module-2 digital control systems

8/18/2019 module-2 digital control systems

1/54

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


2/54

• 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


3/54

• +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# !


4/54

• 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


5/54

• 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


6/54

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


7/54

• 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


8/54

2) 4ase when the pole is midway in theprimary strip and on the ima&inary ais inthe s-plane'

:


9/54

!) 4ase when the pole is near to ws;2 in the primarystrip and on the ima&inary ais in the s-plane

<


10/54

3) 4ase when the pole is e1ual to ws;2and on the ima&inary ais in the s-plane


11/54

6) 4ase when the pole is e1ual to in the s-plane( )2/1 s j s ω σ +−=


12/54

8) 4ase when the poles is e1ual to in the s-plane( ) )2/(111 swith j s ω ω ω σ


13/54

9) 4ase when the pole is at ori&in

in the s-plane !


14/54

:) 4ase when the pole is ne&ative real ais

in the s-plane 3


15/54


16/54

>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


17/54

+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


18/54

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−

→∞ →≤ ≤ ≤ ≤

= −:


19/54

• 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


20/54

• +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


21/54

• 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


22/54

• 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


23/54

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


24/54

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

−=

−=

=

=

−

−


25/54

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


26/54

• 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


27/54

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


28/54

• 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:


29/54

( )

[ ]

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


30/54

• 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

!


31/54


32/54

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


33/54

+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


34/54

• +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 (?$?+)


35/54

!) ?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


36/54

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


37/54

?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


38/54

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

!:


39/54

!<

Flements in table are dened as'


40/54

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


41/54

• 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


42/54

+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


43/54

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


44/54

*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


45/54

) 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


46/54

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


47/54

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


48/54

>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


49/54

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


50/54

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


51/54

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'


52/54

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


53/54

) & 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


54/54

) 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

Documents

module-2 digital control systems