bode stability

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Chapter 14

Frequency Response

Force dynamic process with A sin t ,22

)(

s

AsU

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14.1

1


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Input:

Output:

is the normalized amplitude ratio (AR) is the phase angle, response angle (RA)

AR and are functions of

Assume G(s) known and let

tsinA

tAsin

1 22 2

1 2

2

1

arctan

s j G j K K j

G AR K K

KG

K

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AA /

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Example 14.1:

21 1

( 1)1 1

j

G j jj j

1

1G s

s

2 2 2 21

1 1G j j

K1

K2

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(plot of log |G| vs. log and vs. log )

2 2

0

1

1

arctan

as , 90

G

Use a Bode plot to illustrate frequency response

log coordinates:

1 2 3

1 2 3

1 2 3

1 2 3

1

2

1 2

1 2

log log log log

log log log

G G G G

G G G G

G G G G

G G G G

GG

G

G G G

G G G

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Figure 14.4 Bode diagram for a time delay, e-qs

.

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Example 14.3

0.55(0.5 1)( )

(20 1)(4 1)

ss e

G s

s s

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The Bode plot for a PI controller is shown in next slide.

Note b = 1/I . Asymptotic slope (

0) is -1 on log-log plot.

Recall that the F.R. is characterized by:

1. Amplitude Ratio (AR)

2. Phase Angle ()

F.R. Characteristics of Controllers

For any T.F., G(s)

A) Proportional Controller

B) PI Controller

For

( )

( )

AR G j

G j

( ) , 0C C CG s K AR K

2 2

1

1 1( ) 1 1

1tan

C C C

I I

I

G s K AR K

s

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Series PID Controller. The simplest version of the series PID

controller is

Series PID Controller with a Derivative F il ter. The series

controller with a derivative filter was described in Chapter 8

1

1 (14-50)

Ic c D

I

sG s K s

s

1 1

(14-51) 1

I Dc c

I D

s sG s K

s s

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I deal PID Controll er.

1( ) (1 ) (14 48)c c DI

G s K ss

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Figure 14.6 Bode

plots of ideal parallel

PID controller andseries PID controller

with derivative filter

( = 1).

Ideal parallel:

Series withDerivative Filter:

10 1 4 1

210 0.4 1

c

s sG s

s s

1

2 1 410

cG s ss

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Advantages of FR Analysis for Controller Design:

1. Applicable to dynamic model of any order

(including non-polynomials).

2. Designer can specify desired closed-loop response

characteristics.

3. Information on stability and sensitivity/robustness is

provided.

Disadvantage:

The approach tends to be iterative and hence time-consuming

-- interactive computer graphics desirable (MATLAB)

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Controller Design by Frequency Response

- Stability Margins

Analyze GOL(s) = GCGVGPGM (open loop gain)

Three methods in use:

(1) Bode plot |G|, vs. (open loop F.R.) - Chapter 14

(2) Nyquist plot - polar plot of G(j) - Appendix J(3) Nichols chart |G|, vs. G/(1+G) (closed loop F.R.) - Appendix J

Advantages:

do not need to compute roots of characteristic equation can be applied to time delay systems

can identify stability margin, i.e., how close you are to instability.

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Chapter14

14.8

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Frequency Response Stability Criteria

Two principal results:

1. Bode Stability Criterion

2. Nyquist Stability Criterion

I) Bode stability criterion

A closed-loop system is unstable if the FR of the

open-loop T.F. GOL=GCGPGVGM, has an amplitude ratio

greater than one at the critical frequency, . Otherwise

the closed-loop system is stable.

Note: where the open-loop phase angle

is -1800. Thus,

The Bode Stability Criterion provides info on closed-loop

stability from open-loop FR info.

Physical Analogy: Pushing a child on a swing or

bouncing a ball.

C

value ofC

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Example 1:

A process has a T.F.,

And GV= 0.1, GM= 10 . If proportional control is used, determine

closed-loop stability for 3 values ofKc: 1, 4, and 20.

Solution:

The OLTF is GOL=GCGPGVGM or...

The Bode plots for the 3 values of Kc shown in Fig. 14.9.Note: the phase angle curves are identical. From the Bode

diagram:

KC AROL Stable?

1 0.25 Yes

4 1.0 Conditionally stable

20 5.0 No

3

2( )

(0.5 1)

COL

KG s

s

3

2( )

(0.5 1)

pG s

s

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Figure 14.9 Bode plots forGOL = 2Kc/(0.5s + 1)3.

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For proportional-only control, the ultimate gainKcu is defined to

be the largest value ofKc that results in a stable closed-loop

system.

For proportional-only control, GOL= KcG and G = GvGpGm.

AROL()=Kc ARG() (14-58)

whereARG denotes the amplitude ratio ofG.

At the stability limit, = c,AROL(c) = 1 andKc= Kcu.

1(14-59)

( )cu

G c

KAR

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Example 14.7:

Determine the closed-loop stability of the system,

Where GV= 2.0, GM= 0.25 and GC=KC . Find Cfrom the

Bode Diagram. What is the maximum value ofKc for a stable

system?

Solution:

The Bode plot forKc= 1 is shown in Fig. 14.11.

Note that:

15

4

)(

s

e

sG

s

p

OL

max

1.69rad min

0.235

1 1= 4.25

0.235

C

C

C

OL

AR

KAR

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Ultimate Gain and Ultimate Period

Ultimate Gain: KCU= maximum value of |KC| that results in astable closed-loop system when proportional-only

control is used.

Ultimate Period:

KCUcan be determined from the OLFR when

proportional-only control is used withKC=1. Thus

Note: First and second-order systems (without time delays)

do not have aKCUvalue if the PID controller action is correct.

2

U

C

P

1for 1

C

CU COL

K KAR

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Gain and Phase Margins

The gain margin (GM) and phase margin (PM) provide

measures of how close a system is to a stability limit.

Gain Margin:

Let AC=AROL at = C. Then the gain margin is

defined as: GM= 1/AC

According to the Bode Stability Criterion, GM>1 stability

Phase Margin:

Let g= frequency at which AROL = 1.0 and the

corresponding phase angle is g . The phase margin

is defined as:PM= 180

+ g

According to the Bode Stability Criterion, PM>0 stability

See Figure 14.12.

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Rules of Thumb:

A well-designed FB control system will have:

Closed-Loop FR Characteristics:

An analysis of CLFR provides useful information aboutcontrol system performance and robustness. Typical desired

CLFR for disturbance and setpoint changes and the

corresponding step response are shown in Appendix J.

1.7 2.0 30 45GM PM

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