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Module-2 Time responses of discrete data systems
As the outputs of discrete-data control system are function
of the continuous variable ‘t’, with t = kT, k =, 2, !, "#$t become a must to evaluate the performance of the
system in the time-domain#
Time response of a discrete-data control system can becharacteri%ed by terms as in that of continuous-data
control system
(i) Maximum overshoot
(ii) Rise time
(iii) Delay time
Iv) Settling time
(v) Damping ratio
vi) Damping factor
vii) Natural undamped frequency
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• Time response of a system can be divided into twocate&ories'
() Transient *esponse
(2) +teady state *esponseTransient Response refers to that portion of the
response which is due to the closed loop systempoles#
Transient *esponse depends on the initial conditionof the system#
Transient response of a practical control system,where the output si&nal is continuous-time, oftenehibits damped oscillation before reachin& thesteady state#
Steady-state response refers to that portion ofthe response which is due to the poles of theforcin& function# 2
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• +election of ain . and +amplin& /eriod T
0rom the studies of continuous-data control system'
$ncreasin& the value of ain . &enerally'
(i) would reduce the dampin& ratio#(ii) $ncrease the maimum overshoot#
(iii) $ncrease the natural un-damped fre1uency and thusbandwidth#
iv) reduce the steady-state error if it is nite and non-%ero#
These properties would carry over to discrete-datasystems#
0or samplin& period , for a &iven ., increasin& T wouldcause instability
A smaller samplin& period would re1uire a faster clockrate for di&ital computer, which translates into morecomple hardware and cost# !
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• As the samplin& period becomes smaller, the roots move closer to the % = pointin the rst and fourth 1uadrant of %-plane#
• +election of the samplin& period shouldsuch that if the pole of the di&ital controlsystem would lie in the rst and fourth
1uadrants of %-plane
• very close to the %= point, di&italcontrol system would emulate the
correspondin& continuous-time system#
3
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• 4orrelation between the time response androot location in s-plane and z -plane
4orrelation between the location of the poles of
the system in the s-plane and the transientresponse is well known for the continuous-timesystem
0or eample, comple con5u&ate poles in theleft half of s-plane rise eponentially decayin&sinusoidal responses#
/oles on the ne&ative real ais of s-plane
correspond to monotonically decayin&responses#
+imple con5u&ate roots on the ima&inary ais ofs-plane &ive rise to un-damped constantamplitude sinusoidal oscillations 6
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Multiple-order poles on the ima&inaryais and poles in the ri&ht half of s-plane correspond to unstable responses#
The samplin& operation brin& about anininite number of poles in the s-plane at
net e7ect is e1uivalent to havin& asystem with the poles,
The poles are then mapped onto the %-plane usin& the transformation
,11 s jn j s ω ω σ ++=
,...3,2,1 ±±±=n
( ) sn j s ω ω σ ++= 11
Tse z =
8
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• 0ollowin& 0i&ures illustrates several casesof the root location of a second-ordersystem in the s- and %-plane and their
correspondin& time responses'
() 4ase when poles are ima&inary ais-
9
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2) 4ase when the pole is midway in theprimary strip and on the ima&inary ais inthe s-plane'
:
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!) 4ase when the pole is near to ws;2 in the primarystrip and on the ima&inary ais in the s-plane
<
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3) 4ase when the pole is e1ual to ws;2and on the ima&inary ais in the s-plane
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6) 4ase when the pole is e1ual to in the s-plane( )2/1 s j s ω σ +−=
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8) 4ase when the poles is e1ual to in the s-plane( ) )2/(111 swith j s ω ω ω σ
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9) 4ase when the pole is at ori&in
in the s-plane !
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:) 4ase when the pole is ne&ative real ais
in the s-plane 3
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>hen the poles of the closed loop system&et nearer to the ne&ative real ais of %-plane, the system response becomeoscillatory#
>hen the roots are all real and ne&ativein the %-plane, the response will beoscillatory with alternate positive and
ne&ative values#
8
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+teady +tate /erformance+teady-state error analysis'
The error si&nal of control system is oftendenes the di7erence between the referenceinput and output#
The error si&nal is dened as error
is lost in the %-transformation analysis#
sampled error si&nal is used#
( ) ( ) ( )e t r t b t = −
( )e t * ( )e t
9
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The steady-state error of the di&italsystem is &iven as'
?y the usin& the nal value theorem'
>ith condition that doesnot have any poles on or outside the unitcircle in z -plane#
The steady state error between samplin&instants can be determined by use of themodied z -transform'
* *lim ( ) lim ( ) sst k
e e t e kT →∞ →∞
= =
( )* 11
lim 1 ( ) ss z
e z E z −→
= −
( )11 ( ) z E z −−
[ ] ( )110 1 0 1
lim ( , ) lim 1 ( , )k z m m
e kT m z E z m−
→∞ →≤ ≤ ≤ ≤
= −:
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• where is the modied z -transform of #
The z -transform of error si&nal e(t ) is&iven by'
where @(%) is &iven by'
Therefore'
( , ) E z m ( )e t
( )( )1 ( )
R z E z GH z
= +
( )1 ( ) ( )( ) 1 G s H sGH z z s
− = −
( )* * 11 ( )lim ( ) lim 1 1 ( ) ss t z R z e e t z GH z
−→∞ →
= = − +
( )GH z <
d d i
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• +teady state error due to a step input
+teady state error is &iven by'
et us dene discrete step-errorconstant as'
(/osition error constant)
( ) ( ), ( )1
s
z If r t Au t then R z A
z = =
−
( )* * 11
1
1
1lim ( ) lim 11 ( )
lim1 ( ) 1 lim ( )
sst z
z
z
z A
z e e t z GH z
A A
GH z GH z
−
→∞ →
→→
−= = −+
= =+ +
*
1lim ( ) p z K GH z →= 2
Th A
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• Then
0or due to step input, tends to %ero, asapproaches innity#
which implies that @(%) must be at leastone pole at % = #
+tep error constant is meanin&ful only, ifreference input is a step function
Type of +ystem
*
*1 ss
p
Ae
K =
+*
sse* pK
1 2
(1 )(1 ) (1 )( ) ( )
(1 )(1 ) (1 )
a b m
j
n
K T s T s T sG s H s
s T s T s T s
+ + +=
+ + +A
A
2
T f i d d b f
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• Type of system is dened as number ofpoles of system at ori&in in the s-plane#
The type of system denes as e1ual to 5
( power of the s-term)
• Then
• 0or type system'
( )1 11 2
(1 )(1 ) (1 )( ) 1
(1 )(1 ) (1 )
a b m
j
n
K T s T s T sGH z z
s T s T s T s
−+
+ + += − + + +
A
A
( )1
1( ) 1
in
K Terms due to
GH z z s
non zero poles s domain
− + = −
− −
22
( )
−
=+−−= −
domain z in poles
z nontodueTerms z
z K
z z GH 1
11)(1
1
T d t l i d i
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Terms due to non % = poles in %-domaindoes not contain term (%-) indenominator#
Thus, for a type-, the discrete step-errorconstant is same as step error constant in
the continuous type-data system# 2!
( ) ( )
1
1
1
11
1
1
*
11lim1
1lim
)(lim
K domain z in poles
z nontodueTerms z z
z K z
z GH K
z z
z
p
=−
=−+−
−=
=
−
→
−
→
→
0or Type system
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0or Type- system'
23:
)(
)()1()(
:)(
)()(1)()(
:,
.0)()(
)(
)()(
2
1
Method Fraction arital Appl!in"
s # s
s $ z z G
becomes z G
s s# s $
se sG sG
becomes functiontransfer hold order zerowith
sat rootsha%enot does s #and s $ where
s s#
s $ sGas plant &onsider
Ts
ho
−=
−=
=
=
−
−
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26
( )
( )
∞=
−
=−++−=
=
−
=+−
+−−=
−
→
→
−
domain z in poles
z nontodueTerms z K z K
z GH K
domain z in poles
z nontodueTerms z
z K
z
z K
z z G
z
z p
11)1(lim
)(lim
11)1(1)(
12
3
1
1
*
2
2
3
1
−
−++−= −
domain sin poles
zeronontodueTerms s
K
s
K
z z G3
2
31)1()(
Th f t t t th *
K
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• Thus for system types &reater than ,
+teady state error due to a *amp input
Then steady state error obtained as'
p K = ∞
2( ) ( ), ( ) ( 1) s
Tz If r t Atu t then R z A
z = = −
( )
[ ]
2* * 1
1
1
( 1)lim ( ) lim 1
1 ( )
lim( 1) 1 ( )
sst z
z
Tz A z
e e t z GH z
AT
z GH z
−
→∞ →
→
−= = −+
=− +
28
AT
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et us dene discrete ramp errorconstant as'
(Belocity error constant)
C then
*
1 1
1
lim ( 1) lim( 1) ( )
1lim( 1) ( )
ss
z z
z
AT e
z z GH z
A z GH z
T
→ →
→
=− + −
=−
*
1
1lim( 1) ( )%
z
K z GH z
T →
= −
*
* ss
%
Ae
K
=29
0 th t b l t* 0 *K
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• 0or , the must b e1ual toinnity#
0or which, (%-)@(%) should have at leastone pole at % = #
Dr in other words, @(%) should have atleast two poles at % = #
• meanin&ful only if input is a rampfunction#
+teady state error due to a /arabolic input
0 sse =
% K
*
%
K
22
3( 1)( ) ( ), ( )
2 2( 1) s
A T z z If r t t u t then R z A z
+= =−
2:
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( )
[ ]
2
3* * 1
1
2
21
2
1
2 2
1 1
( 1)
2( 1)
lim ( ) lim 1 1 ( )
( 1)
2lim( 1) 1 ( )
lim( 1)2lim ( 1) lim( 1) ( )
ss t z
z
z
z z
T z z A
z
e e t z GH z
AT z
z GH z
AT
z z z GH z
−
→∞ →
→
→
→ →
+−
= = − +
+
= − +
+=− + −
2<
A
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• et us dene the Discrete ParabolicError constant (acceleration errorconstant) as'
• Then
• 0or , should be innity#
•
0or which @(%) must be at least three=
*
2
2 1
1lim( 1) ( )
ss
z
Ae
z GH z T →
=−
* 2
2 1
1
lim( 1) ( )a z K z GH z T →= −*
* ss
a
Ae
K =
* 0 sse = *
a K
!
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Typeof
+ystem
+tep$nput
*amp $nput /arabolicinput
.
.
2 .
!
∞
∞
∞ ∞
∞
∞
∞
∞
* p K
*1 p
A
K +
* sse * sse* sse*% K *a K
*
%
A
K
*
a
A K
∞!2
+ bili T
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+tability Tests () ?ilinear transformation and Ftended
*outh stability 4riteria (2) Gury stability 4riteria
(!) *oot ocus
0re1uency Eomain analysis' (3) /olar /lot
(6) Hy1uist stability 4riteria
(8) ?ode /lot (?ilinear transformationMethod)
!!
• +tability of inear Ei&ital 4ontrol +ystem
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• +tability of inear Ei&ital 4ontrol +ystem
) Iero-state response
• Dutput of a discrete-data system that is dueto input only is called %ero-state responseJall initial conditions are set to %ero#
2) Iero-input response
• Dutput response of a discrete-data systemthat is due to the initial condition only iscalled %ero-input response#
• >hen a system is sub5ected to both, weapply superposition principle'
• Total response =%ero-state response K
%ero-input response !3
!) ?ounded input ?ounded state stability (?$?+)
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!) ?ounded input ?ounded state stability (?$?+)
+ystem is said to be ?$?+ stable, if, for anybounded input u(k), state (k) is also bounded#
3) ?ounded input ?ounded otput stability (?$?D)+ystem is said to be ?$?D stable, if, for any
bounded input u(k), the output y(k) is alsobounded#
+ince output of a system is a linear combinationof sate variables , system that is ?$?+ stable, isalso a ?$?D stable#
*everse may or may not be true#$f system has a pole that is cancelled by a %ero,
then the system can be ?$?D stable but notnecessary ?$?+ stable#
!6
6) Iero-$nput stability
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6) Iero $nput stability
• +ystem is said to be %ero-input stability, if the %ero-inputresponse y(k) , sub5ected to nite initial conditions,reaches to %ero, as k tend to innity, otherwise the
system is unstable# i#e#
• And secondly
Iero-input stability implies asymptotic stability#
8) Theorem'
L0or linear discrete-data system, ?$?D ,%ero-input andasymptotic stability, all re1uire that root of thecharacteristic e1uation be inside the unit circle in the %-plane#
( ) ! k M ≤ < ∞
lim ( ) 0k
! k →∞
=
!8
• ?ilinear Transformation method
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?ilinear Transformation method
(Ftension of use of *outh-@urwit% criteria)
• Transformation that is al&ebraic and transform
the unit circle in %-plane onto vertical line in acomple variable plane (w-plane) are of followin&form'
Dne such transformation that transform theinterior of unit circle onto left half of w-plane is'
Then *outh-@urwit% criterion can be applied tocharacteristic e1uation of system in w-domain in
normal fashion#
aw b z
cw d
+=
+
1
1
w
z w
+
= −
!9
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Gury’s +tability 4riterion
• iven a nth order characteristic e1uation of a systemin %-domain
•where are realconstants#
+TF/-' 4heck is positive or else it can be made
positive by chan&in& si&n of all the coecients of the0(%)
+TF/-2' 0ollowin& Table is made usin& the coecients
of the 0(%)
1 2
1 2 1 0( ) 0n n
n n F z a z a z a z a z a−
−= + + + + + =
na
0 1 2 1, , , , ,n na a a a and a−A
!:
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!<
Flements in table are dened as'
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Flements in table are dened as'
0
0 1
1
0 3 0 1
0 2
3 0 3 2
;
n k
k
n k
n k
k
n k
a ab
a ab b
cb b
' '
−
− −
−
=
=
= =
M
3
• The necessary and sucient condition for
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• The necessary and sucient condition forpolynomial 0(%) to have no roots on oroutside the unit circle in z -plane are'
• And (n-) constraints'
1) (1) 0
02) ( 1)
0
F
for n e%en F
for n odd
>>
− <
1
2
3
3
o n
o n
o n
o
o
a a
b b
c c
' '
−
−
<
>
>
>
>
M
3
• +in&ular 4ases of Gury 4riteria
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+in&ular 4ases of Gury 4riteria
>hen some or all elements in a row are%ero, tabulation end prematurely#
+uch situation is known as singular case.
$t can remedied by epandin& orcontractin& unit circle innitesimally#
F1uivalent to movin& %eros of 0(%) o7 theunit circle#
Transformation for this purpose is'
= very small realnumber#
, radius of circle epands#
, radius of unit circle reduces#
( )1 , z z whereε ε = +0ε >0ε <
32
• Ei7erence between the number of %eros
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Ei7erence between the number of %erosfound inside (or outside) the circle whenthe circle is epanded or contracted by is
the number of %eroes on the circle#• Transformation of can be
simplify as'
for positiveor ne&ative #
C 4oecient of the term is multiplied by#
( )1 z z ε = +
( ) ( )1 1n n n z n z ε ε + ≈ + ε
n z ( )1 nε +
3!
* t f Ei it l t l t
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*oot ocus of Ei&ital control system
• iven the loop transfer function of a
di&ital control system, @(%) or (%)@(%),havin& the variable parameter . asmultiplyin& factor#
•
Then the characteristic e1uation become'
• is part of loop transfer
function which does not contains .#
• Then loci of points which are roots of
above e1uation, as . varies from innity
11 . ( ) 0 K GH z + =
1 ( )GH z
1
1( )GH z
K
= −
33
• ) Condition on Magnitude
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) Condition on Magnitude
• 2) Condition on Angle
0irst condition is used to nd the values of .on the loci# +econd one are used to nd the
points on the root loci# $n the all procedure of root-locus, all the
an&le are measured in the counter-clock wisedirection with positive -ais as the reference
1
1( )GH z
K =
( )
( )
1
1
0, ( ) (2 1)
0, ( ) 2
0,1, 2, 3...( interger)
for K GH z k
for K GH z k
where k an!
π
π
≥ ∠ = +
< ∠ =
=
36
• 0ollowin& are properties of the root loci in
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0ollowin& are properties of the root loci inthe %-plane for (useful insketchin& the root loci without solvin& for
the roots of the characteristics e1uationas . varies#
) +tartin& and Termination /oints of root
loci(a) . = points (+tartin& points)
These points on the roots loci are at
poles of loop transfer function# The polesincludes those at innity# This are openloop poles#
(b) . = points (Termination points)
0 K ≤ < ∞
∞
38
2) Humber of separate root loci ' e1ual to
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2) Humber of separate root loci ' e1ual tonumber of poles or %eros of @(%),whichever is &reater#
!) +ymmetry' root loci are symmetrical withrespect to real-ais of %-plane#
3) root loci on real-ais'
• loci are found on a &iven section of thereal ais of the %-plane only if the totalnumber of real pole and real %eros of
@(%) to the ri&ht of the section is odd#6) Asymptotes' 0or lar&er value of %, root
loci are asymptotic to the strai&ht lines
which make an&le with real ais, &iven by'.1)(,...,2,1,0,
)(
)12(−−=
−
+= mnk where
mn
k
k
π θ 39
• >here the n and m are number of nite
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>here the n and m are number of nitepoles and %eros of @(%) respectively#
• Humber of asymptotes = (n-m)
8) $ntersection of the asymptotes '
• $ntersection point always lie on the realais of the %-plane#
• the intersection point is &iven by'
)(
)((
)((
1mn
z GH of zerosof part real
z GH of polesof part real
−
− = ∑∑σ
3:
9) ?reakaway or ?reak-in /oints
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9) ?reakaway or ?reak in /oints
Fither lie on real ais or occur on complecon5u&ate pairs#
These points on the loci are points at whichmultiple-order roots lie#
$f the characteristic e1uation is &iven by'
Then the break-away or break-in points can bedetermined from the roots of followin&
e1uation'
)()(
0)(
)(1
0)(1
z ( z A K or
z A
z ( K or
z F
−=
=+=+
3<
0)(')()()('
− z ( z A z ( z AdK
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where the prime denotes the di7erentiation with
respect to %#
$f root locus lies between the two ad5acent openloop poles, then there eists at least one
breakaway point between the two poles#$f root locus lies between the two ad5acent open
loop %eros (one %ero may be at minus innity) ,then there eists at least one break-in point
between the two %eros#$f root locus lies between a open loop pole and a
open loop %ero, then there may eists no onebreak-in or break-away point or both may eists#
0)(
)()()()(2
=
−=
z (dz
6
Then nd the valve of . correspond to the
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Then nd the valve of . correspond to theroots of e1uation d.;d% = ,
And if resultant . is positive then the point is
an actual break-away or a break-in point#Flse the point is neither a breakaway or a
break-in points#
(:) An&le of Arrival ( or An&le of Eeparture) To nd the direction of root loci near the
comple poles or near comple %eros#
An&le of departure is &iven by'
= :- ( sum of an&le contribution of all otherpoles and %eros at the concerned complepole, with appropriate si&ns included)#
6
An&le of arrival is &iven by'
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An&le of arrival is &iven by'
= :- ( sum of an&le contribution of allother poles and %eros at the concerned
comple %ero, with appropriate si&nsincluded)#
0ollowin& &ure shows an&le of departure
calculated at a comple pole'
62
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) & yais
/ut % = 5v in the characteristic e1uation'
+olve for v and . by e1uatin& the real andima&inary part to %ero#
The value of v &ives the location at which root loci
cross the ima&inary ais in the %-plane # The value of . obtained &ives the value of the
correspondin& &ain#
) $ntersection of root loc with the unit circleC /articular value of % and . at an intersect of the
root loci with unit circle can be determined byusin& etended *outh-@urwit% (?ilinear
Transformation criteria) or by Gury‘s Test#
0)(1 1 =+ j% F K
6!
) Balues of . on the roots loci
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) Balues of . on the roots loci
Balues of . can be obtained from thema&nitude condition
• A particular point on the root loci will bea closed pole with a particular value of&ain .#
=
++++++
==
point,...,,
Product
point,...,,
Product
))...()((
))...()((
)(
1
21
21
21
1
concernedtozzzzerosthe
fromdrawn%ectorsof len"thof
concerned to p p p polesthe
fromdrawn%ectorsof len"thof
z z z z z z
p z p z p z
z F K
n
m
n