control system analysis & design by frequency response

CONTROL SYSTEM ANALYSIS & DESIGN

BY FREQUENCY RESPONSE

MAPUA INSTITUTE OF TECHNOLOGY

School of Chemical Engineering & Chemistry

Process Control & Instrumentation

LEARNING OBJECTIVES

• Be able to apply the FREQUENCY RESPONSE method in stability analysis and in the design of controllers

• Be familiar with the BODE Stability Criterion

• Have an understanding of the concept of GAIN and PHASE MARGINS

• Be able to determine initial PID settings

FREQUENCY RESPONSE• The response of an output variable to a sinusoidal input.• Key determinants are the AMPLITUDE and the FREQUENCY

of the sinusoid.

U(s)

Consider a 1st Order System:

Transfer Function = ratio of the Laplace Transform of

the output variable to that of the input variable

• Y(s) = Laplace Transform of the DEVIATION VARIABLE Y

Y(s)Y(s)

where:

• U(s) = Laplace Transform of the DEVIATION VARIABLE U

U(s)=

K

(s + 1)

(5-3)

Suppose a Sinusoidal Change in u is applied:

0 time, t

u

us

A u(t) = us + (A)sint

Y(t) = ?

A

A

where:• A = Amplitude = radian frequency

U(t) = (A)sintPeriod of Oscillation, T

(5-14)

• Radian Frequency = 2fwhere: f = cyclical frequency = 1/T

Parameters of the Sinusoidal Input:

• T = Period of Oscillation = the time it takes to complete one cycle

Sinusoidal Response of a 1st Order System:

0 time, t

y

ys

A

Y(t) = AKe-t/

(22 + 1)

AK

(22 + 1)+ sin(t + )

AK

(22 + 1)

(5-26)

• First Order Sinusoidal Response is attenuated– ratio of the amplitude of Y to that of U is less than 1

• First Order Sinusoidal Response lags the Sinusoidal Input by the phase angle = tan-1(-)• phase lag = ||/ 360f

Properties of 1st Order Sinusoidal Response:

(5-27)

0 time, t

u, y

us, ys

Input (u)

Output (y)

1st Order Frequency Response:• As the frequency increases, the phase angle Φ approaches 90o.

0.00

0.01

0.10

1.00

10.00

0.00

10.

0020.

004

0.00

80.

010.

020.

040.

08 0.1

0.2

0.4

0.8 1 2 4 8 10 20 40 80 10

0

-100.00

-90.00

-80.00

-70.00

-60.00

-50.00

-40.00

-30.00

-20.00

-10.00

0.00

Amplitude Ratio Phase Angle

BODE Plot of a 1st Order Frequency Response:•Log-log plot of Amplitude Ratio and Frequency•Semi-log plot of Phase Angle and Frequency

Lo

gar

ith

m o

f A

MP

LIT

UD

E R

AT

IOP

HA

SE

AN

GL

E

Logarithm of FREQUENCY

• Substitute j for s in the Transfer Function:• G(s) = G(j)

• Multiply the resulting equation by its complex conjugate• Determine the MAGNITUDE and ARGUMENT of the resulting

complex number:• G(j) = Real Part + Imaginary Part• MAGNITUDE = |G(j)| = √[Real Part2 + Imaginary Part2]• ARGUMENT = LG(j) = tan-1 [Imaginary Part/Real Part]

Alternative Way of Determining Frequency (i.e. Sinusoidal) Response from the Transfer Function:

AMPLITUDE RATIO = MAGNITUDEPHASE ANGLE = ARGUMENT

Illustrative Example #1:

Determine the frequency response of a 1st Order System.

Y(s)

U(s)=

K

(s + 1)

• Substitute j for s in the Transfer Function• G(s) = G(j)


G(s) =K

(s + 1)

G(j) =K

(j + 1)

• Multiply the resulting equation by its complex conjugate


G(j) =K

(jτ + 1)x

(-jτ + 1)

(-jτ + 1)

G(j) =(2τ2 + 1)

=K(-jτ + 1) K(1 - τj)

(2τ2 + 1)

G(j) =(2τ2 + 1)

-K Kτ

(2τ2 + 1)j

√[ ]2

[ ]2


|G(j)| = +(2τ2 + 1)

K -Kτ

(2τ2 + 1)

• Determine the MAGNITUDE and ARGUMENT of the resulting complex number:

• G(j) = Real Part + Imaginary Part• MAGNITUDE = |G(j)| = √[Real Part2 + Imaginary Part2]

AMPLITUDE RATIO = |G(j)| =K

√(2τ2 + 1)


LG(j) = tan-1 / K

(2τ2 + 1)

PHASE ANGLE = tan-1(-τ)

• Determine the MAGNITUDE and ARGUMENT of the resulting complex number:

• G(j) = Real Part + Imaginary Part• ARGUMENT = LG(j) = tan-1 [Imaginary Part/Real Part]

(2τ2 + 1)

-Kτ[ ]

AMPLITUDE RATIO (AR) & PHASE ANGLE (Φ)for

Different Systems (1/3)

• 1st Order Systems in Series:• G(s) = K/(τs + 1)n

• AR = K[1/√(τ22 + 1)]n

• Φ = n[tan-1(-τ)]

• Dead Time:• G(s) = e-θs where: θ = Dead Time

• AR = 1

• Φ = -θ[360/(2π)]



• Proportional Controller:• G(s) = Kc

• AR = Kc

• Φ = 0

• Proportional-Integral Controller:• G(s) = Kc[1+1/(τIs)]

• AR = Kc√[1+1/(2τI2)]

• Φ = tan-1[-1/(τI)]



• Proportional-Derivative Controller:• G(s) = Kc[1+τDs]

• AR = Kc√[1+2τD2)]

• Φ = tan-1[τD]

• Proportional-Integral-Derivative Controller:

• G(s) = Kc[1+ 1/(τIs + τDs )]

• AR = Kc√[1+ {τD - 1/(τI)}2]

• Φ = tan-1 [τD - 1/(τI)]

Properties of BODE PLOTS:

• Overall AMPLITUDE RATIO (AR) for a Series of Transfer Functions

• Overall PHASE ANGLE (Φ) for a Series of Transfer Functions

= Product of Individual ARs

= Sum of Individual Φs

BODE STABILITY CRITERION:

A control system is stable if the OPEN-LOOP Frequency Response exhibits an AMPLITUDE RATIO (AR) of less than 1 at its CRITICAL FREQUENCY (c).

• CRITICAL FREQUENCY = Frequency () at which the PHASE ANGLE (Φ)

is -180o

Illustrative Example #2: (P-Control, 1st Order Process)

• Determine if the control system shown below will be stable using the Bode Stability Criterion. The Controller is a Proportional-only controller with the gain set at 1.6. The Process exhibits first order dynamics with a steady state gain of 1 and a time constant of 1. The Measuring Element and the Final Control Element have negligible dynamic lags.

R +Controller

FinalControlElement

Process

MeasuringElement

B

CMV

U+

+

-

e

0.0100

0.1000

1.0000

10.0000

0.001 0.002 0.004 0.008 0.01 0.02 0.04 0.08 0.1 0.2 0.4 0.8 1 2 4 8 10 20 40 80 100

-360

-315

-270

-225

-180

-135

-90

-45

0

45

90


BODE PLOT for Illustrative Example #2: (P-Control, 1st Order Process)

• Determine if the control system shown below will be stable using the Bode Stability Criterion. The Controller is a Proportional-Integral controller with the Gain set at 1.6 and the Integral Time set at 0.2. The Process exhibits first order dynamics with a steady state gain of 1 and a time constant of 1. The Measuring Element and the Final Control Element have negligible dynamic lags.

R +Controller

FinalControlElement

Process

MeasuringElement

B

CMV

U+

+

-

e

Illustrative Example #3: (PI-Control, 1st Order Process)

0.0100

0.1000

1.0000

10.0000

100.0000

1000.0000

10000.0000

0.00

10.

002

0.00

40.

008

0.01

0.02

0.04

0.08 0.

10.

20.

40.

8 1 2 4 8 10 20 40 80 100

-360

-315

-270

-225

-180

-135

-90

-45

0

45

90


BODE PLOT for Illustrative Example #3: (PI-Control, 1st Order Process)


• Consider the feedback control system shown below. The Controller is a Proportional-only controller. The Process, the Measuring Element, and the Final Control Element exhibit first order dynamics. The Process has a gain of 1 and a time constant of 5 seconds. The Measuring Element has a gain of 1 and a time constant of 1 second. The Final Control Element has a gain of 1 and a time constant of 2 seconds. Determine if the loop will be stable for the following Controller Gain settings: (a) Kc = 1, (b) Kc = 12.8, and (c) Kc = 20 . (Adapted from Ex. 11.4/11.10-Seborg, 2ed)

R +Controller

FinalControlElement

Process

MeasuringElement

B

CMV

U+

+

-

e

BODE PLOT for Illustrative Example #4:(a) Kc = 1

0.0000

0.0000

0.0000

0.0000

0.0001

0.0010

0.0100

0.1000

1.0000

10.0000

100.0000

0.001 0.002 0.004 0.008 0.01 0.02 0.04 0.08 0.1 0.2 0.4 0.8 1 2 4 8 10 20 40 80 100

-360

-315

-270

-225

-180

-135

-90

-45

0

45

90


BODE PLOT for Illustrative Example #4:(b) Kc = 12.8

0.0000

0.0000

0.0001

0.0010

0.0100

0.1000

1.0000

10.0000

100.0000

0.001 0.002 0.004 0.008 0.01 0.02 0.04 0.08 0.1 0.2 0.4 0.8 1 2 4 8 10 20 40 80 100

-360

-315

-270

-225

-180

-135

-90

-45

0

45

90


BODE PLOT for Illustrative Example #4:(c) Kc = 20

0.0000

0.0000

0.0001

0.0010

0.0100

0.1000

1.0000

10.0000

100.0000

0.001 0.002 0.004 0.008 0.01 0.02 0.04 0.08 0.1 0.2 0.4 0.8 1 2 4 8 10 20 40 80 100

-360

-315

-270

-225

-180

-135

-90

-45

0

45

90


CONTROL SYSTEM DESIGNBASED ON FREQUENCY RESPONSE

• GAIN MARGIN > 1.7• GAIN MARGIN = 1/ARc

• ARc = AR at the Critical Frequency, c

• 30O < PHASE MARGIN < 45O

• PHASE MARGIN = 180O - Φc

• Φc = Φ at which AR is equal to 1

Source: James B. Riggs, TexasTech University


• Determine the GAIN MARGIN and the PHASE MARGIN of the Control System in Illustrative Example #4 for the following Controller Gain settings:

(a) Kc = 1

(b) Kc = 3.085

(c) Kc = 10

(d) Kc = 12.77. (Adapted from Ex. 11.4/11.10-Seborg, 2ed)

0.0000

0.0000

0.0000

0.0001

0.0010

0.0100

0.1000

1.0000

10.0000

100.0000

0.001 0.002 0.004 0.008 0.01 0.02 0.04 0.08 0.1 0.2 0.4 0.8 1 2 4 8 10 20 40 80 100

-360

-315

-270

-225

-180

-135

-90

-45

0

45

90


BODE Plot for Illustrative Example #5:(b) Kc = 3.085


• Consider the feedback control system shown below. The Controller is a Proportional-only controller. The Process is a first order system with dead time. The process steady state gain is 1, the process time constant is 1 min, and the process dead time is 1.02 min. The Measuring Element and the Final Control Element have negligible dynamic lags. Determine the Controller Gain that will result to a GAIN MARGIN of 1.7.

• (Adapted from Ex. 17.3-Coughanowr, 2ed)

R +Controller

FinalControlElement

Process

MeasuringElement

B

CMV

U+

+

-

e

0.0010

0.0100

0.1000

1.0000

10.0000

100.0000

0.001 0.002 0.004 0.008 0.01 0.02 0.04 0.08 0.1 0.2 0.4 0.8 1 2 4 8 10 20 40 80 100

-360

-315

-270

-225

-180

-135

-90

-45

0

45

90


BODE PLOT for Illustrative Example #6:

CONTROLLER TUNINGBASED ON FREQUENCY RESPONSE

• Prepare a BODE PLOT using a P-only Controller with a Kc of 1.

• Determine the c & ARc

• Compute the ULTIMATE GAIN (Kcu)• ULTIMATE GAIN (Kcu) = Kc at which a Proportional-only

control loop is on the verge of instability.

• Kcu = 1/ARc

• Compute the ULTIMATE PERIOD (Pu)• Pu = 2π/c

CONTROLLER TUNINGBASED ON FREQUENCY RESPONSE

• Determine the PID setting from the ZIEGLER-NICHOLS Tuning Correlation:

Kc τI D

P-Controller

0.5 * Kcu n.a. n.a.

PI Controller

0.45 * Kcu Pu/1.2 n.a.

PID Controller

0.6 * Kcu Pu/2 Pu/8


• Consider the feedback control system shown below. The Controller is a Proportional-Integral controller. The Process is a first order system with dead time. The process steady state gain is 1, the process time constant is 1 min, and the process dead time is 1.02 min. The Measuring Element and the Final Control Element have negligible dynamic lags. Determine the Controller Gain and Integral Time using the Ziegler-Nichols correlation.

• (Adapted from Ex. 17.3-Coughanowr, 2ed)

R +Controller

FinalControlElement

Process

MeasuringElement

B

CMV

U+

+

-

e


• Determine an initial Controller Gain (Kc) setting for the Control System in Illustrative Example #4 using Frequency Response and the Ziegler-Nichols correlation.

. (Adapted from Ex. 11.4/11.10-Seborg, 2ed)

control system analysis & design by frequency response

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