Objectives

• Regression analysis

• Sensor signal processing

Regression analysis

2

Single variable:

Minimum number of points depends on number of variable in the function (3 for the function above).Using the data we can set the system of equation to find the coefficients.

Lagrange interpolation

3

Rewrite:

Find coefficients:

General form:

Regressing analysis for large pool of data (function fitting)

4

From last class

• Does correlation where R2=0.82 represent a good data modeling?

Mean:

Total sum of squares:

Sum of squares of residuals :

Coefficient of determination

Anscombe's quartet • Example of statistical misinterpretation of data

- all data have the same Mean (for x and y), Variance (for x and y)

- correlation R2: 0.816, linear regression: y=3.00+0.500·x

Anscombe's quartet • Example of statistical misinterpretation of data

- all curves have the same Mean (x, y), Variance (x, y)

- correlation R2: 0.816, linear regression

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Data set B

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Data set C

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Data set D

Moral of the story

Francis Anscombe (in 1973) demonstrated • the importance of graphing data before

analyzing it • the effect of outliers on statistical properties

8

Model of complex system based on experimental data

9

Example: chiller modelTOA

water

Building users (cooling coil in AHU)

TCWR=11oCTCWS=5oC

T Condensation

Chiller model

10

EIRFPLCAPFTPP NOMINAL

OACWSOAOACWSCWS TTfTeTdTcTbaCAPTF 12

112

111

PLRcPLRbaEIRFPLR 333 NOMINALQ

QPLR

)(

Impact of temperatures:

Impact of capacity:

Two variable function fitting

Example

12

0 2 4 6 8 100

50

100

150

20025 C35 C45 C

q[kW]

Tevaporator [C]

Fundamentals of Signal Processing

R

I

V

V=I·R

Two approaches:

- Constant Voltage Source- Constant Current Source

Sensor:

RTD, thermistor, hot wire, …..

Cable Losses

Sensor

Signal processing

cableDC signal [mV]

Voltage drop in the cable

Rcable=l·r (l length of cable , r resistance per unit of length) r = f ( voltage, current, diameter, material )

Rcable can be same order of value like DC signal

- Use same length of cables (shorter if possible) - Size diameter of cables to have significantly smaller voltage drop in cable than DC signal

Signal noise

Sensor

Signal processing

cableDC signal [mV]

AC current [120V]

Magnetic field

Current Induction (signal nose)

0 2 4

34

36

38

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42

44

Climate chamber

supply return

T [C]

hour

noise

Signal noise filters

A low pass filter is placed on the signal wires between a signal and an A/D board. It stops frequencies greater than the cut off frequency from entering the A/D board's analog or digital inputs.

A low pass filter may be constructed from on resistor R and one capacitor C. The cut off frequency Fc is determined according to the formula: Fc= 1/2*Pi*C R= 1/2*Pi*C*Fc See the following diagram

The key term in a low pass filter circuit is CUT OFF FREQUENCY. The cut off frequency is the frequency above which no variation of voltage with respect to time may enter the circuit. For example, if a low pass filter had a cut off frequency of 30 Hz, the type of interference associated with line voltage (60Hz) would be filtered out but a signal of 25 Hz would be allowed to pass

Data Acquisition Device

Analog signal collection

Measuring signalto data acquisition

Each Channel has:

- Current source- ± connectors for Voltage measurement

Current source (constant V)

+

-

I (variable A)

Analog signal collectionVoltage measurement ± Voltage measurement

Current measurements

Wheatstone bridge

Wheatstone bridge

Wheatstone bridge

Known resistor

Vo

Vo

R1

Our sensor

R2

+

-

+

-

Calculate R4

Converting Analog signal to Digital signal

Analog-to-digital converter (ADC) - electronic device that converts analog signals to an equivalent digital form- heart of most data acquisition systems

Loss of information in conversion, but no loss in transport and processing