System type, steady state tracking, & Bode plot

11

11)()(

21

sTsTs

sTsTKsGsCsG

Nba

p

C(s) Gp(s)R(s) Y(s)

At very low frequency:gain plot slope = –20N dB/dec.phase plot value = –90N deg

Type = N

Type 0: gain plot flat at very low frequency phase plot approached 0 deg

Kv = 0

Ka = 0

Low freq phase = 0o

Type 1: gain plot -20dB/dec at very low frequency phase plot approached 90 deg

Kp = ∞

Ka = 0

Low freq phase = -90o

=Kv

Low frequency tangent line

N = 2, type = 2Bode gain plot has –40 dB/dec

slope at low freq.Bode phase plot becomes flat

at –180° at low freq.

Kp = DC gain → ∞Kv = ∞ also

Ka = value of straight line at ω = 1

= s0dB^2

Back to general theory

log20log20log20 NKjG

0

0

Asymptotic straight line:

20 log 20log

At =1: 20log 20log

When y=1, straight line cross hor. axis.

The crossing frequency is:

0 dB 20log 20log

20 log 2 20log

a

s dB

a s dB

y K N

y K K

K N

K

20

20

20 log 20log

a s dB

a s dB

K

K

Type 1: gain plot -40dB/dec at very low frequency phase plot approached 180 deg

Kp = ∞

Kv = ∞

Low freq phase = -180o

Low frequency tangent line

Example

Ka

s0dB=Sqrt(Ka)

How should the phase plot look like?

2 :freq lowAt

j

KjG

2

:linestraight

KjG

16,142

KK

4at dB 0 isit

1||

vK pK

16 aK

Example continued

Suppose the closed-loop system is stable:If the input signal is a step, ess would be =

If the input signal is a ramp, ess would be =

If the input signal is a unit acceleration, ess would be =

Example continued

System type, steady state tracking, & Bode plot

At very low frequency:gain plot slope = –20N dB/dec.phase plot value = –90N deg

If LF gain is flat, N=0, Kp = DC gain, Kv=Ka=0

If LF gain is -20dB/dec, N=1, Kp=inf, Kv=LFg_tan_c , Ka=0

If LF gain is -40dB/dec, N=2, Kp=Kv=inf, Ka=(LFg_tan_c)2

System type, steady state tracking, & Nyquist plot

11

11

21

jTjTj

jTjTKjG N

ba

C(s) Gp(s)

Nj

KjG

As ω → 0

Type 0 system, N=0

Kp=lims0 G(s) =G(0)=K

0+

Kp

G(j)

Type 1 system, N=1Kv=lims0 sG(s) cannot be determined easily from Nyquist plot

0+

infinity

G(j) -j∞

Type 2 system, N=2

Ka=lims0 s2G(s) cannot be determined easily from Nyquist plot

0+

infinity

G(j) -∞

System type on Nyquist plot

Kp

System relative order

Examples

System type =

Relative order =

System type =

Relative order =

Margins on Bode plots

In most cases, stability of this closed-loop

can be determined from the Bode plot of G:– Phase margin > 0– Gain margin > 0

G(s)

freq.over -crossgain :gc dB0or 1 ,at jGgc

margin phase :PM

gcjG 180

freq.over -cross phase :pc

180pcjG

dB log20margingain : pcjGGM

in value 1 pcjG

If never cross 0 dB line (always below 0 dB line), then PM = ∞.

If never cross –180° line (always above –180°), then GM = ∞.

If cross –180° several times, then there are several GM’s.

If cross 0 dB several times, then there are several PM’s.

jG

jG

jG

jG

52

1100

ss

ssGExample:

Bode plot on next page. 11

110

51

21

ss

s

100near line dB 0 cross .1 jG100 gc

_______PM

______at 180 cross .2 pcωjG _______GM

254

252

sss

sGExample:

Bode plot on next page.

______near line dB 0 cross .1 jG

______ gc

______about is at gcωjG

_______PM

1

1

2542

251

sss

1. Where does cross the –180° lineAnswer: __________

at ωpc, how much is

2. Closed-loop stability: __________

_______jG

________ pc

jG

________GM

1

120

2

40 :Example

21

ssss

sG

1. crosses 0 dB at __________

at this freq,

2. Does cross –180° line? ________

3. Closed-loop stability: __________

_______ jG

________ gc jG

________GM

________PM

jG

Margins on Nyquist plot

Suppose:• Draw Nyquist plot G(jω) & unit circle

• They intersect at point A• Nyquist plot cross neg.

real axis at –k

in value1kGM

indicated angle :Then PM

-100 -50 0 50-200

-150

-100

-50

0

50

100

150

200 Nyquist Diagram

Real Axis

Imag

inary

Axis

-2 -1.5 -1 -0.5 0-2

-1.5

-1

-0.5

0

0.5

1

1.5

2 Nyquist Diagram

Real Axis

Imagin

ary Ax

is

-4 -2 0 2 4-10

-5

0

5

10 Nyquist Diagram

Real Axis

Imagin

ary Ax

is

-2 -1.5 -1 -0.5 0-2

-1.5

-1

-0.5

0

0.5

1

1.5

2 Nyquist Diagram

Real Axis

Imagin

ary Ax

is