15

Sampled-Data Control Systems

15.1 INTRODUCTION [1–5]

The methods presented in the previous chapters deal with the fundamental

properties of continuous systems.However,digital computers are available

not only for the design of control systems but also to perform the control

function.Digital computers and microprocessors are used in many control

systems.The size and cost advantages associated with the microcomputer

make its use as a controller economical and practical.Using such a digital

processor with a plant that operates in the continuous-time domain requires

that its input signal be discrete,which requires the sampling of the signals used

by the controller.Such sampling may be an inherent characteristic of the

system.For example,a radar tracking systemsupplies information on an air-

plane’s position and motion to a digital processor at discrete periods of time.

This information is therefore available as a succession of data points.When

there is no inherent sampling,an analog-to-digital (A/D) converter must be

incorporated in a digital or sampled-data (S-D) control system.The sampling

process can be performed at a constant rate or at a variable rate,or it may be

random.The discussion in this chapter is based on a constant-rate sampling.

The output of thecontroller must thenbe convertedfromdiscreteformintoan

analog signal by a digital-to-analog (D/A) converter.A functional block

diagramof such a systemis shown in Fig.15.1,in which the signals e* and u*

Copyright © 2003 Marcel Dekker, Inc.

are in discrete form,u is piecewise continuous,and the remaining signals are

continuous.The notation e* and u* denotes that these signals are sampled at

specified time intervals and are therefore in discrete form.Systems that

include a digital computer are known as digital control systems.The synthesis

techniques for such systems are based on representing the entire system as an

equivalent sampled or discrete or an equivalent pseudo-continuous-time

(PCT) system.

There are various approaches that may be used in analyzing the stability

and time-response characteristics of S-Dsystems.These approaches may be

divided into two distinct categories:(1) direct (DIR) and (2) digitization

(DIG) or discrete digital control analysis techniques.In the first (DIR) cate-

gory the analysis is performed entirely in the discrete domain (z plane).In the

second (DIG) category the analysis and synthesis of the sampled-data system

is carried out entirely by transformation to the w

0

plane or,by use of the Pade¤

approximation (see Ref.1,App.C),entirely in the s plane.The use of the Pade¤

approximation,along with the information provided by a Fourier analysis of

the sampled signal,results in the modeling of a sampled-data control system

by a PCTcontrol system.The w

0

plane analysis is not covered in this text but

the reader is referredtoRef.1where anoverviewof the DIRandDIGmethods

is presented.

The analysis anddesignof sampled-data control systems,as discussedin

this chapter,is expedited by the use of CADpackages,such as MATLAB or

TOTAL-PC(see Sec.10.6 and Appendixes Cand D).

15.2 SAMPLING

Sampling may occur at one or more places in a system.The sampling opera-

tion is represented in a block diagramby the symbol for a switch.Figure 15.2

shows a systemwith sampling of the actuating signal.Note that the output c (t)

is a continuous function of time.A fictitious sampler that produces the

mathematical function c*(t) (see Sec.15.6) is also shown.The sampling

FIGURE 15.1 Control system incorporating a digital computer.

568 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

process can be considered a modulation process in which a pulse train p(t),

with magnitude 1/g and period T,multiplies a continuous-time function f (t)

and produces the sampled function f

p

ðtÞ.This is represented by

f

p

ðtÞ ¼ pðtÞf ðtÞ ð15:1Þ

These quantities are shown in Fig.15.3.AFourier-series expansion of p(t) is

pðtÞ ¼

1

g

X

þ1

n¼1

C

n

e

jno

s

t

ð15:2Þ

where the sampling frequency is o

s

¼ 2p=T and the Fourier coefficients C

n

are given by

C

n

¼

1

T

Z

T

0

pðtÞe

jo

s

t

dt ¼

1

T

sinðno

s

g=2Þ

no

s

g=2

e

jno

s

g=2

ð15:3Þ

The sampled function is therefore

f

p

ðtÞ ¼

X

þ1

n¼1

C

n

f ðtÞe

jno

s

t

ð15:4Þ

If the Fourier transformof the continuous function f (t) is F( jo),the Fourier

transformof the sampled function is [2,3,5]

F

p

ð joÞ ¼

X

þ1

n¼1

C

n

Fð joþjno

s

Þ ð15:5Þ

Acomparison of the Fourier spectra of the continuous and sampled functions

is shown inFig.15.4.It is seenthat the sampling process produces afundamen-

tal spectrum similar in shape to that of the continuous function.It also

FIGURE 15.2 Block diagram of a sampled-data control system involving the

sampling of the actuating signal.

Sampled-Data Control Systems 569

Copyright © 2003 Marcel Dekker, Inc.

FIGURE 15.3 (a) Continuous function f(t);(b) sampling train p(t);(c) sampled

function f

p

ðtÞ.

FIGURE 15.4 Frequency spectra for (a) a continuous function f(t) and (b) a

pulse-sampled function f

p

ðtÞ.

570 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

produces a succession of spurious complementary spectra that are shifted

periodically by a frequency separation no

s

.

If the sampling frequency is sufficiently high,there is very little overlap

between the fundamental and complementary frequency spectra.In that case

a low-pass filter could extract the spectrumof the continuous input signal by

attenuating the spurious higher-frequency spectra.The forward transfer

function of a control system generally has a low-pass characteristic,so the

systemresponds with more or less accuracy to the continuous signal.

15.3 IDEAL SAMPLING

To simplify the mathematical analysis of sampled-data control systems,the

concept of an ideal sampler (impulse sampling) is introduced.If the duration

of g of the sampling pulse is much less than the sampling time Tand much

smaller than the smallest time constant of e(t),then the pulse train p(t) can be

represented by an ideal sampler.Since the area of each pulse of p(t) has unit

value,the impulse train also has a magnitude of unity.The ideal sampler pro-

duces the impulse train

T

(t),which is shown in Fig.15.5 and is represented by

T

ðtÞ ¼

X

þ1

k¼1

ðt kT Þ ð15:6Þ

where ðt kT Þ is the unit impulse which occurs at t ¼T.The frequency

spectrumof the function that is sampled by the ideal impulse train is shown

in Fig.15.6.It is similar to that shown in Fig.15.4,except that the complemen-

tary spectra have the same amplitude as the fundamental spectrum [1].

Because the forward transfer function of a control system attenuates the

higher frequencies,the overall system response is essentially the same with

the idealized impulse sampling as with the actual pulse sampling.Note

in Figs.15.4 and 15.6 that sampling reduces the amplitude of the spectrum

by the factor 1/T.The use of impulse sampling simplifies the mathematical

analysis of sampled systems and is therefore used extensively to represent the

sampling process.

FIGURE 15.5 An impulse train.

Sampled-Data Control Systems 571

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When f (t) ¼0 for t <0,the impulse sequence is

f

ðtÞ ¼ f ðtÞ

T

ðtÞ ¼

X

þ1

k¼0

f ðtÞ

T

ðt kT Þ ð15:7Þ

and the Laplace transformis given by the infinite series

F

ðsÞ ¼

X

þ1

k¼0

f ðkT Þe

kTs

ð15:8Þ

where f (kT) represents the functionf (t) at the sampling times kT.Because the

expression F*(s) contains the terme

Ts

,it is not an algebraic expression but a

transcendental one.Therefore,a change of variable is made:

z e

Ts

ð15:9Þ

FIGURE 15.6 Frequency spectra for (a) a continuous function f(t);(b) an impulse-

sampled function f*(t),when o

s

>2o

c

[1];(c) an impulse-sampled function f*(t),when

o<2o

c

[1].

572 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

where s ¼(1/T) ln z.Equation (15.8) can nowbe written as

½F

ðsÞ

s¼ð1=T Þ lnz

¼ FðzÞ ¼

X

þ1

k¼0

f ðktÞz

k

ð15:10Þ

For functions f (t) which have zerovalue for t <0,the one-sidedZtransformis

the infinite series

FðzÞ ¼ f ð0Þ þf ðT Þz

1

þf ð2T Þz

2

þ ð15:11Þ

If the Laplace transform of f (t) is a rational function,it is possible to

write F *(s) in closed form.When the degree of the denominator of F(s) is at

least 2 higher than the degree of the numerator,the closed form can be

obtained from

F

ðsÞ ¼

X

at poles of Fð pÞ

residues of Fð pÞ

1

1 e

ðspÞT

ð15:12Þ

where F(p) is the Laplace transformof f (t) with s replaced by p.The Z trans-

formin closed formmay be obtained fromEq.(15.12) and is written as [4]

FðzÞ ¼

^

FFðzÞ þ ¼

X

at poles of Fð pÞ

residues of Fð pÞ

1

1 e

pT

z

1

þb

ð15:13Þ

where

b ¼ lim

s!1

sFðxÞ lim

z!1

^

FFðzÞ ð15:14Þ

The value of b given by Eq.(15.14) ensures that the initial value f (0)

represented by F(s) and F(z) are identical.F(z) is called the Z transform of

f *(t).The starred and z forms of the impulse response transfer function are

easily obtained for the ordinary Laplace transfer function.

Example 15.1.Consider the transfer function

GðsÞ ¼

K

sðs þaÞ

ð15:15Þ

The corresponding impulse transfer function in the s domain is

G

ðsÞ ¼

Ke

sT

ð1 e

at

Þ

að1 e

sT

Þð1 e

ðsþaÞT

Þ

ð15:16Þ

and in the z domain,

GðzÞ ¼

ð1 e

aT

ÞKz

1

að1 z

1

Þð1 e

aT

z

1

Þ

ð15:17Þ

Sampled-Data Control Systems 573

Copyright © 2003 Marcel Dekker, Inc.

The transformation of Eq.(15.16) into the z domain,as given by Eq.(15.17),

results in the primary strip ðjo

s

=2<jo<jo

s

=2Þ and the infinite number of

complementary strips ð...j5o

s

=2<jo<j3o

s

=2;j3o

s

=2<jo<

jo

s

=2,and jo

s

=2<jo<j3o

s

=2,j3o

s

=2<jo<j5o

s

=2;...Þ in the s plane.

The pole for k ¼ 0 is said to lie in the primary strip in the s plane and the

remaining poles,for k 6¼ 0,are said to lie in the complementary strips in the s

plane.Note that they are uniformly spaced with respect to their imaginary

parts with a separation jo

s

.The poles of G*(s) in Eq.(15.16) are infinite in

number.These poles exist at s ¼jko

s

and s ¼a þjko

s

for all values of

1<k<þ1.The complementary strips are transformed into the same

(overlapping) portions in the z plane.Thus,by contrast,there are just two

poles of Eq.(15.17),located at z ¼1 and z ¼e

aT

.The root-locus method can

therefore be applied easily in the z plane,whereas for sampled functions

it is not very convenient to use in the s plane because of the infinite number

of poles.

The transformation of the s plane into the z plane can be investigated by

inserting s ¼sþjo

d

into

z ¼ e

Ts

¼ e

T

e

jo

d

T

¼ e

T

e

j2po

d

=o

s

ð15:18Þ

where o

d

¼ o

n

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 z

2

p

.

1.Lines of constant sin the s plane map into circles of radius equal to

e

sT

in the z plane as illustrated in Fig.15.7 where s¼zo

n

.

In Fig.15.8 are shown loci of constant z and loci of constant o

n

.

Specifically,the segment of the imaginary axis in the s plane of

width o

s

maps into the circle of unit radius [unit circle (UC)] in

the z plane as shown in Fig.15.17b;successive segments map into

FIGURE 15.7 Transformation from the s plane to the z plane.

574 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

overlapping circles.This fact shows that a proper consideration

of sampled-data (S-D) systems in the z plane requires the use of a

multiple-sheeted surface,i.e.,a Riemann surface.But by virtue of

the uniform repetition of the roots of the characteristic equation,

the condition for stability is that all roots of the characteristic equa-

tioncontainedinthe principal branch lie withinthe unit circle inthe

principal sheet of the z plane.

2.Lines of constant o

d

in the s plane map into radial rays drawn at the

angle oTin the z plane.The portion of the constant o

d

line in the left

half of the s plane becomes the radial ray withinthe unit circle in the

z plane.The negative part of real axis 1<<0 in the s plane is

mapped on the segment of the real axis defined by 0 <z 1.

3.The constant-damping-ratio ray in the s plane is defined by the

equation

s ¼ þjo

d

¼ o

d

cot þjo

d

when ¼ cos

1

z.Therefore,

z ¼ e

sT

¼ e

o

d

T cot þjo

d

T

¼ e

o

d

T cot

ﬀo

d

T ð15:19Þ

The corresponding map describes a logarithmic spiral in the z plane.

FIGURE 15.8 Plots of loci of constant z and loci of constant o

n

.(From Ref.1,with

permission of the McGraw-Hill Companies.)

Sampled-Data Control Systems 575

Copyright © 2003 Marcel Dekker, Inc.

In summary,the strips in the left-half s plane (s<0) map into the region

inside the UCinthe z plane,and the strips inthe right-half s plane (s>0) map

into the region outside the UCin the z plane.

15.4 Z-TRANSFORM THEOREMS

Ashort list of Z transforms is given inTable 15.1.Several simple properties of

the one-sided Z transform permit the extension of the table and facilitate

its use in solving difference equations.These properties are presented as

theorems.They apply when the Z transform of f

(t),denoted by Z[f

(t)],is

F(z) and the sampling time isT.

Theorem 1.Translation in Time (Time Shift).Shifting to the right (delay)

yields

Z½ f

ðt pT Þ ¼ z

p

FðzÞ ð15:20Þ

Shifting to the left (advance) yields

Z½ f

ðt þpT Þ ¼ z

p

FðzÞ

X

p1

i¼0

f ðiT Þz

pi

ð15:21Þ

Theorem 2.Final Value.If F(z) converges for jzj >1 and all poles of

(1 z)F(z) are inside the unit circle,then

lim

k!1

f ðkT Þ ¼ lim

z!1

½ð1 z

1

ÞFðzÞ ð15:22Þ

Theorem 3.Initial Value.If lim

z!1

FðzÞ exists,then

lim

k!0

f ðkT Þ ¼ lim

z!1

FðzÞ ð15:23Þ

15.5 DIFFERENTIATION PROCESS [1]

An approximation to the continuous derivative cðtÞ ¼ _rrðtÞ uses the first-

backward difference or Euler’s technique:

cðkTÞ ¼

frðkTÞ r½ðk 1ÞTg

T

ð15:24Þ

Using this numerical analysis approach to approximate the derivative results

inananticipatorysolutionrequiring future inputs tobeknown inorder togen-

erate the current solution (noncausal situation).This future knowledge is

usually difficult to generate in a real-time control system.These numerical

576 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

analysis concepts are very important in realizing a digital control law or

equation that will prove to be a linear difference equation.

15.5.1 First Derivative Approximation

The backward-difference discretization of the differentiation process is

defined by

_ccðtÞ DcðtÞ

dcðtÞ

dt

lim

T!0

cðkT Þ c½ðk 1ÞT

T

ð15:25Þ

T

ABLE

15.1

Table of z Transforms

Continuous time

function

Laplace

transform

z transform

of f *(t) F(z)

Discrete time function

f(kt) for k 0

1.

0

(t) 1 1

0

(0)

2. (t kT)

k is any integer

e

kTs

z

1

(t kT)

3.u

1

(t)

1

s

z

z 1

(t kT)

4.tu

1

(t)

1

s

2

Tz

ðz 1Þ

2

kT (t kT)

5:

t

2

2

u

1

ðtÞ

1

s

3

T

2

zðz þ1Þ

2ðz 1Þ

3

kT

2

(t kT)

6.e

at

1

s þa

z

z e

aT

e

akT

7:

e

bt

e

at

a b

1

ðs þaÞðs þbÞ

1

a b

z

z e

bT

z

z e

aT

1

a b

ðe

bkT

e

akT

Þ

8.u

1

(t) e

at

a

sðs þaÞ

a

ð1 e

aT

Þz

ðz 1Þðz e

aT

Þ

(t kT)e

akT

9:t

1 e

aT

a

a

s

2

ðs þaÞ

Tz

ðz 1Þ

2

ð1 e

aT

Þz

aðz 1Þðz e

aT

Þ

kT

1 e

akT

a

10.sin at

a

s

2

þa

2

z sinaT

z

2

2 z cos aT þ1

sin akT

11.cos at

s

s

2

þa

2

zðz cos aTÞ

z

2

2 z cos aT þ1

cos akT

12.e

at

sin bt

b

ðs þaÞ

2

þb

2

ze

aT

sinbT

z

2

2 ze

aT

cos bT þe

2aT

e

akT

sin bkT

13.e

at

cos bt

s þa

ðs þaÞ

2

þb

2

z

2

ze

aT

cos bT

z

2

2 ze

aT

cos bT þe

2aT

e

akT

cos bkT

14.te

at

1

ðs þaÞ

2

Tze

aT

z e

aT

2

kT e

akT

Sampled-Data Control Systems 577

Copyright © 2003 Marcel Dekker, Inc.

where c (t) has the appropriate continuity characteristics and D is the deriva-

tive operator.This equation may be approximated by the first-backward

difference as follows:

DcðkT Þ ¼

dcðtÞ

dt

t¼kT

cðkT Þ c½ðk 1ÞT

T

¼

1

T

rcðkT Þ ð15:26Þ

15.5.2 Second Derivative Approximation

The second derivation backward-difference discretization is defined as

D DcðtÞ D

2

cðtÞ

d

2

cðtÞ

dt

2

lim

T!0

DcðkT Þ Dc½ðk 1ÞT

T

ð15:27Þ

which may be approximated by

D

2

cðkTÞ ¼

d

2

cðtÞ

dt

2

t¼kT

DcðkT Þ Dc½ðk 1ÞT

T

ð15:28Þ

Note that the term Dc(kT) in Eq.(15.28) is given by Eq.(15.26).The term

Dc[(k 1)T ] is expressed in the format of Eq.(15.26) by replacing k by (k 1)

to yield

1

T

rc½ðk 1ÞT ¼

c½ðk 1ÞT c½ðk 2ÞT

T

ð15:29Þ

Substituting Eqs.(15.26) and (15.29) into Eq.(15.28) yields the second-

backward difference

D

2

cðkT Þ

1

T

2

frcðkT Þ rc½ðk 1ÞTg

1

T

2

fcðkT Þ 2c½ðk 1ÞTg

1

T

2

r

2

cðkT Þ ð15:30Þ

15.5.3 r th Derivative Approximation

The approximation of the r th-backward difference of the r th derivative is

D

r

cðkT Þ

1

T

r

fr

r1

cðkT Þ r

r1

c½ðk 1ÞT þc½ðk 2ÞTg

¼

1

T

r

r

r

cðkT Þ ð15:31Þ

where r ¼1,2,3,...and k ¼0,1,2,....For a given value of r,Eq.(15.31) can be

expanded in a similar manner as is done for the second derivative backward

difference approximation to obtain the expanded representation of D

r

c (kT).

578 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

15.6 SYNTHESIS IN THE z DOMAIN (DIRECT METHOD)

For the block diagramof Fig.15.2,the systemequations are

CðsÞ ¼ GðsÞE

ðsÞ ð15:32Þ

EðsÞ ¼ RðsÞ BðsÞ ¼ RðsÞ GðsÞHðsÞE

ðsÞ ð15:33Þ

The starred transformof Eq.(15.33) is

E

ðsÞ ¼ R

ðsÞ GH

ðsÞE

ðsÞ ð15:34Þ

where GH

(s) [G(s)H(s)].Note that,in general,GH

(s) 6¼G

(s)H

(s).

Solving for E

(s) fromEq.(15.34) and substituting into Eq.(15.32) yields

CðsÞ ¼

GðsÞR

ðsÞ

1 þGH

ðsÞ

ð15:35Þ

The starred transformC

(s) obtained fromEq.(15.35) is

C

ðsÞ ¼

G

ðsÞR

ðsÞ

1 þGH

ðsÞ

ð15:36Þ

The Z transform of this equation is obtained by replacing each starred

transformby the corresponding function of z:

CðzÞ ¼

GðzÞRðzÞ

1 þGHðzÞ

ð15:37Þ

Although the output c (t) is continuous,the inverse of C(z) in Eq.(15.37) yields

only the set of values c (kT),k ¼0,1,2,...,corresponding to the values at

the sampling instants.Thus,c (kT) is the set of impulses from an ideal

fictitious sampler located at the output,as shown in Fig.15.2,which operates

in synchronismwith the sampler of the actuating signal.

Example 15.2.Derive the expressions (1) C

(s),(2) C(z),and (3) the control

ratio C(z)/R(z) if possible for the S-Dcontrol systemof Fig.15.9.

Part (1).To derive C

(s),complete the following two steps.

Step 1:The expressions that relate all the external and internal variables

shown in Fig.15.9 are:

CðsÞ ¼ G

2

ðsÞE

1

ðsÞ ðaÞ

E

1

ðsÞ ¼ G

1

ðsÞEðsÞ ðbÞ

EðsÞ ¼ RðsÞ BðsÞ ðcÞ

BðsÞ ¼ H

1

ðsÞI

ðsÞ ðdÞ

IðsÞ ¼ H

2

ðsÞCðsÞ ðeÞ

Sampled-Data Control Systems 579

Copyright © 2003 Marcel Dekker, Inc.

Step 2:The objectives are (i) to manipulate these equations in such

a manner that all the internal variables are eliminated;and (ii) to

obtain a relationship between the input and output systemvariables

only.Substituting equation (a) into equation (e) and equation (c) into

equation (b) yields,respectively:

IðsÞ ¼ G

2

ðsÞH

2

ðsÞE

1

ðsÞ ðf Þ

E

1

ðsÞ ¼ G

1

ðsÞRðsÞ G

1

ðsÞBðsÞ ðgÞ

Equation (d) is substituted into equation (g) to yield:

E

1

ðsÞ ¼ G

1

ðsÞRðsÞ G

1

ðsÞH

1

ðsÞI

ðsÞ ðhÞ

The starred transformof equations (a),( f ),and (h) are,respectively:

C

ðsÞ ¼ G

2

ðsÞE

1

ðsÞ ðiÞ

I

ðsÞ ¼ G

2

H

2

ðsÞE

1

ðsÞ ð jÞ

E

1

ðsÞ ¼ G

1

R

ðsÞ G

1

H

1

ðsÞI

ðsÞ ðkÞ

Notethat the input forcing functionr (t) is nowincorporatedas part of

the starred transformG

1

R

(s).

Substituting I

(s) fromequation( j) intoequation(k) andthensolving

for E

1

ðsÞ yields:

E

1

ðsÞ ¼

G

1

R

ðsÞ

1 þG

1

H

1

ðsÞG

2

H

2

ðsÞ

ðlÞ

This equation is substituted into equation (i) to yield:

C

ðsÞ ¼

G

2

ðsÞG

1

R

ðsÞ

1 þG

1

H

1

ðsÞG

2

H

2

ðsÞ

ð15:38Þ

FIGURE 15.9 A nonunity feedback sampled-data control system.(From Ref.1,

with permission of the McGraw-Hill Companies.)

580 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

Part (2).The Z transformof Eq.(15.38) is obtained by replacing each

starred transformby the corresponding function of z.

CðzÞ ¼

G

2

ðzÞG

1

RðzÞ

1 þG

1

H

1

ðzÞG

2

H

2

ðzÞ

ð15:39Þ

Part (3).Note that the input R(z) is incorporated as part of G

1

R(z);thus

this equation cannot be manipulated to obtain the desired control

ratio.Apseudo control ratio can be obtained as follows:

CðzÞ

RðzÞ

p

¼

1

RðzÞ

½CðzÞ ¼

1

RðzÞ

G

2

ðzÞG

1

RðzÞ

1 þG

1

H

1

ðzÞG

2

H

2

ðzÞ

ð15:40Þ

15.6.1 z-Plane Stability

The control ratio for the systemof Fig.15.2,obtained fromEq.(15.37),is

CðzÞ

RðzÞ

¼

GðzÞ

1 þGHðzÞ

¼

WðzÞ

QðzÞ

ð15:41Þ

The characteristic equation of a S-Dcontrol systemis obtained by setting to

zero the denominator of the control system’s control ratio.For example,the

characteristic equation for the closed-loop system as represented by

Eq.(15.41) is given by

QðzÞ ¼ 1 þGHðzÞ ¼ 0 ð15:42Þ

Ingeneral,for a closed-loop S-Dcontrol system,the characteristic equa-

tion has the form:

QðzÞ 1 þPðzÞ ¼ 0 ð15:43Þ

As mentioned previously,the nature and location of the roots of the

characteristic equation (15.42) determine the stability and the dynamic

behavior of the closed-loop system in a manner similar to that for analog

systems.For a continuous-time system,the root locus for the closed-

loop system is based on the characteristic equation 1 þG(s)H(s) ¼0 or,

equivalently,G(s)H(s) ¼"

j(lþ2h)p

,where h ¼0,

1,

2,....Similarly,a root-

locus analysis can be made for a S-Dsystemwhose closed-loop characteristic

equation is given by Eq.(15.42).

15.6.2 System Stability [5]

The fundamental concepts of stability are presented in the previous chapters.

For the discrete model c (kT þ1) ¼f [c (kT),kT],the system is stable if all

solutions remain within a small distance of eachother after a specific discrete

value of time,k

n

T.Asymptotic stability refers to the condition that all the

Sampled-Data Control Systems 581

Copyright © 2003 Marcel Dekker, Inc.

solutions to the discrete model are stable and the distance between themgoes

to zero as k!1.Asymptotic stability then implies stability.For a discrete

time-invariant linear system model where F(x,kT)!F(z),asymptotic

stability requires that all characteristic roots of the characteristic equation

[or poles for F(z)] are within the unit circle (UC).Another stability relation-

ship defined as bounded-input bounded output (BIBO) refers to the application

of a bounded input generating a bounded output in a discrete time-invariant

linear model.Asymptotic stability for this model implied BIBOstability.

To determine if a given discrete time-invariant linear model is stable

(asymptotically stable),a number of techniques [1] are available:genera-

tion of characteristic values,Nyquist’s method,Jury’s stability test,root-

locus,Bode diagram,and Lyapunov’s second method.The characteristic

values for a given z domain transfer function can be determined by using a

computer-aided-design (CAD) package with the appropriate accuracy.

Systemstability is determined by the location of the roots of the system

characteristic equationinthe z domain.InSec.15.3it is statedthat the primary

and complementary strips are transformed into the same (overlapping)

portions of the z plane.That is,the strips in the left-half s plane (s<0) map

into the region inside the UC in the z plane,and the strips in the right-half s

plane (s>0) map into the region outside the UC in the z plane.Figure 15.10

illustrates the mapping of the primary strip into the inside of the unit circle in

the z plane.Therefore,a sample-data control systemis stable if the roots of the

z domain characteristic equation lie inside the UC.In Chapter 16 it is shown

that by transforming a sample-data control systemto its corresponding PCT

analog control system its stability can be determined in the same manner as

for analog systems.

FIGURE 15.10 The mapping of the s plane into the z plane by means of z ¼e

Ts

.

(From Ref.1,with permission of the McGraw-Hill Companies.)

582 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

15.6.3 System Analysis

Analysis of the performance of the sampled system,corresponding to the

response given by Eq.(15.37),can be performed by the frequency-response or

root-locus method.It should be kept in mind that the geometric boundary for

stability in the z plane is the unit circle.As an example of the root-locus

method,consider the sampled feedback systemrepresented by Fig.15.2 with

H(s) ¼1.Using G(s) given by Eq.(15.15) with a ¼1 and a sampling timeT¼1,

the equation for G(z) is

GðzÞ ¼ GHðzÞ ¼

0:632Kz

ðz 1Þðz 0:368Þ

ð15:44Þ

The characteristic equation,as obtained fromEq.(15.42),is 1 þGH(z) =0 or

GHðzÞ ¼ 1 ð15:45Þ

The usual root-locus techniques can be used to obtain a plot of the roots of

Eq.(15.45) as a function of the sensitivity K.The root locus is drawn in

Fig.15.11.The maximumvalue of K for stability is obtained fromthe magni-

tude conditionwhichyields K

max

¼2.73.This occurs at the crossingof the root

locus and the unit circle.The selection of the desired roots canbe based on the

damping ratio z desired.For the specified value of z,the spiral given by

Eq.(15.19) and shown in Fig.15.8,must be drawn.The intersection of this

curve with the root locus determines the roots.Alternatively,it is possible to

specify the setting time.This determines the value of s in the s plane,so the

circle of radius e

sT

can be drawn in the z plane.The intersection of this circle

and the root locus determines the roots.For the value z ¼0.48 the roots are

FIGURE 15.11 Root locus for Eqs.(15.44) and (15.45).

Sampled-Data Control Systems 583

Copyright © 2003 Marcel Dekker, Inc.

z ¼0.368 þj0.482,shown on the root locus in Fig.15.11,and the value of K¼1.

The control ratio is therefore

CðzÞ

RðzÞ

¼

0:632z

z 0:368

j0:482

¼

0:632z

z

2

0:736z þ0:368

ð15:46Þ

For a unit-step input the value of R(z) is

RðzÞ ¼

z

z 1

ð15:47Þ

so that

CðzÞ ¼

0:632z

2

ðz 1Þðz

2

0:736z þ0:368Þ

ð15:48Þ

The expression for C(z) can be expanded by dividing its denominator into its

numerator to get a power series in z

1

.

CðzÞ ¼ 0:632z

1

þ1:096z

2

þ1:205z

3

þ1:120z

4

þ1:014z

5

þ0:98z

6

þ ð15:49Þ

The inverse transformof C(z) is

cðkT Þ ¼ 0 ðtÞ þ0:632 ðt T Þ þ1:096 ðt 2T Þ þ ð15:50Þ

Hence,comparing Eq.(15.49) with Eq.(15.11),the values of c (kT) at the

sampling instants are the coefficients of the terms in the series of Eq.(15.50)

at the corresponding sampling instants.Aplot of the values c (kT) is shown in

Fig.15.12.The curve of c (t) is drawn as a smooth curve through these plotted

points.

FIGURE 15.12 Plot of c(nT) for Eq.(15.50).

584 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

This section presents several basic characteristics in the analysis of

sampled-data (S-D) systems.Additional topics of importance include the

reconstruction of the continuous signal from the sampled signal by use of

hold circuits and the problem of compensation to improve performance.

These topics are presented in the following sections.

15.7 THE INVERSE Z TRANSFORM

In the example of Sec.15.6 the inverse of C(z) is obtained by expanding this

function into an infinite series in terms of z

k

.This is referred to as the

power-series method.Then the coefficients of z

k

are the values of c (kT).The

division required to obtain C(z) as an infinite series (open form) is easily

performed on a digital computer.However,if an analytical expression in

closed form for c

(t) is desired,C(z) can be expanded into partial fractions

which appear inTable15.1.Since the Z transforms inTable15.1contain a zero

at the origin in the numerator,the Heaviside partial fractionexpansion is first

performed on C(z)/z;i.e.,functions of C(z) that contain a zero at the origin

must first be put into proper form.

Example 15.3.For C(z) given by Eq.(15.48) withT¼1,

CðzÞ

z

¼

A

z 1

þ

Bz þC

z

2

0:736z þ0:368

ð15:51Þ

The coefficients are evaluated by the usual Heaviside partial-fraction

method and yield A¼1,B¼1,and C¼0.368.Inserting these values into

Eq.(15.51) and using the form of entries 3 and 13 of Table 15.1,the response

transformC(z) is

CðzÞ ¼

z

z 1

zðz e

a

cos bÞ

z

2

2ze

a

cos b þe

2a

ð15:52Þ

wherea ¼0.5andb ¼1.Thus,fromTable15.1thesystemoutput of Eq.(15.52) in

closed formis

cðkT Þ ¼ 1 e

0:5k

cos k ð15:53Þ

The open form introduces an error that is propagated in the division due to

round off.This does not occur with the closed form.Remember that C(z)

represents the sampled values c

(t) of c (t) at the sample instants kT.Thus,the

inversion from the z domain to the time domain yields c

(t) or the values

c(kT).Thus,the inversion process yields values of c (t) only at the sampling

instants.In other words,C(z) does not contain any information about the

values in between the sampling instants.This is an inherent limitation of the

Z transformmethod.

Sampled-Data Control Systems 585

Copyright © 2003 Marcel Dekker, Inc.

Inorder toobtainthe inverse Ztransformof functions of C(z) that donot

contain a zero at the origin,the reader is referred to Ref.1.

15.8 ZERO-ORDER HOLD

The function of a zero-order hold (ZOH) is to reconstruct a piecewise-

continuous signal from the sampled function f

(t).It holds the output

amplitude constant at the value of the impulse for the durationof the sampling

period T.Thus,it has the transfer function

G

zo

ðsÞ ¼

1 e

Ts

s

ð15:54Þ

The action of the ZOH is shown in Fig.15.13.When the sampling time T is

small or the signal is slowly varying,the output of the ZOHis frequently used

in digital control systems.It is used to convert the discrete signal obtained

from a digital computer into a piecewise-continuous signal that is the input

to the plant.The ZOHis a low-pass filter that,together with the basic plant,

FIGURE 15.13 Input and output signals for a zero-order hold:(a) continuous input

signal e(t) and the sampled signal e*(t):(b) continuous signal e(t) and the piecewise-

constant output m(t) of the zero-order hold.

586 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

attenuates the complementary frequency spectra introduced by the sampling

process.The ZOHdoes introduce a lag angle into the system,and therefore

it can affect the stability and time-response characteristic of the system.

Example 15.4.Figure 15.14 shows a unity-feedback system in which a

zero-order hold is placed after the sampler and before the plant G

x

(s),which

is given by Eq.(15.14) where a ¼1.The Z transform of the forward transfer

function G

z

(z) is

G

z

ðzÞ ¼

CðzÞ

EðzÞ

¼Z½GðsÞ ¼Z½G

zo

ðsÞG

x

ðsÞ ¼Z

1e

Ts

s

K

sðs þ1Þ

ð15:55Þ

Because Z(1 e

Ts

) ¼1 z

1

,this term can be factored from Eq.(15.55) to

give,forT¼1,

G

z

ðzÞ ¼ð1z

1

ÞZ

K

s

2

ðs þ1Þ

¼

Kðze

1

þ12e

1

Þ

ðz 1Þðz e

1

Þ

¼

0:368Kðz þ0:717Þ

ðz 1Þðz 0:368Þ

ð15:56Þ

Acomparison with the Z transformwithout the ZOHas given by Eq.(15.44)

shows that the zero is moved fromthe origin to z ¼0.717.

Theroot locusfor thesystemof Fig.15.14isshowninFig.15.15andmaybe

compared with the root locus without the ZOH in Fig.15.11.The maximum

value of the gain K for stability is K

max

¼0.8824.With z ¼0.707,the

closed-loop control ratio is

CðzÞ

RðzÞ

¼

0:118ðz þ0:717Þ

ðz 0:625 þj0:249Þðz 0:625 j0:249Þ

¼

0:118ðz þ0:17Þ

z

2

1:25z þ0:453

ð15:57Þ

FIGURE 15.14 The uncompensated sampled-data (S-D) control system.

Sampled-Data Control Systems 587

Copyright © 2003 Marcel Dekker, Inc.

For this control ratio,e

sT

¼0.673 so that the corresponding real part of the

closed-loop poles in the s domain is s¼0.396 and the approximate setting

time isT

s

¼4/0.396¼10.1s.

This example illustrates that a second-order sampled system becomes

unstable for large values of gain.This is in contrast to a continuous second-

order system,which is stable for all positive values of gain.The advantage of

a discrete control systemis the greater flexibility of compensation that can be

achieved with a digital compensator.

15.9 LIMITATIONS

Likeany methodof analysis,there arelimitations tothe use of the Z-transform

method.In spite of these limitations,however,this method is a very useful and

convenient analytical tool because of its simplicity.Nevertheless,the follow-

ing limitations must be kept in mind when applying and interpreting the

results of the Z-transformmethod:

1.The use of an ideal sampler in the analysis of a discrete-data system

is baseduponthe model inwhichthe strengths (area) of the impulses

inthe impulse trainc

(t) are equal tothe corresponding values of the

input signal c (t) at the sampling instants kT.For this mathematical

model to be ‘‘close’’ to correct,it is necessary for the sampling

duration (or pulse width) g to be very small in comparison with the

smallest time constant of the system.It must also be very much

smaller than the sampling timeT.

FIGURE 15.15 Root locus for the system of Fig.15.14.

588 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

2.While the inverse ZtransformZ

1

[C(z)] yields only c (kT) and not a

unique analytical function c (t),the modified Z-transform method

Z

m

,[5] removes this limitation.

3.The impulse responses of rational C(s) functions (exclusive of a hold

device) experience a jumpbehavior at t ¼0 [at t ¼kTfor c

(t)] unless

these functions have at least two more poles than zeros.For these

functions,

lim

t!kT

þ

cðtÞ ¼ L

þ

6¼ lim

t!kT

cðtÞ ¼ L

i.e.,the value of c (kT),as t approaches the value of kTfromthe low

side,is not identical to the value obtained when approaching kT

from the high side.In other words,a discontinuity occurs in the

value of c (kT) at t ¼kT.Fortunately,the C(z) encountered in the

analysis of most practical control systems have at least two more

poles than zeros (n wþ2),and this limitation does not apply for

such systems.

15.10 STEADY-STATE ERROR ANALYSIS FOR

STABLE SYSTEMS

The three important characteristics of a control system are (1) stability,

(2) steady-state performance,and (3) transient response.The first itemof interest

in the analysis of a systemis its stability characteristics.If there is a range of

gain for which a system yields a stable performance,then the next item of

interest is the system’s steady-state error characteristics;i.e.,can the system

output c (kT) follow a given input r (kT) with zero or a small value of error

[e(kT) ¼r (kT) c (kT)].If the first two items are satisfactory,a transient

time-response analysis is then made.Section 15.6.2 and 15.6.3 deal with the

first item.This section discusses the second characteristic,and the remaining

sections deal,mainly,with the transient response.

Fortunately,the analysis of the steady-state error characteristics of

a unity-feedback S-D system parallels the analysis for a unity-feedback

continuous-time stable system based upon system types and upon

specific forms of the system input function.For the unity-feedback S-D

system of Fig.15.14,the control ratio and the output and error signals

expressed in the z domain are,respectively,

CðzÞ

RðzÞ

¼

G

z

ðzÞ

1 þG

z

ðzÞ

ð15:58Þ

CðzÞ ¼

G

z

ðzÞ

1 þG

z

ðzÞ

RðzÞ ð15:59Þ

Sampled-Data Control Systems 589

Copyright © 2003 Marcel Dekker, Inc.

and

EðzÞ ¼ RðzÞ CðzÞ ¼

RðzÞ

1 þG

z

ðzÞ

ð15:60Þ

Thus,if the system is stable,the final-value theorem can be applied to

Eqs.(15.59) and (15.60) to obtain the steady-state or final value of the output

and the error at the sampling instants:i.e.,

c

ð1Þ ¼ lim

t!1

c

ðtÞ ¼ lim

z!1

ð1 z

1

ÞG

z

ðzÞ

1 þG

z

ðzÞ

RðzÞ

"#

ð15:61Þ

e

ð1Þ ¼ lim

t!1

e

ðtÞ ¼ lim

z!1

ð1 z

1

ÞRðzÞ

1 þG

z

ðzÞ

"#

ð15:62Þ

The steady-state error analysis for the stable nonunity feedback system

of Fig.15.16a may be analyzed by determining its equivalent z domain-stable

unity-feedback system shown in Fig.15.16b.The control ratios of the

respective configurations of Fig.15.16 are,respectively,

CðzÞ

RðzÞ

¼

G

z

ðzÞ

1 þG

ZO

G

x

HðzÞ

¼

NðzÞ

DðzÞ

ð15:63Þ

CðzÞ

RðzÞ

¼

G

eq

ðzÞ

1 þG

eq

ðzÞ

ð15:64Þ

where N(z) and D(z) are the numerator and denominator polynomials,

respectively,of Eq.(15.63).G

eq

(z) is determined by equating Eqs.(15.63) and

(15.64) and manipulating to obtain

G

eq

ðzÞ ¼

NðzÞ

DðzÞ NðzÞ

¼

NðzÞ

D

G

ðzÞ

¼

NðzÞ

ðz 1Þ

m

D

0

G

ðzÞ

ð15:65Þ

FIGURE 15.16 (a) A stable nonunity-feedback sampled-data system;(b) the

equivalent z domain stable unity feedback system of Fig.15.16a.(From Ref.1,with

permission of the McGraw-Hill Companies.)

590 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

where D

0

G

ðzÞ is the resultant polynomial remaining after the term (z 1)

m

is

factored out of D

G

(z) ¼D(z) N(z).A similar procedure can be utilized

to determine G

eq

(z) for other stable nonunity-feedback S-D systems.Thus,

for Fig.15.16b,the steady or final value of the output and the error at the

sampling instants,for this stable system,are given respectively as follows:

c

ð1Þ ¼ lim

t!1

c

ðtÞ ¼ lim

z!1

ð1 z

1

ÞG

eq

ðzÞ

1 þG

eq

ðzÞ

RðzÞ

"#

ð15:66Þ

e

ðtÞ ¼ lim

t!1

e

ðtÞ ¼ lim

z!1

ð1 z

1

ÞRðzÞ

1 þG

eq

ðzÞ

"#

ð15:67Þ

15.10.1 Steady-State Error-Coefficients

The steady-state error coefficients have the same meaning and importance for

sampled-data systems as for continuous timesystems;i.e.,howwell the system

output can followa given type of input forcing function.The following deriva-

tions of the error coefficients are independent of the systemtype.They apply to any

systemtype and are defined for specific forms of the input.These error coefficients

are useful only for a stable systemand are defined in this text for a unity-feedback

system.

Step Input.R(z) ¼R

o

z/(z 1).The step error coefficient K

p

is defined as

K

p

c

ð1Þ

e

ð1Þ

ð15:68Þ

Substituting fromEqs.(15.61) and (15.62) into Eq.(15.68) yields

K

p

¼ lim

z!1

GðzÞ ð15:69Þ

which applies only for a step input,r(t) ¼R

0

u

1

(t).

Ramp Input.R(z) ¼R

1

zl(z 1)

2

.The ramp error coefficient K

v

is defined as

K

v

steady-state value of derivative of output

e

ð1Þ

ð15:70Þ

Since Eq.(15.26) represents the derivative of c (t) in the discrete-time domain,

use of the translation and final-value theorems [Eqs.(15.20) and (15.22),

respectively] permits Eq.(15.70) to be written as

K

v

¼

lim

z!1

ð1z

1

Þ

2

T

CðzÞ

h i

e

ð1Þ

ð15:71Þ

Sampled-Data Control Systems 591

Copyright © 2003 Marcel Dekker, Inc.

Substituting fromEqs.(15.61) and (15.62) into Eq.(15.71) yields

K

v

¼

1

T

lim

z!1

z 1

z

G

z

ðzÞ

s

1

ð15:72Þ

which applies only for a ramp input,r (t) ¼R

1

tu

1

(t).

Parabolic Input.RðzÞ ¼ R

2

T

2

zðz þ1Þ=2ðz 1Þ

3

.The parabolic error

coefficient K

a

is defined as

K

a

steady-state value of second derivative of output

e

ð1Þ

ð15:73Þ

Since Eq.(15.30) represents the secondderivative of the output inthe discrete-

time domain,use of the translation and final-value theorems permits

Eq.(15.73) to be written as

K

a

¼

lim

z!1

ð1z

1

Þ

3

T

2

h i

e

ð1Þ

ð15:74Þ

Substituting fromEqs.(15.59) and (15.67) into Eq.(15.74) yields

K

a

¼

1

T

2

lim

z!1

ðz 1Þ

2

z

2

G

z

ðzÞ

"#

s

2

ð15:75Þ

which applies only for a parabolic input,r(t) ¼R

2

t

2

u

1

(t)/2.

15.10.2 Evaluation of Steady-State Error Coefficients

The forward transfer function of a sampled-data unity-feedback systemin the

z domain has the general form

GðzÞ ¼

Kz

d

ðz a

1

Þðz a

2

Þ ðz a

i

Þ

ðz 1Þ

m

ðz b

1

Þðz b

2

Þ ðz b

j

Þ

ð15:76Þ

where a

i

and b

j

may be real or complex,d and mare positive integers,m¼0,1,

2,...,and mrepresents the systemtype.Note that the (z 1)

m

termin Eq.(15.76)

corresponds tothe s

m

terminthe denominator of theforwardtransfer function

of a continuous-time Type m system.Substituting from Eq.(15.76) into

Eqs.(15.69),(15.72),and (15.75),respectively,yields the following values of the

592 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

steady-state error coefficients for the variousType msystems:

K

p

¼

lim

z!1

Kz

d

ðz a

1

Þðz a

2

Þ

ðz b

1

Þðz b

2

Þ

¼ K

0

Type 0 ð15:77Þ

1 Type 1 ð15:78Þ

1 Type 2 ð15:79Þ

8

>

>

>

>

<

>

>

>

>

:

K

v

¼

1

T

lim

z!1

Kz

d

ðz 1Þðz a

1

Þðz a

2

Þ

zðz b

1

Þðz b

2

Þ

¼ 0 Type 0 ð15:80Þ

1

T

lim

z!1

Kz

d

ðz a

1

Þðz a

2

Þ

zðz b

1

Þðz b

2

Þ

¼ K

1

Type 1 ð15:81Þ

1 Type 2 ð15:82Þ

8

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

:

K

a

¼

1

T

2

lim

z!1

Kz

d

ðz 1Þ

2

ðz a

1

Þðz a

2

Þ

z

2

ðz b

1

Þðz b

2

Þ

¼ 0 Type 0 ð15:83Þ

1

T

2

lim

z!1

Kz

d

ðz 1Þðz a

1

Þðz a

2

Þ

z

2

ðz b

1

Þðz b

2

Þ

¼ 0 Type 1 ð15:84Þ

1

T

2

lim

z!1

Kz

d

ðz a

1

Þðz a

2

Þ

z

2

ðz b

1

Þðz b

2

Þ

¼ K

2

Type 2 ð15:85Þ

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

In applying Eqs.(15.77) to (15.85) it is required that the denominator of G(z)

be in factored form to ascertain if it contains any z 1 factor(s).Table 15.2

summarizes the results of Eqs.(15.77) to (15.85).

15.10.3 Use of Steady-State Error Coefficients

The importance of the steady-state error coefficients is illustratedby means of

an example.

T

ABLE

15.2

Steady-State Error Coeﬃcients for Stable Systems

System

type

Step error

Coeﬃcient K

p

Ramp error

Coeﬃcient K

v

Parabolic error

Coeﬃcient K

a

0 K

0

0 0

1 1 K

1

0

2 1 1 K

2

Sampled-Data Control Systems 593

Copyright © 2003 Marcel Dekker, Inc.

Example 15.5.For the system of Fig.15.14,consider G

z

(z) of the form of

Eq.(15.76) with m¼1 (a Type 1 system).Determine the value of e

ð1Þ for

each of the three standard inputs (step,ramp,and parabolic),assuming that

the systemis stable.

For the definitions of Eqs.(15.68),(15.70),and(15.73) andfromTable15.2,

the following results are obtained:

e

ð1Þ ¼

c

ð1Þ

K

p

¼

c

ð1Þ

1

¼ 0 ð15:86Þ

e

ð1Þ ¼

steady-state value of derivative of output

K

1

¼ E

0

ð15:87Þ

e

ð1Þ ¼

steady-state value of second derivative of output

0

¼ 1

ð15:88Þ

Thus,a Type 1 sampled-data stable system (1) can follow a step input with

zero steady-state error,(2) can follow a ramp input with a constant error E

0

,

and (3) cannot follow a parabolic input.Equation (15.87) indicates that the

value of E

0

,for a given value of R

1

(for the ramp input),may be made smaller

by making K

1

larger.This assumes that the desired degree of stability and the

desired transient performance are maintained while K

1

is increasing in value.

Asimilar analysis can be made forTypes 0 and 2 systems (see problems).

Example 15.6.For the system of Fig.15.17,consider that G

eq

(z),given by

Eq.(15.65),is of the form of Eq.(15.76) with m¼2 (an equivalent Type 2

system).Determine the steady-state error coefficients for this system.

FIGURE 15.17 A nonunity-feedback sampled-data control system.(From Ref.1,

with permission of the McGraw-Hill Companies.)

594 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

Applying the definitions of Eqs.(15.69),(15.71),and (15.75) yields,

respectively,

K

p

¼ 1 K

v

¼ 1 K

a

¼

1

T

2

NðzÞ

z

2

D

0

G

ðzÞ

¼ K

2

Example 15.7.The expression for the output of the control systemshown in

Fig.15.17 is

CðzÞ ¼

G

2

ðzÞG

1

ðzÞRðzÞ

1 þG

2

G

1

HðzÞ

ð15:89Þ

For a specified input r (t) (step,ramp,or parabolic),obtain R(z) ¼Z[r (t)],

corresponding to a sampled r (t),and then obtain the pseudo control ratio

[C(z)/R(z)]

p

;i.e.,divide both sides of Eq.(15.89) by R(z) to obtain

CðzÞ

RðzÞ

p

¼

1

RðzÞ

G

2

ðzÞG

1

ðzÞRðzÞ

1 þG

2

G

1

HðzÞ

ð15:90Þ

By setting Eq.(15.90) equal to Eq.(15.64),it is then possible to obtain an

expression for G

eq

(z) that represents the forward transfer function of an

equivalent unity-feedback system represented by Fig.15.16b,where

G

z

(z) G

eq

(z).This expression of G

eq

(z) can then be used to determine the

‘‘effective’’ Type m system that Fig.15.17 represents and to solve for K

p

,K

v

,

and K

a

.

As a specific example,let

CðzÞ

RðzÞ

p

¼

Kðz

2

þaz þbÞ

z

3

þcz

2

þdz þe

¼

NðzÞ

DðzÞ

ð15:91Þ

where r (t) ¼u

1

(t),K¼0.129066,a ¼0.56726,b ¼0.386904,c ¼1.6442,

d ¼1.02099,and e ¼0.224445.For these values of the coefficients the

systemis stable.Setting Eq.(15.91) equal to Eq.(15.65) and solving for G

eq

(z)

yields

G

eq

ðzÞ ¼

Kðz

2

þaz þbÞ

z

3

1:773266z

2

þ0:97776z 0:174509

¼

NðzÞ

D

G

ðzÞ

Applying the final-value theoremto Eq.(15.91) yields cð1Þ ¼ 1.Therefore,the

nonunity feedback system of Fig.15.17 effectively acts at least as a Type 1

system.Based upon Eq.(15.76) this implies that D

G

(z) contains at least one

factor of the form z 1.Dividing D

G

(z) by z 1 yields z

2

0.773266z þ

0.17451,which does not contain z 1as a factor.Thus,the nonunity-feedback

Sampled-Data Control Systems 595

Copyright © 2003 Marcel Dekker, Inc.

systemis aType1system.The equation for G

eq

(z) is rewritten as follows:

G

eq

ðzÞ ¼

Kðz

2

þaz þbÞ

ðz 1Þðz

2

0:773266z þ0:17451Þ

For the ramp input r (t) ¼R

1

tu

1

(t) with R

1

¼0.5 and forT¼0.1,then applying

Eqs.(15.72) and (15.87) yields,respectively,

K

1

¼

1

T

lim

z!1

z 1

z

G

eq

ðzÞ

¼

Kð1 þa þbÞ

Tð0:401244Þ

s

1

and

e

ð1Þ ¼

R

1

K

1

¼

0:5

3:796

¼ 0:1317

Thus,the systemfollows a ramp input with a steady-state error.

15.11 ROOT-LOCUS ANALYSIS FOR SAMPLED-DATA

CONTROL SYSTEMS

The first thing that a designer wants to know about a given S-D system is

whether or not it is stable.This can be determined by examining the

roots obtained from the characteristic equation 1 þG(z)H(z) ¼0.Thus,the

root-locus method is used to analyze the performance of a S-D control

system in the same manner as for a continuous-time control system.For

either type of system,the root locus is a plot of the roots of the characteristic

equation of the closed-loop system as a function of the gain constant.This

graphical approach yields a clear indication of gain-adjustment effects

with relatively small effort compared with other methods.The underlying

principle is that the poles of C(z)/R(z) or C(z) (transient-response modes)

are related to the zeros and poles of the open-loop transfer function

G(z)H(z) and also the gain.An important advantage of the root-locus

method is that the roots of the characteristic equation of the system can

be obtained directly,which results in a complete and accurate solution

of the transient and steady-state response of the controlled variable.

Another important feature is that an approximate control solution can be

obtained with a reduction of the required work.With the help of a CAD

package,it is possible to synthesize a compensator,if one is required,with

relative ease.

This section presents a detailed summary of the root-locus method.The

first subsection details a procedure for obtaining the root locus,the next

subsection defines the root-locus construction rules for negative feedback,

and the last subsection contains examples of this method.

596 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

15.11.1 Procedure Outline

The procedure to be followed in applying the root-locus method is outlined in

this subsection.This procedure is easier when a CAD program is used to

obtain the root locus.Such a program can provide the desired data for

the root locus in plotted or tabular form.This procedure,which is a modified

version of the one applicable for the continuous-time systems,is summarized

as follows

Step 1.Derive the open-loop transfer function G(z)H(z) of the system.

Step 2.Factorize the numerator and denominator of the transfer

function into linear factors of the formz þa,where a may be real or

complex.

Step 3.Plot the zeros and poles of the open-loop transfer function in the

z plane,where z ¼s

z

þjo

z

.

Step 4.The plotted zeros and poles of the open-loop function

determine the roots of the characteristic equation of the closed-loop

system [1 þG(z)H(z) ¼0].By use of geometrical shortcuts or a

digital-computer program,determine the locus that describes the

roots of the closed-loop characteristic equation.

Step 5.Calibratethelocus interms of theloopsensitivity K.If thegainof

the open-loop system is predetermined,the location of the exact

roots of 1 þG(z)H(z) ¼0 is immediately known.If the location of the

roots (or z) is specified,the required value of Kcan be determined.

Step 6.Once the roots have been found in step 5,the system’s time

response can be calculated by taking the inverse Z transform,either

manually or by use of a computer program.

Step 7.If the response does not meet the desired specifications,

determine the shape that the root locus must have to meet these

specifications.

Step 8.Synthesize the network that must be inserted into the system,

if other than gain adjustment is required,to make the required

modificationon the original locus.This process,called compensation,

is described in Chaps.10 to12.

15.11.2 Root-Locus Construction Rules for

Negative Feedback

The systemcharacteristic equation,Eq.(15.43),is rearranged as follows:

PðzÞ ¼ 1 ð15:92Þ

Sampled-Data Control Systems 597

Copyright © 2003 Marcel Dekker, Inc.

Assume that P(z) represents the open-loop function

PðzÞ ¼ GðzÞHðzÞ ¼

Kðz z

1

Þ ðz z

i

Þ ðz z

w

Þ

ðz p

1

Þ ðz p

c

Þ ðz p

n

Þ

ð15:93Þ

where z

i

andp

c

are the open-loopzeros and poles,respectively,n is the number

of poles,w is the number of zeros,and K is defined as the static loop sensitivity

(gain constant) when P(z) is expressed in this format.Equation (15.92) falls

into the mathematical format for root-locus analysis;i.e.,

Magnitude condition:jPðzÞj ¼ 1 ð15:94Þ

Angle condition:b ¼

ð1 þ2hÞ180

for K > 0

h360

for K<0

ð15:95Þ

Thus,the construction rules for continuous-time systems,with minor modifi-

cations since the plot is in the z plane,are applicable for S-Dsystems and are

summarized as follows:

Rule 1.The number of branches of the root locus is equal tothe number

of poles of the open-loop transfer function.

Rule 2.For positive values of K,the root exist on those portions of the

real axis for which the sum of the poles and zeros to the right is an

odd integer.For negative values of K,the root locus exists on those

portions of the real axis for which the sumof the poles and zeros to

the right is an even integer (including zero).

Rule 3.The root locus starts (K¼0) at the open-loop poles and

terminates ðK ¼

1Þ at the open-loop zeros or at infinity.

Rule 4.The angles of the asymptotes of the root locus that end at

infinity are determined by

g ¼

ð1 þ2hÞ180

½no:of poles of GðzÞHðzÞ ½no:of zeros of GðzÞHðzÞ

for k>0 ð15:96Þ

and

g ¼

h360

½no:of poles of GðzÞHðzÞ ½no:of zeros of GðzÞHðzÞ

for k<0 ð15:97Þ

Rule 5.The real-axis intercept of the asymptotes is

z

0

¼

P

n

c¼1

Re p

c

P

w

i¼1

Re z

i

n w

ð15:98Þ

598 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

Rule 6.Thebreakaway point for thelocus betweentwopoles onthe real

axis (or the break-in point for the locus between two zeros on the

real axis) can be determined by taking the derivative of the loop

sensitivity K with respect to z.Equate this derivative to zero and

find the roots of the resulting equation.The root that occurs between

the poles (or the zero) is the breakaway (or break-in) point.

Rule 7.For K>0 the angle of departure froma complex pole is equal

to180

minus the sumof the angle fromthe other poles plus the sum

of the angle from the zeros.Any of these angles may be positive

or negative.For K<0 the departure angle is 180

fromthat obtained

for K>0.

For K>0 the angle of approachtoa complex zerois equal tothe sum

of theangles fromthe poles minus the sumof theangles fromthe other

zeros minus 180

.For K<0 the approach angle is 180

from that

obtained for K>0.

Rule 8.The root loci are symmetrical about the real axis.

Rule 9.The static loop sensitivity calibration K of the root locus can

be made by applying the magnitude condition given by Eq.(15.94)

as follows:

K ¼

jz p

1

j jz p

2

j jz p

c

j jz p

n

j

jz z

1

j jz z

2

j jz z

w

j

ð15:99Þ

Rule 10.The selection of the dominant roots of the characteristic

equation is based upon the specification that give the required

system performance;i.e.,it is possible to evaluate s,o

d

,and z,

fromEqs.(3.60),(3.61),and (3.64).

These values in turn are mapped into the z domain to determine the

location of the desired dominant roots in the z plane.The loop sensi-

tivity for these roots is determined by means of the magnitude condi-

tion.The remaining roots are then determined to satisfy the same

magnitude condition.

A root-locus CAD digital-computer program produces an accurate

calibrated root locus.This considerably simplifies the work

required for the system design.By specifying z for the dominant

roots or K,use of a computer program can yield all the roots of the

characteristic equation.

y

y

Agood engineering design rule as a first estimate for a calculation step size in a CADprogramis

T/10 in order to generate accurate results.

Sampled-Data Control Systems 599

Copyright © 2003 Marcel Dekker, Inc.

It should be remembered that the entire unbounded left-hand s plane is

mapped into the unit circle (UC) in the z plane.The mappings of the poles

and zeros in the left-half s plane into the z plane migrate toward the vicinity of

the 1 þj0 point as T!0.Thus,in plotting the poles and zeros of

G(z) ¼Z[G(s)],they approach the 1 þj0 point as T!0.For a ‘‘small-enough’’

value of T and an inappropriate plot scale,some or all of these poles and zeros

‘‘appear tolie ontopof one another.’’ Therefore,cautionshouldbe exercisedin

selecting the scale for the root-locus plot and the degreeof accuracy that needs to

be maintained for an accurate analysis and design of a S-Dsystem.

15.11.3 Root-Locus Design Examples

Example 15.8.The second-order characteristic equation,for a givencontrol

ratio,z

2

0.2Az þ0.1A¼0 is partitioned to put it into format of Eq.(15.92) as

follows:

z

2

¼ 0:2Az 0:1A ¼ Kðz 0:5Þ ð15:100Þ

where K¼0.2A.Equation (15.100) is rearranged to yield

PðzÞ ¼

Kðz 0:5Þ

z

2

¼ 1 ð15:101Þ

that is of the mathematical format of Eq.(15.92).The poles and zero of

Eq.(15.101) are plotted in the z plane as shown in Fig.15.18.The construction

rules applied to this example yield the following information.

Rule 1.Number of branches of the root locus is given by n ¼2.

Rule 2.For K>0 (A<0) the real-axis locus exists between z ¼0.5 and

z ¼ 1,and for K<0 (A>0) the real-axis locus exists between

z ¼0.5 and z ¼ þ1.

FIGURE 15.18 Poles and zero of Eq.(15.101).(From Ref.1,with permission of the

McGraw-Hill Companies.)

600 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

Rule 3.The root-locus branches start,withK¼0,at the poles of P(z).One

branch ends at the zero,z ¼0.5,and one branch ends at infinity for

K ¼

1.

Rule 4.Asymptotes

For K>0:

g ¼

ð1 þ2hÞ180

2 1

¼ ð1 þ2hÞ180

thus g ¼ 180

For K<0:

g ¼

h360

2 1

¼ h360

thus g ¼ 0

ðor 360

Þ

Rule 5.Not applicable for the asymptotes determined by Rule 4.

Rule 6.For this example there is no breakaway point on the real axis for

K>0.For K<0,Eq.(15.101) is rearranged to obtain the function

WðzÞ ¼

z

2

z þ0:5

¼ K ð15:102Þ

Taking the derivative of this function and setting it equal to zero

yields the break-in and breakaway points.Thus,

dWðzÞ

dz

¼

zðz 1Þ

ðz þ0:5Þ

2

¼ 0

which yields z

1,2

¼0,1.Therefore,z

1

¼0 is the breakaway point and

z

2

¼1is the break-in point.

Rule 7.Not applicable for this example.

Rule 8.The root locus is symmetrical about the real axis.

Rule 9.The static loop sensitivity calibration of the root locus can be

made by evaluating Eq.(15.102) for various values of z that lie on the

root locus.

Rule 10.Not applicable for this example.

The root locus for Eq.(15.101) is shown in Fig.15.19.The UC intersections

of the root locus as determined by a CAD programoccur for the static loop

sensitivity values of K¼2/3 and K¼2.

Example 15.9.Given the unity-feedback sampled-data system shown in

Fig.15.14,where G

x

(s) ¼K

G

/[s(s þ2)],the objectives of this example are as

follows:(a) determine C(z)/R(z) in terms of K

G

and T (i.e.,the values of K

G

and Tare unspecified);(b) determine the root locus and the maximumvalue

of K

G

for a stable response withT¼0.1s;(c) determine the steady-state error

characteristics with various inputs for this systemfor those values of K

G

and

T that yield a stable system response;and (d) determine the roots of the

Sampled-Data Control Systems 601

Copyright © 2003 Marcel Dekker, Inc.

characteristic equation for z ¼0.6,the corresponding time response c (kT),

and the figure of merit (FOM).

Solution.

a.The forward transfer function of the open-loop systemis

G

z

ðsÞ ¼ G

zo

ðsÞG

x

ðsÞ ¼

K

G

ð1 e

sT

Þ

s

2

ðs þ2Þ

¼ ð1 e

sT

Þ

K

G

s

2

ðs þ2Þ

¼ G

e

ðsÞ

K

G

s

2

ðs þ2Þ

Thus,using entry 8 inTable15.1yields

G

z

ðzÞ ¼ G

e

ðzÞZ

K

G

s

2

ðs þ2Þ

¼ ð1 z

1

ÞZ

K

G

s

2

ðs þ2Þ

¼

K

G

½T þ0:5e

2T

0:5Þz þð0:5 0:5e

2T

Te

2T

Þ

2ðz 1Þðz e

2T

Þ

ð15:103Þ

Substituting Eq.(15.103) into

CðzÞ

RðzÞ

¼

G

z

ðzÞ

1 þG

z

ðzÞ

¼

NðzÞ

DðzÞ

FIGURE 15.19 Root-locus plot of Example 15.8.(From Ref.1,with permission of

the McGraw-Hill Companies.)

602 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

yields

CðzÞ

RðzÞ

¼

0:5K

G

½ðT 0:5þ0:5e

2T

Þz þð0:50:5e

2T

Te

2T

Þ

z

2

½ð1þe

2T

Þ 0:5K

G

ðT 0:5þ0:5e

2T

Þz

þe

2T

þ0:5K

G

ð0:50:5e

2T

Te

2T

Þ

ð15:104Þ

b.From Eq.(15.104),the characteristic equation is given by

Q(z) ¼1 þG

z

(z) ¼0,which yields

G

z

ðzÞ ¼

K z þ

0:50:5e

2T

Te

2T

Tþ0:5e

2T

0:5

ðz 1Þðz e

2T

Þ

¼ 1 ð15:105Þ

where

K ¼ 0:5K

G

ðT þ0:5e

2T

0:5Þ

ForT¼0.1s,

G

z

ðzÞ ¼

Kðz þ0:9355Þ

ðz 1Þðz 0:81873Þ

¼ 1 ð15:106Þ

and K¼0.004683 K

G

.The root locus for Eq.(15.106) is shown

in Fig.15.20.For T¼0.1 the maximum value of K for a stable

FIGURE 15.20 A root-locus sketch for Eq.(15.106) where T¼0.1 s.(From Ref.1,

with permission of the McGraw-Hill Companies.)

Sampled-Data Control Systems 603

Copyright © 2003 Marcel Dekker, Inc.

response is K0.1938,which results in K

G

max

41:38.An analysis

of Eq.(15.105) reveal that:

1.The pole at e

2T

approaches the UC as T!0 and approaches the

origin as T!1.

2.The zero approaches 2 as T!0 (this can be determined by

applying L’Ho“pital’s rule twice) and approaches the origin as

T!1.

3.Based upon the plot scale chosen,it may be difficult to interpret

or secure accurate values from the root locus in the vicinity

of the 1 þj0 point as T!0.Both poles will ‘‘appear’’ to be

superimposed.

As a consequence of items 1 and 2 and considering only the root

locus,one may jump to the conclusion that the range of K

G

for

a stable systemdecreases asT!0.This is not the case for this exam-

plesince Kis a functionof T,as shown inthe next section.As pointed

out in item3,for an open-loop transfer function having a number of

poles and zeros inthe vicinity of z ¼1,it may be difficult toobtain an

accurate root-locus plot in the vicinity of z ¼1if the plotting area is

too large.This accuracy aspect can best be illustrated if the z con-

tours of Fig.15.8 are used graphically to locate a set of dominant

complex roots p

1,2

corresponding to a desired z.Trying to graphi-

cally determine the values of the roots at the intersection of

the desired z contours and the dominant root-locus branches is

most difficult.Any slight error in the values of p

1,2

may result in

a pair of dominant roots having a value of z that is larger or smaller

than the desired value.This problemis also involved even if a com-

puter-aided program is used to locate this intersection,especially

if the program is not implemented with the necessary degree of

calculation accuracy.Also,the word length (number of binary

digits) of the selected digital control processor may not be sufficient

to provide the desired damping performance without extended

precision.It may be necessary to reduce the plotting area to

a sufficiently small regionabout z ¼1and thentoreduce the calcula-

tion step size in order to obtain an accurate picture of the range of

K for a stable system performance.This aspect of accuracy is

amplified at the end of this section.

c.For a step input [R(z) ¼z/(z 1)],C(z) is solved from Eq.(15.104).

Applying the final-value theoremto C(z) yields

cð1Þ ¼ lim

z!1

½ð1 z

1

ÞCðzÞ ¼ 1

Therefore,eð1Þ ¼ 0 for a stable system.

604 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

For other than step inputs,the steady-state performance charac-

teristics of E(z) for a unity-feedback systemmust be analysed.Thus,

EðzÞ ¼

1

1 þG

z

ðzÞ

RðzÞ ð15:107Þ

Consideringtherampinput R(z) ¼Tz/(z 1)

2

,E(z)for thisexampleis

EðzÞ ¼

2ðz 1Þðz e

2T

Þ

2ðz 1Þðz e

2T

ÞþK

G

½ðT þ0:5e

2T

0:5Þz

þð0:50:5e

2T

Te

2T

Þ

8

>

>

>

>

<

>

>

>

>

:

9

>

>

>

>

=

>

>

>

>

;

Tz

ðz 1Þ

2

Thus,for a stable system,

e

ð1Þ ¼ lim

z!1

½ð1 z

1

ÞEðzÞ ¼

1

K

K>0

> 0

Therefore,a sampled-data unity-feedback stable control system

whose plant G

x

(s) is Type 1 has the same steady-state performance

characteristics as does a stable unity-feedback continuous-time

control system.In a similar manner,an analysis with other

polynomial inputs can be made for otherType mplants.

d.The roots of the characteristic equation for z ¼0.6 are

p

1,2

¼0.90311

j0.12181 (where K¼0.01252).The output time-

response function is

cðkT Þ ¼0:01252000r½ðk 1ÞT þ0:01171246r½ðk 2ÞT þ

1:80621000c½ðk 1ÞT 0:83044246c½ðk 2ÞT

and the FOM are M

p

1.113,t

p

2.35 s,t

s

¼3.6 s,and

K

1

¼1.3368 s

1

.Note that K

G

¼2.6735.In addition,it should be

noted that the roots can also be determined graphically by the

use of the z contours of Fig.15.8 as shown in Fig.5.9 with limited

accuracy.

As discussedpreviously inthis section,Figs.15.21and15.22illustrate the

care that must be exercised in performing a mathematical analysis for a

sample-data control system for small values of T.For the value of T¼0.01s,

the three poles of G

z

(z) in Fig.15.21appear to be on top of one another (for the

plotting scale used in figure).Thus,it is most difficult to locate accurately the

dominant poles p

1,2

for z ¼0.45 in this figure.An error in the graphical

interpretation of the values of p

1,2

can easily put these poles outside the

UC or on another z contour.The root-locus plot in Fig.15.22 corresponds

to that portion of the root locus in Fig.15.21in the vicinity of the 1 þj0 point.

Sampled-Data Control Systems 605

Copyright © 2003 Marcel Dekker, Inc.

That is,Fig.15.22 is an enlargement of the area about the 1 þj0 point of

Fig.15.21.By ‘‘blowing up’’ this region,one can plot the root locus accurately

in the vicinity of the1 þj0 point and accurately determine p

1,2

.

15.12 SUMMARY

Inthis chapter thesampling process associatedwithanLTIsystemis analyzed

by using the impulse-function representation of the sampled quantity.Linear

difference equations are introduced in this chapter and used to model a

continuous-time or a sampled-data (S-D) control system based upon the

approximation of differentiation.The effective use of the concept of an ideal

FIGURE 15.21 Root-locus for G

z

ðzÞ ¼ K

z

ðz þ0:26395Þðz þ3:6767Þ=ðz 1Þ

ðz 0:99005Þðz 0:95123Þ where T ¼ 0:001 s:(From Ref.1,with permission of the

McGraw-Hill Companies.)

606 Chapter 15

Copyright © 2003 Marcel Dekker, Inc.

sampler in analyzing sampled-data systems is introduced.An important

aspect of a S-Dcontrol systemis the data conversion process (reconstruction

or construction process),which is modelled by a zero-order hold (ZOH)

device and the analog-to-digital (A/D) and the digital-to-analog (D/A)

conversion devices.

The synthesis in the z domain and the associated stability analysis in the

z domain is presented in this chapter.This is followed by the steady-state ana-

lysis of stable S-Dcontrol systems.The chapter concludes with the root-locus

guidelines for the design of S-Dcontrol systems.Numerous examples are also

presented.

REFERENCES

1.Houpis,C.H.,and Lamont,G.B.:Digital Control Systems,Theory,Hardware,

Software,2nd ed.,McGraw-Hill,NewYork,1992.

2.Franklin,G.G.,and Powell,J.D.:Digital Control of Dynamic Systems,2nd ed.,

Addison-Wesley,Reading,Mass.,1990.

3.Philips,C.L.,andNagle,H.T.:Digital Control SystemAnalysis andDesign,Prentice-

Hall,Englewood Cliffs,N.J.,1984.

4.Ogata,K.:Discrete-Time Computer Control Systems,Prentice-Hall,Englewood

Cliffs,N.J.,1987.

5.Houpis,C.H.,and Lamont,G.B.:Digital Control Systems:Theory,Hardware,

Software,McGraw-Hill,NewYork,1988.

FIGURE 15.22 Enlargement of the 1 þj 0 area of the root-locus plot of Fig.15.21

(T¼0.01 s).(From Ref.1,with permission of the McGraw-Hill Companies.)

Sampled-Data Control Systems 607

Copyright © 2003 Marcel Dekker, Inc.

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