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LESSON 21: METHODS OF SYSTEM

ANALYSIS

ET 438a Automatic Control Systems Technology

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LEARNING OBJECTIVES

After this presentation you will be able to:

Compute the value of transfer function for given

frequencies.

Compute the open loop response of a control system.

Compute and interpret the closed loop response of a

control system.

Compute and interpret the error ratio of a control system.

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FREQUENCY RESPONSE OF CONTROL SYSTEMS

Control limits determined by comparing the open loop response of

system to closed loop response.

Open loop response of control system: )s(H)s(G)s(SP

)s(Cm

G(s) SP

+ -

H(s)

Cm(s)

C(s)

Where:

Cm(s) = measurement feedback

SP(s) = setpoint signal value

G(s) = forward path gain

H(s) = feedback path gain

Measurement

disconnected from

feedback

Error = SP Controller

Manipulating element

Process

Measurement

System 3

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FREQUENCY RESPONSE OF CONTROL SYSTEMS

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Closed loop response of control system:

G(s) SP

+ -

H(s)

Cm(s)

C(s)

)s(H)s(G1

)s(H)s(G

)s(SP

)s(Cm

Output

Input

Note: this is not the I/O relationship that

was used earlier. (C(s)/SP(s))

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FREQUENCY RESPONSE OF CONTROL SYSTEMS

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Frequency response of system divided into three ranges:

Zone 1- controller decreases error

Zone 2 - controller increases error

Zone 3 - controller has no effect on error

SP change frequency determines what zone is activated. Overall system

frequency determines values of zone transition frequencies

Error Ratio (ER) plot determines where zones occur

)s(H)s(G1)s(H)s(G11

ER

|MagnitudeError loop-Open|

|MagnitudeError loopClosed|ER

Replace s with

jw and compute

magnitude of

complex number

FREQUENCY RESPONSE OF CONTROL SYSTEMS

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Typical Error Ratio plot showing operating zones and controller action

Error reduced

Error increased

No Effect

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COMPUTING TRANSFER FUNCTION VALUES

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Example 21-1: Given the forward gain, G(s), and the feedback system

gain, H(s) shown below, find 1) open loop transfer function, 2) closed

loop transfer function, 3) error ratio.

s478.01

356.0)s(H

s0063.0s379.01

8.21)s(G

2

4) compute the values of the open/closed loop transfer functions when

w=0.1, 1, 10 and 100 rad/sec. 5) compute the value of the error ratio

when w=0.1, 1, 10 and 100 rad/sec. 6) Use MatLAB to plot the open

and closed loop transfer function responses on the same axis.

EXAMPLE 21-1 SOLUTION (1)

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1) Open Loop Transfer Function

Expand the denominator

Ans

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EXAMPLE 21-1 SOLUTION (2)

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2) Find closed loop transfer function

Multiply numerator and denominator by 1+0.857s+0.18746s2+0.00301s3

and simplify

Ans

EXAMPLE 21-1 SOLUTION (3)

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3) Find the error ratio

Magnitude only

Expand denominator and

simplify

Substitute in G(s)H(s) from part 1 into above

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EXAMPLE 21-1 SOLUTION (4)

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Error ratio calculations

Ans

EXAMPLE 21-1 SOLUTION (5)

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4) Compute the values of the open and closed loop transfer functions for

w=0.1 1 10 100 rad/s Substitute jw for s

Open loop

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EXAMPLE 21-1 SOLUTION (6)

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Now for w=1 rad/sec

EXAMPLE 21-1 SOLUTION (7)

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Now for w=10 rad/sec

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EXAMPLE 21-1 SOLUTION (8)

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For w=100 rad/sec

EXAMPLE 21-1 SOLUTION (9)

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EXAMPLE 21-1 SOLUTION (10)

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Convert all gain values into dB

4) Compute the close loop response using the previously calculated values of

G(s)H(s)

EXAMPLE 21-1 SOLUTION (11)

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EXAMPLE 21-1 SOLUTION (12)

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EXAMPLE 21-1 SOLUTION (13)

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Convert all gain values into dB

5) Compute the values of the error ratio

Use open loop values to compute

values of ER at given frequencies

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EXAMPLE 21-1 SOLUTION (14)

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Now for w=1 rad/sec

EXAMPLE 21-1 SOLUTION (15)

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For w=10 rad/sec

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EXAMPLE 21-1 SOLUTION (16)

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Convert all gain values into dB

System becomes

uncontrollable between

these two frequencies

Error ratio magnitude increases as frequency increases. It peak and becomes a

constant value of 1 (0 dB)

INTERPRETING ERROR RATIO PLOTS

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Define control zones

Zone 1: ER< 0 dB good

control. Controller

decreases error

Zone 2: ER> 0 dB poor

control. Controller

increases error

Zone 3: ER-=0 dB no

control. Controller does

not change error

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GENERATING PLOTS USING MATLAB

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Use MatLAB script to create open and closed loop Bode plots of example

system

% Example bode calculations

clear all;

close all;

% define the forward gain numerator and denominator coefficients

numg=[21.8];

demg=[0.0063 0.379 1];

% define the feedback path gain numerator and denominators

numh=[0.356];

demh=[0.478 1];

% construct the transfer functions

G=tf(numg,demg);

H=tf(numh,demh);

% find GH(s)

GH=G*H

% find the closed loop transfer function

GHc=GH/(1+GH)

% The value in curly brackets are freq. limits

bode(GH,'go-',GHc,'r-‘,{0.1,100})

BODE PLOTS OF EXAMPLE 21-1

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-60

-40

-20

0

20

Magnitude (

dB

)

10-1

100

101

102

-270

-225

-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/s)

Open Loop

Closed Loop

1s857.0s1875.0s00301.0

761.7)s(H)s(G

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761.8s365.8s564.2s351.0s1013.1s1007.9

761.7s651.6s455.1s0234.0

)s(H)s(G1

)s(H)s(G235366

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MATLAB CODE FOR ERROR PLOT EXAMPLE 21-1

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% Example Error Ratio calculations

clear all;

close all;

% define the forward gain numerator and denominator coefficients

numg=[21.8];

demg=[0.0063 0.379 1];

% define the feedback path gain numerator and denominators

numh=[0.356];

demh=[0.478 1];

% construct the transfer functions

G=tf(numg,demg);

H=tf(numh,demh);

% find GH(s)

GH=G*H;

% find the error ratio

ER=1/((1+GH)*(1-GH));

[mag,phase,W]=bode(ER,{0.1,100}); %Use bode plot with output sent to arrays

N=length(mag); %Find the length of the array

gain=mag(1,1:N); %Extract the magnitude from the mag array

db=20.*log10(gain); % compute the gain in dB and plot on a semilog plot

semilogx(W,db);

grid on; %Turn on the plot grid and label the axis

xlabel('Frequency (rad/s)');

ylabel('Error Ratio (dB)');

ERROR RATIO PLOT

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10-1

100

101

102

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Frequency (rad/s)

Err

or

Ratio (

dB

) Zone 1

Stable to 6.5 rad/s

Zone 2

Error

increases

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ET 438a Automatic Control Systems Technology

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END LESSON 21: METHODS OF SYSTEM

ANALYSIS