Voltage Control System: Proportional Integral (PI) Controllers Team Purple: John Pangle Jessica...

Voltage Control System: Proportional Integral

(PI) Controllers

Team Purple:John Pangle

Jessica Raymond

Justin WhittENGR 329

November 30, 2005

Outline

Voltage System Background Previous Work: SSOC, Step, Sine,

Bode, P-Only PI Model Results PI Experimental Results Conclusions Luyben Method Review Recent Results Conclusions

Purple Team 9/14/05

Voltage System Schematic

Block Diagram

M(t) is measured in % and C(t) is in Volts.

Purple Team 10/25/05

Voltage System SSOC

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60 70 80 90 100

m(t), Input (%)

c(t

), O

utp

ut

(V) Operating Range:

Output = 50V to 140V

Operating Range:Input = 70% to 100%

75%-100% InputSlope = 2 V/%

Background: Step Input

50

120

40 50 60

Time (sec)

Ou

tpu

t (v

olt

s)

70

90

40 50 60

Time (sec)

Inp

ut

(%)

Voltage System

K – Gain (Volt/%)t0 – Dead Time (s)τ – Time Constant (s)

Background: FOPDT Fit

Voltage System 70% Base, 20% Step

Trial 1

70

75

80

85

90

45 47 49 51 53 55

Time (sec)

40

60

80

100

120

Output, c(t) (V)

Input, m(t) (%)

Fit 3 Data:K= 3 V / %τ= 0.9 secto = 0 sec

FOPDT Model

50

55

60

65

70

75

80

85

90

95

10 15 20 25 30

Time (sec)

Input (%)

50

60

70

80

90

100

110

120

Experimental Input

Model Input

Experimental Output

Model Output

Output (V)

Voltage SystemExperimental vs. Model

90%-70% Step(Trial 1)

Team Purple09/20/05

K = 2 V / %

to = 0 s

τ = 0.5 s

Background: Sinusoidal Input

80

90

100

11 11.5 12

Time (sec)

Ou

tpu

t (V

olt

s)

74

80

86

11 11.5 12

Time (sec)

Inp

ut

(%)

K – Gain (Volt/%)t0 – Dead Time (s)τ – Time Constant (s)

Values taken from Point A AR = 1.5 f = 4 Hz

Bode Plot K = 2 V / %

= 0.04 sec

to = 0.08 sec

)(tan 1

ot

Voltage System Experimental Bode Plot80% Base 5 Amplitude

0.1

1

10

0.1 1 10 100

Asp

ect

Rat

io (

V/%

)

fu = 5 Hz

order = 2 Hz

K = 2 V / %

1/Kcu = 1 % / V

Kcu = 1 V / %

-450

-360

-270

-180

-90

0

0.1 1 10 100

Frequency (Hz)

Ph

ase

An

gle

JDP

tau = 0.04 secto = 0.08 sec

Model Bode Plot, 80% Base 5 Amplitude

0.1

1

10

0.1 1 10 100

Frequency (Hz)

Asp

ect

Rati

o (

V/%

) K = 2 V/%

order = 2 Hz

fu = 5 Hz

1/Kcu = 1 %/V

Kcu = 1 V/%

-450

-400

-350

-300

-250

-200

-150

-100

-50

0

0.1 1 10 100

Ph

ase A

ng

le

Experimental Model

Model ValuesK = 2.0 V/%tau = 0.04 secto = 0.08 sec

JDP

Background: Sinusoidal InputVoltage Sine Input 75-85%

Frequency of 0.1 Hz

70

75

80

85

90

20 25 30 35 40

Time (sec)

Input (%)

80

90

100

110

120

Output (V)

Input Value(%) Model Input Output(Volts) Model Output

Background: FOPDT Averages

K (v / %) 2

τ (sec) 0.04

to (sec) 0.06

FOPDT ParametersAverages of

Background: Feedback Loops

Kc

1

0

s

Ke st

R(volts)

E(volts)

M(%)

C(volts)

+ -

ste 0

ste 0

st

st

e st

21

21

0

0

0

CE =

Background: P-Only ControllersVoltage System P-Only Root Locus

-60

-40

-20

0

20

40

60

-40 -35 -30 -25 -20 -15 -10 -5 0 5

REAL AXIS

IMA

GIN

AR

Y A

XIS

KQD = .8 KCU = 1

KCD = .01

K1/500 = .2 K1/10 = .6

fu = 8 Hz

Units of KC = %/ V

Background: P-Only ControllersVoltage System

Proportional Only Model

105

110

115

120

125

5 5.5 6

TIME (sec)

OU

TP

UT

/SE

T P

OIN

T (

V)

JWTeam Purple10/28/05

Kccd = 0.01 % / V

Offset = - 9 v

Voltage System Proportional Only

Experimental

70

75

80

85

90

95

100

9 9.5 10 10.5 11 11.5 12

Time (sec)

Inp

ut

(%)

100

105

110

115

120

125

130

Ou

tpu

t (V

)

Input Value(%) Output(Volts) SET-P(Volts)

Kccd = .01 % / V

Decay Ratio: 0MonotonicSettling Time: 0.1 secOffset: - 23 V

Background: P-Only ControllersVoltage System SSOC & Controller Operating Line

90

100

110

120

130

91.8 91.84 91.88 91.92 91.96 92 92.04 92.08

Input (%)

Ou

tpu

t (

V )

Kccd = 0.01 % / V

Delta R = - 20 VOffset = - 23 V



105

110

115

120

125

5 5.5 6

TIME (sec)

OU

TP

UT

/SE

T P

OIN

T (

V)


Kc1/500 = 0.2 % / VOffset = - 7 v

Voltage System Proportional Only Experimental

70

75

80

85

90

95

100

9 9.5 10 10.5 11 11.5 12

Time (sec)

Inp

ut

(%)

100

105

110

115

120

125

130

Ou

tpu

t (V

)


Kc1/500 = 0.2 % / V

Decay Ratio: 0OscillatorySettling Time: 0.5 secOffset: - 16 V

Background: P-Only ControllersVoltage System SSOC & Controller Operating Line

100

105

110

115

120

125

87 88 89 90 91 92 93

Input (%)

Ou

tpu

t (

V )

Kc1/500 = 0.2 % / V




105

110

115

120

125

5 5.5 6

TIME (sec)

OU

TP

UT

/SE

T P

OIN

T (

V)


Kc1/10 = 0.6 % / V

Offset = - 4 v

Background: P-Only Controllers

RESULTS FOR PROPORTIONAL FEEDBACK CONTROLLERSET-P(Volts)

Voltage System Proportional Only Experimental

70

75

80

85

90

95

100

9 9.5 10 10.5 11 11.5 12

Time (sec)

Inp

ut

(%)

100

105

110

115

120

125

130

Ou

tpu

t (V

)


Kc1/10 = 0.6 % / V

Decay Ratio: 0.7OscillatorySettling Time: 0.6 secOffset: - 12 V

Voltage System SSOC & Controller Operating Line

100

105

110

115

120

125

80 82 84 86 88 90 92 94

Input (%)

Ou

tpu

t (

V )

Kc1/10 = 0.6 % / V




105

110

115

120

125

5 5.5 6

TIME (sec)

OU

TP

UT

/SE

T P

OIN

T (

V)


KQD = 0.8 % / V

Offset = - 3 v

Background: P-Only ControllersVoltage System Proportional Only Experimental

70

75

80

85

90

95

100

9 9.5 10 10.5 11 11.5 12

Time (sec)

Inp

ut

(%)

100

105

110

115

120

125

130

Ou

tpu

t (V

)


KcQD = 0.8 % / V

Decay Ratio: 1OscillatorySettling Time: neverOffset: - 8 V


100

105

110

115

120

125

80 82 84 86 88 90 92 94

Input (%)

Ou

tpu

t (

V )

KcQD = 0.8 % / V




100

110

120

130

140

5 5.5 6

TIME (sec)

OU

TP

UT

/SE

T P

OIN

T (

V)


Kcu = 1 % / V

Background: P-Only ControllersVoltage System Proportional Only Experimental

70

75

80

85

90

95

100

9 9.5 10 10.5 11 11.5 12

Time (sec)

Inp

ut (

%)

100

105

110

115

120

125

130


Kc = 1 % / V

Decay Ratio: 1OscillatorySettling Time: neverOffset: - 5 V


100

105

110

115

120

125

84 86 88 90 92 94

Input (%)

Ou

tpu

t (

V )

Kc = 1 % / VDelta R = - 20 VOffset = - 5 V


Agree

Moderately AgreeDisagree

Symbol

RL Value for Kc (%/V) Model Experimental

KCD 0.01 0 0K1/500 0.2 0 0K1/10 0.6 0.1 0.7KQD 0.8 0.2 1Kcu 1 1 1

Decay Ratio


Agree


SymbolRL Value for

Kc (%/V) Model ( sec ) Experimental ( sec )KCD 0.01 0.1 0.1

K1/500 0.2 0.2 0.5K1/10 0.6 0.4 1.2KQD 0.8 0.5 neverKcu 1 never never

Settling Time


Agree


SymbolRL Value for

Kc (%/V) Model ( V ) Experimental ( V )Controller

Line (V) FVT ( V )KCD 0.01 -20 -23 -23 -20

K1/500 0.2 -14 -16 -14 -14K1/10 0.6 -9 -8 -8 -9KQD 0.8 -8 -8 -7 -8Kcu 1 CBD -5 -5 -6

Offset

PI Controllers

M(s)

1

0

s

Ke st

R(s) E(s) C(s)

+ -

sK

Ic

11

PI ControllersVoltage System PI Controller I= .2 s

-60

-40

-20

0

20

40

60

-40 -30 -20 -10 0REAL

IMA

GIN

AR

YKcCD= 0.01 % / V

KcQD= 0.7 % / VKcu= 1 % / V

fu= 8 Hz

Kc1/500= 0.2 % / V

Kc1/10= 0.6 % / V

FOPDT:K= 2 V / %Tau=0.04 sto=0.06 s

PI ControllersVoltage System PI Controller i = 0.1

-60

-40

-20

0

20

40

60

-40 -30 -20 -10 0

IMA

GIN

AR

Y

KcCD = 0.01

Kc-500 = 0.2

Kc-10 = 0.6

Kcu = 1

KcQD = 0.7

fu = 7 Hz


PI ControllersVoltage System PI Controller I= .01 s

-40

-20

0

20

40

-10 -8 -6 -4 -2 0REAL

IMA

GIN

AR

Y

KcCD= 0.02 % / V

KcQD= 0.07 % / V

Kcu= 0.1 % / V fu= 3.2 Hz

Kc1/500= 0.03 % / V

Kc1/10= 0.05 % / V


Voltage System PI Controller I= .01 s

-60

-40

-20

0

20

40

60

-40 -30 -20 -10 0REAL

IMA

GIN

AR

Y


PI Controllers (Kc = 0.8)Voltage System

PI ControllerExperimental

0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140Output (volt)

Kc = 0.8 % / v

τi = 0.2

DR = 1UltimateSettling Time = Never


Voltage SystemPI ControllerExperimental

0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140Output (volt)


Kc = 0.8 % / vτi = 0.1



0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140Output (volt)


Kc = 0.8 % / vτi = 0.05



0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140Output (volt)


Kc = 0.8 % / vτi = 0.03

DR = 1UnstableSettling Time = Never


0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.8 % / vτi = 0.01




0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140Output (volt)


Kc = 0.5 % / vτi = 0.2 s

DR = 0.63Under DampedSettling Time = 1 s


0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.5 % / vτi = 0.1 s

DR = 0.54Under DampedSettling Time = 0.9 s


0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

80

90

100

110

120

130

140

Output (volt)


Kc = 0.5 % / vτi = 0.05 s


`


0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.5 % / vτi = 0.025 s



0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.5 % / vτi = 0.01 s




0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140Output (volt)


Kc = 0.4 % / vτi = 0.2 s



0

25

50

75

100

25 25.2 25.4 25.6 25.8 26

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.4 % / vτi = 0.1 s



0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.4 % / vτi = 0.05 s



0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140Output (volt)


Kc = 0.4 % / vτi = 0.03 s



0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.4 % / vτi = 0.01 s




0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140Output (volt)


Kc = 0.2 % / vτi = 0.2 s

DR = 0.29Quarter DecaySettling Time = 0.9 s


0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.2 % / vτi = 0.1 s



0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.2 % / vτi = 0.05 s

DR = 0Critically DampedSettling Time = 0.4 s


0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.2 % / vτi = 0.03 s



0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.2 % / vτi = 0.01 s




0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.1 % / v

τi = 0.2 sDR = 0Critcally DampedSettling Time = 2 s


0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.1 % / vτi = 0.1 s

DR = 0Critically DampedSettling Time = 1 s


0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.1 % / vτi = 0.05 s

DR = 0Over DampedSettling Time = 0.6 s


0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.1 % / vτi = 0.03 s



0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.1 % / vτi = 0.01 s




0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140Output (volt)


Kc = 0.05 % / vτi = 0.2 s

DR = 0Over DampedSettling Time = 5 s


0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.05 % / vτi = 0.1 s

DR = 0Over DampedSettling Time = 2 s


0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.05 % / vτi = 0.05 s



0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.05 % / vτi = 0.03 sDR = 0Over DampedSettling Time = 0.6 s


0

25

50

75

100

25 25.5 26 26.5 27

Time (sec)

Input (%)

100

110

120

130

140

Output (volt)


Kc = 0.05 % / vτi = 0.01 s


PI ControllersVoltage System


8

0

1

2

3

4

5

6

7

8

0.8 0.5 0.4 0.2 0.1 0.05

Kc ( % / volt)

Se

ttli

ng

Tim

e (

se

c)

0.2

0.1

0.05

0.03

0.0188 8

τI



0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.8 0.5 0.4 0.2 0.1 0.05

Kc ( % / volt)

Dec

ay R

atio

0.2

0.1

0.05

0.03

0.01

τI



0.00.1

0.20.3

0.40.5

0.60.7

0.80.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Kc (% / Volt)

Dec

ay R

atio

0.2

0.1

0.05

0.03

0.01

τI



0

25

50

75

100

0 5 10 15 20 25 30 35 40 45

Time (sec)

Input (% )

0

20

40

60

80

100

120

140

m(t) % c(t) volts Set Point (volts)

Output (volt)

Kc = 0.2 % / v

τi = 0.03


Conclusions

Kc (% / volt)

Decay Ratio

Monotonic/ Oscillatory

Settling Time (sec)

Description

0.8 1.0 Oscillatory Never Ultimate0.5 1.0 Oscillatory 0.9 Under Damped0.4 0.3 Oscillatory 0.4 Quarter Decay0.2 0.1 Oscillatory 0.9 Tenth Decay0.1 0.0 Monotonic 1 Critically Damped

0.05 0.0 Monotonic 3 Over Damped

Recommend PI over P-only Recommend τI = 0.1 Yields widest range of desirable decay ratios. Provides acceptable settling times.

Luyben Method

Presented by

William L. Luyben, Ph.D

at Reno AIChE Meeting

Nov. 6, 2001

Luyben Method

1. Insert relay into feedback loop.

Relay

1

0

s

Ke st

R(volts)

E(volts)

M(%)

C(volts)

+ -

Luyben Method

2. Specify upper & lower limits of m(t)

Luyben Method Proportional Feedback Voltage Station Enter this Information about you:

Name Justin Whitt

Location emcs

E-mail address j_w hitt@com

Enter these Parameters for the Experiment:

Length of experiment (sec): 30

Lab Wizard

Controller output bias value (%): (=Baseline input value (%))

80

Lab Wizard

Controller tuning parameter and Set Point

Proportional controller gain, Kc (%/Volt): 9999999999

Lab Wizard

Set-point value (Volts): 100

Lab Wizard

Controller Output Clamping Specifications

Floor (%): 75

Lab Wizard

Ceiling (%): 85

Lab Wizard

RUN EXPERIMENT

Luyben MethodVoltage System

Relay Feedback TestInput:95%, A=5

85

90

95

100

105

110

22.5 22.6 22.7 22.8 22.9 23.0 23.1

Time (sec)

Input (%)

80859095100105110115120125130


Input m(t) (%)

Output c(t) (Volts)

Setpoint r(t) (Volts)

Luyben Method

3.Find Tu from c(t)

4.Calculate Kcu

a

hK cu

4



70

75

80

85

90

22.5 22.6 22.7 22.8 22.9 23.0 23.1

Time (sec)

Inp

ut

(%)


Input m(t) (%)

h

h = 5%

a

Kcu%)5(4



90

95

100

105

22.5 22.6 22.7 22.8 22.9 23.0 23.1

Time (sec)

Ou

tpu

t (V

olt

s)


Output c(t) (Volts)

a

a = 5.1 Volts Tu

Tu = 0.19s

voltvoltsK cu

%3.1

)_1.5(

%)5(4

HzsT

fu

u 3.519.0

11

Luyben Method

Voltage System Ultimate Frequency

0

2

4

6

8

75%-85% 90%-100%

fu (Hz)

Bode Luyben P-Only RL

Luyben Method

Voltage System Ultimate Controller Gain

0

0.4

0.8

1.2

1.6

75%-85% 90%-100%

Kcu (%/V)

Bode Luyben P-Only RL

Luyben Method

What about to and τ? I still have to run another test right?

Luyben Method

uP

bF

4Curvature Factor =

Y

0

a

a/2

b

Pu

time

t1 t2

Luyben Method

Y0

aa/2

b

Pu

time1)(

s

eKG

stp

s

o

)/(

)(

o

stps

t

eKG o

)0/(

)/1()(

o

st

s

ts

eG

o

time

Y

0

a

-a

aPu

22

1

F

Pb u

Y 0

Pu

time

a

5.08

1

F

Pb u

t1 t2

F=2

F=0.5

Luyben Method

)(6788.2)(8974.9

)(7147.122783.5)/(log32

10

FF

Fto

F=1 for Voltage System

to/τ = 2

Luyben Method

to = 2τ

)(tan 1

ot

)(tan2

)(tan1

10

uu

uut

τ = 0.01 and to= 0.02

Luyben Method

Voltage System Time Constant

0

0.01

0.02

0.03

0.04

0.05

75%-85% 90%-100%

Tau (sec)

Bode Luyben

Luyben Method

Voltage System Dead Time

0

0.02

0.04

0.06

0.08

0.1

75%-85% 90%-100%

To (sec)

Bode Luyben

Conclusion

Only one simple test required (7 steps)

Works well for any order systems

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