15
SampledData 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 continuoustime 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 analogtodigital (A/D) converter must be
incorporated in a digital or sampleddata (SD) 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 constantrate sampling.
The output of thecontroller must thenbe convertedfromdiscreteformintoan
analog signal by a digitaltoanalog (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 pseudocontinuoustime
(PCT) system.
There are various approaches that may be used in analyzing the stability
and timeresponse characteristics of SDsystems.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 sampleddata 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 sampleddata 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 sampleddata control systems,as discussedin
this chapter,is expedited by the use of CADpackages,such as MATLAB or
TOTALPC(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 continuoustime 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.AFourierseries 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 sampleddata control system involving the
sampling of the actuating signal.
SampledData Control Systems 569
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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
pulsesampled 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 lowpass filter could extract the spectrumof the continuous input signal by
attenuating the spurious higherfrequency spectra.The forward transfer
function of a control system generally has a lowpass characteristic,so the
systemresponds with more or less accuracy to the continuous signal.
15.3 IDEAL SAMPLING
To simplify the mathematical analysis of sampleddata 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.
SampledData 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 impulsesampled 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 onesidedZtransformis
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
SampledData 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 rootlocus 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 sampleddata (SD) systems in the z plane requires the use of a
multiplesheeted 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 constantdampingratio 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 McGrawHill Companies.)
SampledData Control Systems 575
Copyright © 2003 Marcel Dekker, Inc.
In summary,the strips in the lefthalf s plane (s<0) map into the region
inside the UCinthe z plane,and the strips inthe righthalf s plane (s>0) map
into the region outside the UCin the z plane.
15.4 ZTRANSFORM THEOREMS
Ashort list of Z transforms is given inTable 15.1.Several simple properties of
the onesided 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 realtime 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 backwarddifference 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
SampledData 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 firstbackward
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 backwarddifference 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 thbackward 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 SDcontrol 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
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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 sampleddata control system.(From Ref.1,
with permission of the McGrawHill 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 zPlane 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 SDcontrol systemis obtained by setting to
zero the denominator of the control system’s control ratio.For example,the
characteristic equation for the closedloop system as represented by
Eq.(15.41) is given by
QðzÞ ¼ 1 þGHðzÞ ¼ 0 ð15:42Þ
Ingeneral,for a closedloop SDcontrol 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 closedloop system in a manner similar to that for analog
systems.For a continuoustime 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 SDsystemwhose closedloop 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
SampledData 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
timeinvariant 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 boundedinput bounded output (BIBO) refers to the application
of a bounded input generating a bounded output in a discrete timeinvariant
linear model.Asymptotic stability for this model implied BIBOstability.
To determine if a given discrete timeinvariant 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
computeraideddesign (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 lefthalf s plane (s<0) map
into the region inside the UC in the z plane,and the strips in the righthalf 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 sampledata 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 sampledata 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 McGrawHill 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 frequencyresponse or
rootlocus 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 rootlocus
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 rootlocus 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).
SampledData 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 unitstep 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
sampleddata (SD) 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
powerseries 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 partialfraction
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.
SampledData 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 ZEROORDER HOLD
The function of a zeroorder 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 piecewisecontinuous signal that is the input
to the plant.The ZOHis a lowpass filter that,together with the basic plant,
FIGURE 15.13 Input and output signals for a zeroorder 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 zeroorder 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 timeresponse characteristic of the system.
Example 15.4.Figure 15.14 shows a unityfeedback system in which a
zeroorder 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
closedloop 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 sampleddata (SD) control system.
SampledData Control Systems 587
Copyright © 2003 Marcel Dekker, Inc.
For this control ratio,e
sT
¼0.673 so that the corresponding real part of the
closedloop poles in the s domain is s¼0.396 and the approximate setting
time isT
s
¼4/0.396¼10.1s.
This example illustrates that a secondorder 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 Ztransform
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 Ztransformmethod:
1.The use of an ideal sampler in the analysis of a discretedata 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 Ztransform 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 STEADYSTATE ERROR ANALYSIS FOR
STABLE SYSTEMS
The three important characteristics of a control system are (1) stability,
(2) steadystate 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 steadystate 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
timeresponse 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 steadystate error characteristics of
a unityfeedback SD system parallels the analysis for a unityfeedback
continuoustime stable system based upon system types and upon
specific forms of the system input function.For the unityfeedback SD
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
SampledData 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 finalvalue theorem can be applied to
Eqs.(15.59) and (15.60) to obtain the steadystate 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 steadystate error analysis for the stable nonunity feedback system
of Fig.15.16a may be analyzed by determining its equivalent z domainstable
unityfeedback 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 nonunityfeedback sampleddata system;(b) the
equivalent z domain stable unity feedback system of Fig.15.16a.(From Ref.1,with
permission of the McGrawHill 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 nonunityfeedback SD 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 SteadyState ErrorCoefficients
The steadystate error coefficients have the same meaning and importance for
sampleddata 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 unityfeedback
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
steadystate value of derivative of output
e
ð1Þ
ð15:70Þ
Since Eq.(15.26) represents the derivative of c (t) in the discretetime domain,
use of the translation and finalvalue 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
SampledData 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
steadystate 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 finalvalue 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 SteadyState Error Coefficients
The forward transfer function of a sampleddata unityfeedback 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 continuoustime 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.
steadystate 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 SteadyState Error Coefficients
The importance of the steadystate error coefficients is illustratedby means of
an example.
T
ABLE
15.2
SteadyState 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
SampledData 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
steadystate value of derivative of output
K
1
¼ E
0
ð15:87Þ
e
ð1Þ ¼
steadystate value of second derivative of output
0
¼ 1
ð15:88Þ
Thus,a Type 1 sampleddata stable system (1) can follow a step input with
zero steadystate 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 steadystate error coefficients for this system.
FIGURE 15.17 A nonunityfeedback sampleddata control system.(From Ref.1,
with permission of the McGrawHill 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 unityfeedback 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 finalvalue 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 nonunityfeedback
SampledData 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 steadystate error.
15.11 ROOTLOCUS ANALYSIS FOR SAMPLEDDATA
CONTROL SYSTEMS
The first thing that a designer wants to know about a given SD 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
rootlocus method is used to analyze the performance of a SD control
system in the same manner as for a continuoustime control system.For
either type of system,the root locus is a plot of the roots of the characteristic
equation of the closedloop system as a function of the gain constant.This
graphical approach yields a clear indication of gainadjustment effects
with relatively small effort compared with other methods.The underlying
principle is that the poles of C(z)/R(z) or C(z) (transientresponse modes)
are related to the zeros and poles of the openloop transfer function
G(z)H(z) and also the gain.An important advantage of the rootlocus
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 steadystate 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 rootlocus method.The
first subsection details a procedure for obtaining the root locus,the next
subsection defines the rootlocus 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 rootlocus 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 continuoustime systems,is summarized
as follows
Step 1.Derive the openloop 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 openloop transfer function in the
z plane,where z ¼s
z
þjo
z
.
Step 4.The plotted zeros and poles of the openloop function
determine the roots of the characteristic equation of the closedloop
system [1 þG(z)H(z) ¼0].By use of geometrical shortcuts or a
digitalcomputer program,determine the locus that describes the
roots of the closedloop characteristic equation.
Step 5.Calibratethelocus interms of theloopsensitivity K.If thegainof
the openloop 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 RootLocus Construction Rules for
Negative Feedback
The systemcharacteristic equation,Eq.(15.43),is rearranged as follows:
PðzÞ ¼ 1 ð15:92Þ
SampledData Control Systems 597
Copyright © 2003 Marcel Dekker, Inc.
Assume that P(z) represents the openloop 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 openloopzeros 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 rootlocus 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 continuoustime systems,with minor modifi
cations since the plot is in the z plane,are applicable for SDsystems and are
summarized as follows:
Rule 1.The number of branches of the root locus is equal tothe number
of poles of the openloop 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 openloop poles and
terminates ðK ¼
1Þ at the openloop 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 realaxis 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 breakin 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 breakin) 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 rootlocus CAD digitalcomputer 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.
SampledData Control Systems 599
Copyright © 2003 Marcel Dekker, Inc.
It should be remembered that the entire unbounded lefthand s plane is
mapped into the unit circle (UC) in the z plane.The mappings of the poles
and zeros in the lefthalf 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 ‘‘smallenough’’
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 rootlocus plot and the degreeof accuracy that needs to
be maintained for an accurate analysis and design of a SDsystem.
15.11.3 RootLocus Design Examples
Example 15.8.The secondorder 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 realaxis locus exists between z ¼0.5 and
z ¼ 1,and for K<0 (A>0) the realaxis 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
McGrawHill Companies.)
600 Chapter 15
Copyright © 2003 Marcel Dekker, Inc.
Rule 3.The rootlocus 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 breakin 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 breakin 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 unityfeedback sampleddata 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 steadystate 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
SampledData 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 openloop 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 Rootlocus plot of Example 15.8.(From Ref.1,with permission of
the McGrawHill 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 rootlocus sketch for Eq.(15.106) where T¼0.1 s.(From Ref.1,
with permission of the McGrawHill Companies.)
SampledData 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 openloop transfer function having a number of
poles and zeros inthe vicinity of z ¼1,it may be difficult toobtain an
accurate rootlocus 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 rootlocus 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
puteraided 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 finalvalue 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 steadystate performance charac
teristics of E(z) for a unityfeedback 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 sampleddata unityfeedback stable control system
whose plant G
x
(s) is Type 1 has the same steadystate performance
characteristics as does a stable unityfeedback continuoustime
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
sampledata 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 rootlocus plot in Fig.15.22 corresponds
to that portion of the root locus in Fig.15.21in the vicinity of the 1 þj0 point.
SampledData 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 impulsefunction representation of the sampled quantity.Linear
difference equations are introduced in this chapter and used to model a
continuoustime or a sampleddata (SD) control system based upon the
approximation of differentiation.The effective use of the concept of an ideal
FIGURE 15.21 Rootlocus 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
McGrawHill Companies.)
606 Chapter 15
Copyright © 2003 Marcel Dekker, Inc.
sampler in analyzing sampleddata systems is introduced.An important
aspect of a SDcontrol systemis the data conversion process (reconstruction
or construction process),which is modelled by a zeroorder hold (ZOH)
device and the analogtodigital (A/D) and the digitaltoanalog (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 steadystate ana
lysis of stable SDcontrol systems.The chapter concludes with the rootlocus
guidelines for the design of SDcontrol systems.Numerous examples are also
presented.
REFERENCES
1.Houpis,C.H.,and Lamont,G.B.:Digital Control Systems,Theory,Hardware,
Software,2nd ed.,McGrawHill,NewYork,1992.
2.Franklin,G.G.,and Powell,J.D.:Digital Control of Dynamic Systems,2nd ed.,
AddisonWesley,Reading,Mass.,1990.
3.Philips,C.L.,andNagle,H.T.:Digital Control SystemAnalysis andDesign,Prentice
Hall,Englewood Cliffs,N.J.,1984.
4.Ogata,K.:DiscreteTime Computer Control Systems,PrenticeHall,Englewood
Cliffs,N.J.,1987.
5.Houpis,C.H.,and Lamont,G.B.:Digital Control Systems:Theory,Hardware,
Software,McGrawHill,NewYork,1988.
FIGURE 15.22 Enlargement of the 1 þj 0 area of the rootlocus plot of Fig.15.21
(T¼0.01 s).(From Ref.1,with permission of the McGrawHill Companies.)
SampledData Control Systems 607
Copyright © 2003 Marcel Dekker, Inc.
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