Feedback Control Systems (FCS)

Dr. Imtiaz Hussainemail: imtiaz.hussain@faculty.muet.edu.pk

URL :http://imtiazhussainkalwar.weebly.com/

Lecture-32-33Closed Loop Frequency Response

Introduction

• One of the important problems in analyzing a control system is to find all closed-loop poles or at least those closes to the jω axis (or the dominant pair of closed-loop poles).

• If the open-loop frequency-response characteristics of a system are known, it may be possible to estimate the closed-loop poles closest to the jω axis.

Closed Loop Frequency Response• For a stable, unity-feedback closed-loop system, the closed-loop

frequency response can be obtained easily from that of the open loop frequency response.

• Consider the unity-feedback system shown in following figure. The closed-loop transfer function is

)()(

)()(

sG

sG

sR

sC

1

Closed Loop Frequency Response• Following figure shows the polar plot of G(s).

• The vector OA represents G(jω1), where ω1 is the frequency at point A.

• The length of the vector OA is

• And the angle is

)( 1jG

)( 1jG

Closed Loop Frequency Response

• The vector PA, the vector from -1+j0 point to Nyquist locus represents 1+G(jω1).

• Therefore, the ratio of OA, to PA represents the closed loop frequency response.

)()(

)()(

1

1

1

1

1

jR

jC

jG

jG

PA

OP

Closed Loop Frequency Response• The magnitude of the closed loop

transfer function at ω=ω1 is the ratio of magnitudes of vector OA to vector PA.

• The phase of the closed loop transfer function at ω=ω1 is the angle formed by OA to PA (i.e Φ-θ).

• By measuring the magnitude and phase angle at different frequency points, the closed-loop frequency-response curve can be obtained.

Closed Loop Frequency Response

• Let us define the magnitude of the closed-loop frequency response as M and the phase angle as α, or

jMejR

jC

)()(

ieZ

Closed Loop Frequency Response

• Let us define the magnitude of the closed-loop frequency response as M and the phase angle as α, or

• From above equation we can find the constant-magnitude loci and constant-phase-angle loci.

• Such loci are convenient in determining the closed-loop frequency response from the polar plot or Nyquist plot.

jMejR

jC

)()(

Constant Magnitude Loci (M circles)• To obtain the constant-magnitude loci, let us first note

that G(jω) is a complex quantity and can be written as follows:

• Then the closed loop magnitude M is given as

• And M2 is

jYXjG )(

jYX

jYXM

1 )(

)()()(

sG

sG

sR

sC

1

22

222

1 YX

YXM

)(

Constant Magnitude Loci (M circles)

• Hence

• If M=1 then,

• This is the equation of straight line parallel to y-axis

and passing through (-0.5,0) point.

22

222

1 YX

YXM

)(

22222 21 YXYXXM

02 22222222 YXYMXMXMM

0121 222222 MYMXMXM )()(

012 X

2

1X

Constant Magnitude Loci (M circles)

• If M≠1 then,

• Add to both sides

0121 222222 MYMXMXM )()(

011

22

22

2

22

M

MYX

M

MX

22

2

1M

M

22

2

22

2

2

22

2

22

1111

2

M

M

M

M

M

MYX

M

MX

22

22

22

4

2

22

111

2

M

MY

M

MX

M

MX

Constant Magnitude Loci (M circles)

• This is the equation of a circle with

22

22

22

4

2

22

111

2

M

MY

M

MX

M

MX

22

22

2

2

22

11

M

MY

M

MX

1

01

2

2

2

M

Mradius

M

Mcentre ,

Constant Magnitude Loci (M circles)

• The constant M loci on the G(s) plane are thus a family of circles.

• The centre and radius of the circle for a given value of M can be easily calculated.

• For example, for M=1.3, the centre is at (–2.45, 0) and the radius is 1.88.

Constant Phase Loci (N circles)

• The phase angle of closed loop transfer function is

• The phase angle α is

jYX

jYX

jR

jC

1)(

)(

jYX

jYXe j

1

)(tan)(tanX

Y

X

Y

111

Constant Phase Loci (N circles)

• If we define

• then

• We obtain

)(tan)(tanX

Y

X

Y

111

Ntan

)(tan)(tantan

X

Y

X

YN

111

X

Y

X

YX

Y

X

Y

N

11

1

Constant Phase Loci (N circles)

X

Y

X

YX

Y

X

Y

N

11

1

22 YXX

YN

YYXXN )( 22

0122 YN

YXX

Constant Phase Loci (N circles)

0122 YN

YXX

Adding to both sides 24

1

4

1

N

2222

4

1

4

1

4

11

4

1

NNY

NYXX

2

22

4

1

4

1

2

1

2

1

NNYX

This is an equation of circle with

24

1

4

1

2

1

2

1

Nradius

Ncentre

,

Closed Loop Frequency Response

Closed Loop Frequency Response

END OF LECTURES-32-33

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