Chapter 14

Frequency Response

Force dynamic process with A sin ωt , 22

)(ωω+

=s

AsU

Chapter 14

14.1

1

Input:

Output:

is the normalized amplitude ratio (AR)

φ is the phase angle, response angle (RA)

AR and φ are functions of ω

Assume G(s) known and let

tsinA ω

( )φω +tAsinˆ

( ) 1 2

2 2

1 2

2

1

arctan

s j G j K K j

G AR K K

KG

K

ω ω

φ

= = +

= = +

= ∠ =

Chapter 14

AA /ˆ

2

Example 14.1:

( ) 21 1( 1)

1 1

jG j j

j j

τ ωω

τ ω τ ω−

= ⋅ = −+ −

( ) 1

1G s

sτ=

+

( ) 2 2 2 2

1

1 1G j j

ωτω

ω τ ω τ= −

+ +

K1 K2

Chapter 14

3

Chapter 14

4

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

( )

2 2

0

1

1

arctan

as , 90

Gω τ

φ ωτ

ω φ

=+

= −

→∞ → −

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

= ⋅ ⋅

= ⋅ ⋅

= + +

∠ = ∠ + ∠ + ∠

=

= −

∠ = ∠ −∠

Chapter 14

5

Figure 14.4 Bode diagram for a time delay, e-θs.

Chapter 14

6

Chapter 14

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 ωb = 1/τI . Asymptotic slope (ω→ 0) is -1 on log-log plot.

Recall that the F.R. is characterized by:

1. Amplitude Ratio (AR)

2. Phase Angle (φ)

F.R. Characteristics of Controllers

For any T.F., G(s)

A) Proportional Controller

B) PI Controller

For

( )

( )

AR G j

G j

ω

φ ω

=

=∠

( ) , 0C C CG s K AR K φ= ∴ = =

2 2

1

1 1( ) 1 1

1tan

C C C

I I

I

G s K AR Ksτ ω τ

φτ ω

−

= + = +

= −

Chapter 14

8

Chapter 14

9

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)

τ

Ic c D

I

sG s K s

s

+= +

( ) τ 1 τ 1(14-51)

τ ατ 1

I Dc c

I D

s sG s K

s s

+ += +

Chapter 14

Ideal PID Controller.

1( ) (1 ) (14 48)

c c D

I

G s K ss

ττ

= + + −

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 1c

s sG s

s s

+ + = +

( ) 12 1 4

10cG s s

s

= + + C

hapter 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)

Chapter 14

12

Controller Design by Frequency Response

- Stability Margins

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

Three methods in use:

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

(2) Nyquist plot - polar plot of G(jω) - Appendix J

(3) Nichols chart |G|, φ 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.

Chapter 14

13

Chapter 14

14.8

14

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.

Cω

value of C

ω ω≡

Chapter 14

15

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:

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)

C

OL

KG s

s=

+

3

2( )

(0.5 1)p

G ss

=+

Chapter 14

16

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

Chapter 14

17

• 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 )cu

G c

KAR

=Chapter 14

18

Example 14.7:

Determine the closed-loop stability of the system,

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

Bode Diagram. What is the maximum value of Kc for a stable

system?

Solution:

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

Note that:

15

4)(

+=

−

s

esG

s

p

OL

max

1.69rad min

0.235

1 1= 4.25

0.235

C

C

C

OL

AR

KAR

ω ω

ω

=

=

=

∴ = =

Chapter 14

19

Chapter 14

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

Pπω

≡

1for 1

C

CU C

OL

K KAR

ω ω=

= =

Chapter 14

21

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 ω = ωC. Then the gain margin is

defined as: GM = 1/AC

According to the Bode Stability Criterion, GM >1 ⇔ stability

• Phase Margin:

Let ωg = frequency at which AROL = 1.0 and the

corresponding phase angle is φg . The phase margin

is defined as: PM = 180° + φg

According to the Bode Stability Criterion, PM >0 ⇔ stability

See Figure 14.12.

Chapter 14

22

Chapter 14

23

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≤ ≤ ≤ ≤o o

Chapter 14

24

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Chapter 14

25