obtained directly for the
second order system in a few steps and the Bodé Diagram
can be plotted for different values of ωTp and ζ.
Bode Diagram
- for Second Order System
Note that ωn = 1/Tp ,
the natural frequency
of the system, so the
X-axis is the relative
frequency ω/ωn
Notes on Second Order Frequency Response
2nd order response has maximum lag of 180° - twice a 1st order
system. If phase angle exceeds 90°, then it is a higher order.
The real problem faced by 2nd and higher order systems shows up at
the natural frequency ωn . The amplitude ratio can be >>>> 1.0 and
if ζ approaches 0, the response is amplified to  . With a 3rd order
system, two critical frequencies may exist while for a 4th order
system there can be three. Clearly such system responses are not
desired or tolerable and must be avoided.
As well the phase lag of higher order systems for high frequency
inputs is equal to -90n where n is the system order. With such high
lags, these systems are more difficult to control and design.
Bode Diagram for PI Controller
• PI Controller
Bode Diagram for PI Controller
• PI Controller with Kc = 2.0 and TI = 10
Bode Diagram for PD Controller
• PD Controller
Bode Diagram for PD Controller
• PD Controller with Kc = 2.0 and TD = 10
Bode Diagram for PID Controller
• PID Controller
Bode Diagram for PD Controller
• PID Controller with Kc = 2.0, TI = 10 and TD = 4
Notch filters are often used in control systems to
eliminate the effects of mechanical resonance.
The general transfer function for many types of
filters, including notch filters, is equivalent to
the transfer function of a PID controller.
Controller Design using Frequency Response
• Advantages
1. Applicable to all orders & types of dynamic models
2. Designer can specify desired closed-loop response
3. Information on stability and sensitivity is obtained
• Disadvantages
1. Approach is iterative and hence, is time-consuming–
interactive computer graphics are necessary
2. Results often taken to be "gospel" (far from reality of most
processes and plants)
3. Diagrams represent expected results for a model – not
necessarily the real process
Frequency Response Stability Criteria
Two principal methods to assess stability:
- Bode Stability Criterion
- Nyquist Stability Criterion
The Bode Stability Criterion
The Bode stability criterion states that a closed-loop
system is unstable if the Frequency Response of the
open-loop Transfer Function has an amplitude ratio
greater than 1.0 at the critical frequency ωc. Otherwise
the closed-loop system is stable. For the analysis, ωc is
the value of ω where open-loop phase angle is -180°
Bode Stability Criteria
• The Bode stability criterion
– a closed-loop system is unstable if the Frequency
Response of the open-loop Transfer Function has an
amplitude ratio > 1.0 at the critical frequency Cω.
– Otherwise the closed-loop system is stable.
– Cω is where the open-loop phase angle = 180°
Bode Stability Criteria
Example:
• A process has a Transfer Function:
G p (s) 
2
( 0 . 5 s  1)
3
, G v  1, G c  K c
• For proportional control, determine the closedloop stability for three values of Kc = 1, 4, and 20.
Bode Stability Criteria
Solution:
• The Open Loop Transfer Function
G p (s) 
2Kc
( 0 . 5 s  1)
3
• The Bode plots for the three values of Kc are shown
on the next slide. Note that the phase angle curves
are identical for all three cases. (Explain?)
Bode Stability Criteria
AROL
φOL
ωc
ω/ωn
Ultimate Gain and Ultimate Period
The Ultimate Controller Gain, KcU, is the maximum
value of Kc that results in stable closed loop system
with proportional-only control (continuous oscillation).
The Ultimate Period is the reciprocal of ωc as follows:
KcU is determined from the Open Loop Frequency
Response with P-only control and Kc =1. So:
Gain Margin
GM is determined from the Amplitude Ratio Bodé Diagram.
1. Determine critical frequency (ωc) where phase lag = 180°.
2. Determine the value of the Amplitude Ratio (Ac) at ωc.
3. The Gain Margin is then defined as:
GM = 1/Ac
4. The Bode Stability Criterion states:
If the value of GM < 1.0, the system is stable
for all input sine wave frequencies.
Phase Margin
PM is determined from the Phase Lag Bodé Diagram.
1. Find the frequency ωg where the Amplitude Ratio = 1.0.
2. Determine the value of the phase lag φg at this ωg value
3. The phase margin is defined as
PM = 180° + φg
The Bode Stability Criterion states
If PM is less than 0 , the system is stable
for all sine wave input frequencies
Gain and Phase margin
AROL
ωg
ω/ωn
ω
ωcc
φOL φg
ωg
ω/ωn
ωc
Gain Margin and Phase Margin
1.7 or 2.0 ≤ GM
30° or 45°≤ PM
So, one can adjust Kc such that the GM is
at least 1.7 or the PM must be more than 30°
Comparison of Transient Response
• System response to a step change in set point for a PI
controller tuned by frequency response (
) and by
Zeigler-Nichols rules (
).