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[.]

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 First-Order Process

14.3.2 Integrating Process

14.3.3 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

trans-fer 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

analy-sis, 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 rel-ative stability, namely gain and phase margins These metrics indicate how close a control system is to insta-bility 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 con-sidered 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 Sinusoidal Forcing of a First-Order Process 245

the term frequency response The responses for

first-and second-order processes forced by a sinusoidal input

were presented in Chapter 5 Recall that these responses

consisted of sine, cosine, and exponential terms

Specifi-cally, 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

ω2τ2+ 1(ωτe

−t∕τ − ωτ cos ωt + sin ωt) (5-23)

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

y𝓁(t) = √ KA

ω2τ2+ 1sin (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 ̂ A that is a function

of the forcing frequency:

̂

ω2τ2+ 1 (14-2) Dividing both sides of Eq 14-2 by the input signal

amplitude A yields the amplitude ratio (AR)

AR = A ̂

ω2τ2+ 1 (14-3a) which can, in turn, be divided by the process gain

to yield the normalized amplitude ratio (AR N):

AR N= AR

ω2τ2+ 1 (14-3b) Next we examine the physical significance of the

pre-ceding equations, with specific reference to the blending

A

P

A

Time

shift, Δt

Output, y 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

process example discussed earlier In Chapter 4, the transfer function model for the stirred-tank blending system was derived as

X′(s) = K1

τs + 1 X

′

1(s) + K2

τs + 1 W

′

2(s) + K3

τs + 1 W

′

1(s) (4-84) 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

sinu-soidal 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 pres-ence 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 fre-quency 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

devi-ations in w2are 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

pro-cesses 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 cor-responding periodic variation in the output is difficult (perhaps impossible) to detect or measure

Input–output phase shift and attenuation (or amplifi-cation) 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 pro-cess, 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

because of the repeated experiments for different

val-ues 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 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

imagi-nary (I) terms (j indicates an imagiimagi-nary component):

Similar to Eq 14-1, we can express the long time

response for a linear system (cf Eq 14-1) as

y𝓁(t) = ̂ A sin(ωt + ϕ) (14-5)

̂

A and ϕ are related to I(ω) and R(ω) by the following

relations (Seborg et al., 2004):

̂

A = A√

Both ̂ A 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

AR = A ̂

A = |G| =√

R2+ I2 (14-7)

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

ϕ = ∠G = tan−1(I∕R) (14-8)

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

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

com-plex conjugate multiplication

Step 3 The output sine wave has amplitude

̂

A = A√

R2+ I2and phase angle ϕ = tan−1(I/R) The amplitude ratio is AR =√

R2+ I2and is

independent of the value of A.

EXAMPLE 14.1

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

SOLUTION

First substitute s = jω in the transfer function

τjω + 1=

1

Then multiply both numerator and denominator by the

complex conjugate of the denominator, that is, −jωτ + 1

(jωτ + 1)(−jωτ + 1)=

−jωτ + 1

ω2τ2+ 1

ω2τ2+ 1+ j

(−ωτ)

ω2τ2+ 1= R + jI (14-11) where

and

From Eq 14-7,

AR = |G(jω|) =

√(

1

ω2τ2+ 1

)2

+

(

−ωτ

ω2τ2+ 1 )2

Simplifying,

AR =

√ (1 + ω2τ2) (ω2τ2+ 1)2 = √ 1

ϕ = ∠G(jω) = tan−1(−ωτ) = −tan−1(ωτ) (14-13b)

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

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

14.3 Bode Diagrams 247

From this example, we conclude that direct analysis

of the complex transfer function G(jω) is

computation-ally 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) = G a (s)G b (s)G c (s) · · ·

G1(s)G2(s)G3(s) · · · (14-15) G(s) is converted to the complex form G(jω) by the

sub-stitution s = jω:

G(jω) = G a(jω)Gb(jω)Gc(jω) · · ·

G1(jω)G2(jω)G3(jω) · · · (14-16) The magnitude and phase angle of G(jω) are as follows:

|G(jω)| = | G a(jω)‖Gb(jω)‖Gc(jω)| · · ·

|G1(jω)‖G2(jω)‖G3(jω)| · · · (14-17a)

∠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

com-putation 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

soft-ware such as MATLAB, there is no need to factor the

numerator or denominator, as discussed in Section 14.3

EXAMPLE 14.2

Calculate the amplitude ratio and phase angle for the

over-damped second-order transfer function

(τ1s + 1)(τ2s + 1)

SOLUTION

Using Eq 14-15, let

G a = K

G1= τ1s + 1

G2= τ2s + 1

Substituting s = jω

G a (jω) = K

G1(jω) = jωτ1+ 1

G2(jω) = jωτ2+ 1 The magnitudes and angles of each component of the

com-plex transfer function are

|G1| =√ω2τ2+ 1 ∠G1= tan−1(ωτ1)

|G2| =√ω2τ2+ 1 ∠G2= tan−1(ωτ2)

Combining these expressions via Eqs 14-17a and 14-17b yields

|G1(jω)‖G2(jω)|

ω2τ2+ 1√

ϕ = ∠G(jω) = ∠G a (jω) − (∠G1(jω) + ∠G2(jω))

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 develop-ment, 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 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 spread-sheet software can be used to generate Bode plots, it is much more convenient to use software designed specif-ically 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

0.01 0.1 1 10 100 –90

–60 –30 0

–120

ωτ

0.01 0.1 1

ωτ

ωb = 1/τ

ωb = 1/τ

Normalized

amplitude

ratio, AR N

Phase angle

ϕ (deg)

Figure 14.2 Bode diagram for a first-order process.

AR dbis defined as

AR db = 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

G(s) = Y(s) U(s) =

K

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

sta-ble 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:

AR = |G(jω)| =||

||jω K||

|| = Kω (14-20)

ϕ = ∠G(jω) = ∠K − tan−1(j∞) = −90∘ (14-21)

14.3.3 Second-Order Process

A general transfer function for a second-order system without numerator dynamics is

τ2s2+ 2ζτs + 1 (14-22) Substituting s = jω and rearranging into real and

imagi-nary parts (see Example 14.1) yields

(1 − ω2τ2)2+ (2ζωτ)2 (14-23a)

ϕ = tan−1

[ −2ζωτ

1 − ω2τ2

]

(14-23b) 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

low-frequency limits of the second-order system are iden-tical to those of the first-order system However, the

limits are different at high frequencies, ωτ ≫ 1.

For overdamped systems, the normalized amplitude

ratio is attenuated ( ̂ A/KA < 1) for all ω For

under-damped systems, the amplitude ratio plot exhibits a maximum (for values of 0< ζ <√2∕2) at the resonant frequency

ωr=

√

1 − 2ζ2

(ARN)max= 1

2ζ√

1 − ζ2 (14-26) These expressions can be derived by the interested

reader The resonant frequency ω r 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 (AR N)maxdepend on ξ This behavior is used in designing organ pipes to cre-ate 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 Bode Diagrams 249

2

AR N

–135 –90 –45 0

–180

ωτ

ϕ

(deg)

ϕ (deg)

0.001 0.01 0.1 1

0.0001

ωτ

ζ = 1

ζ = 1

5

Slope = –2

2

2

5 5

1

0.4

AR N

–135 –90 –45 0

–180

ωτ

0.01 0.1 1 10

0.001

ωτ

ζ = 0.2

Slope = –2

0.4 0.8

ζ = 0.2 0.8

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

trans-fer function is sometimes retrans-ferred 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

ampli-tude ratio and phase angle for this term

Substituting s = jω into G(s) = τs + 1 gives

from which

AR = |G(jω)| =√

ω2τ2+ 1 (14-28a)

ϕ = ∠G(jω) = tan−1(ωτ) (14-28b)

Therefore, a process zero contributes a positive phase

angle that varies between 0 and +90∘ The output

sig-nal amplitude becomes very large at high frequencies

(i.e., AR → ∞ as ω → ∞), which is a physical

impossi-bility 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

ω2τ2+ 1 (14-29a)

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

charac-teristics can be obtained by substituting s = jω:

which can be written in rational form by substitution of the Euler identity

G(jω) = cos ωθ − j sin ωθ (14-31)

0.01 0.1 1 10 0.1

1 10

–540

–360

–180

0

AR

ωθ

ωθ

ϕ

(deg)

Figure 14.4 Bode diagram for a time delay, e −θs

From Eq 14.6,

AR = |G(jω)| =

√ cos2ωθ + sin2ωθ = 1 (14-32)

ϕ = ∠G(jω) = tan−1

(

−sin ωθ cos ωθ )

or

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

dis-cussed in Section 14.6

EXAMPLE 14.3

Generate the Bode plot for the transfer function

G(s) = 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 (AR N)

AR

ω (rad/min)

ω (rad/min)

ϕ (deg)

Figure 14.5 Bode plot of the transfer function in

Example 14.3

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

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 Frequency Response Characteristics of Feedback Controllers 251

14.4 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 (K c > 0) If a

controller is direct-acting (K c < 0), the AR plot does not

change, because |K c| is used in calculating the

magni-tude However, the phase angle is shifted by −180∘ when

K cis negative For example, a direct-acting proportional

controller (K c < 0) has a constant phase angle of −180∘.

As a practical matter, it is possible to use the

absolute value of K c to calculate ϕ when designing

closed-loop control systems, because stability

consider-ations (see Chapter 11) require that K c < 0 only when

K v K p K m < 0 This choice guarantees that the open-loop

gain (K OL = Kc K v K p K m) 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

con-troller with positive gain

In this case, |G c(jω)| = Kc, which is independent of ω.

Therefore,

and

Proportional-Integral Controller A

proportional-integral (PI) controller has the transfer function,

G c(s) = Kc

(

1 + 1

τIs

)

= Kc

( τIs + 1

τIs

) (14-37)

Substituting s = jω gives

G c(jω) = Kc

(

1 + 1

τI jω

)

= Kc

(

1 − j

ωτI

) (14-38)

Thus, the amplitude ratio and phase angle are

AR = |G c(jω)| = Kc

√

1 + 1 (ωτI)2 = Kc

√ (ωτI)2+ 1

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 = K cand ϕ = 0∘;

nei-ther is a function of ω in this region (cf the proportional

controller)

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

G c (s) = K c(1 + τD s) (14-41) The frequency response characteristics are similar to those of an LHP zero:

AR = K c√

(ωτD)2+ 1 (14-42)

Proportional-Derivative Controller with Filter As

indicated in Chapter 8, the PD controller is most often realized by the transfer function

G c(s) = Kc

( τDs + 1 ατDs + 1

)

(14-44)

where α has a value in the range 0.05–0.2 The frequency response for this controller is given by

AR = K c

√ (ωτD)2+ 1

ϕ = tan−1(ωτD) − tan−1(αωτD) (14-46) The pole in Eq 14-44 bounds the high-frequency

asymp-tote of the AR

lim

ω→∞AR = lim

ω→∞|G c (jω)| = K c ∕α = 2∕0.1 = 20 (14-47)

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

deriva-tive 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):

G c(s) = Kc

(

1 + 1

τI s+ τDs

)

= Kc

(

1 + τIs + τIτD s2

τI s

)

(14-48)

Substituting s = jω and rearranging gives

G c (jω) = K c

(

1 + 1

jωτ I + jωτ D

)

= K c

[

1 + j

(

ωτD− 1 ωτI )] (14-49)

101

100

AR

ϕ

(deg)

With derivative filter Ideal

ω (rad/min) 100

50

0

–50

–100

ω (rad/min)

Figure 14.6 Bode plots of ideal parallel PID controller and

ideal parallel PID controller with derivative filter (α = 0.1)

Ideal parallel: G c (s) = 2

(

10s + 4s

)

Parallel with derivative filter: G c (s) = 2

(

0.4s + 1

)

Parallel PID Controller with a Derivative Filter The

parallel controller with a derivative filter was described

in Chapter 8 and Table 8.1

G c(s) = Kc

(

1 + 1

τI s+ τDs

ατD s + 1

)

(14-50)

Figure 14.6 shows a Bode plot for an ideal PID

con-troller, with and without a derivative filter (see Table

8.1) The controller settings are K c= 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

fre-quencies, in contrast to the controller with the derivative

filter (Eq 14-50), which has a bounded AR at all

fre-quencies 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

ω→∞|G c (jω)| = K c ∕α = 20K c (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 τIand 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∕√

τIτD where ϕ = 0∘ and AR = Kc Varying K c

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

Series PID Controller The simplest version of the

series PID controller is

G c(s) = Kc

(

τ1s + 1

τ1s

) (τDs + 1) (14-52) 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 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

parame-ter 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 advan-tage 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 charac-teristic equation was defined in Chapter 11 as

1 + G OL (s) = 0 (14-53) where the open-loop transfer function in Eq 14-53 is

G OL (s) = G c (s)G v (s)G p (s)G m (s).

14.6 Bode Stability Criterion 253

Before stating the Bode stability criterion, we

intro-duce two important definitions:

1 A critical frequency ω c 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 ω g is a value of ω for

which AR OL(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

trans-fer function G OL = Gc G v G p G m 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 ω c and a single gain crossover frequency ω g

Then the closed-loop system is stable if the open-loop

amplitude ratio AR OL(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 K c 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:

The corresponding magnitude and argument are

|G OL (jω)| = 1 and ∠G OL (jω) = −180∘ (14-55)

For a marginally stable system, ωc= ωg and the

fre-quency of the sustained oscillation, ωc, is caused by a pair

of roots on the imaginary axis at s = ±ω c j Substituting

this expression for s into Eq 14-55 gives the following

expressions for a conditionally stable system:

ϕOL(ωc) = ∠GOL(jωc) = −180∘ (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 G OL (s) that does not satisfy the required

con-ditions, the Nyquist stability criterion discussed in

Appendix J can be applied

10000 100 1 0.01 0 –90 –180 –270 –360

ω (radians/time)

AR OL

ϕOL (deg)

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 ωccan 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 K c can actually improve the stability of the closed-loop system

for certain ranges of K c 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 anal-ogy 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., ampli-tude) 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 pro-duce 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, y sp (t) = A sin (ω c t), for a long period of

time Assume that during this period, the measured

output, y m, is disconnected, so that the feedback loop

is broken before the comparator After the initial

transient dies out, y m 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