Ebook Process dynamics and control (4th edition): Part 2

Chapter

14

Frequency Response Analysis
and Control System Design
CHAPTER CONTENTS

14.1

Sinusoidal Forcing of a First-Order Process

14.2

Sinusoidal Forcing of an nth-Order Process

14.3

Bode Diagrams
14.3.1
14.3.2
14.3.3

First-Order Process
Integrating Process
Second-Order Process

14.3.4 Process Zero
14.3.5 Time Delay
14.4

Frequency Response Characteristics of Feedback Controllers

14.5

Nyquist Diagrams

14.6

Bode Stability Criterion

14.7

Gain and Phase Margins

Summary

In previous chapters, Laplace transform techniques
were used to calculate transient responses from transfer functions. This chapter focuses on an alternative
way to analyze dynamic systems by using frequency
response analysis. Frequency response concepts and
techniques play an important role in stability analysis, control system design, and robustness assessment.
Historically, frequency response techniques provided
the conceptual framework for early control theory and
important applications in the field of communications
(MacFarlane, 1979).
We introduce a simplified procedure to calculate the
frequency response characteristics from the transfer
function model of any linear process. Two concepts,
the Bode and Nyquist stability criteria, are generally
applicable for feedback control systems and stability

244

analysis. Next we introduce two useful metrics for relative stability, namely gain and phase margins. These
metrics indicate how close a control system is to instability. A related issue is robustness, that is, the sensitivity
of control system performance to process variations and
to uncertainty in the process model.
The design of robust feedback control systems is considered in Appendix J.

14.1

SINUSOIDAL FORCING OF
A FIRST-ORDER PROCESS

We start with the response properties of a first-order
process when forced by a sinusoidal input and show
how the output response characteristics depend on
the frequency of the input signal. This is the origin of


14.1

the term frequency response. The responses for firstand second-order processes forced by a sinusoidal input
were presented in Chapter 5. Recall that these responses
consisted of sine, cosine, and exponential terms. Specifically, for a first-order transfer function with gain K and
time constant τ, the response to a general sinusoidal
input, x(t) = A sin ωt, is
KA
(ωτe−t∕τ − ωτ cos ωt + sin ωt) (5-23)
ω2 τ2 + 1

where y is in deviation form.
If the sinusoidal input is continued for a long time,
the exponential term (ωτe−t/τ ) becomes negligible. The
remaining sine and cosine terms can be combined via a
trigonometric identity to yield
y(t) =

y𝓁 (t) = √

KA
ω2 τ2 + 1

sin (ωt + ϕ)

(14-1)

where ϕ = −tan−1 (ωτ). The long-time response y𝓁 (t) is
called the frequency response of the first-order system
and has two distinctive features (see Fig. 14.1).
1. The output signal is a sine wave that has the same
frequency, but its phase is shifted relative to the
input sine wave by the angle ϕ (referred to as the
phase shift or the phase angle); the amount of phase
shift depends on the forcing frequency ω.
2. The sine wave has an amplitude  that is a function
of the forcing frequency:
KA
 = √
ω2 τ2 + 1

(14-2)

Dividing both sides of Eq. 14-2 by the input signal
amplitude A yields the amplitude ratio (AR)
AR =

Â
K
=√
A
ω2 τ2 + 1

(14-3a)

which can, in turn, be divided by the process gain
to yield the normalized amplitude ratio (ARN ):
ARN =

AR
1
=√
2
K
ω τ2 + 1

(14-3b)

Next we examine the physical significance of the preceding equations, with specific reference to the blending
P
A

A
Output, y

Time
shift, Δt

Input, x
Time, t

Figure 14.1 Attenuation and time shift between input and
output sine waves. The phase angle ϕ of the output signal is
given by ϕ = Δt/P × 360∘, where Δt is the time shift and P is
the period of oscillation.

Sinusoidal Forcing of a First-Order Process

245

process example discussed earlier. In Chapter 4, the
transfer function model for the stirred-tank blending
system was derived as
X ′ (s) =

K3
K1
K2
X ′ (s) +
W ′ (s) +
W ′ (s) (4-84)

τs + 1 1
τs + 1 2
τs + 1 1

Suppose flow rate w2 is varied sinusoidally about
a constant value, while the other inlet conditions
are kept constant at their nominal values; that is,
w′1 (t) = x′1 (t) = 0. Because w2 (t) is sinusoidal, the output
composition deviation x′ (t) eventually becomes sinusoidal according to Eq. 5-24. However, there is a phase
shift in the output relative to the input, as shown in
Fig. 14.1, owing to the material holdup of the tank. If
the flow rate w2 oscillates very slowly relative to the
residence time τ(ω ≪ 1/τ), the phase shift is very small,
approaching 0∘ , whereas the normalized amplitude
̂
ratio (A/KA)
is very nearly unity. For the case of a
low-frequency input, the output is in phase with the
input, tracking the sinusoidal input as if the process
model were G(s) = K.
On the other hand, suppose that the flow rate is
varied rapidly by increasing the input signal frequency.
For ω ≫ 1/τ, Eq. 14-1 indicates that the phase shift
approaches a value of −π/2 radians (−90∘ ). The presence of the negative sign indicates that the output lags
behind the input by 90∘ ; in other words, the phase lag
is 90∘ . The amplitude ratio approaches zero as the frequency becomes large, indicating that the input signal
is almost completely attenuated; namely, the sinusoidal
deviation in the output signal is very small.
These results indicate that positive and negative deviations in w2 are essentially canceled by the capacitance
of the liquid in the blending tank if the frequency is high

enough. High frequency implies ω ≫ 1/τ. Most processes behave qualitatively similar to the stirred-tank
blending system, when subjected to a sinusoidal input.
For high-frequency input changes, the process output
deviations are so completely attenuated that the corresponding periodic variation in the output is difficult
(perhaps impossible) to detect or measure.
Input–output phase shift and attenuation (or amplification) occur for any stable transfer function, regardless
of its complexity. In all cases, the phase shift and
amplitude ratio are related to the frequency ω of the
sinusoidal input signal. In developments up to this
point, the expressions for the amplitude ratio and phase
shift were derived using the process transfer function.
However, the frequency response of a process can also
be obtained experimentally. By performing a series of
tests in which a sinusoidal input is applied to the process, the resulting amplitude ratio and phase shift can
be measured for different frequencies. In this case, the
frequency response is expressed as a table of measured
amplitude ratios and phase shifts for selected values
of ω. However, the method is very time-consuming


246

Chapter 14

Frequency Response Analysis and Control System Design

because of the repeated experiments for different values of ω. Thus other methods, such as pulse testing
(Ogunnaike and Ray, 1994), are utilized, because only a
single test is required.
In this chapter, the focus is on developing a powerful

analytical method to calculate the frequency response
for any stable process transfer function. Later in this
chapter, we show how this information can be used to
design controllers and analyze the properties of the
closed loop system responses.

14.2

The shortcut method can be summarized as follows:
Step 1. Substitute s = jω in G(s) to obtain G(jω).
Step 2. Rationalize G(jω), i.e., express G(jω) as the
sum of real (R) and imaginary (I) parts R + jI,
where R and I are functions of ω, using complex conjugate multiplication.
Step 3. The √
output sine wave has amplitude
−1
̂
A = A R2 + I 2 and phase angle
√ϕ = tan (I/R).
2
2
The amplitude ratio is AR = R + I and is
independent of the value of A.

SINUSOIDAL FORCING OF AN
nTH-ORDER PROCESS

This section presents a general approach for deriving the
frequency response of any stable transfer function. The
physical interpretation of frequency response is not

valid for unstable systems, because a sinusoidal input
produces an unbounded output response instead of a
sinusoidal response. A rather simple procedure can be
employed to find the sinusoidal response.
After setting s = jω in G(s), by algebraic manipulation
we can separate the expression into real (R) and imaginary (I) terms (j indicates an imaginary component):
G(jω) = R(ω) + jI(ω)

ϕ = tan (I∕R)

G(s) =

(14-6b)

Both  and ϕ are functions of frequency ω. A simple
but elegant relation for the frequency response can be
derived, where the amplitude ratio is given by
√
Â
AR =
(14-7)
= |G| = R2 + I 2
A
The absolute value denotes the magnitude of G, and
the phase shift (also called the phase angle or argument
of G, ∠G) between the sinusoidal output and input is
given by
(14-8)
ϕ = ∠G = tan−1 (I∕R)
Because R(ω) and I(ω) (and hence AR and ϕ) can be

obtained without calculating the complete transient
response y(t), these characteristics provide a convenient
shortcut method to determine the frequency response
of transfer functions.
Equations 14-7 and 14-8 can calculate the frequency
response characteristics of any stable G(s), including
those with time-delay terms.

1
τs + 1

(14-9)

SOLUTION
First substitute s = jω in the transfer function
G(jω) =

1
1
=
τjω + 1
jωτ + 1

(14-10)

Then multiply both numerator and denominator by the
complex conjugate of the denominator, that is, −jωτ + 1
−jωτ + 1
−jωτ + 1
= 2 2

(jωτ + 1)(−jωτ + 1)
ω τ +1

G(jω) =

(14-5)

 and ϕ are related to I(ω) and R(ω) by the following
relations (Seborg et al., 2004):
√
(14-6a)
 = A R2 + I 2
−1

Find the frequency response of a first-order system, with

(14-4)

Similar to Eq. 14-1, we can express the long time
response for a linear system (cf. Eq. 14-1) as
y𝓁 (t) = Â sin(ωt + ϕ)

EXAMPLE 14.1

(−ωτ)
1
+j 2 2
= R + jI
ω2 τ2 + 1
ω τ +1

=
where

(14-11)

R=

1
ω2 τ2 + 1

(14-12a)

I=

−ωτ
ω2 τ2 + 1

(14-12b)

and
From Eq. 14-7,
√
AR = |G(jω|) =

(

)2

1

ω2 τ2

+1

(
+

−ωτ
+1

)2

ω2 τ2

Simplifying,
√
AR =

1
(1 + ω2 τ2 )
= √
(ω2 τ2 + 1)2
ω2 τ2 + 1

ϕ = ∠G(jω) = tan−1 (−ωτ) = −tan−1 (ωτ)

(14-13a)
(14-13b)

If the process gain had been a positive value K instead of 1,

AR = √

K
ω2 τ2 + 1

(14-14)

and the phase angle would be unchanged (Eq. 14-13b).
Both the amplitude ratio and phase angle are identical to those values calculated in Section 14.1 using the
time-domain derivation.


14.3

From this example, we conclude that direct analysis
of the complex transfer function G(jω) is computationally easier than solving for the actual long-time output
response j𝓁 (t). The computational advantages are even
greater when dealing with more complicated processes,
as shown in the following. Start with a general transfer
function in factored form
G (s)Gb (s)Gc (s) · · ·
(14-15)
G(s) = a
G1 (s)G2 (s)G3 (s) · · ·
G(s) is converted to the complex form G(jω) by the substitution s = jω:
G (jω)Gb (jω)Gc (jω) · · ·
G(jω) = a
(14-16)
G1 (jω)G2 (jω)G3 (jω) · · ·
The magnitude and phase angle of G(jω) are as follows:

|Ga (jω)‖Gb (jω)‖Gc (jω)| · · ·
|G(jω)| =
(14-17a)
|G1 (jω)‖G2 (jω)‖G3 (jω)| · · ·
∠G(jω) = ∠Ga (jω) + ∠Gb (jω) + ∠Gc (jω) + · · ·
− [∠G1 (jω) + ∠G2 (jω) + ∠G3 (jω) + · · ·]

(14-17b)

Equations 14-17a and 14-17b greatly simplify the computation of |G(jω)| and ∠G(jω) and, consequently, AR
and ϕ, for factored transfer functions. These expressions
eliminate much of the complex algebra associated with
the rationalization of complicated transfer functions.
Hence, the factored form (Eq. 14-15) may be preferred
for frequency response analysis. On the other hand, if
the frequency response curves are generated using software such as MATLAB, there is no need to factor the
numerator or denominator, as discussed in Section 14.3.

Calculate the amplitude ratio and phase angle for the overdamped second-order transfer function
G(s) =

K
(τ1 s + 1)(τ2 s + 1)

SOLUTION
Using Eq. 14-15, let
Ga = K
G1 = τ1 s + 1
G2 = τ2 s + 1
Substituting s = jω

Ga (jω) = K
G1 (jω) = jωτ1 + 1
G2 (jω) = jωτ2 + 1
The magnitudes and angles of each component of the complex transfer function are
|Ga | = K
√
|G1 | = √ω2 τ21 + 1
|G2 | = ω2 τ22 + 1

∠Ga = 0
∠G1 = tan−1 (ωτ1 )
∠G2 = tan−1 (ωτ2 )

247

Combining these expressions via Eqs. 14-17a and 14-17b
yields
|Ga (jω)|
|G1 (jω)‖G2 (jω)|
K
√
= √
ω2 τ21 + 1 ω2 τ22 + 1

AR = |G(jω)| =

(14-18a)

ϕ = ∠G(jω) = ∠Ga (jω) − (∠G1 (jω) + ∠G2 (jω))
= −tan−1 (ωτ1 ) − tan−1 (ωτ2 )

(14-18b)

14.3 BODE DIAGRAMS
The Bode diagram (or Bode plot) provides a convenient
display of the frequency response characteristics in
which AR and ϕ are each plotted as a function of ω.
Ordinarily, ω is expressed in units of radians/time to
simplify inverse tangent calculations (e.g., Eq. 14-18b)
where the arguments must be dimensionless, that is,
in radians. Occasionally, a cyclic frequency, ω/2π, with
units of cycles/time, is used. Phase angle ϕ is normally
expressed in degrees rather than radians. For reasons
that will become apparent in the following development, the Bode diagram consists of: (1) a log–log plot
of AR versus ω and (2) a semilog plot of ϕ versus ω.
These plots are particularly useful for rapid analysis of
the response characteristics and stability of closed-loop
systems.

14.3.1
EXAMPLE 14.2

Bode Diagrams

First-Order Process

In the past, when frequency response plots had to be
generated by hand, they were of limited utility. A much
more practical approach now utilizes spreadsheets or
control-oriented software such as MATLAB to simplify

calculations and generate Bode plots. Although spreadsheet software can be used to generate Bode plots, it is
much more convenient to use software designed specifically for control system analysis. Thus, after describing
the qualitative features of Bode plots of simple transfer
functions, we illustrate how the AR and ϕ components
of such a plot are generated by a MATLAB program in
Example 14.3.
For a first-order model, K/(τs + 1), Fig. 14.2 shows a
general log–log plot of the normalized amplitude ratio
versus ωτ, for positive K. For a negative valve of K, the
phase angle is decreased by −180∘ . A semilog plot of ϕ
versus ωτ is also shown. In Fig. 14.2, the abscissa ωτ has
units of radians. If K and τ are known, ARN (or AR) and
ϕ can be plotted as a function of ω. Note that, at high
frequencies, the amplitude ratio drops to an infinitesimal
level, and the phase lag (the phase angle expressed as a
positive value) approaches a maximum value of 90∘ .
Some books and software define AR differently,
in terms of decibels. The amplitude ratio in decibels


248

Chapter 14

Frequency Response Analysis and Control System Design
ωb = 1/τ

ϕ = ∠G(jω) = ∠K − tan−1 (j∞) = −90∘

(14-21)

1

14.3.3
Normalized
amplitude 0.1
ratio, ARN

Second-Order Process

A general transfer function for a second-order system
without numerator dynamics is
G(s) =

0.01
0.01

0.1

1
ωτ

10

100

0
–30

ωb = 1/τ

Phase angle
–60
ϕ (deg)
–90
–120
0.01

0.1

1
ωτ

10

100

Figure 14.2 Bode diagram for a first-order process.

ARdb is defined as
ARdb = 20 log AR

(14-19)

The use of decibels merely results in a rescaling of the
Bode plot AR axis. The decibel unit is employed in
electrical communication and acoustic theory and is
seldom used today in the process control field. Note that
the MATLAB bode routine uses decibels as the default
option; however, it can be modified to plot AR results,

as shown in Fig 14.2. In the rest of this chapter, we only
derive frequency responses for simple transfer functions
(integrator, first-order, second-order, zeros, time delay).
Software should be used for calculating frequency
responses of more complicated transfer functions.

14.3.2

Integrating Process

The transfer function for an integrating process was
given in Chapter 5.
Y(s)
K
G(s) =
=
(5-32)
U(s)
s
Because of the single pole located at the origin, this
transfer function represents a marginally stable process.
The shortcut method of determining frequency response
outlined in the preceding section was developed for stable processes, that is, those that converge to a bounded
oscillatory response for a sinusoidal input. Because the
output of an integrating process is bounded when forced
by a sinusoidal input, the shortcut method does apply
for this marginally stable process:
|K| K
(14-20)
AR = |G(jω)| = || || =

| jω | ω

K
τ2 s2 + 2ζτs + 1

(14-22)

Substituting s = jω and rearranging into real and imaginary parts (see Example 14.1) yields
K
AR = √
(14-23a)
2
2
(1 − ω τ )2 + (2ζωτ)2
[
]
−2ζωτ
(14-23b)
ϕ = tan−1
1 − ω2 τ2
Note that, in evaluating ϕ, multiple results are obtained
because Eq. 14-23b has infinitely many solutions, each
differing by n180∘ , where n is a positive integer. The
appropriate solution of Eq. 14-23b for the second-order
system yields −180∘ < ϕ < 0.
Figure 14.3 shows the Bode plots for overdamped
(ξ > 1), critically damped (ξ = 1), and underdamped
(0 < ξ < 1) processes as a function of ωτ. The lowfrequency limits of the second-order system are identical to those of the first-order system. However, the
limits are different at high frequencies, ωτ ≫ 1.
ARN ≈ 1∕(ωτ)2

ϕ ≈ −180o

(14-24a)
(14-24b)

For overdamped systems, the normalized amplitude
̂
ratio is attenuated (A/KA
< 1) for all ω. For underdamped systems, the amplitude√ratio plot exhibits a
maximum (for values of 0 < ζ < 2∕2) at the resonant
frequency
√
1 − 2ζ2
(14-25)
ωr =
τ
(ARN )max =

1
√
2ζ 1 − ζ2

(14-26)

These expressions can be derived by the interested
reader. The resonant frequency ωr is that frequency for
which the sinusoidal output response has the maximum
amplitude for a given sinusoidal input. Equations 14-25
and 14-26 indicate how ωr and (ARN )max depend on ξ.
This behavior is used in designing organ pipes to create sounds at specific frequencies. However, excessive

resonance is undesirable, for example, in automobiles,
where a particular vibration is noticeable only at a
certain speed. For industrial processes operated without
feedback control, resonance is seldom encountered,
although some measurement devices are designed to
exhibit a limited amount of resonant behavior. On the
other hand, feedback controllers can be tuned to give
the controlled process a slight amount of oscillatory


14.3
1

249

10

ζ=1

ζ = 0.2
1

0.1
2

ARN

5

0.01

0.001
0.0001
0.01

Bode Diagrams

ARN

0.01

Slope = –2
0.1

1
ωτ

10

0.4 0.8
0.1

Slope = –2

0.001
0.01

100

0

0.1

1
ωτ

10

100

0
1

–45
ϕ
(deg)

–90

–45

2
5
2

–135
–180
0.01

ϕ
(deg)

5

1
ωτ

0.4

–135

ζ=1
0.1

–90

ζ = 0.2
10

100

–180
0.01

0.1

1
ωτ

0.8
10

100

Figure 14.3 Bode diagrams for second-order processes. Right: underdamped. Left: overdamped and critically damped.

or underdamped behavior in order to speed up the
controlled system response (see Chapter 12).

14.3.4

Process Zero

A term of the form τs + 1 in the denominator of a transfer function is sometimes referred to as a process lag,
because it causes the process output to lag the input (the
phase angle is negative). Similarly, a process zero of the
form τs + 1 (τ > 0) in the numerator (see Section 6.1)
causes the sinusoidal output of the process to lead the
input (ϕ > 0); hence, a left-half plane (LHP) zero often is
referred to as a process lead. Next we consider the amplitude ratio and phase angle for this term.
Substituting s = jω into G(s) = τs + 1 gives
G(jω) = jωτ + 1
from which
AR = |G(jω)| =

√
ω2 τ2 + 1

ϕ = ∠G(jω) = tan−1 (ωτ)

(14-27)

(14-28a)
(14-28b)

Therefore, a process zero contributes a positive phase
angle that varies between 0 and +90∘ . The output signal amplitude becomes very large at high frequencies
(i.e., AR → ∞ as ω → ∞), which is a physical impossibility. Consequently, in practice a process zero is always
found in combination with one or more poles. The order
of the numerator of the process transfer function must
be less than or equal to the order of the denominator, as
noted in Section 6.1.

Suppose that the numerator of a transfer function
contains the term 1 − τs, with τ > 0. As shown in
Section 6.1, a right-half plane (RHP) zero is associated
with an inverse step response. The frequency response
characteristics of G(s) = 1 − τs are
√
(14-29a)
AR = ω2 τ2 + 1
ϕ = −tan−1 (ωτ)

(14-29b)

Hence, the amplitude ratios of LHP and RHP zeros
are identical. However, an RHP zero contributes phase
lag to the overall frequency response because of the
negative sign. Processes that contain an RHP zero or
time delay are sometimes referred to as nonminimum
phase systems because they exhibit more phase lag

than another transfer function that has the same AR
characteristics (Franklin et al., 2014). Exercise 14.11
illustrates the importance of zero location on the phase
angle.

14.3.5

Time Delay

The time delay e−θs is the remaining important process
element to be analyzed. Its frequency response characteristics can be obtained by substituting s = jω:
G(jω) = e−jωθ

(14-30)

which can be written in rational form by substitution of
the Euler identity
G(jω) = cos ωθ − j sin ωθ

(14-31)


250

Chapter 14

Frequency Response Analysis and Control System Design

10

AR

AR

1

0.1
0.01

0.1

ωθ

1

10

ω (rad/min)

0

–180

ϕ
(deg)

ϕ
(deg)
–360

–540
0.01

0.1

1

10

ωθ
−θs

Figure 14.4 Bode diagram for a time delay, e

Figure 14.5 Bode plot of the transfer function in
Example 14.3.

.

From Eq. 14.6,

√
AR = |G(jω)| = cos2 ωθ + sin2 ωθ = 1
(
)
sin ωθ
ϕ = ∠G(jω) = tan−1 −
cos ωθ

(14-32)

or
ϕ = −ωθ

(14-33)

Because ω is expressed in radians/time, the phase angle
in degrees is −180ωθ/π. Figure 14.4 illustrates the Bode
plot for a time delay. The phase angle is unbounded, that
is, it approaches −∞ as ω becomes large. By contrast, the
phase angles of all other process elements are smaller in
magnitude than some multiples of 90∘ . This unbounded
phase lag is an important attribute of a time delay and
is detrimental to closed-loop system stability, as is discussed in Section 14.6.
EXAMPLE 14.3
Generate the Bode plot for the transfer function
G(s) =

ω (rad/min)

5(0.5s + 1)e−0.5s
(20s + 1)(4s + 1)

where the time constants and time delay have units of
minutes.

SOLUTION
The Bode plot is shown in Fig. 14.5. The steady-state gain
(K = 5) is the value of AR when ω → 0. The phase angle
at high frequencies is dominated by the time delay. The

MATLAB code for generating a Bode plot of the transfer
function is shown in Table 14.1. In this code the normalized
AR is used (ARN ).

Table 14.1 MATLAB Program to Calculate and Plot the
Frequency Response in Example 14.3
%Make a Bode plot for G = 5 (0.5s + 1)e^–0.5s/(20s + 1)
%(4s + 1)
close all
gain = 5;
tdead = 0.5;
num = [0.5 1];
den = [80 24 1];
G = tf (gain∗ num, den) %Define the system as a transfer
%function
points = 500;
%Define the number of points
ww = logspace (−2, 2, points); %Frequencies to be evaluated
[mag, phase, ww] = bode (G,ww); % Generate numerical
%values for Bode plot
AR = zeros (points, 1); % Preallocate vectors for Amplitude
%Ratio and Phase Angle
PA = zeros (points, 1);
for i = 1 : points
AR(i) = mag (1,1,i)/gain; %Normalized AR
PA(i) = phase (1,1,i) – ((180/pi) ∗ tdead∗ ww(i));
end
figure
subplot (2,1,1)
loglog(ww, AR)

axis ([0.01 100 0.001 1])
title (‘Frequency Response of a SOPTD with Zero’)
ylabel(‘AR/K’)
subplot (2,1,2)
semilogx(ww,PA)
axis ([0.01 100 −270 0])
ylabel(‘Phase Angle (degrees)’)
xlabel(‘Frequency (rad/time)’)


14.4

14.4

Frequency Response Characteristics of Feedback Controllers

FREQUENCY RESPONSE
CHARACTERISTICS OF
FEEDBACK CONTROLLERS

In order to use frequency response analysis to design
control systems, the frequency-related characteristics
of feedback controllers must be known for the most
widely used forms of the PID controller discussed in
Chapter 8. In the following derivations, we generally
assume that the controller is reverse-acting (Kc > 0). If a
controller is direct-acting (Kc < 0), the AR plot does not
change, because |Kc | is used in calculating the magnitude. However, the phase angle is shifted by −180∘ when
Kc is negative. For example, a direct-acting proportional
controller (Kc < 0) has a constant phase angle of −180∘.

As a practical matter, it is possible to use the
absolute value of Kc to calculate ϕ when designing
closed-loop control systems, because stability considerations (see Chapter 11) require that Kc < 0 only when
Kv Kp Km < 0. This choice guarantees that the open-loop
gain (KOL = Kc Kv Kp Km ) will always be positive. Use
of this convention conveniently yields ϕ = 0∘ for any
proportional controller and, in general, eliminates the
need to consider the −180∘ phase shift contribution of
the negative controller gain.
Proportional Controller. Consider a proportional controller with positive gain
Gc (s) = Kc

(14-34)

In this case, |Gc (jω)| = Kc , which is independent of ω.
Therefore,
(14-35)
AR = Kc
and

ϕ = 0∘

(14-36)

Proportional-Integral Controller. A proportionalintegral (PI) controller has the transfer function,
(
(
)
)
τI s + 1

1
= Kc
(14-37)
Gc (s) = Kc 1 +
τI s
τI s
Substituting s = jω gives
(
(
)
)
1
j
= Kc 1 −
Gc (jω) = Kc 1 +
τI jω
ωτI

(14-38)

Thus, the amplitude ratio and phase angle are
√
√
(ωτI )2 + 1
1
AR = |Gc (jω)| = Kc 1 +
=
K
c
(ωτI )2

ωτI
(14-39)
ϕ = ∠Gc (jω) = tan−1 (−1∕ωτI ) = tan−1 (ωτI ) − 90∘
(14-40)
Based on Eqs. 14-39 and 14-40, at low frequencies,
the integral action dominates. As ω → 0, AR → ∞, and
ϕ → −90∘ . At high frequencies, AR = Kc and ϕ = 0∘ ; neither is a function of ω in this region (cf. the proportional
controller).

251

Ideal Proportional-Derivative Controller. The ideal
proportional-derivative (PD) controller (cf. Eq. 8-11)
is rarely implemented in actual control systems but is a
component of PID control and influences PID control
at high frequency. Its transfer function is
Gc (s) = Kc (1 + τD s)

(14-41)

The frequency response characteristics are similar to
those of an LHP zero:
√
AR = Kc (ωτD )2 + 1
(14-42)
ϕ = tan−1 (ωτD )

(14-43)

Proportional-Derivative Controller with Filter. As

indicated in Chapter 8, the PD controller is most often
realized by the transfer function
(
)
τD s + 1
Gc (s) = Kc
(14-44)
ατD s + 1
where α has a value in the range 0.05–0.2. The frequency
response for this controller is given by
√
(ωτD )2 + 1
AR = Kc
(14-45)
(αωτD )2 + 1
ϕ = tan−1 (ωτD ) − tan−1 (αωτD )

(14-46)

The pole in Eq. 14-44 bounds the high-frequency asymptote of the AR
lim AR = lim |Gc (jω)| = Kc ∕α = 2∕0.1 = 20 (14-47)

ω→∞

ω→∞

Note that this form actually is an advantage, because
the ideal derivative action in Eq. 14-41 would amplify
high-frequency input noise, due to its large value of AR
in that region. In contrast, the PD controller with derivative filter exhibits a bounded AR in the high-frequency

region. Because its numerator and denominator orders
are both one, the high-frequency phase angle returns
to zero.
Parallel PID Controller. The PID controller can be
developed in both parallel and series forms, as discussed
in Chapter 8. Either version exhibits features of both
the PI and PD controllers. The simpler version is the
following parallel form (cf. Eq. 8-14):
)
(
(
)
1 + τI s + τI τD s2
1
Gc (s) = Kc 1 +
+ τD s = Kc
τI s
τI s
(14-48)
Substituting s = jω and rearranging gives
(
)
[
(
)]
1
1
Gc (jω) = Kc 1 +
+ jωτD = Kc 1 + j ωτD −
jωτI

ωτI
(14-49)


252

Chapter 14

Frequency Response Analysis and Control System Design

102

moves the amplitude ratio curve up or down, without
affecting the width of the notch. Generally, the integral
time τI is larger than τD , typically τI ≈ 4τD .

Ideal
With derivative filter

AR 101
100
10–3

10–2

10–1

100

101

ω (rad/min)
100
50
ϕ
(deg)

0
–50
–100
10–3

10–2

10–1

100

101

ω (rad/min)

Parallel PID Controller with a Derivative Filter. The
parallel controller with a derivative filter was described
in Chapter 8 and Table 8.1.
(14-50)

Figure 14.6 shows a Bode plot for an ideal PID controller, with and without a derivative filter (see Table
8.1). The controller settings are Kc = 2, τI = 10 min,
τD = 4 min, and α = 0.1. The phase angle varies from

−90∘ (ω → 0) to +90∘ (ω → ∞).
A comparison of the amplitude ratios in Fig. 14.6
indicates that the AR for the controller without the
derivative filter in Eq. 14-48 is unbounded at high frequencies, in contrast to the controller with the derivative
filter (Eq. 14-50), which has a bounded AR at all frequencies. Consequently, the addition of the derivative
filter makes the series PID controller less sensitive to
high-frequency noise. For the typical value of α = 0.10,
Eq. 14-50 yields at high frequencies:
ARω→∞ = lim |Gc (jω)| = Kc ∕α = 20Kc
ω→∞

This controller transfer function can be interpreted as
the product of the transfer functions for PI and PD
controllers. Because the transfer function in Eq. 14-52
is physically unrealizable and amplifies high-frequency
noise, a more practical version includes a derivative
filter.

14.5

Figure 14.6 Bode plots of ideal parallel PID controller and
ideal parallel PID controller
filter (α = 0.1)
( with derivative
)
1
Ideal parallel: Gc (s) = 2 1 +
+ 4s
10s
(

)
1
4s
Parallel with derivative filter: Gc (s) = 2 1 +
+
10s 0.4s + 1

)
(
τD s
1
Gc (s) = Kc 1 +
+
τI s ατD s + 1

Series PID Controller. The simplest version of the
series PID controller is
(
)
τ1 s + 1
(14-52)
Gc (s) = Kc
(τD s + 1)
τ1 s

(14-51)

When τD = 0, the parallel PID controller with filter is
the same as the PI controller of Eq. 14-37.
By adjusting the values of τI and τD , one can prescribe

the shape and location of the notch in the AR curve.
Decreasing τI and increasing τD narrows the notch,
whereas the opposite changes broaden it. Figure 14.6
indicates that the center of the notch is located at
√
ω = 1∕ τI τD where ϕ = 0∘ and AR = Kc . Varying Kc

NYQUIST DIAGRAMS

The Nyquist diagram is an alternative representation of
frequency response information, a polar plot of G(jω)
in which frequency ω appears as an implicit parameter. The Nyquist diagram for a transfer function G(s)
can be constructed directly from |G(jω)| and ∠G(jω)
for different values of ω. Alternatively, the Nyquist
diagram can be constructed from the Bode diagram,
because AR = |G(jω)| and ϕ = ∠G(jω). The advantages
of Bode plots are that frequency is plotted explicitly as
the abscissa, and the log–log and semilog coordinate
systems facilitate block multiplication. The Nyquist
diagram, on the other hand, is more compact and is
sufficient for many important analyses, for example,
determining system stability (see Appendix J). Most
of the recent interest in Nyquist diagrams has been in
connection with designing multiloop controllers and
for robustness (sensitivity) studies (Maciejowski, 1989;
Skogestad and Postlethwaite, 2005). For single-loop
controllers, Bode plots are used more often.

14.6

BODE STABILITY CRITERION

The Bode stability criterion has an important advantage in comparison with the alternative of calculating
the roots of the characteristic equation in Chapter 11.
It provides a measure of the relative stability rather
than merely a yes or no answer to the question “Is the
closed-loop system stable?”
Before considering the basis for the Bode stability
criterion, it is useful to review the General Stability
Criterion of Section 11.1: A feedback control system is
stable if and only if all roots of the characteristic equation
lie to the left of the imaginary axis in the complex
plane.
Thus, the imaginary axis divides the complex plane
into stable and unstable regions. Recall that the characteristic equation was defined in Chapter 11 as
1 + GOL (s) = 0

(14-53)

where the open-loop transfer function in Eq. 14-53 is
GOL (s) = Gc (s)Gv (s)Gp (s)Gm (s).


14.6

Before stating the Bode stability criterion, we introduce two important definitions:
1. A critical frequency ωc is a value of ω for which
ϕOL (ω) = −180∘ . This frequency is also referred to
as a phase crossover frequency.
2. A gain crossover frequency ωg is a value of ω for

which AROL (ω) = 1.
The Bode stability criterion allows the stability of a
closed-loop system to be determined from the open-loop
transfer function.
Bode Stability Criterion. Consider an open-loop transfer function GOL = Gc Gv Gp Gm that is strictly proper
(more poles than zeros) and has no poles located on
or to the right of the imaginary axis, with the possible
exception of a single pole at the origin. Assume that the
open-loop frequency response has only a single critical
frequency ωc and a single gain crossover frequency ωg .
Then the closed-loop system is stable if the open-loop
amplitude ratio AROL (ωc ) < 1. Otherwise, it is unstable.
The root locus diagrams of Section 11.5 (e.g.,
Fig. 11.27) show how the roots of the characteristic
equation change as controller gain Kc changes. By
definition, the roots of the characteristic equation are
the numerical values of the complex variable, s, that
satisfy Eq. 14-53. Thus, each point on the root locus
also satisfies Eq. 14-54, which is a rearrangement of
Eq. 14-53:
(14-54)
GOL (s) = −1
The corresponding magnitude and argument are
(14-55)
|G (jω)| = 1 and ∠G (jω) = −180∘
OL

OL

For a marginally stable system, ωc = ωg and the frequency of the sustained oscillation, ωc , is caused by a pair

of roots on the imaginary axis at s = ±ωc j. Substituting
this expression for s into Eq. 14-55 gives the following
expressions for a conditionally stable system:
AROL (ωc ) = |GOL (jωc )| = 1
ϕOL (ωc ) = ∠GOL (jωc ) = −180∘

(14-56)
(14-57)

for some specific value of ωc > 0. Equations 14-56 and
14-57 provide the basis for the Bode stability criterion.
Some of the important properties of the Bode stability
criterion are
1. It provides a necessary and sufficient condition for
closed-loop stability, based on the properties of the
open-loop transfer function.
2. The Bode stability criterion is applicable to systems
that contain time delays.
3. The Bode stability criterion is very useful for a wide
variety of process control problems. However, for
any GOL (s) that does not satisfy the required conditions, the Nyquist stability criterion discussed in
Appendix J can be applied.

Bode Stability Criterion

253

10000
100

AROL

1
0.01
0

–90
ϕOL
–180
(deg)
–270
–360
0.001

0.01

1
0.1
ω (radians/time)

10

100

Figure 14.7 Bode plot exhibiting multiple critical frequencies.

For many control problems, there is only a single ωc
and a single ωg . But multiple values for ωc can occur, as
shown in Fig. 14.7. In this somewhat unusual situation,
the closed-loop system is stable for two different ranges

of the controller gain (Luyben and Luyben, 1997).
Consequently, increasing the absolute value of Kc can
actually improve the stability of the closed-loop system
for certain ranges of Kc . For systems with multiple ωc
or ωg , the Bode stability criterion has been modified by
Hahn et al. (2001) to provide a sufficient condition for
stability.
As indicated in Chapter 11, when the closed-loop
system is marginally stable, the closed-loop response
exhibits a sustained oscillation after a set-point change
or a disturbance. Thus, the amplitude neither increases
nor decreases.
In order to gain physical insight into why a sustained
oscillation occurs at the stability limit, consider the analogy of an adult pushing a child on a swing. The child
swings in the same arc as long as the adult pushes at
the right time and with the right amount of force. Thus
the desired sustained oscillation places requirements on
both timing (i.e., phase) and applied force (i.e., amplitude). By contrast, if either the force or the timing is not
correct, the desired swinging motion ceases, as the child
will quickly protest. A similar requirement occurs when
a person bounces a ball.
To further illustrate why feedback control can produce sustained oscillations, consider the following
thought experiment for the feedback control system
shown in Fig. 14.8. Assume that the open-loop system is
stable and that no disturbances occur (D = 0). Suppose
that the set-point is varied sinusoidally at the critical
frequency, ysp (t) = A sin (ωc t), for a long period of
time. Assume that during this period, the measured
output, ym , is disconnected, so that the feedback loop
is broken before the comparator. After the initial

transient dies out, ym will oscillate at the excitation
frequency ωc , because the response of a linear system
to a sinusoidal input is a sinusoidal output at the same


254

Chapter 14

Frequency Response Analysis and Control System Design

Figure 14.8 Sustained oscillation in a
feedback control system.

Gd

D=0

Yd
Ysp

Km

Ysp

+

E
–

P

Gc

Gv

Ym

frequency (see Section 14.2). Suppose the two events
occur simultaneously: (i) the set-point is set to zero, and
(ii) ym is reconnected. If the feedback control system
is marginally stable, the controlled variable y will then
exhibit a sustained sinusoidal oscillation with amplitude
A and frequency ωc .
To analyze why this special type of oscillation occurs
only when ω = ωc , note that the sinusoidal signal E in
Fig. 14.8 passes through transfer functions Gc , Gv , Gp ,
and Gm before returning to the comparator. In order to
have a sustained oscillation after the feedback loop is
reconnected, signal Ym must have the same amplitude
as E and a 180∘ phase shift relative to E. Note that the
comparator also provides a −180∘ phase shift because of
its negative sign. Consequently, after Ym passes through
the comparator, it is in phase with E and has the same
amplitude, A. Thus, the closed-loop system oscillates
indefinitely after the feedback loop is closed because
the conditions in Eqs. 14-56 and 14-57 are both satisfied.
But what happens if Kc is increased by a small amount?
Then, AROL (ωc ) is greater than one, the oscillations
grow, and the closed-loop system becomes unstable.

In contrast, if Kc is reduced by a small amount, the
oscillation is damped and eventually dies out.

EXAMPLE 14.4
A process has the third-order transfer function (time constant in minutes),
Gp (s) =

2
(0.5s + 1)3

Also, Gv = 0.1 and Gm = 10. For a proportional controller,
evaluate the stability of the closed-loop control system
using the Bode stability criterion and three values of Kc : 1,
4, and 20.

SOLUTION
For this example,
GOL = Gc Gv Gp Gm = (Kc )(0.1)

2Kc
2
(10) =
(0.5s + 1)3
(0.5s + 1)3

U

Yu

Gp

+

+

Y

Gm

100

Kc = 20

10

AROL

Kc = 4

1

Kc = 1

0.1
0.01
0
ϕOL

–90

(deg) –180
–270
0.01

0.1

ωc
1
ω (rad/min)

10

100

Figure 14.9 Bode plots for GOL = 2Kc /(0.5s + 1)3 .
Figure 14.9 shows a Bode plot of GOL for three values of Kc .
Note that all three cases have the same phase angle plot,
because the phase lag of a proportional controller is zero
for Kc > 0.
From the phase angle plot, we observe that ωc =
3.46 rad/min. This is the frequency of the sustained
oscillation that occurs at the stability limit, as discussed
previously. Next, we consider the amplitude ratio AROL for
each value of Kc . Based on Fig. 14.9, we make the following
classifications:
Kc

AROL (for ω = ωc )

1

4
20

0.25
1
5

Classification
Stable
Marginally stable
Unstable

In Section 12.5.1, the concept of the ultimate gain was
introduced. For proportional-only control, the ultimate
gain Kcu was defined to be the largest value of Kc that
results in a stable closed-loop system. The value of Kcu
can be determined graphically from a Bode plot for
transfer function G = Gv Gp Gm . For proportional-only
control, GOL = Kc G. Because a proportional controller
has zero phase lag, ωc is determined solely by G. Also,
AROL (ω) = Kc ARG (ω)

(14-58)


14.6

where ARG denotes the amplitude ratio of G. At the
stability limit, ω = ωc , AROL (ωc ) = 1 and Kc = Kcu . Substituting these expressions into Eq. 14-58 and solving for
Kcu gives an important result:

Kcu =

1
ARG (ωc )

Bode Stability Criterion

100

b

10
AR

(14-59)

255

a

1

c

0.1
0.01
90

The stability limit for Kc can also be calculated for PI and
PID controllers and is denoted by Kcm , as demonstrated

by Example 14.5.

0
ϕ
(deg)

a

–90

c
b

–180

EXAMPLE 14.5
Consider PI control of an overdamped second-order process (time constants in minutes),
Gp (s) =

5
(s + 1)(0.5s + 1)

Gm = Gv = 1
(a) Determine the value of Kcu .
(b) Use a Bode plot to show that controller settings
of Kc = 0.4 and τI = 0.2 min produce an unstable
closed-loop system.
(c) Find Kcm , the maximum value of Kc that can be used
with τI = 0.2 min and still have closed-loop stability.
(d) Show that τI = 1 min results in a stable closed-loop system for all positive values of Kc .

SOLUTION
(a) In order to determine Kcu , we set Gc = Kc . The
open-loop transfer function is GOL = Kc G where
G = Gv Gp Gm . Because a proportional controller does
not introduce any phase lag, G and GOL have identical
phase angles.
(b) Consequently, the critical frequency can be determined
graphically from the phase angle plot for G. However,
curve a in Fig. 14.10 indicates that ωc does not exist
for proportional control, because ϕOL is always greater
than −180∘ . As a result, Kcu does not exist, and thus Kc
does not have a stability limit. Conversely, the addition
of integral control action can produce closed-loop instability. Curve b in Fig. 14.10 indicates that an unstable
closed-loop system occurs for Gc (s) = 0.4(1 + 1/0.2s),
because AROL > 1 when ϕOL = −180∘ .
(c) To find Kcm for τI = 0.2 min, we note that ωc depends on
τI but not on Kc , because Kc has no effect on ϕOL . For
curve b in Fig. 14.10, ωc = 2.2 rad/min, and the corresponding amplitude ratio is AROL = 1.38. To find Kcm ,
multiply the current value of Kc by a factor, 1/1.38. Thus,
Kcm = 0.4/1.38 = 0.29.
(d) When τI is increased to 1 min, curve c in Fig. 14.10
results. Because curve c does not have a critical frequency, the closed-loop system is stable for all positive
values of Kc .

–270
0.01

1 ωc

0.1

10

100

ω (rad/min)

Figure 14.10 Bode plots for Example 14.5
Curve a: G(s)

(
)
1
Curve b: GOL (s): Gc (s) = 0.4 1 +
0.2s
(
)
1
Curve c: GOL (s): Gc (s) = 0.4 1 +
s

EXAMPLE 14.6
Find the critical frequency for the following process and
PID controller, assuming Gv = Gm = 1.
(
)
e−0.3s
1
Gp (s) =

Gc (s) = 20 1 +
+s
(9s + 1)(11s + 1)
2.5s

SOLUTION
Figure 14.7 shows the open-loop amplitude ratio and phase
angle plots for GOL . Note that the phase angle crosses −180∘
at three points. Because there is more than one value of ωc ,
the Bode stability criterion cannot be applied.

EXAMPLE 14.7
Evaluate the stability of the closed-loop system for:
4e−s
5s + 1
The time constant and time delay have units of minutes
and,
Gv = 2, Gm = 0.25, Gc = Kc
Gp (s) =

Obtain ωc and Kcu from a Bode plot. Use an initial value of
Kc = 1.

SOLUTION
The Bode plot for GOL and Kc = 1 is shown in Fig. 14.11.
For ωc = 1.69 rad/min, ϕOL = −180∘ , and AROL = 0.235.
For Kc = 1, AROL = ARG and Kcu can be calculated from


256

Chapter 14

Frequency Response Analysis and Control System Design

Eq. 14-59. Thus, Kcu = 1/0.235 = 4.25. Setting Kc = 1.5 Kcu
gives Kc = 6.38. A larger value of Kc causes the closed-loop
system to become unstable. Only values of Kc less than Kcu
result in a stable closed-loop system.

AROL
ARc =

1

1
GM
ωg

100

AROL

1

ω

0

0.235

0.1

ϕOL

0.01
90
0
ϕOL

ωc

ϕg
Phase
margin

(deg)
–180

–90

ωg

(deg) –180

ω

ωc

–270
–360

0.01

Figure 14.12 Gain and phase margins on a Bode plot.
0.1

1

10
100
ωc = 1.69 rad/min
ω (rad/min)

Figure 14.11 Bode plot for Example 14.7, Kc = 1.

14.7

GAIN AND PHASE MARGINS

Rarely does the model of a chemical process stay
unchanged for a variety of operating conditions and disturbances. When the process changes or the controller
is poorly tuned, the closed-loop system can become
unstable. Thus, it is useful to have quantitative measures
of relative stability that indicate how close the system
is to becoming unstable. The concepts of gain margin
(GM) and phase margin (PM) provide useful metrics
for relative stability.
Let ARc be the value of the open-loop amplitude
ratio at the critical frequency ωc . Gain margin GM is
defined as:
1

(14-60)
GM ≜
ARc
According to the Bode stability criterion, ARc must
be less than one for closed-loop stability. An equivalent stability requirement is that GM > 1. The gain
margin provides a measure of relative stability, because
it indicates how much any gain in the feedback loop
component can increase before instability occurs. For
example, if GM = 2.1, either process gain Kp or controller gain Kc could be doubled, and the closed-loop
system would still be stable, although probably very
oscillatory.
Next, we consider the phase margin. In Fig. 14.12, ϕg
denotes the phase angle at the gain-crossover frequency
ωg where AROL = 1. Phase margin PM is defined as
PM ≜ 180 + ϕg

(14-61)

The phase margin also provides a measure of relative
stability. In particular, it indicates how much additional
time delay can be included in the feedback loop before
instability will occur. Denote the additional time delay
as Δθmax . For a time delay of Δθmax , the phase angle is
−Δθmax ω (see Section 14.3.5). Thus, Δθmax can be calculated from the following expression,
(
)
180∘
PM = Δθmax ωg
(14-62)
π

or
(
)(
)
PM
π
(14-63)
Δθmax =
∘
ωg
180
where the (π/180∘ ) factor converts PM from degrees to
radians. Graphical representations of the gain and phase
margins in a Bode plot are shown in Fig. 14.12.
The specification of phase and gain margins requires
a compromise between performance and robustness. In
general, large values of GM and PM correspond to sluggish closed-loop responses, whereas smaller values result
in less sluggish, more oscillatory responses. The choices
for GM and PM should also reflect model accuracy and
the expected process variability.
Guideline. In general, a well-tuned controller should
have a gain margin between 1.7 and 4.0 and a phase margin between 30∘ and 45∘ .
Recognize that these ranges are approximate and that
it may not be possible to choose PI or PID controller
settings that result in specified GM and PM values.
Tan et al. (1999) have developed graphical procedures
for designing PI and PID controllers that satisfy GM
and PM specifications. The GM and PM concepts are
easily evaluated when the open-loop system does not
have multiple values of ωc or ωg . However, for systems

with multiple ωg , gain margins can be determined from
Nyquist plots (Doyle et al., 2009).


14.7

Gain and Phase Margins

257

EXAMPLE 14.8
For the FOPTD model of Example 14.7, calculate the PID
controller settings for the following approaches:
(a) IMC (Table 12.1 with τc = 1)
(b) Continuous Cycling: Use the Tyreus–Luyben tuning relations (Luyben and Luyben, 1997), which are
Kc = 0.45 Kcu ; τI = 2.2 Pu ; τD = Pu /6.3
Assume that the two PID controllers are implemented
in the parallel form with a derivative filter (α = 0.1) in
Table 8.1. Plot the open-loop Bode diagram and determine
the gain and phase margins for each controller.
For the Tyreus–Luyben settings, determine the maximum increase in the time delay Δθmax that can occur while
still maintaining closed-loop stability.

SOLUTION
GOL = Gc Gv Gp Gm = Gc

Controller settings

Kc

τI (min)

τD (min)

IMC
Tyreus–Luyben

1.83
1.91

5.5
8.2

0.45
0.59

The open-loop transfer function are
(
) ( −s )
1
0.45s
2e
GOL, IMC = 1.83 1 +
+
5.5s 0.045s + 1
5s + 1
(
) ( −s )
1
0.59s

2e
GOL, T−L = 1.91 1 +
+
8.18s 0.059s + 1
5s + 1
Figure 14.13 shows the frequency response of GOL for the
two controllers. The gain and phase margins can be determined by inspection of the Bode diagram or by using the
MATLAB command margin.

2e−s
5s + 1

(a) IMC tuning:
Based on Table 12.1, (line H) for τc = 1, we have
θ
τ+
5 + 0.5
2
= 1.83;
Kc = (
) =
θ
2(1
+ 0.5)
K
τc +
2
θ
τI = τ + = 5.5 min;
2

τθ
5
τD =
=
= 0.45 min
2τ + θ
10 + 1
(b) Tyreus–Luyben:
From Example 14.7, the ultimate gain is Kcu = 4.25, and
2π
= 3.72 min. Therefore,
the ultimate period is Pu =
1.69
the PID controller settings are

Controller
IMC
Tyreus–Luyben

GM

PM

ωc (rad/min)

2.2
1.8

68.5∘
76∘

2.38
2.51

Based on Fig. 14.13, the Tyreus–Luyben controller settings
are close to the IMC tuning result. The value of Δθmax is
calculated from Eq. 14-63, and the information in the preceding table:
(76∘ )(π rad)
= 1.7 min
Δθmax =
(0.79 rad∕min)(180∘ )
Thus, time delay θ can increase by as much as 70% and still
maintain closed-loop stability.
IMC
Tyreus–Luyben

102
AR
100

10–2

10–1

100

101

102

101

102

ω (rad/min)
(a)
0
–200
ϕ
–400
(deg)
–600
–800 –2
10

IMC
Tyreus–Luyben
10–1

100

ω (rad/min)
(b)

Figure 14.13 Comparison of GOL Bode plots for Example 14.8.


258

Chapter 14

Frequency Response Analysis and Control System Design

SUMMARY
Frequency response techniques are powerful tools for
the design and analysis of feedback control systems. The
frequency response characteristics of a process, its amplitude ratio AR and phase angle, characterize the dynamic
behavior of the process and can be plotted as functions
of frequency in Bode diagrams. The Bode stability criterion provides exact stability results for a wide variety of

control problems, including processes with time delays.
It also provides a convenient measure of relative stability, such as gain and phase margins. Control system
design involves trade-offs between control system performance and robustness (see Appendix J). Modern control systems are typically designed using a model-based
technique, such as those described in Chapter 12.

REFERENCES
Doyle, J. C., B. A. Francis, and A. R. Tannenbaum, Feedback Control
Theory, Macmillan, New York, 2009.
Franklin, G. F., J. D. Powell, and A. Emami-Naeini, Feedback Control
of Dynamic Systems, 7th ed., Prentice Hall, Upper Saddle River, NJ,
2014.
Hahn, J., T. Edison, and T. F. Edgar, A Note on Stability Analysis Using
Bode Plots, Chem. Eng. Educ. 35(3), 208 (2001).
Luyben, W. L., and M. L. Luyben, Essentials of Process Control,
McGraw-Hill, New York, 1997, Chapter 11.
MacFarlane, A. G. J., The Development of Frequency Response Methods in Automatic Control, IEEE Trans. Auto. Control, AC-24, 250
(1979).

Maciejowski, J. M., Multivariable Feedback Design, Addison-Wesley,
New York, 1989.

Ogunnaike, B. A., and W. H. Ray, Process Dynamics, Modeling, and
Control, Oxford University Press, New York, 1993.
Seborg, D. E., T. F. Edgar, and D. A., Mellichamp, Process Dynamics
and Cantrol, 2nd ed., John Wiley and Sons, Hoboken, NJ, 2004.
Skogestad, S., and I. Postlethwaite, Multivariable Feedback Design:
Analysis and Design, 2d ed., John Wiley and Sons, Hoboken, NJ,
2005.
Tan, K. K., Q.-G. Wang, C. C. Hang, and T. Hägglund, Advances in
PID Control, Springer, New York, 1999.

EXERCISES
14.1 A heat transfer process has the following transfer function between a temperature T (in ∘ C) and an inlet flow rate q
where the time constants have units of minutes:
3(1 − s)
T ′ (s)
=
Q′ (s)
s(2s + 1)
If the flow rate varies sinusoidally with an amplitude of 2 L/min
and a period of 0.5 min, what is the amplitude of the temperature signal after the transients have died out?
14.2 Using frequency response arguments, discuss how
well e−θs can be approximated by a two-term Taylor series
expansion, 1 − θs. Compare your results with those given in
Section 6.2.1 for a 1/1 Padé approximation.
14.3 A data acquisition system for environmental monitoring
is used to record the temperature of an air stream as measured
by a thermocouple. It shows an essentially sinusoidal variation
after about 15 s. The maximum recorded temperature is 128 ∘ F,
and the minimum is 120 ∘ F at 1.8 cycles per min. It is estimated
that the thermocouple has a time constant of 5 s. Estimate the

actual maximum and minimum air temperatures.
14.4 A perfectly stirred tank is used to heat a flowing liquid.
The dynamic model is shown in Fig. E14.4.

2
0.1s

Figure E14.4

where:
P is the power applied to the heater
Q is the heating rate of the system
T is the actual temperature in the tank
Tm is the measured temperature
time constants have units of min
A test has been made with P′ varied sinusoidally as
P′ = 0.5 sin 0.2t
For these conditions, the measured temperature is
Tm′ = 3.464 sin(0.2t + ϕ)
Find a value for the maximum error bound between T ′ and Tm′
if the sinusoidal input has been applied for a long time.
14.5 Determine if the following processes can be made unstable by increasing the gain of a proportional controller Kc to a
sufficiently large value using frequency response arguments:
2
(a) Gv Gp Gm =
s+1
3
(b) Gv Gp Gm =
(s + 1)(2s + 1)
4

(c) Gv Gp Gm =
(s + 1)(2s + 1)(3s + 1)
5e−s
(d) Gv Gp Gm =
2s + 1
14.6 Two engineers are analyzing step-test data from a bioreactor. Engineer A says that the data indicate a second-order
overdamped process, with time constants of 2 and 6 min but
no time delay. Engineer B insists that the best fit is a FOPTD
model, with τ = 7 min and θ = 1 min. Both engineers claim a
proportional controller can be set at a large value for Kc to


Exercises
control the process and that stability is no problem. Based
on their models, who is right, who is wrong, and why? Use a
frequency-response argument.
14.7

Plot the Bode diagram (0.1 ≤ ω ≤ 100) of the third-order
transfer function,

4
(10s + 1)(2s + 1)(s + 1)
Find both the value of ω that yields a −180∘ phase angle and
the value of AR at that frequency.
G(s) =

Using MATLAB, plot the Bode diagram of the following
transfer function:
6(s + 1)e−2s

G(s) =
(4s + 1)(2s + 1)
Repeat for the situation where the time-delay term is replaced
by a 1/1 Padé approximation. Discuss how the accuracy of the
Padé approximation varies with frequency.
14.8

14.9 Two thermocouples, one of them a known standard, are
placed in an air stream whose temperature is varying sinusoidally. The temperature responses of the two thermocouples
are recorded at a number of frequencies, with the phase angle
between the two measured temperatures as shown below. The
standard is known to have first-order dynamics and a time
constant of 0.15 min when operating in the air stream. From
the data, show that the unknown thermocouple also is a first
order and determine its time constant.

259

14.12 Develop expressions for the amplitude ratio as a
function of ω of each of the two forms of the PID
controller:
(a) The parallel controller of Eq. 8-13.
(b) The series controller of Eq. 8-15.
Plot AR/Kc vs. ωτD for each AR curve. Assume τ1 = 4τD and
α = 0.1.
For what region(s) of ω are the differences significant?
14.13 You are using proportional control (Gc = Kc ) for a pro4
0.6
and Gp =
(time constants in s).

cess with Gv =
2s + 1
50s + 1
You have a choice of two measurements, both of which exhibit
2
2
or Gm2 =
.
first-order dynamic behavior, Gm1 =
s+1
0.4s + 1
Can Gc be made unstable for either process?
Which measurement is preferred for the best stability and performance properties? Why?

Frequency
(cycles/min)

Phase Difference
(deg)

14.14 For the following statements, discuss whether they are
always true, sometimes true, always false, or sometimes false.
Cite evidence from this chapter.
(a) Increasing the controller gain speeds up the response for a
set-point change.
(b) Increasing the controller gain always causes oscillation in
the response to a setpoint change.
(c) Increasing the controller gain too much can cause instability in the control system.
(d) Selecting a large controller gain is a good idea in order to
minimize offset.

0.05
0.1
0.2
0.4
0.8
1.0
2.0
4.0

4.5
8.7
16.0
24.5
26.5
25.0
16.7
9.2

14.15 Use arguments based on the phase angle in frequency
response to determine if the following combinations of
G = Gv Gp Gm and Gc become unstable for some value of Kc .
1
Gc = Kc
(a) G =
(4s + 1)(2s + 1)
)
(
1
1

(b) G =
Gc = Kc 1 +
(4s + 1)(2s + 1)
5s

14.10 Exercise 5.19 considered whether a two-tank liquid
surge system provided better damping of step disturbances
than a single-tank system with the same total volume. Reconsider this situation, this time with respect to sinusoidal
disturbances; that is, determine which system better damps
sinusoidal inputs of frequency ω. Does your answer depend
on the value of ω?
14.11 A process has the transfer function of Eq. 6-14 with
K = 2, τ1 = 10, τ2 = 2. If τa has the following values,
Case i: τa = 20
Case ii: τa = 4

Case iii: τa = 1
Case iv: τa = −2

Plot the composite amplitude ratio and phase angle curves
on a single Bode plot for each of the four cases of numerator
dynamics. What can you conclude concerning the importance
of the zero location for the amplitude and phase characteristics
of this second-order system?

(c) G =

s+1
(4s + 1)(2s + 1)

Gc = Kc

(d) G =

1−s
(4s + 1)(2s + 1)

Gc = Kc

(e) G =

e−s
4s + 1

(2s + 1)
s

Gc = Kc

14.16 Plot the Bode diagram for a composite transfer function
consisting of G(s) in Exercise 14.8 multiplied by that of a
parallel-form PID controller with Kc = 0.21, τI = 5, and
τD = 0.42.
Repeat for a series PID controller with filter that employs
the same settings. How different are these two diagrams? In
particular, by how much do the two amplitude ratios differ
when ω = ωc ?
14.17 For the process described by the transfer function
12
G(s) =

(8s + 1)(2s + 1)(0.4s + 1)(0.1s + 1)


260

Chapter 14

Frequency Response Analysis and Control System Design

(a) Find second-order-plus-time-delay models that approximate G(s) and are of the form
̂
G(s)
=

Ke−θs
(τ1 s + 1)(τ2 s + 1)

One of the approximate models can be found by using the
method discussed in Section 6.3; the other, by using a method
from Chapter 7.
(b) Compare all three models (exact and approximate) in the
frequency domain and a FOPTD model.
14.18 Obtain Bode plots for both the transfer function:
G(s) =

10(2s + 1)e−2s
(20s + 1)(4s + 1)(s + 1)

and a FOPTD approximation obtained using the method
discussed in Section 6.3. What do you conclude about the

accuracy of the approximation relative to the original transfer
function?
(a) Using the process, sensor, and valve transfer functions in Exercise 11.21, find the ultimate controller gain
Kcu using a Bode plot. Using simulation, verify that values of Kc > Kcu cause instability.
(b) Next fit a FOPTD model to G and tune a PI controller for
a set-point change. What is the gain margin for the controller?
14.19

14.20 A process that can be modeled as a time delay (gain = 1)
is controlled using a proportional feedback controller. The
control valve and measurement device have negligible dynamics and steady-state gains of Kv = 0.5 and Km = 1, respectively.
After a small set-point change is made, a sustained oscillation
occurs, which has a period of 10 min.
(a) What controller gain is being used? Explain.
(b) How large is the time delay?
14.21 The block diagram of a conventional feedback control
system contains the following transfer functions:
)
(
1
Gv = 1
Gc = Kc 1 +
5s
1
Gm =
s+1
5e−2s
Gp = Gd =
10s + 1
(a)

(b)
(c)
(d)

Plot the Bode diagram for the open-loop transfer function.
For what values of Kc is the system stable?
If Kc = 0.2, what is the phase margin?
What value of Kc will result in a gain margin of 1.7?

14.22 Consider the storage tank with sightglass in Fig. E14.22.
The parameter values are R1 = 0.5 min/ft2 , R2 = 2 min/ft2 ,
A1 = 10 ft2 , Kv = 2.5 cfm/mA, A2 = 0.8 ft2 , Km = 1.5 mA/ft,
and τm = 0.5 min.
(a) Suppose that R2 is decreased to 0.5 min/ft2 . Compare the
old and new values of the ultimate gain and the critical frequency. Would you expect the control system performance to
become better or worse? Justify your answer.
(b) If PI controller settings are calculated using the ZieglerNichols rules, what are the gain and phase margins? Assume
R2 = 2 min/ft.

Figure E14.22

14.23 A process (including valve and sensor-transmitter) has
the approximate transfer function, G(s) = 2e−0.2s /(s + 1)
with time constant and time delay in minutes. Determine PI controller settings and the corresponding gain
margins by two methods:
(a) Direct synthesis (τc = 0.3 min).
(b) Phase margin = 40∘ (assume τI = 0.5 min).
(c) Simulate these two control systems for a unit step
change in set point. Which controller provides the better
performance?

14.24 Consider the feedback control system in Fig. 14.8, and
the following transfer functions:
)
(
2
2s + 1
Gv =
Gc = Kc
0.1s + 1
0.5s + 1
0.4
3
Gp =
Gd =
s(5s + 1)
5s + 1
Gm = 1
(a) Plot a Bode diagram for the open-loop transfer function.
(b) Calculate the value of Kc that provides a phase margin
of 30∘ .
(c) What is the gain margin when Kc = 10?
14.25 Hot and cold liquids are mixed at the junction of two
pipes. The temperature of the resulting mixture is to
be controlled using a control valve on the hot stream.
The dynamics of the mixing process, control valve, and
temperature sensor/transmitter are negligible and the sensortransmitter gain is 6 mA/mA. Because the temperature sensor
is located well downstream of the junction, an 8 s time delay
occurs. There are no heat losses/gains for the downstream
pipe.
(a) Draw a block diagram for the closed-loop system.

(b) Determine the Ziegler–Nichols settings (continuous
cycling method) for both PI and PID controllers.
(c) For each controller, simulate the closed-loop responses for
a unit step change in set point.
(d) Does the addition of derivative control action provide a
significant improvement? Justify your answer.
14.26 For the process in Exercise 14.23, the measurement
is to be filtered using a noise filter with transfer function
GF (s) = 1/(0.1s + 1). Would you expect this change to result
in better or worse control system performance? Compare the
ultimate gains and critical frequencies with and without the
filter. Justify your answer.


Exercises
14.27 The dynamic behavior of the heat exchanger shown in
Fig. E14.27 can be described by the following transfer functions
(H. S. Wilson and L. M. Zoss, ISA J., 9, 59 (1962)):

Process:

261

2 ∘ F∕lb∕min
T′
=
′
Ws
(0.432s + 1)(0.017s + 1)

Control valve:
0.047 in∕psi
X′
=
P′
0.083s + 1

Ws′
lb
= 112
X′
min in

Temperature sensor-transmitter:
0.12 psi∕∘ F
P′
=
′
T
0.024s + 1
The valve lift x is measured in inches. Other symbols are
defined in Fig. E14.27.
(a) Find the Ziegler–Nichols settings for a PI controller.
(b) Calculate the corresponding gain and phase margins.
14.28 Consider the control problem of Exercise 14.28 and a PI
controller with Kc = 5 and τI = 0.3 min.
(a) Plot the Bode diagram for the open-loop system.
(b) Determine the gain margin from the Bode plot.

Figure E14.27

Chapter

15

Feedforward and Ratio Control
CHAPTER CONTENTS

15.1

Introduction to Feedforward Control

15.2

Ratio Control

15.3

Feedforward Controller Design Based on Steady-State Models
15.3.1 Blending System

15.4

Feedforward Controller Design Based on Dynamic Models

15.5

The Relationship Between the Steady-State and Dynamic Design Methods
15.5.1

Steady-State Controller Design Based on Transfer Function Models

15.6

Configurations for Feedforward–Feedback Control

15.7

Tuning Feedforward Controllers

Summary

In Chapter 8 it was emphasized that feedback control is
an important technique that is widely used in the process
industries. Its main advantages are.

2. It does not provide predictive control action to
compensate for the effects of known or measurable
disturbances.

1. Corrective action occurs as soon as the controlled
variable deviates from the set point, regardless of
the source and type of disturbance.

3. It may not be satisfactory for processes with large
time constants and/or long time delays. If large
and frequent disturbances occur, the process may
operate continuously in a transient state and never
attain the desired steady state.

4. In some situations, the controlled variable cannot
be measured on-line, so feedback control is not
feasible.

2. Feedback control requires minimal knowledge
about the process to be controlled; in particular, a
mathematical model of the process is not required,
although it can be very useful for control system
design.
3. The ubiquitous PID controller is both versatile and
robust. If process conditions change, re-tuning the
controller usually produces satisfactory control.
However, feedback control also has certain inherent disadvantages:
1. No corrective action is taken until after a deviation
in the controlled variable occurs. Thus, perfect control, where the controlled variable does not deviate
from the set point during disturbance or set-point
changes, is theoretically impossible.

262

For situations in which feedback control by itself is not
satisfactory, significant improvement can be achieved
by adding feedforward control. But feedforward control requires that the disturbances be measured (or
estimated) on-line.
In this chapter, we consider the design and analysis of feedforward control systems. We begin with an
overview of feedforward control. Then ratio control,
a special type of feedforward control, is introduced.
Next, design techniques for feedforward controllers
are developed based on either steady-state or dynamic

15.1

models. Then alternative configurations for combined
feedforward–feedback control systems are considered. This chapter concludes with a section on tuning
feedforward controllers.

15.1

Introduction to Feedforward Control

263

LC

LT

INTRODUCTION TO FEEDFORWARD
CONTROL

The basic concept of feedforward control is to measure
important disturbance variables and take corrective
action before they upset the process. In contrast, a
feedback controller does not take corrective action until
after the disturbance has upset the process and generated a nonzero error signal. Simplified block diagrams
for feedforward and feedback control are shown in
Fig. 15.1.
Feedforward control has several disadvantages:
1. The disturbance variables must be measured
on-line. In many applications, this is not feasible.

2. To make effective use of feedforward control, at
least an approximate process model should be
available. In particular, we need to know how the
controlled variable responds to changes in both
the disturbance variable and the manipulated variable. The quality of feedforward control depends
on the accuracy of the process model.
3. Ideal feedforward controllers that are theoretically
capable of achieving perfect control may not be
physically realizable. Fortunately, practical approximations of these ideal controllers can provide very
effective control.
Feedforward control was not widely used in the process industries until the 1960s (Shinskey, 1996). Since

D

Ysp

Feedforward
controller
U

Process

Y

D

Ysp

Feedback
controller

U

Process

Y

Figure 15.1 Simplified block diagrams for feedforward and
feedback control.

Steam
Boiler
drum

Feedwater

Hot
gas

Figure 15.2 Feedback control of the liquid level in a boiler
drum.

then, it has been applied to a wide variety of industrial
processes. However, the basic concept is much older
and was applied as early as 1925 in the three-element
level control system for boiler drums. We will use this
control application to illustrate the use of feedforward
control.
A boiler drum with a conventional feedback control
system is shown in Fig. 15.2. The level of the boiling

liquid is measured and used to adjust the feedwater flow
rate. This control system tends to be quite sensitive to
rapid changes in the disturbance variable, steam flow
rate, as a result of the small liquid capacity of the boiler
drum. Rapid disturbance changes are produced by
steam demands made by downstream processing units.
Another difficulty is that large controller gains cannot be used because level measurements exhibit rapid
fluctuations for boiling liquids. Thus a high controller
gain would tend to amplify the measurement noise
and produce unacceptable variations in the feedwater
flow rate.
The feedforward control scheme in Fig. 15.3 can
provide better control of the liquid level. The steam
flow rate is measured, and the feedforward controller
adjusts the feedwater flow rate so as to balance the
steam demand. Note that the controlled variable, liquid
level, is not measured. As an alternative, steam pressure
could be measured instead of steam flow rate.
Feedforward control can also be used advantageously
for level control problems where the objective is surge
control (or averaging control), rather than tight level
control. For example, the input streams to a surge
tank will be intermittent if they are effluent streams
from batch operations, but the tank exit stream can
be continuous. Special feedforward control methods
have been developed for these batch-to-continuous
transitions to balance the surge capacity requirement


264

Chapter 15

Feedforward and Ratio Control

outputs of the feedforward and feedback controllers are
added together and the combined signal is sent to the
control valve.

Feedforward
controller
FFC

15.2
FT

Steam
Boiler
drum

Feedwater

Hot
gas

Figure 15.3 Feedforward control of the liquid level in a
boiler drum.

for the measured inlet flow rates with the surge control
objective of gradual changes in the tank exit stream

(Blevins et al., 2003).
In practical applications, feedforward control is
normally used in combination with feedback control.
Feedforward control is used to reduce the effects of measurable disturbances, while feedback trim compensates
for inaccuracies in the process model, measurement
errors, and unmeasured disturbances. The feedforward
and feedback controllers can be combined in several
different ways, as will be discussed in Section 15.6.
A typical configuration is shown in Fig. 15.4, where the

Feedback
controller
+
+

LC
FFC

Feedforward
controller

RATIO CONTROL

Ratio control is a special type of feedforward control
that has had widespread application in the process
industries. Its objective is to maintain the ratio of two
process variables at a specified value. The two variables
are usually flow rates, a manipulated variable u and a
disturbance variable d. Thus, the ratio
u

(15-1)
R≜
d
is controlled rather than the individual variables. In
Eq. 15-1, u and d are physical variables, not deviation
variables.
Typical applications of ratio control include (1) specifying the relative amounts of components in blending
operations, (2) maintaining a stoichiometric ratio of
reactants to a reactor, (3) keeping a specified reflux
ratio for a distillation column, and (4) holding the
fuel-air ratio to a furnace at the optimum value.
Ratio control can be implemented in two basic
schemes. For Method I in Fig. 15.5, the flow rates for
both the disturbance stream and the manipulated stream
are measured, and the measured ratio, Rm = um /dm , is
calculated. The output of the divider element is sent
to a ratio controller (RC) that compares the calculated
ratio Rm to the desired ratio Rd and adjusts the manipulated flow rate u accordingly. The ratio controller is
typically a PI controller with the desired ratio as its
set point.
The main advantage of Method I is that the measured
ratio Rm is calculated. A key disadvantage is that a

Disturbance stream, d

LT
FT

FT
dm

Ratio controller

Steam
Boiler
drum

Feedwater

Divider

Rm
um

RC

p

FT

Hot
gas

Figure 15.4 Feedforward–feedback control of the boiler
drum level.

Manipulated stream u

Figure 15.5 Ratio control, Method I.

Ratio set point

Rd


15.2

divider element must be included in the loop, and this
element makes the process gain vary in a nonlinear
fashion. From Eq. 15-1, the process gain
( )
1
∂R
=
Kp =
(15-2)
∂u d d
is inversely related to the disturbance flow rate d.
Because of this significant disadvantage, the preferred
scheme for implementing ratio control is Method II,
which is shown in Fig. 15.6.
In Method II, the flow rate of the disturbance stream
is measured and transmitted to the ratio station (RS),
which multiplies this signal by an adjustable gain, KR ,
whose value is the desired ratio. The output signal from
the ratio station is then used as the set point usp for the
flow controller, which adjusts the flow rate of the manipulated stream, u. The chief advantage of Method II is
that the process gain remains constant. Note that disturbance variable d is measured in both Methods I and II.
Thus, ratio control is, in essence, a simple type of feedforward control.
A disadvantage of both Methods I and II is that
the desired ratio may not be achieved during transient
conditions as a result of the dynamics associated with

the flow control loop for u. Thus, after a step change
in disturbance d, the manipulated variable will require
some time to reach its new set point, usp . Fortunately,
flow control loops tend to have short settling times
and this transient mismatch between u and d is usually
acceptable. For situations where it is not, modified
versions of Method II have been proposed by Hägglund
(2001) and Visioli (2005a,b).

Ratio Control

265

Regardless of how ratio control is implemented, the
process variables must be scaled appropriately. For
example, in Method II the gain setting for the ratio
station KR must take into account the spans of the two
flow transmitters. Thus, the correct gain for the ratio
station is
S
(15-3)
KR = Rd d
Su
where Rd is the desired ratio, and Su and Sd are the
spans of the flow transmitters for the manipulated
and disturbance streams, respectively. If orifice plates
are used with differential-pressure transmitters, then
the transmitter output is proportional to the flow rate
squared. Consequently, KR should then be proportional
to R2d rather than Rd , unless square root extractors are

used to convert each transmitter output to a signal that
is proportional to flow rate (see Exercise 5.2).

EXAMPLE 15.1
A ratio control scheme is to be used to maintain a stoichiometric ratio of H2 and N2 as the feed to an ammonia
synthesis reactor. Individual flow controllers will be used
for both the H2 and N2 streams. Using the information given
below,
(a) Draw a schematic diagram for the ratio control scheme.
(b) Specify the appropriate gain for the ratio station, KR .
Available information:
(i) The electronic flow transmitters have built-in square
root extractors. The spans of the flow transmitters are
30 L/min for H2 and 15 L/min for N2 .
(ii) The control valves have pneumatic actuators.
(iii) Each required current-to-pressure (I/P) transducer has
a gain of 0.75 psi/mA.

Disturbance stream, d

(iv) The ratio station is an electronic instrument with
4–20 mA input and output signals.
FT

SOLUTION

dm
RS Ratio station

Set point usp

FC

um
p
FT

Manipulated stream u

Figure 15.6 Ratio control, Method II.

The stoichiometric equation for the ammonia synthesis
reaction is
3H2 + N2 ⇄ 2NH3
In order to introduce a feed mixture in stoichiometric proportions, the ratio of the molar flow rates (H2 /N2 ) should
be 3:1. For the sake of simplicity, we assume that the ratio
of the molar flow rates is equal to the ratio of the volumetric flow rates. But, in general, the volumetric flow rates also
depend on the temperature and pressure of each stream (cf.
the ideal gas law).
(a) The schematic diagram for the ammonia synthesis reaction is shown in Fig. 15.7. The H2 flow rate is considered
to be the disturbance variable, although this choice is
arbitrary, because both the H2 and N2 flow rates are


266

Chapter 15

Feedforward and Ratio Control

d

(H2)
FT

I/P

FC

dm

RS

Ratio
station

usp
I/P

NH3
synthesis
reactor

N2, H2, NH3

FC

um
FT

u
(N2)

Figure 15.7 Ratio control scheme for an ammonia synthesis reactor of Example 15.1.
controlled. Note that the ratio station is merely a device
with an adjustable gain. The input signal to the ratio station is dm , the measured H2 flow rate. Its output signal
usp serves as the set point for the N2 flow control loop.
It is calculated as usp = KR dm .
(b) From the stoichiometric equation, it follows that
the desired ratio is Rd = u/d = 1/3. Substitution into
Eq. 15-3 gives
( )(
)
30 L∕min
1
2
=
KR =
3
15 L∕min
3

Distillate
D, y

Feed
F, z

Bottoms
B, x

Figure 15.8 A simplified schematic diagram of a distillation

column.

15.3

FEEDFORWARD CONTROLLER
DESIGN BASED ON STEADY-STATE
MODELS

A useful interpretation of feedforward control is that
it continually attempts to balance the material or
energy that must be delivered to the process against
the demands of the disturbance (Shinskey, 1996). For
example, the level control system in Fig. 15.3 adjusts the
feedwater flow so that it balances the steam demand.
Thus, it is natural to base the feedforward control calculations on material and energy balances. For simplicity,
we will first consider designs based on steady-state
balances using physical variables rather than deviation
variables. Design methods based on dynamic models
are considered in Section 15.4.
To illustrate the design procedure, consider the distillation column shown in Fig. 15.8, which is used to
separate a binary mixture. Feedforward control has
gained widespread acceptance for distillation column
control owing to the slow responses that typically occur

with feedback control. In Fig. 15.8, the symbols B, D, and
F denote molar flow rates, while x, y, and z are the mole
fractions of the more volatile component. The objective
is to control the distillate composition y despite measurable disturbances in feed flow rate F and feed composition z, by adjusting distillate flow rate D. It is assumed
that measurements of x and y are not available.
The steady-state mass and component balances for

the distillation column are
F =D+B
Fz = D y + B x

(15-4)
(15-5)

where the bar over a variable denotes a steady-state
value. Solving Eq. 15-4 for B and substituting into
Eq. 15-5 gives
F(z − x)
D=
(15-6)
y−x
Because x and y are not measured, we replace x and y
by their set points and replace D, F, and z by D(t), F(t),


15.3

Feedforward Controller Design Based on Steady-State Models

and z(t), respectively. These substitutions yield a feedforward control law:
F(t) [z(t) − xsp ]
D(t) =
(15-7)
ysp − xsp
Thus, the feedforward controller calculates the required
value of the manipulated variable D from the measurements of the disturbance variables, F and z, and the
knowledge of the composition set points xsp and ysp .

Note that Eq. 15-7 is based on physical variables, not
deviation variables.
The feedforward control law is nonlinear due to the
product of two process variables, F(t) and z(t). Because
the control law was designed based on the steady-state
model in Eqs. 15-4 and 15-5, it may not perform well
for transient conditions. This issue is considered in
Sections 15.4 and 15.7.

15.3.1

two input signals: the x1 measurement x1m , and the set
point for the exit composition xxp .
The starting point for the feedforward controller
design is the steady-state mass and component balances
that were considered in Chapter 1,
w = w1 + w2
w x = w1 x1 + w2 x2

w2 =

(15-10)

w1 [xsp − x1 (t)]
x2 − xsp

(15-11)

Note that this feedforward control law is also based on
physical variables rather than deviation variables.

The feedforward control law in Eq. 15-11 is not in the
final form required for actual implementation, because it
ignores two important instrumentation considerations:
First, the actual value of x1 is not available, but its measured value x1m is. Second, the controller output signal
is p rather than inlet flow rate, w2 . Thus, the feedforward
control law should be expressed in terms of x1m and p,
rather than x1 and w2 . Consequently, a more realistic
feedforward control law should incorporate the appropriate steady-state instrument relations for the w2 flow
transmitter and the control valve, as shown below.

Suppose that the sensor/transmitter for x1 is an electronic instrument with negligible dynamics and a
standard output range of 4–20 mA. In analogy with
Section 9.1, if the calibration relation is linear, it can be
written as

FFC
p

AT

x2
w2

w1

w1 (x − x1 )
x2 − x

Composition Measurement for x1

xsp

x1

(15-9)

In order to derive a feedforward control law, we replace
x by xsp and w2 and x1 by w2 (t) and x1 (t), respectively:

Blending System

x1m

(15-8)

These equations are the steady-state version of the
dynamic model in Eqs. 2-12 and 2-13. Substituting
Eq. 15-8 into Eq. 15-9 and solving for w2 gives

w2 (t) =

To further illustrate the design method, consider the
blending system and feedforward controller shown
in Fig. 15.9. We wish to design a feedforward control
scheme to maintain exit composition x at a constant
set point xsp , despite disturbances in inlet composition,
x1 . Suppose that inlet flow rate w1 and the composition
of the other inlet stream x2 are constant. It is assumed
that x1 is measured but that x is not. (If x were measured, then feedback control would also be possible.)
The manipulated variable is inlet flow rate w2 . The

flow-head √
relation for the valve on the exit line is given
by w = Cv h. Note that the feedforward controller has

267

x1m (t) = Kt [x1 (t) − (x1 )0 ] + 4

(15-12)

where (x1 )0 is the zero of this instrument and Kt is its
gain. From Eq. 9.1,
Kt =

output range 20 − 4 mA
=
input range
St

(15-13)

where St is the span of the instrument.

h

Control Valve and Current-to-Pressure Transducer
x
w

Figure 15.9 Feedforward control of exit composition in the

blending system.

Suppose that the current-to-pressure transducer and the
control valve are designed to have linear input–output
relationships with negligible dynamics. Their input
ranges (spans) are 4–20 mA and 3–15 psi, respectively.
Then in analogy with Eq. 9-1, the relationship between


268

Chapter 15

Feedforward and Ratio Control

the controller output signal p(t) and inlet flow rate w2 (t)
can be written as
w2 (t) = Kv KIP [ p(t) − 4] + (w2 )0

x1m (t) − 4
+ (x1 )0
Kt

x1m

(15-14)

where Kv and KIP are the steady-state gains for the control valve and I/P transducer, respectively, and (w2 )0 is
the minimum value of the w2 flow rate that corresponds
to the minimum controller output value of 4 mA. Note

that all of the symbols in Eqs. 15-8 through 15-14 denote
physical variables rather than deviation variables.
Rearranging Eq. 15-12 gives
x1 (t) =

xsp

(15-15)

C1 ≜ 4 −
C2 ≜

(w2 )0
Kv KIP

w1
Kv KIP Kt

C3 ≜ 4 + Kt (x1 )0

w2,sp

FC

w2m

p

FT

AT

x1

x2
w2

w1

Substituting Eqs. 15-14 and 15-15 into Eq. 15-11 and rearranging the resulting equation provides a feedforward
control law that is suitable for implementation:
[
]
Kt xsp − x1m (t) + C3
p(t) = C1 + C2
(15-16)
x2 − xsp
where

FFC

h

x

(15-17)
(15-18)
(15-19)

An alternative feedforward control scheme for the

blending system is shown in Fig. 15.10. Here the feedforward controller output signal serves as a set point to
a feedback controller for flow rate w2 . The advantage
of this configuration is that it is less sensitive to valve
sticking and upstream pressure fluctuations. Because
the feedforward controller calculates the w2 set point
rather than the signal to the control valve p, it would
not be necessary to incorporate Eq. 15-14 into the
feedforward control law.
The blending and distillation column examples illustrate that feedforward controllers can be designed using
steady-state mass and energy balances. The advantages
of this approach are that the required calculations
are quite simple, and a detailed process model is not
required. However, a disadvantage is that process
dynamics are neglected, and consequently the control
system may not perform well during transient conditions. The feedforward controllers can be improved by
adding dynamic compensation, usually in the form of a
lead–lag unit. This topic is discussed in Section 15.7. An
alternative approach is to base the controller design on
a dynamic model of the process, as discussed in the next
section.
In many feedforward control applications (e.g., the
two previous examples), the controller output is the
desired value of a flow rate through a control valve.
Because control valves tend to exhibit hysteresis and

w

Figure 15.10 Feedforward control of exit composition using
an additional flow control loop.

other nonlinear behavior (Chapter 9), the controller
output is usually the set-point for the flow control loop,
rather than the signal to the control valve. This strategy
provides more assurance that the calculated flow rate is
actually implemented.

15.4

FEEDFORWARD CONTROLLER
DESIGN BASED ON DYNAMIC
MODELS

In this section, we consider the design of feedforward control systems based on dynamic, rather than
steady-state, process models. We will restrict our attention to design techniques based on linear dynamic
models. But nonlinear process models can also be used
(Smith and Corripio, 2006).
As the starting point, consider the block diagram
in Fig. 15.11. This diagram is similar to Fig. 11.8 for
feedback control, but an additional signal path through
transfer functions, Gt and Gf , has been added. The
disturbance transmitter with transfer function Gt sends
a measurement of the disturbance variable to the feedforward controller Gf . The outputs of the feedforward
and feedback controllers are then added together, and
the sum is sent to the control valve. In contrast to the
steady-state design methods of Section 15.3, the block
diagram in Fig. 15.11 is based on deviation variables.
The closed-loop transfer function for disturbance
changes in Eq. 15-20 can be derived using the block