ME 322: InstrumentationLecture 39

April 27, 2015Professor Miles Greiner

Integral Control program, Proportional Control response model

Announcements/Reminders• This week: Lab 12 Feedback Control• HW 13 Will accept Wednesday• HW 14 Due Friday, X3 (Last HW assignment)• Review Labs 10, 11, and 12: Wednesday and Friday• Open Lab Practice Session: Saturday and Sunday• Lab Practicum Finals (Starting a week from today)

– Schedule on WebCampus– Guidelines, – http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Tests/Index.htm

• Drop extra-credit LabVIEW Workshop and Lab 12.1• I will add 6 points to everyone’s Midterm 2 (1.5% of grade)

• Wind tunnel CTA• We are working to adjust the CTA feedback system so they will work correctly for the

final• You are graded on LabVIEW programming, data acquisition, calculations, clear

plots and tables, conclusions (not on equipment failures)

Modify Proportional Control

• Use Control-U to make block diagram clearer• Shift register, input DTi

– Add to FTOp • Display FTOi (bar and numerical indicators)• Add 10log(DTi) and log(DTi) to plots• Add Write to Measurement File VI

– Use next available file name– No Headers– One time column– Microsoft Excel

Lab 12 Integral Control Block Diagram

Write To Measurement File File Format: Microsoft Excel (.xlsx) File Path:C:\Users\Miles Greiner\Documents\LabVIEW Data\test.xlsx Mode: Save to one file Ask user to choose file: False If a file already exists: Use next available filename X value(time) columns: One column only Description:

• Modify proportional VI– http://

wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2012%20Thermal%20Control/Lab%20Index.htm

Figure 1 VI Front Panel

• Plots help the user monitor the time-dependent measured and set-point temperatures T and TSP, temperature error T–TSP, and control parameters

Process Sample Data• http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2

012%20Thermal%20Control/Lab%20Index.htm

• Add time scale in minutes– Calculate difference, general format, times 24*60

• Figure 3– Plot T, TSP, DT and 10log(DTi) versus time

• Figure 4– Plot T-TSP, -DT, 10log(DTi) and 0 versus time

• Table 1– Determine time periods when behavior reaches “steady state,” and find

and during those times • Be sure to use an integer number of cycles

• Figure 5– Plot versus DT and DTi

• Figure 6 – Plot steady state error versus DT and DTi

Figure 3 Measured, Set-Point, Lower-Control Temperatures and DTi versus Time

• Data was acquired for 40 minutes with a set-point temperature of 85°C.• Measure response to each setting for 10 minutes or less

• The time-dependent water temperature is shown with different values of the control parameters DT and DTi.

• Proportional control is off when DT = 0 • Integral control is effectively off when DTi = 107 (10log(DTI) = 70)

Figure 4 Temperature Error, DT and DTi versus Time

• The temperature oscillates for DT = 0, 5, and 15°C, but was nearly steady for DT = 20°C • Sometimes not steady until DT = 30°C• May dependent on TC location and water level

• DTi was set to 100 from roughly t = 25 to 30 minutes, but the systems oscillated, and so it was increased to 1000.

• The controlled-system behavior depends on the relative locations of the heater, thermocouple, and side of the beaker, and the amount of water in the beaker. These parameters were not controlled during the experiment (Lab 12.1 investigates these rich behaviors).

Table 1 Controller Performance Parameters

• This table summarizes the time periods when the system exhibits steady state behaviors for each DT and DTi.

• During each steady state period– TA is the average temperature– TA – TSP is an indication of the average controller error – The Root-Mean-Squared temperature TRMS is an indication of controller

unsteadiness

DT [°C] Dti

Time Range [min] TA [°C]

TRMS

[°C]TA-TSP

[°C]0 1.E+07 4.43 to 7.50 88.22 3.42 3.22

5 1.E+07 9.45 to 14.48 85.85 2.79 0.85

15 1.E+07 17.62 to 22.34 83.01 0.62 -1.99

20 1.E+07 23.61 to 25.41 82.48 0.10 -2.52

20 1000 35.51 to 39.44 85.06 0.23 0.06

Figure 5 Controller Unsteadiness versus Proportionality Increment and Set-Point Temperature

• TRMS is and indication of thermocouple temperature unsteadiness

• Unsteadiness decreased as DT increased, and was not strongly affected by DTi.

Figure 6 Average Temperature Error versus Set-Point Temperature and Proportionality Increment

• The average temperature error– Is positive for DT = 0, but decreases and becomes

negative as DT increases. – Is significantly improved by Integral control.

Proportional-Control Thermal Analysis

• Is TW = TTC? Is TW uniform?• Energy Balance for Water

–

• For proportional control:

TW

HeaterQIN = FTO(QMAX)

QOUT = hA(TW -TEnv)TEnvTTC

For Large DT • We observed that TTC is steady when DT is

sufficiently large – Under that condition, assume TTC = TW =

–

– Steady State Error

• – How does this prediction compare to measurements?

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 1 2 3 4 5 6 7 8 9 10

T RM

S[C

]

DT [C]

TSP = 65°C

TSP = 85°C

Measured Proportional-Control Steady-State Error

• The temperature is steady (TRMS becomes small) once DT is sufficiently large

• Prediction: • Measurements show the error magnitude increases as increases

– When no need for control• Steady state error decreases as increases (DT decreases)

– But when DT is too small the temperature oscillates (TRMS)• Why does this happen? (

𝑇 𝑅𝑀𝑆

𝑒𝑆𝑆

Predict time-dependent TTC(t) for TC

• TC Energy Balance (assuming )–

• (Eqn. A)

• Constants

• Dynamic relationship between (t) and

TW

TTCQIN = hA(TW -TTC)

Water Energy-Balance

• For proportional control:

• Where the controller gain is

– , , and are constants– In addition to , ,

TW

HeaterQIN = FTO(QMAX)

QOUT = hA(TW -TENV)TENVTTC

Collect Terms

– (Eqn. B)

– Dynamic relationship between and • Couple with Eqn. A: • What do we have?

– Two, 1st-order, coupled, constant-coefficient liner-differential equations for (t) and (t)

System Solution• Solve Eqn. A for • Plug into Eqn. B and collect terms

Collect Terms– More Constant

– 2nd order, linear, Constant-coefficient differential-equation, non-homogeneous (RHS= Constant)

• Solution • Particular Solution for RHS = Constant,

– – Same as for simple 1st order system

Homogeneous Solution

– Assumed solution – Characteristic Equation

• If DT is small enough, then – will be large enough so that – , – b will be imaginary and – will be oscillatory

• We observed oscillations for small DT

What is the largest minimum value of DT that will have a steady behavior?

– , – ,

Problem X3• Problem X3: A 200-Watt heater and a 1.5-mm-diameter

thermocouple are placed in a water-filled beaker of diameter 3 cm and height 5 cm. If the heat transfer coefficient between the beaker and air is 5 W/m2K, and between the water and thermocouple is 1000 W/m2K, estimate the lowest proportional control temperature increment DTMin, for which the control system will be steady (not oscillatory). Assume the thermocouple properties to be that of iron, and evaluate water properties at 30°C.

• Heater: Q = 200 W• Beaker: D = 3 cm, H = 5 cm, hAir = 5 W/m2K• TC: D = 1.5 mm, hTC = 1000 W/m2K, iron

Proportional Control

1st Law

Proportional ControlFind

Want DT to be small, but that leads to oscillation.

Proportional Control

No way for a 1st order constant coefficient deferential equation to give oscillation.

How to predict/model oscillations?

(Twater ≠ TTC)

Better Model

1st Law

TC

Water

Unknown: T, T W

HAHeater HT MW

Homogenous Solutions:

A B

Characteristic Equation

If

Then set complex

If DT is small enough then get oscillations.