Preview • To show how frequency response methods can be used to investigate stability. – Gain margin, phase margin, and bandwidth in the context of Bode plots and Nyquist stability criterion. – Time delays in the system on both stability and performance. – Phase lag introduced by the time delay

Chap. 9 Stability in the Frequency Domain

2

Introduction • Stability – Routh-Hurwitz method (Ch. 6) – Root Locus (Ch. 7) – The stability of a system in the real frequency domain, in terms of the frequency response. (Ch. 8-9)

• The frequency response of a system provides sufficient information for the determination of the relative stability of the system. – The response can readily be obtained by exciting the system with sinusoidal input signal. – A frequency domain stability criterion would be useful for determining suitable approaches to adjusting the parameters of a system in order to increase its relative stability.

Chap. 9 Stability in the Frequency Domain

3

Introduction • Nyquist stability criterion – A fundamental approach to the investigation of the stability of linear control systems. – It is based on the theorem in the theory of the function of a complex variable : mapping contours. Char. eq

F (s)  1  L(s)  0

where

L(s)  G(s) H (s)

F (s)  (s)  1   Ln   Lm Lq   0

– All zeros of F(s) lie in the left-hand s-plane. • Mapping of the right-hand s-plane into the F(s)-plane.

Chap. 9 Stability in the Frequency Domain

4

Mapping Contours in the s-Plane • A contour map – A contour or trajectory in one plane mapped or translated into another plane by a relation F(s).

F (s)  2s  1  2(s  1 / 2) u  jv  F (s)  2s  1  2(  j )  1 u  2  1, v  2 Chap. 9 Stability in the Frequency Domain

5

Mapping Contours in the s-Plane • Conformal mapping – The mapping which retains the angles of the s-plane contour on the F(s)-plane.

• Area enclosed – The area with a contour to the right of the traversal of the contour is considered to be the area enclosed by the contour. – We will assume clockwise traversal of a contour to be positive and the area enclosed within the contour to be on the right.

F (s)  s /( s  2) Chap. 9 Stability in the Frequency Domain

6

Mapping Contours in the s-Plane • Cauchy’s theorem is concerned with mapping a function F(s) that has a finite number of poles and zeros within the contour, n F ( s )  1  L( s )  1 

N ( s ) D( s )  N ( s )   D( s ) D( s )

K  ( s  si ) i 1 M

 (s  s )

where L( s) 

N ( s) D( s )

k

k 1

– The poles of L(s) are the poles of F(s). – It is the zeros of F(s) that are the characteristic roots of the system and that indicate its response. Y ( s )  T ( s ) R( s ) 

P  k

( s )

k

R( s ) 

P  k

F ( s)

k

R( s )

Chap. 9 Stability in the Frequency Domain

7

Mapping Contours in the s-Plane • Cauchy’s theorem (the principle of the argument) If a contour in the s-plane encircles Z zeros and P poles of F(s) and does not pass through and poles or zeros of F(s) and the traversal is in the clockwise direction along the contour, the corresponding contour in the F(s)-plane encircles the origin of the F(s)-plane N=Z - P times in the clockwise direction. – The encirclement of the poles and zeros of F(s) can be related to the encirclement of the origin in the F(s)-plane.

F ( s)  Chap. 9 Stability in the Frequency Domain

s s  12 8

Mapping Contours in the s-Plane • Cauchy’s theorem F ( s) 

( s  z1 )(s  z2 ) ( s  p1 )(s  p2 )

F ( s)  F ( s) F ( s) 

s  z1 s  z 2 s  p1 s  p2

(( s  z1 )  ( s  z2 )  ( s  p1 )  ( s  p2 ))

 F ( s) ( z1   z2   p1   p2 )

Chap. 9 Stability in the Frequency Domain

9

Mapping Contours in the s-Plane • Cauchy’s theorem – If Z zeros were enclosed with in s, then the net angle would be equal to z=2(Z) rad. – If Z zeros and P poles are encircled as s is traversed, z=2(Z) 2(P) is the net resultant angle of F(s).  F  Z   P 2N  2Z  2P

– And the net number of encirclements of the origin of the F(s) is N=Z-P.

Chap. 9 Stability in the Frequency Domain

10

Mapping Contours in the s-Plane

N  3  1  2

N  Z  P  1

Chap. 9 Stability in the Frequency Domain

11

The Nyquist Criterion • For a system to be stable, all the zeros of the characteristic equation, F(s), must lie in the lefthand s-plane. n F ( s )  1  L( s ) 

K  ( s  si ) i 1 M

 (s  s ) k

k 1

– We choose a contour in the s-plane that encloses the entire right-hand splane, and we determine whether any zeros of F(s) lie within the contour.

N Z P • If P = 0, the number of unstable roots of the system is equal to N.

Chap. 9 Stability in the Frequency Domain

12

The Nyquist Criterion • We may define the function F (s)  F (s)  1  L(s)

• Nyquist stability criterion A feedback system is stable if and only if the contour L in the L(s)plane does not encircle the (-1,0) point when the number of poles of L(s) in the right-hand s-plane is zero (P=0).

• When the number of poles of L(s) in the right-hand splane is other than zero, the Nyquist criterion is stated: A feedback system is stable if and only if, for the contour L, the number of counterclockwise encirclements of the (-1,0) point is equal to the number of poles of L(s) with positive real parts.

Chap. 9 Stability in the Frequency Domain

13

The Nyquist Criterion • Example 9.1 (system with two real poles) GH ( s) 

100 ( s  1)(0.1s  1)

– The contour does not encircle the –1 point, and the system is always stable for all K greater than zero.

Chap. 9 Stability in the Frequency Domain

14

The Nyquist Criterion • Example 9.2 (system with a pole at the origin) GH ( s) 

K s(s  1)

Chap. 9 Stability in the Frequency Domain

15

The Nyquist Criterion • Example 9.2 (system with a pole at the origin) – cont. – The origin of the s-plane • The small semicircular detour around the pole at the origin can be represented by setting s  e j and allowing  to vary from –90 at =0- to + 90 at =0.  K  K lim GH ( s)  lim  j   lim  e  j  0  0 e    0    • The angle of the contour in the GH(s)-plane changes from 90 at =0- to -90 at =0+, passing through 0 at =0. • The radius of the contour in the GH(s)-plane for this portion of the contour is infinite.

Chap. 9 Stability in the Frequency Domain

16

The Nyquist Criterion • Example 9.2 (system with a pole at the origin) – cont. – The Portion from =0+ to  =+ GH (s) s  j  GH ( j ) K    j ( j  1)

lim GH ( j )  lim

 

 lim

 

K

 2

  ( / 2)  tan 1 

• Therefore, the magnitude approaches zero at an angle of -180.

Chap. 9 Stability in the Frequency Domain

17

The Nyquist Criterion • Example 9.2 (system with a pole at the origin) – cont. – The Portion from =+ to =- K 2 j e r  r 2

lim GH ( s) s re j  lim r 

• As  changes from +90 at =+ to -90 at =-. Thus the contour moves from an angle of -180 at =+ to +180 at  =-. • The magnitude of the GH(s) contour when r is infinite is always zero or a constant.

Chap. 9 Stability in the Frequency Domain

18

The Nyquist Criterion • Example 9.2 (system with a pole at the origin) – cont. – The Portion from =- to =0GH (s) s  j  GH ( j ) • The polar plot from =- to  =0- is symmetrical to the polar plot from =+ to  =0+.

– To investigate the stability of this system, • • • •

P within the right-hand s-plane is zero. We require N=Z=0 for this system to be stable. And the contour GH must not encircle the –1 point in the GH-plane. We find that irrespective of the gain K and the time constant , the contour does not encircle the –1 point, and the system is always stable.

Chap. 9 Stability in the Frequency Domain

19

The Nyquist Criterion • Example 9.3 (System with three poles) GH ( s) 

K s( 1s  1)( 2 s  1)

K 2

Chap. 9 Stability in the Frequency Domain

20

The Nyquist Criterion • Example 9.5 (System with a pole in the right-hand s-plane) GH ( s) 

K1 , when K 2  0 s( s  1)

The system is unstable.

Chap. 9 Stability in the Frequency Domain

21

The Nyquist Criterion • Example 9.5 (System with a pole in the right-hand s-plane) GH ( s) 

K1 (1  K 2 s) , when K 2  0 s( s  1)

The system is stable when K1K2>1.

Chap. 9 Stability in the Frequency Domain

22

Relative Stability and The Nyquist Criterion • The Nyquist criterion can be utilized to define and ascertain the relative stability of a system. – The Nyquist criterion is defined in terms of the (-1,0) point on the polar plot or the 0-dB, -180 point on the Bode diagram. GH ( j ) 

u

K j ( j 1  1)( j 2  1)

 K 1 2  1   2 Intersection point

The system has roots on the j-axis when u= -1

Chap. 9 Stability in the Frequency Domain

23

Relative Stability and The Nyquist Criterion • Gain margin: – The reciprocal of the gain |GH(j)| at the frequency at which the phase angle reaches –180. The gain margin is the increase in the system gain when phase = -180 that will result in a marginally stable system with intersection of the –1+j0 point on the Nyquist diagram.

• Phase margin: – The phase angle through which the GH(j) locus must be rotated so that the unity magnitude |GH(j)| = 1 point will pass through the (-1,0) point in the GH(j) plane. The phase margin is the amount of phase shift of the GH(j) at unity magnitude that will result in a marginally stable system with intersection of the –1+j0 point on the Nyquist diagram. Chap. 9 Stability in the Frequency Domain

24

Relative Stability and The Nyquist Criterion • The gain and phase margins from the Bode diagram. – The critical point, u=-1, v=0, in the GH(j) –plane is equivalent to a logarithmic magnitude of 0 dB and a phase angle of 180 on the Bode diagram.

GH ( j ) 

1 j ( j  1)(0.2 j  1)

Phase margin = 180 -137 =43 Gain margin = 15 dB

Chap. 9 Stability in the Frequency Domain

25

Relative Stability and The Nyquist Criterion • Magnitude-phase curve – The critical stability point is the 0 dB, -180 point. GH1 ( j ) 

1 j ( j  1)(0.2 j  1)

Phase margin =43 Gain margin = 15 dB GH 2 ( j ) 

1 j ( j  1) 2

Phase margin = 20

Gain margin = 5.7 dB Chap. 9 Stability in the Frequency Domain

26

Relative Stability and The Nyquist Criterion • The phase margin of a second-order system and relate the phase margin to the damping ratio of an underdamped system.  n2 Loop-transfer function of a second-order system GH (s)  s( s  2 n )

Characteristic equation for the system The closed-loop roots

s 2  2 n s   n2  0

s1, 2   n  j n 1   2

In the frequency domain:  n2 GH ( j )  j ( j  2 n )

The magnitude of the frequency response is equal to 1 at c  n2 1  c ( c2  4 2 n2 )1/ 2

 

2 2 c

 4 2 n2 ( c2 )   n4  0

Chap. 9 Stability in the Frequency Domain

27

Relative Stability and The Nyquist Criterion Solving for c

1/ 2  c2 4 2        4 1 2  n2

The phase margin for this system is Correlation between the frequency response and the time response

 c    2 n  

 pm  180  90  tan 1 

 1 1/ 2  (4 4  1)1/ 2  2 2   90  tan 1   2  1/ 2     1 1   tan 2    (4 4  1)1/ 2  2 2    

  0.01 pm

Chap. 9 Stability in the Frequency Domain

28

Time-domain Performance Criteria Specified in the Frequency Domain • The transient performance of a feedback system can be estimated from the closed-loop frequency response. G( j ) Y ( j )  T ( j )  1  GH ( j ) R( j )

In unity feedback system, T ( j )  M ( )e j ( ) 

G( j ) 1  G ( j )

The relationship between T(j) and G(j) G( j )  u  jv

G ( j ) u  jv u 2  v2   M 1  G ( j ) 1  u  jv (1  u ) 2  v 2

2

1/ 2

(1  M 2 )u 2  (1  M 2 )v 2  2M 2u  M 2

1/ 2

2 M 2u  M 2   M 2   M 2  2 2     u v   1  M 2  1  M 2   1  M 2   1  M 2 

2

2

 M2   M  2  u    v   2  2  1  M 1  M    

Chap. 9 Stability in the Frequency Domain

29

2

Time-domain Performance Criteria Specified in the Frequency Domain • We can plot several circles of constant magnitude M in the G(j)-plane.

Chap. 9 Stability in the Frequency Domain

30

Time-domain Performance Criteria Specified in the Frequency Domain • The open-loop frequency response for a system for two gain values, where K2 > K1, and closed-loop frequency response magnitude curves.

Chap. 9 Stability in the Frequency Domain

31

Time-domain Performance Criteria Specified in the Frequency Domain – Circles of constant closed-loop phase angles T ( j )  M ( )e

j ( )

G( j )  1  G ( j )

  T ( j )  

u  jv 1  u  jv

v  v   tan 1    tan 1   u 1 u 

taking tangent of both sides

v  0, where N  tan   const. adding 1/4[ 1+(1/N2)] to both sides N 2 1 1 1    (u  0.5) 2   v    1  2  2N  4 N  

u 2  v2  u 

– The constant phase angle curves can be obtained for various values of N.

• Nichols chart – The constant M and N circles to the log-magnitude-phase diagram.

Chap. 9 Stability in the Frequency Domain

32

Time-domain Performance Criteria Specified in the Frequency Domain • Nichols chart

Chap. 9 Stability in the Frequency Domain

33

Time-domain Performance Criteria Specified in the Frequency Domain • Example 9.7 – Stability using the Nichols chart G( j ) 

K j ( j  1)(0.2 j  1)

Maximum magnitude : +2.5dB at r =0.8 Closed-loop phase angle at r : -72 -3-dB closed-loop bandwidth : B =1.33 The closed-loop phase angle at B : -142

Chap. 9 Stability in the Frequency Domain

34

System Bandwidth • The bandwidth of the closed-loop system is an excellent measure of the range of fidelity of response of the system. – In systems where the low-frequency magnitude is 0dB on the Bode diagram, the bandwidth is measured at the –3dB frequency. – The speed of response to a step input will be roughly proportional to B , and the settling time is inversely proportional to B .

Chap. 9 Stability in the Frequency Domain

35

System Bandwidth T1 ( s) 

1 s 1

T2 ( s) 

1 5s  1

Chap. 9 Stability in the Frequency Domain

36

System Bandwidth T3 ( s) 

100 s 2  10s  100

T4 ( s) 

900 s 2  30s  900

Chap. 9 Stability in the Frequency Domain

37

The Stability of Control Systems with Time Delays • Many control systems have a time delay within the closed loop of the system that affects the stability of the system. • Time delay – The time interval between the start of an event at one point in a system and its resulting action at another point in the system.

• The Nyquist criterion can be utilized to determine the effect of the time delay on the relative stability of the feedback system. – A pure time delay, without attenuation Gd (s)  e sT , where T is the time delay

– The Nyquist criterion remains valid for a system with a time delay because the factor does not introduce any additional poles or zeros within the contour.

Chap. 9 Stability in the Frequency Domain

38

The Stability of Control Systems with Time Delays d T v

Loop-transfer function G( s)Gc ( s)e  sT The frequency response of the system is obtained from the loop transfer function  jT

GH ( j )  GGc( j )e

The delay factor results in a phase shift

 ( )  T

Chap. 9 Stability in the Frequency Domain

39

PID Controllers in the Frequency Domain • PID controller • PI controller • PD controller

K2  K3s s K Gc ( s)  K1  2 s

Gc ( s)  K1 

Gc (s)  K1  K3 s

• In general, PID controllers are particularly useful for reducing steady-state error and improving the transient response when G(s) has one or two poles.

Chap. 9 Stability in the Frequency Domain

40

PID Controllers in the Frequency Domain • PID controller

 K 3 2 K1   K2  s  s  1 K 2 s  1  s  1 K2 K2      Gc ( s)   s s

for K2=2 and =10

Chap. 9 Stability in the Frequency Domain

41

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