chapter 6 stability criteria_nise

8/17/2019 Chapter 6 Stability Criteria_nise

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Chapter 6: Stability1

Chapter 6Stability


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introduction

• C(t)=Cforced (t)+Cnatural (t)• For a bounded input, depend upon

natural response, the system can bestable, unstable or marginally stable• A system is stable if every bounded

input yields a bounded output ( ! ")• !f input is bounded but the totalresponse is not bounded, the system

is unstable


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Stability

• A system is stable if the naturalresponse approaches #ero as timeapproaches infinity$

• A system is marginally stable if thenatural response neither decays norgro%s, but remains constant oroscillates$


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&arginally stable

'ote that theroots are on theimaginary a is


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Figure 6.1

Closed-looppolesand response:a. stablesystem;b. unstablesystem


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Figure 6.2Common causeof problems infinding closed-loop poles:a. originalsystem;b. equivalent

system

(R-C)G=C C(1+G)=GR, C=[ G/(1+G )]R

(s)

(s)* + (s)-


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Chapter 6: Stability

Figure 6.3

Equivalentclosed-looptransferfunction


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Chapter 6: Stability!

Table 6.1Initial layout for Routh table


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Chapter 6: Stability"

Table 6.2Completed Routh table

!f no sign changes, stable


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Chapter 6: Stability1#

Figure 6.4

a. Feedbacksystem for E ample !"#;b. equivalentclosed-loopsystem

1#3#311#1###

1###)5)(3)(2(1###

)5)(3)(2(1###

1

)5)(3)(2(1###

)(1)(

23 +++=

++++=

++++

+++=

+

s s s s s s

s s s

s s s

sG

sG


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Table 6.3

Completed Routh table for E ample !"#

.%o sign changes, t%ounstable poles


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Table 6.4Completed Routh table for E ample !"$

356321#

)( 2345 +++++=

s s s s s sT


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Table 6.5%etermining signs in first column of aRouth table &ith 'ero as first element ina ro&

.%o sign changes/ t%o unstable poles


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Table 6.6Routh table forE ample !"(

123653)(

t$%$&'erte be%a&%riteria,1

356321#

)(

2345

2345

+++++=

=

+++++=

s s s s s s Dd

s

s s s s s sT


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Table 6.7

Routh table for E ample !")

56!4261#

)( 2345 +++++=

s s s s s sT

All#ero

0(s)=s1+2s3+4,05(s)=1s6+ 3s

0(s)05(s)


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C$&tr$l Sy te*16

*he au iliary polynomial

• *he equation that immediately precedesthe 'ero entry in the Routh array"


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Figure 6.5Root positionsto generate evenpolynomials:

A +B +C +or anycombination

!f t%o + realpoles, there mustbe t%o 7 realroots, other%iset%o roots are on

89a is


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Chapter 6: Stability1!

Table 6.8 Routh table for E ample !",

2#3!4!5"3"22122#

)( 23456! ++++++++=

s s s s s s s s sT


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Figure 6.6Feedbackcontrol systemfor E ample !"!

2##61162##

)6116(2##

1)6116(

2##

)(1)()(

234

23

23

++++=

++++

+++=+=

s s s s

s s s s

s s s s sG

sG sT


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Table 6.10

Routh table for E ample !"!


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Figure 6.7Feedback controlsystem for E ample !".

123232

1

)23232(1

1

)23232(1

)(1)(

)(

2345

234

234

+++++=

+++++

++++=+

=

s s s s s

s s s s s

s s s s s sG

sG sT


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Table 6.11 Routh table for E ample !".

.%o poles in rhs/unstable

.hree poles in lhs


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Table 6.12 /lternative Routh table for E ample !".

23232it Rearra&+e

1232321

)(

2345

2345

+++++

+++++=

s s s s s

s s s s s sT

.he sameresult/ .%opoles in rhs/.hree poles in

lhs

0ut s= *d/ s and d

sho%s the samesign


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Figure 6.8 Feedback control system for E ample !"0

12!1"212!"64!241#31

)1"212!"64!241#3(12!1

)1"212!"64!241#3(12!

)(1

)()(

23456!

23456

23456

++++++++=

++++++++

+++++++=+

=

s s s s s s s s

s s s s s s s s

s s s s s s s s

sG

sG sT


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Table 6.13

Routh table for E ample !"0

( )

0(s)=s2+4s1+63s3+21/ t%o rhs (t%o sign changes), sot%o lhs, the remaining t%o on 89a is


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Table 6.14ummary of pole locations for E ample !"0


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Figure 6.9Jason is anunder&ater+ remote-controlled vehicle thathas been used toe plore the &reckage ofthe Lusitania " *hemanipulator andcamera comprisesome of the vehicle1scontrol systems"

Courtesy of Woods ole !"ea#ogra$%i" stitutio#.


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Chapter 6: Stability2"

Figure 6.10

Feedback controlsystem for E ample!"2

K s s s

K

s s s

K

s s s

K

sG

sG sT

+++=

+++

++=

+=

1!

)11)((1

)11)(()(1

)()(

23


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Table 6.15


!f no sign changeat the first column/

unstable!f there is a #eroro%, 8% is possible Stable

:;


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Table 6.16

Routh table for E ample !"2 &ith K 3 #(0!

0(s)= 4s3+ 64205(s)=62s

As there is no

sign changebelo%, there aret%o 8% poles

&arginally stable


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Figure 6.11*he F/45CRobot 6- )77can beconfigured for )-or ,-a is ofmotion" It is

seen heremoving andstacking bo es"

Courtesy of Fa#u" 'oboti"s.


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Table 6.17Routh table for E ample !"#7

s1

+6s6

+6:s3

+3::

( )

( )= 2+1# s1+6s6+6:s 3+3::=( 2+1#)

( 2+3y+2#)


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Stability in State Space

• System poles are e ual to the eigenvalues of the system matri A

• A =λ• (λ !9A) =:• = (λ !9A)9, :

• = ad8(λ !9A)

*det ( λ !9A)-:• det ( λ !9A)=:• >se det (s!9A)=:


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[ ]

526)et(

251#

1!2

13

251#

1!2

13#

##

##

##

)(

##1

#

#

1#

251#

1!2

13#

23 −−−=−

+−−−−−

=

−−−−=−

=

+

−−−

=

s s s A sI

s

s

s

s

s

s

A sI

x y

u x x

s(s94)(s+3)+6:+ :+ :(s94)+?s92(s+3)

=s692s39@s9?3

36


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Table 6.18Routh table for E ample !"##

&e rh , t.$ lh 0& table

3


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Table 6.19

Routh table for antenna controlcase study

3!


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Table 6.20Routh table for 5F case study

3"


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Figure (6.1

4#


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Figure (6.2

41


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Figure (6.3

42


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Figure (6.4

43


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Figure (6.5

44


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Figure (6.6

45


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Figure (6.7

h b l46


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Figure (6.8

Ch 6 S bili4


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Figure (6.9

Ch 6 S bili4!


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Figure (6.10Closed-loopsystem&ith pole plot

Ch t 6 St bilit4"


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Figure (6.11

/ircraft pitch loopmodel

Ch t 6 St bilit5#


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Figure (6.128lock diagram of achemical process-controlsystem

Ch t 6 St bilit51


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Figure (6.13a. Soft Arm used for feeding;b. simplified

blockdiagram

Courtesy of )a*u%i+o )a,a-ura /a#derbilt #i ersity.



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Figure (6.14

*o&ed vehicle rollcontrol



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Figure (6.15Cutting force controlsystem

1997 .



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Figure (6.16a. / magneticlevitationtransportationsystem;b. simplifiedblock

diagram 9 #220IEEE

a$a# ir i#es (%oto 'esear"%ers.


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Table 6.2Completed Routh table



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Table 6.3Completed Routh table for E ample !"#



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Table 6.4Completed Routh table for E ample !"$


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Table 6.7

Routh table for E ample !")



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Table 6.8Routh table for E ample !",


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p y

Table 6.11 Routh table for E ample !".


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p y

Table 6.13




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Table 6.14ummary of pole locations for E ample !"0



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Table 6.15Routh table forE ample !"2

Chapter 6: Stability#


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Table 6.16Routh table for E ample !"2 &ith K 3 #(0!



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Table 6.17

Routh table for E ample !"#7



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Table 6.18Routh table for E ample !"##



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Table 6.19Routh table for antenna controlcase study



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Table 6.20Routh table for 5F case study

chapter 6 stability criteria_nise

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