b_lecture11 extension of the root locus automatic control system

The effects on the systems performance adding the open zeros or open poles

Adding a open zero in the left s-plane

For example )3)(2(

)()(

)3)(2()( 12

11

sss

asKsGH

sss

KsGH

Generally, adding a open zero in the left s-plane will lead the root loci to be bended to the left. And the more closer to the imaginary axis the open zero is,

the more prominent the effect on the systems performance is.

Re

Im

23

Re

6

Im

23

Re5.2

Im

23

Re4

Im

23

5.0a

5.0a 25.1a

5.2a

Adding a open pole in the left s-plane

For example ))(2(

)3()(

)2(

)3()( 12

11

asss

sKsGH

ss

sKsGH

Generally, adding a open pole in the left s-plane will lead the root loci to be bended to the right. And the more closer to the imaginary axis the open

pole is, the more prominent the effect on the systems performance is.

The effects on the systems performance adding the open zeros or open poles

Re

Re

23

Im

Re

Im

5

Re

ImIm

23 23

1 23

2a

0a 5.0a

Dominant poles and zeros of transfer functions

The location of the poles and zeros of a transfer function in the s-plane greatly affects the transient response of the system.

For analysis and design purposes, it is important to sort out the poles that have a dominant effect on the transient response and call these

the dominant poles.

Because most control systems in practice are of orders higher than two, it would be useful to establish guidelines on the approximation of high-order systems by lower-order ones insofar as the transient response is concerned.

In design, we can use the dominant poles to control the dynamic performance of the system, whereas the insignificant poles are used for the purpose of ensuring that the controller transfer function can be realized by physical components.

For all practical purposes, we can divide the s-plane into regions in which the dominant and insignificant poles can lie.

We intentionally do not assign specific values to the coordinates, since these are all relative to a given system.


Im

Re

unstable

region

unstable

region

region of

dominant

poles

region of

insignificant

poles

D

The poles that are close to the

imaginary axis in the left-half

s-plane give rise to transient

responses that will decay relatively

slowly, whereas the poles that are

far away from the axis (relative to

the dominant poles) correspond to

fast-decaying time responses.

The distance D between the

dominant region and the least

significant region will be subject to

discussion. Regions of dominant and insignificant poles in the s-plane.

The question is: How large a pole is considered to be really large? It has been recognized in practice and in the literature that if the magnitude of the real part of a pole is at least 5 to 10 times that of a dominant pole or a pair of complex dominant poles, then the pole may be regarded as insignificant insofar as the transient response is concerned.

The zeros that are close to the imaginary axis in the left-half s-plane

affect the transient responses more significantly, whereas the zeros that are far away from the axis (relative to the dominant poles) have a smaller effect on the time response.


We must point out that the regions shown in Fig. are selected merely for the definitions of dominant and insignificant poles.

For design purposes, such as in pole placement design, the dominant poles and the insignificant poles should most likely be located in the red regions.

Again, we do not show any

absolute coordinates. except

that the desired region of the

dominant poles is centered

around the line that corresponds

to = 0.707.

It should also be noted that,

while designing, we cannot place the

insignificant poles arbitrarily far to the

left in the s-plane or these may require

unrealistic system parameter values.

Im

Re

unstable

region

unstable

region

region of

dominant

poles

region of

insignificant

poles

D

450

450

Regions of dominant and insignificant poles

in the s-plane for design purpose.


The proper way of neglecting the insignificant poles with

consideration of the steady-state response

(a) )22)(10(

20

)(

)(2

ssssR

sCThe pole at s=-10 is 10 times the

real part of the complex conjugate

poles, which are at -1 j 1 . (b)

)22)(110/(10

20

)(

)(2

ssssR

sC

Then we reason that when the absolute value of s is much smaller than 10,

because of the dominant nature of the complex poles. The term s/10 can be neglected

when compared with 1. Then, Eq.(b) is approximated by

(c) )22(10

20)(

2

sssM

the third-order system described by Eq. (a) and the second-order system

approximated by Eq. (c) all have a final value of unity when a unit-step input is applied.

On the other hand, if we simply throw away the term (s + 10) in Eq. (a), the

approximating second-order system will have a different steady-state value when a

unit-step input is applied.

110/ s

Example

Conventional root locus

system. theof ) ( the toalproportion

is and system a of locusroot plot the toparameter

variable theas )( select the weGenerally

*

*

gainloopopenK

K

gainroot locusK

ed.investigat be to

need isparameter theof locus-root then thecases,

many in variableis system theofparameter other Maybe

*no-K

We illustrate the parameter root locus and its sketching approaches by following example:

Parameter root locus --- the variable parameter of the control

systems is another parameter besides K * .

Parameter root locus

Extension of The Root Locus

. system theof locus-root sketch the ,0 from varing If

1

4

: is system a offunction transfer loopopen The

))(ss(sG(s)H(s)

:Solution

0141 041

:is system theofequation sticcharacteri

2 )s(s)(ss))(ss(s

0)(12(2

11

41

11

22

sG

)ss)(s

)s(s

)(ss

)s(s eq

Loci versus Other Parameters

Example 1

functiontransfer

loop-open equivalent The

)ss)(s

)s(ssGeq

2(2

1)(

2

The procedure of sketching root locus is shown as following:

2 ,2

7

2

1

:poles loopOpen

1 0

:zeros loopOpen (1)

32,1

21

pjp

z,z Re

Im

12

2

7

2

1j

2

7

2

1j


Example 1


4 1

1

s s

1

TK

The open-loop transfer function of the system is

which is not in the standard form as:

n

j

j

m

i

i

ps

zs

KsHsG

1

1

)(

)(

)()(

)1(

4

TKss

Example 2

The characteristic equation of the system is

042 sKss T

Now, equation is in suitable form for a root-locus study. We

need to identify open-loop transfer function, which we do

by writing the equivalent to the equation as

04

12

ss

sKT


Thus, for root-locus purposes,

the zeros are at s=0, And the poles are at -1/2+j1.94 and -1/2-j1.94.

Example 2

Zero-Degree Root Loci

The open loop transfer function

n

j

j

m

i

i

ps

zs

KsHsG

1

1

)(

)(

)()(

The characteristic equation is

0

)(

)(

1

1

1

n

j

j

m

i

i

ps

zs

K 1

)(

)(

1

1

n

j

j

m

i

i

ps

zs

K

)(sG

)(sH

1)(

)(

)()(

1

1

n

j

j

m

i

i

ps

zs

KsHsG

The magnitude and angle requirement for the

zero-degree root locus are

criterionMagnitude

ps

zs

Kn

j

j

m

i

i

1

||

||

1

1

criterionAnglekkpszsn

j

j

m

i

i ,2 ,1 ,0 2)()(11

The characteristic equation is )(sG

)(sH


The Zero-Degree (00) Root Locus

In summary, all rules are the same, except:

All 1800s become 00s.

Odd becomes even in Rule 4.

If we substitute k2 for , )12( k

the sketching rules of the conventional root locus

are also suitable to thezero degree root locus (only related to the rule 4, 5 and 9 ).


b_lecture11 extension of the root locus automatic control system

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