Chapter 14Frequency Response

Force dynamic process with A sin t , 22)(

sA

sU

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14.1

1

Input:

Output: is the normalized amplitude ratio (AR) is the phase angle, response angle (RA)AR and are functions of ωAssume G(s) known and let

tsinA

tAsinˆ

1 2

2 21 2

2

1

arctan

s j G j K K j

G AR K K

KG

K

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AA /ˆ

2

Example 14.1:Example 14.1:

21 1( 1)

1 1

jG j j

j j

1

1G s

s

2 2 2 2

1

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-s.

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Example 14.3

0.55(0.5 1)( )

(20 1)(4 1)

ss eG s

s s

7

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 CI I

I

G s K AR Ks

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Series PID Controller. The simplest version of the series PID controller is

Series PID Controller with a Derivative Filter. The series controller with a derivative filter was described in Chapter 8

τ 1τ 1 (14-50)

τI

c c DI

sG s K s

s

τ 1 τ 1(14-51)

τ ατ 1I D

c cI D

s sG s K

s s

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Ideal PID Controller.

1( ) (1 ) (14 48)c c D

I

G s K ss

10

Figure 14.6 Bode plots of ideal parallel PID controller and series PID controller with derivative filter (α = 1).

Ideal parallel:

Series with Derivative Filter:

10 1 4 12

10 0.4 1cs s

G ss s

12 1 4

10cG s ss

C

hap

ter

14

11

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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14

Frequency Response Stability Criteria

Two principal results:1. Bode Stability Criterion2. Nyquist Stability Criterion

I) Bode stability criterionA 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 of C

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Example 1: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 of Kc: 1, 4, and 20.

Solution: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 Bodediagram:KC AROL Stable?1 0.25 Yes4 1.0 Conditionally stable20 5.0 No

3

2( )

(0.5 1)C

OL

KG s

s

3

2( )

(0.5 1)pG ss

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

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• For proportional-only control, the ultimate gain Kcu is defined to be the largest value of Kc that results in a stable closed-loop system.

• For proportional-only control, GOL= KcG and G = GvGpGm.AROL(ω)=Kc ARG(ω) (14-58)

where ARG denotes the amplitude ratio of G.

• At the stability limit, ω = ωc, AROL(ωc) = 1 and Kc= Kcu.

1(14-59)

(ω )cuG c

KAR

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Example 14.7:Example 14.7:

Determine the closed-loop stability of the system,

Where GV = 2.0, GM = 0.25 and GC =KC . Find C from theBode Diagram. What is the maximum value of Kc for a stablesystem?

Solution:Solution:

The Bode plot for Kc= 1 is shown in Fig. 14.11. Note that:

15

4)(

s

esG

s

p

OL

max

1.69 rad min

0.235

1 1= 4.25

0.235

C

C

COL

AR

KAR

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14.11

20

Ultimate Gain and Ultimate Period

• Ultimate Gain: KCU = maximum value of |KC| that results in a stable closed-loop system when proportional-only control is used.

• Ultimate Period:

• KCU can be determined from the OLFR when proportional-only control is used with KC =1. Thus

• Note: First and second-order systems (without time delays) do not have a KCU value if the PID controller action is correct.

2U

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 about control 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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