Chap. 9 Stability in the Frequency Domain

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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.

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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.

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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

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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

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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 )

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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 )

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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 2N 2Z 2P

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

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Mapping Contours in the s-Plane

N 3 1 2

N Z P 1

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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.

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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.

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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.

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

K s(s 1)

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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.

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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.

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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.

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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.

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

K s( 1s 1)( 2 s 1)

K 2

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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.

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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.

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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

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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

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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

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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

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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

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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

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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

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2

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

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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.

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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.

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

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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

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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 .

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System Bandwidth T1 ( s)

1 s 1

T2 ( s)

1 5s 1

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System Bandwidth T3 ( s)

100 s 2 10s 100

T4 ( s)

900 s 2 30s 900

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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.

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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 jT

GH ( j ) GGc( j )e

The delay factor results in a phase shift

( ) T

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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.

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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

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