Stability from Nyquist plot

• Plot G(jω) for ω > 0• Flip about real axis to get plot of G(jω) for

ω < 0• If G(s) has pole(s) on jω-axis, make a

small detour around these pole(s), so that G(jω) is connected

• Count the #encirclement of –1 by G(jω)• Nyquist criterion: Z=P+N

Example:

G(s) stable, P = 0

G(jω) for ω > 0 as given.

1. Get G(jω) forω < 0 by conjugating

2. Connect ω = 0– to ω = 0+.But how?

Choice a) :

Where’s “–1” ?

# encirclement N = _______

Z = P + N = _______

Make sense? _______

Incorrect

Choice b) :

Where is“–1” ?

# encir.N = _____

Z = P + N= _______

closed-loopstability _______

Correct

Incorrect

Example: G(s) stable, P = 0

1. Get conjugatefor ω < 0

2. Connect ω = 0–

to ω = 0+.

Needs to goone full circlewith radius ∞.

Choice a) :

N = 0

Z = P + N = 0

closed-loopstable

Incorrect!

Choice b) :

N = 2

Z = P + N= 2

Closedloop has two unstable poles

Correct!

Example: G(s) has one unstable pole

P = 1, no unstable zeros

1. Get conjugate

2. Connectω = 0–

to ω = 0+.How?One unstablepole/zeroIf connect in c.c.w.

# encirclement N = ?

If “–1” is to the left of A

i.e. A > –1

then N = 0

Z = P + N = 1 + 0 = 1

but if a gain is increased, “–1” could be inside, N = –2

Z = P + N = –1

c.c.w. is impossible

If connect c.w.:

For A > –1N = ______

Z = P + N

= ______

For A < –1N = ______

Z = ______

No contradiction. This is the correct way.

Example: G(s) stable, minimum phase

P = 0

G(jω) as given:

get conjugate.

Connect ω = 0–

to ω = 0+,00 Kdirection c.w.

If A < –1 < 0 :N = ______Z = P + N = ______stability of c.l. : ______

If B < –1 < A : A=-0.2, B=-4, C=-20N = ______Z = P + N = ______closed-loop stability:

______

Gain margin: gain can be varied between (-1)/(-0.2) and (-1)/(-4),

or can be less than (-1)/(-20)

If C < –1 < B :N = ______Z = P + N = ______closed-loop stability: ______

If –1 < C :N = ______Z = P + N = ______closed-loop stability: ______

G(s)

Open vs Closed Loop Frequency Response And Frequency Domain Specifications

C(s)

Goal: 1) Define typical “good” frequency response shape for closed-loop 2) Relate closed-loop freq response shape to step response shape 3) Relate closed-loop freq shape to open-loop freq response shape 4) Design C(s) to make C(s)G(s) into “good” shape.

Mr and BW are widely used

Closed-loop phase resp. rarely used

10-0.6

10-0.4

10-0.2

100

100.2

-15

-10

-5

0

5

10

15 =0.1

=0.2

=0.3

=0.4

=0.1

0.2

0.3

No resonancefor <= 0.7

Mr=0.3dB for =0.6

Mr=1.2dB for =0.5

Mr=2.6dB for =0.4

For small zeta,resonance freqis about n

BW ranges from0.5n to 1.5n

For good range,BW is 0.8~1.25 n

So take BW ≈ n

Prototype 2nd order system closed-loop frequency response

n

-3dB

BW

1.580.63

0.79 1.26

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Closed-loop BW to n ratio

BW≈n

BW/n

100

-20

-15

-10

-5

0

5

10

15

=0.1

0.2

0.3

No resonancefor <= 0.7

Mr<0.5 dB for =0.6

Mr=1.2 dB for =0.5

Mr=2.6 dB for =0.4

When >=0.6no visible resonance peak

Prototype 2nd order system closed-loop frequency response

n

When <=0.5 visible resonance peak near n

n

Since we design for >=0.5, Mr and r are of less value

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.750

0.5

1

1.5

2

2.5

3

Mr in value

Mr in dB

Prototype 2nd order system closed-loop frequency response

Mr vs

0.4 0.45 0.5 0.55 0.6 0.650

0.5

1

1.5

2

2.5

3

Mr in dB

20.7

50.4

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.750

5

10

15

20

25

30

Percentage Overshoot in closed-loop step response

> 0.5 is good

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.40

5

10

15

20

25

30

Mr

Percentage Overshoot in closed-loop step response

Mr < 15% is good,

>40% not tolerable

0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

30

Percentage Overshoot in closed-loop step response

Mr in dB

Mr < 1 dB is good, >3 dB not tolerable

10-2

10-1

100

101

102

-200

-150

-100

-50

0

50

100

150

n

=0.1 0.2 0.3

0.4

gc

In the range of good zeta,gc is about 0.7 times n

Open loop frequency response

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Open-loop gc to n ratio

gc≈0.7n

10-2

10-1

100

101

102

-180

-170

-160

-150

-140

-130

-120

-110

-100

-90

n

=0.10.2

0.3

0.4

In the range of good zeta,PM is about 100*

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.930

35

40

45

50

55

60

65

70

Phase Margin

PM = 100

30 35 40 45 50 55 60 65 700

5

10

15

20

25

30

35

40

Phase Margin in degrees: PM in deg

Percentage Overshoot in closed-loop step response

PM+Mp =70 line

Pe

rce

nta

ge

ove

rsh

oot:

Mp

Important relationships• Closed-loop BW are very close to n

• Open-loop gain cross over gc ≈ (0.65~0.8)*

n,

• When <= 0.6, r and n are close

• When >= 0.7, no resonance• determines phase margin and Mp:

0.4 0.5 0.6 0.7

PM 44 53 61 67 deg ≈100 Mp 25 16 10 5 %

PM+Mp ≈70

Mid frequency requirements• gc is critically important

– It is approximately equal to closed-loop BW

– It is approximately equal ton • Hence it determines tr, td directly

• PM at gc controls – Mp 70 – PM

• PM and gc together controls and d

– Determines ts, tp

• Need gc at the right frequency, and need sufficient PM at gc

Low frequency requirements• Low freq gain slope and/or phase

determines system type

• Height of at low frequency determine error constants Kp, Kv, Ka

• Which in turn determine ess

• Need low frequency gain plot to have sufficient slope and sufficient height

High frequency requirements• Noise is always present in any system

• Noise is rich in high frequency contents

• To have better noise immunity, high frequency gain of system must be low

• Need loop gain plot to have sufficient slope and sufficiently small value at high frequency

Desired Bode plot shape

Ess requirement

Noise requirement

0

-90

-180

0dB

gcd

High low-freq-gain for steady state trackingLow high-freq-gain for noise attenuationSufficient PM near gc for stability

PMd

Mid frequency

C(s) Gp(s)

21

21)(psps

zszsKsC

Controller design with Bode

From specs: => desired Bode shape of Gol(s)Make Bode plot of Gp(s) Add C(s) to change Bode shape as desiredGet closed loop systemRun step response, or sinusoidal responseModify controller as needed