Stability in the Frequency Domain

In previous chapters, we discussed stability and developed various tools to determine stability.

For a control system, it is necessary to determine whether the system is stable.

To determine the stability of a closed-loop system, we must investigate the characteristic equation of the system.

Introduction

0)()(1)( sHsGsF

)(sG

)(sH

Characteristic Equation

To ensure stability we must ascertain that all the zeros of F(s) lie in the left hand of s-plane.

A frequency domain stability criterion was developed by H.Nyquist in 1932.

The Nyquist stability criterion is based on a theorem in the theory of the function of a complex variable due to CauchyThe argument principle (Cauchys theorem)

Introduction

The Nyquist Criterion

A closed loop system is stable if Z=0, where Z=P-2N P is the number of open-loop poles of G(s)H(s) in the right-hand of s-plane. N is the number of counterclockwise

encirclements of the (-1, j0) point

for the positive frequency

value( ). Z is unstable poles of the system.

G(s)H(s)

0 -1

Re

Im

0

A method to investigate the stability of a system in terms of the open-loop frequency response.

It is sufficient and necessary condition of the stability of the linear systems.

Consider a unity-gain feedback system whose open-loop transfer function is as follows

1

2)(

ssG

Stable

1P2

1N

02 NPZ

2 1

G(s)

0

Example 1

A system is stable if Z=0, where Z=P-2N P is the number of open-loop poles of G(s)H(s) in the right-hand of s-plane. N is the number of counterclockwise encirclements of the (-1, j0) point

for the positive frequency value. Z is unstable poles of the system.

Consider a unity-gain feedback system whose open-loop transfer function is as follows

1

5.0)(

ssG

Unstable

1P 0N

12 NPZ

1

5.0

G(s)

0

Example 2

There is one unstable pole of the system is in the right-hand of s-plane.

Consider a single-loop control system, where open-loop transfer function is

)11.0)(1(

100)()(

sssHsG

0 ,0 NP

02 NPZ

Stable

G(s)H(s)

1

0

Example 3

Consider a single-loop control system, where open-loop transfer function is

)42()(

2

sss

KsG )(sG

Determine the limiting value of K in order to maintain a stable system.

Example 4

2n000 1809090)( njG

142

)2(

K

jG

In order to maintain a stable system

8K

The changes of the open-loop gain only alter the magnitude of G(j)H(j).

1

G(j)H(j) locus traverses the left real axis of the point (-1, j0) in G(j)H(j)-plane L()0dB and () =180o in Bode diagram

)()(

GHL lg20

1 2 3

GH

)0( LNNN

1

2 3 0

1

)( )(

NNN

Application of the Nyquist criterion in the Bode diagram

Application of the Nyquist criterion in the Bode diagram

We have the Nyquist criterion in the Bode diagram : The sufficient and necessary condition of the stability of the linear closed loop systems is : When vary from 0+ , the number of the net positive traversing is P/2.

Here: the net positive traversing Nthe difference between the number of the positive traversing and the number of the negative traversing in all L()0dB ranges of the open-loop systems Bode diagram. N=N+-N-

positive traversingN+ () traverses the -180o line from below to above in the open-loop systems Bode diagram; negative traversing N- () traverses the -180o line from above to below.

)()(

GHL lg20

1 2 3

GH

)0( LNNN

1

2 3 0

1

)( )(

NNN

The Bode diagram of a open-loop stable system is shown in Fig., determine whether the closed loop system is stable.

In terms of the Nyquist criterion in the Bode diagram:

Because the open-loop system is stable, P = 0 .

The number of the net positive traversingis 0 ( P/2 = 0 ).

The closed loop system is stable .

Solution

0dB, 0o

)(L20

40

60

40

20

40

60

270o

90o

L()

()

180o )(

)(

Application of the Nyquist criterion in the Bode diagram

The Nyquist criterion provides us with suitable information concerning the absolute stability and, furthermore, can be utilized to define and ascertain the relative stability of a system.

The Nyquist stability criterion is defined in terms of (-1,0) point on the polar plot or the (0dB, -1800) point on the Bode diagram. Clearly the proximity of the -locus to this stability point is a measure of the relative stability of a system.

In frequency domain, the relative stability could be described by the gain margin and the phase margin.

Relative Stability and the Nyquist Criterion

)()( jHjG

The gain margin is defined as

|)()(|

1

gg jHjGh

The phase margin is

Gain margin and phase margin

(dB) )()(lg20lg20 ggn jHjGhL or

)()(1800 cc jHjG

GHg

j

0

[GH]

gGHlg 20

L

0

GHGH

L

(a) (b)

c

c

-1

g

uencysover freqPhase-crosjHjGg

180)()( : 0g

encyover frequGain-crossjHjGc

c 1)()( :

The geometrical meanings is shown in this Figure

Re 1

Im

1/h

stable

Critical stability

unstable

The physical signification : h amount of the open-loop gain that can be allowed to increase before the closed-loop system reaches to be unstable.

For the minimum phase system: h>1 the closed loop system is stable .

amount of the phase shift of G(j)H(j) to be allowed before the closed-loop system reaches to be unstable.

For the minimum phase system: >0 the closed loop system is stable.

Gain margin and phase margin

)12.0)(1(

1)()(

jjjjHjG

Gain margin

Phase margin

Example

Example

A unity-gain feedback system whose open-loop transfer function is :

try to find gain margin and phase margin.

)102.0)(15.0(

10)(

ssssG

2

50c

[-20][-40]

[-60]

180

270

90

dbjG )(lg20 )( / jG

)( jG

)(lg20 jG

)/1( s

hLg

(1) , 20lg ( ) 0 4.47c c cwhen G j

180 ( )

180 90 (0.5 ) (0.02 )

19

c

c c

G j

arctg arctg

(2) = , ( ) 180g gwhen G j

90 (0.5 ) (0.02 ) 180g garctg arctg

[ (0.5 ) (0.02 )] 90g gtg arctg arctg tg

10g

20lg ( ) 14h gL G j db

gain margin is 190

phase margin is 14 dB