APPLICATION OF PRONY ANALYSIS TO …sas03013/docs/MSThesis_satnam.pdfAPPLICATION OF PRONY ANALYSIS...
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APPLICATION OF PRONY ANALYSIS TO CHARACTERIZE PULSED CORONA REACTOR MEASUREMENTS by Satnam Singh A thesis submitted to the Department of Electrical and Computer Engineering and The Graduate School of The University of Wyoming in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in ELECTRICAL ENGINEERING Laramie, Wyoming August, 2003 1
Transcript of APPLICATION OF PRONY ANALYSIS TO …sas03013/docs/MSThesis_satnam.pdfAPPLICATION OF PRONY ANALYSIS...
APPLICATION OF PRONY ANALYSIS TO CHARACTERIZE PULSED CORONA REACTOR MEASUREMENTS
by
Satnam Singh
A thesis submitted to the Department of Electrical and Computer Engineering
and The Graduate School of The University of Wyoming
in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE
in
ELECTRICAL ENGINEERING
Laramie, Wyoming
August, 2003
1
Singh, Satnam, Application of Prony Analysis to Characterize Pulsed Corona Reactor
Measurements, M.S., Department of Electrical and Computer
Engineering, August 2003.
Pulsed Corona Reactor (PCR) techniques have been recently shown to offer a cost-
effective and environmentally friendly way to remove pollutants such as nitrogen oxides
(NOx) and sulfur dioxide (SO2) from flue gas exhausts associated with coal-burning
power plants, automotive engines, steel plants and paper mills.
Recent conjectures note that the voltage and current measurements at the PCR
electrode terminals appear to be well described by a sum of damped sinusoidal functions.
Apart from PCR, these kinds of damped signals are found in several engineering and
medical problems. There are several mathematical models and techniques available for
such systems. In this thesis, we investigated the variation in measurement composition
via the classical signal processing technique known as “Prony analysis (PA).” Our
hypothesis is that the system health and reactor contents of the PCR can be detected by
inspecting the roots of the Prony fit.
We have implemented the PA in a graphical user interface we call the “Prony
Toolbox,” built around MATLAB functions. Prony Toolbox contains all the necessary
features in the application setting of PCR characterization. Using Prony Toolbox, PA
was performed to the PCR measurements obtained from experiments carried out in the
Chemical Engineering Department at the University of Wyoming.
PA results display that a specific clustering of poles occurs at a specific model
order for each of the analyzed reactor feed gases. The clustering of poles occurs for
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pulsed current as well as for pulsed voltage. This is an important finding as a specific
pole clustering may describe the presence of a particular gas. This effect might be
viewed similar to chemical structure sensing techniques such as NMR.
These preliminary gas studies are precursors to PCR pollution abatement
applications. The observations made from of PA results are encouraging and promising
for future research work.
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TABLE OF CONTENTS
CHAPTER 1INTRODUCTION ............................................................................................................ 10
1.1 Motivation............................................................................................................... 10 1.2 System Identification Using Prony Analysis .......................................................... 12 1.3 Organization of Thesis............................................................................................ 15 1.4 Synopsis of Thesis .................................................................................................. 15
CHAPTER 2 PRONY ANALYSIS ........................................................................................................ 17
2.1 Comparison Between Fourier Series and Prony Analysis ...................................... 17 2.2 Original Prony Analysis.......................................................................................... 18 2.3 Prony Analysis for Time-Domain Design of IIR Filters ........................................ 24 2.4 Modified Prony Analyses ....................................................................................... 26
CHAPTER 3 DESIGN BASIS OF THE PRONY TOOLBOX.............................................................. 28
3.1 Data Input Format ................................................................................................... 29 3.2 Reduction in Data Length ....................................................................................... 29 3.3 Data Range.............................................................................................................. 30 3.4 Data Preprocessing and Filtering ............................................................................ 30 3.5 Model Order Determination.................................................................................... 30 3.6 Graphic Mode of Prony Analysis Fit ...................................................................... 31 3.7 Signal-Noise Separation.......................................................................................... 32 3.8 Root Inspection ....................................................................................................... 32 3.9 Accuracy of Prony Fit............................................................................................. 32 3.10 Numeric Summary ................................................................................................ 33 3.11 Simultaneous Display of Several Prony Analysis Sessions.................................. 33 3.12 Graphical User Interface ....................................................................................... 33
CHAPTER 4 IMPLEMENTING THE FEATURES OF THE PRONY TOOLBOX............................. 35
4.1 GUI Development Process...................................................................................... 35 4.2 Introduction to GUIDE ........................................................................................... 37
4.2.1 Callback Programming .................................................................................... 43 4.3 Installation of PTbox................................................................................................... 45
4.4 Graphical User Interfaces in PTbox........................................................................ 45 4.5 PTbox Splash GUI .................................................................................................. 47 4.6 Prepare Prony Data GUI ......................................................................................... 48
4
TABLE OF CONTENTS (CONT'D)
4.7 Perform PA GUI ..................................................................................................... 51 4.8 Compare PA Sessions GUI..................................................................................... 56 4.9 Export Data GUI ..................................................................................................... 58 4.10 Common Features ................................................................................................. 60 4.11 The Source Code................................................................................................... 62
CHAPTER 5 RESULTS AND DISCUSSION....................................................................................... 63
5.1 The Measured Data ................................................................................................. 63 5.2 File and Variable Nomenclature ............................................................................. 64 5.3 Data Preprocessing with PTbox.............................................................................. 66 5.4 Prony Analysis with PTbox .................................................................................... 68 5.5 Refining the Model Order in PTbox ....................................................................... 74
CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH .............................................................. 84
6.1 The Features of PTbox............................................................................................ 84 6.2 PTbox Applied to the PCR Data............................................................................. 85 6.3 Future Research Areas ............................................................................................ 86
6.3.1 Breadth of Data Analysis ................................................................................. 86 6.3.2 Validation of the consistency of experimental process.................................... 87 6.3.3 Geometry of Pulsed Corona Reactor ............................................................... 87 6.3.4 Pole Clustering................................................................................................. 87 6.3.5 Enhancement Features of PTbox ..................................................................... 88
6.4 Conclusions............................................................................................................. 89
REFERENCES ................................................................................................................. 90
APPENDIX A PA RESULTS OF PULSED VOLTAGE FOR ARGON GAS........................................ 92
A.1 MO vs. MSE Analysis of Pulsed Voltage.............................................................. 92 A.2 Pole Diagrams in the Vicinity of the MO/MSE Saddle Point for Pulsed Voltage. 95
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TABLE OF CONTENTS (CONT'D)
APPENDIX B PA RESULTS OF PULSED VOLTAGE AND PULSED CURRENT FOR HELIUM GAS................................................................................................................................... 98
B.1 MO vs. MSE Analysis of Pulsed Current .............................................................. 98 B.3 MO vs. MSE Analysis of Pulsed Voltage ............................................................ 103 B.4 Pole Diagrams in the Vicinity of the MO/MSE Saddle Point for Pulsed Voltage106
APPENDIX C PA RESULTS OF PULSED VOLTAGE AND PULSED CURRENT FOR NITROGEN GAS................................................................................................................................. 109
C.1 MO vs. MSE Analysis of Pulsed Current ............................................................ 109 C.2 Pole Diagrams in the Vicinity of the MO/MSE Saddle Point for Pulsed Current112 C.3 MO vs. MSE Analysis of Pulsed Voltage ............................................................ 114 C.4 Pole Diagrams in the Vicinity of the MO/MSE Saddle Point for Pulsed Voltage117
APPENDIX D................................................................................................................. 120 MATLAB CODE OF PRONY TOOLBOX................................................................... 120
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LIST OF TABLES
Table 2.1.1 Comparison Between Fourier Series and Prony Analysis ............................. 18 Table 5.1.1a Case List for Pulsed Current ........................................................................ 63 Table 5.1.1b Case List for Pulsed Voltage ....................................................................... 64 Table 5.2.1 Acronym Glossary for Variable and File Nomenclature ............................... 65 Table 5.4.1a Data of MO vs. MSE of Pulsed Current for Argon Gas at 60 SCFH .......... 71 Table 5.4.1b Data of MO vs. MSE of Pulsed Current for Argon Gas at 100 SCFH ........ 71 Table 5.4.2 Data of MO vs. NMSE of Pulsed Current for Argon Gas at 60 SCFH ......... 73 Table 5.5.1 List of PA Sessions of Pulsed Current for Argon Gas at MO 12 .................. 77 Table 5.5.2 MO/MSE Saddle Points and Clustering Points for All the CASES .............. 83 Table A.1.1a Data of MO vs. MSE of Pulsed Voltage for Argon Gas at 60 SCFH......... 94 Table A.1.1b Data of MO vs. MSE of Pulsed Voltage for Argon Gas at 100 SCFH....... 94 Table B.1.1a Data of MO vs. MSE of Pulsed Current for Helium Gas at 60 SCFH...... 100 Table B.1.1b Data of MO vs. MSE of Pulsed Current for Helium Gas at 100 SCFH.... 100 Table B.3.1a Data of MO vs. MSE of Pulsed Voltage for Helium Gas at 60 SCFH ..... 105 Table B.3.1b Data of MO vs. MSE of Pulsed Voltage for Helium Gas at 100 SCFH ... 105 Table C.1.1a Data of MO vs. MSE of Pulsed Current for Nitrogen Gas at 60 SCFH.... 111 Table C.1.1b Data of MO vs. MSE of Pulsed Current for Nitrogen Gas at 100 SCFH . 111 Table C.3.1a Data of MO vs. MSE of Pulsed Voltage for Nitrogen Gas at 60 SCFH ... 116 Table C.3.1b Data of MO vs. MSE of Pulsed Voltage for Nitrogen Gas at 100 SCFH. 116 Table D.1 List of M-files in PTbox ................................................................................ 120
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LIST OF FIGURES
Figure 1.1.1 PCR Input Voltage for Argon Gas ............................................................... 11 Figure 1.2.1 Pulsed Corona Reactor System Identification Model .................................. 13 Figure 1.2.2 Prony Analysis System Identification Loop................................................. 14 Figure 1.3.1 Organization Chart of Thesis........................................................................ 15 Figure 2.2.1 Pulsed Corona Reactor LTI System ............................................................. 19 Figure 4.1.1 GUI Development Process Road Map ......................................................... 36 Figure 4.2.1 Layout Editor in GUIDE .............................................................................. 38 Figure 4.2.2 Alignment Tool in GUIDE........................................................................... 39 Figure 4.2.3 Property Inspector in GUIDE....................................................................... 40 Figure 4.2.4 Object Browser in GUIDE ........................................................................... 41 Figure 4.2.5 Menu Editor in GUIDE ................................................................................ 42 Figure 4.2.1.1 Example of Callback Programming .......................................................... 44 Figure 4.4.1 Data Flow Path in PTbox ............................................................................. 46 Figure 4.5.1 PTbox Splash GUI........................................................................................ 47 Figure 4.6.1 Prepare Prony Data GUI............................................................................... 48 Figure 4.6.2 File Open Dialog Window............................................................................ 49 Figure 4.7.1 Perform PA GUI........................................................................................... 52 Figure 4.8.1 Compare PA Sessions GUI .......................................................................... 57 Figure 4.9.1 Export Data GUI........................................................................................... 59 Figure 4.10.1 Toolbar in PTbox........................................................................................ 60 Figure 4.10.2 Context Menu for an Axis in PTbox .......................................................... 60 Figure 4.10.3 Status Text Message Examples in PTbox .................................................. 61 Figure 4.10.4 Navigation Feature: Disable/Enable of Plot Push Button in PTbox........... 61 Figure 4.10.5 Help Menu in PTbox .................................................................................. 62 Figure 5.3.1 Application Settings in Prepare Prony Data GUI for MO vs. MSE Analysis
................................................................................................................................... 67 Figure 5.4.1 Application Settings in Perform PA GUI for MO vs. MSE Analysis .......... 68 Figure 5.4.2a Plot of MO vs. MSE of Pulsed Current for Argon Gas at 60 SCFH .......... 70 Figure 5.4.2b Plot of MO vs. MSE of Pulsed Current for Argon Gas at 100 SCFH........ 70 Figure 5.4.3 Plot of MO vs. NMSE of Pulsed Current for Argon Gas at 60 SCFH......... 72 Figure 5.5.1 Application Settings in Compare Sessions GUI........................................... 75 Figure 5.5.2 Poles of Pulsed Current for Argon at MO 12............................................... 77 Figure 5.5.3 Poles of Pulsed Current for Argon at MO 14............................................... 77 Figure 5.5.4 Poles of Pulsed Current for Argon at MO 16 (Saddle/Clustering Point) ..... 78 Figure 5.5.5 Poles of Pulsed Current for Argon at MO 18............................................... 78 Figure 5.5.6 Poles of Pulsed Current for Argon at MO 22............................................... 79 Figure 5.5.7 Poles of Pulsed Current for Helium at MO 16 (Clustering Point) ............... 80 Figure 5.5.8 Poles of Pulsed Current for Nitrogen at MO 16 (Clustering Point) ............. 81 Figure 5.5.9 Poles of Pulsed Voltage for Argon at MO 17 (Clustering Point)................. 81 Figure 5.5.10 Poles of Pulsed Voltage for Helium at MO 15 (Clustering Point) ............. 82 Figure 5.5.11 Poles of Pulsed Voltage for Nitrogen at MO 12 (Clustering Point)........... 82 Figure A.1.1a Plot of MO vs. MSE of Pulsed Voltage for Argon Gas at 60 SCFH......... 93
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LIST OF FIGURES (CONT'D)
Figure A.1.1b Plot of MO vs. MSE of Pulsed Voltage for Argon Gas at 100 SCFH....... 93 Figure A.2.1 Poles of Pulsed Voltage for Argon at MO 12.............................................. 95 Figure A.2.2 Poles of Pulsed Voltage for Argon at MO 14.............................................. 96 Figure A.2.3 Poles of Pulsed Voltage for Argon at MO 17 (Clustering Point)................ 96 Figure A.2.4 Poles of Pulsed Voltage for Argon at MO 18 (Saddle Point)...................... 97 Figure A.2.5 Poles of Pulsed Voltage for Argon at MO 22.............................................. 97 Figure B.1.1a Plot of MO vs. MSE of Pulsed Current for Helium Gas at 60 SCFH........ 99 Figure B.1.1b Plot of MO vs. MSE of Pulsed Current for Helium Gas at 100 SCFH ..... 99 Figure B.2.1 Poles of Pulsed Current for Helium at MO 10 .......................................... 101 Figure B.2.2 Poles of Pulsed Current for Helium at MO 12 (Saddle Point)................... 102 Figure B.2.3 Poles of Pulsed Current for Helium at MO 16 (Clustering Point)............. 102 Figure B.2.4 Poles of Pulsed Current for Helium at MO 18 .......................................... 103 Figure B.3.1a Plot of MO vs. MSE of Pulsed Voltage for Helium Gas at 60 SCFH ..... 104 Figure B.3.1a Plot of MO vs. MSE of Pulsed Voltage for Helium Gas at 100 SCFH ... 104 Figure B.4.1 Poles of Pulsed Voltage for Helium at MO 10 .......................................... 106 Figure B.4.2 Poles of Pulsed Voltage for Helium at MO 12 .......................................... 107 Figure B.4.3 Poles of Pulsed Voltage for Helium at MO 15 (Clustering Point) ............ 107 Figure B.4.4 Poles of Pulsed Voltage for Helium at MO 17 (Saddle Point) .................. 108 Figure C.1.1a Plot of MO vs. MSE of Pulsed Current for Nitrogen Gas at 60 SCFH ... 110 Figure C.1.1b Plot of MO vs. MSE of Pulsed Current for Nitrogen Gas at 100 SCFH . 111 Figure C.2.1 Poles of Pulsed Current for Nitrogen at MO 12 ........................................ 112 Figure C.2.2 Poles of Pulsed Current for Nitrogen at MO 15 ........................................ 113 Figure C.2.3 Poles of Pulsed Current for Nitrogen at MO 16 (Clustering Point)........... 113 Figure C.2.4 Poles of Pulsed Current for Nitrogen at MO 17 ........................................ 114 Figure C.3.1a Plot of MO vs. MSE of Pulsed Voltage for Nitrogen Gas at 60 SCFH... 115 Figure C.3.1b Plot of MO vs. MSE of Pulsed Voltage for Nitrogen Gas at 100 SCFH. 115 Figure C.4.1 Poles of Pulsed Voltage for Nitrogen at MO 10........................................ 117 Figure C.4.2 Poles of Pulsed Voltage for Nitrogen at MO 11........................................ 118 Figure C.4.3 Poles of Pulsed Voltage for Nitrogen at MO 12 (Clustering Point) .......... 118 Figure C.4.4 Poles of Pulsed Voltage for Nitrogen at MO 14 (Saddle Point)................ 119
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CHAPTER 1
INTRODUCTION
1.1 Motivation
Today our environment is facing serious threats due to exhaust gases from coal-
burning power plants, automotive engines, steel plants, paper mills and chemical plants.
Increasing levels of nitrogen oxide (NOx) and sulfur dioxide (SO2) are hindering the air
quality (Kim 1999). Environmental engineering is facing new challenges to control air
quality. Several new gas treatment methods have been investigated in recent years, and
Pulsed Corona Reactor (PCR) methods represent one of the very promising technology
paradigms with application to improving air quality. The main advantage of PCR
techniques is that they are modular, so they can be used on any scale, large or small. In
addition, they have lower capital cost and typically don’t require x-ray radiation shielding
(Kim 1999). This new approach to clean gas and water works by creating active species
in situ so transport losses can be avoided.
A pulsed corona is produced by applying a pulsed voltage to an electrode
separated from a grounded body by an appropriate distance. It is identical to a power
source that creates a discharge which is switched on and off on a microsecond time scale
to obtain high electric fields without sparking. The pulsed corona discharge contains
high-energy electrons that tend to dissociate molecules and create radicals such as O-,
OH-, N and indirectly HO2, O3 and others (Veldhuizen, Rutgers 2001). These radicals
initiate chemical reactions to oxidize the impurities of air: NOx and SO2 into acids and
hydrocarbons into CO2 and H2O, which are non-pollutants.
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Recent conjectures note that the voltage and current measurements at the PCR
electrode terminals appear to be well described by a sum of damped sinusoidal functions.
Figure 1.1.1 shows a plot of a voltage measurement at the PCR electrode terminals for a
reactor containing Argon gas.
Figure 1.1.1 PCR Input Voltage for Argon Gas
The plot of Figure 1.1.1 demonstrates that the gross features of a PCR terminal
voltage are consistent with damped sinusoidal form. Analysis of the PCR signals is
important in order to control the reactor yield and outlet gas composition. Apart from
PCR, these kinds of damped signals are found in several engineering and medical
problems. For example, in the medical literature, human heart sounds have been shown
to consist of transient, damped sinusoidal components with closely spaced frequency
characteristics (Sava, McDonnell 1995). Spectral analysis of human heart sounds is of
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paramount importance to detect different cardiac abnormalities, especially those related
to valvular origin as well as malfunctioning of implanted artificial heart valves (Sava,
McDonnell 1995). Electrical power systems are another area of study in which modal
content of power oscillations are best estimated as transient damped sinusoidal
components from measured inputs or disturbances to the power system (Trudnowski,
Johnson, Hauer 1998). It has also been shown that the dominant complex natural
responses of radar targets are best described by transient damped oscillatory signals
(Chuang, Moffatt 1976).
1.2 System Identification Using Prony Analysis
Characterization of PCR measurements is a system identification problem. Rates
of removal of NOx and SO2 might be key parameters that are related to the impulse
voltage and impulse current applied to the corona reactor. The particular composition of
PCR voltage and current measurements tends to vary with the chemical species present in
the reactor. The relationships among the above stated variables can be examined by
applying a suitable mathematical and analytical model. There are several mathematical
models and techniques available for such systems. In this thesis, we investigate the
variation in measurement composition via the classical signal processing technique
known as “Prony analysis.” Figure 1.2.1 shows a general model of system identification,
adapted to our particular system, using Prony analysis (Hsia 1977).
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Clean Air
Pulsed Corona Reactor
Prony Analysis
Measuring Instrument Noise
Flue Gas / Air Pollutants
Input Voltage & Current
appro
predic
we ha
Toolb
system
impor
(Ljun
Measured Voltage
& Current
Pulsed Corona Reactor Model
Figure 1.2.1 Pulsed Corona Reactor System Identification Model
In fact, a major aspect of the system identification process is development of
priate models (Ljung 2001). A good model is essential for accurate simulation and
tion. Many different methods are available to construct the models. In this thesis,
ve implemented the Prony analysis in a graphical user interface we call the “Prony
ox,” built around MATLAB functions. The Prony Toolbox, or PTbox, follows the
identification loop as shown in the Figure 1.2.2. It has a natural logical flow:
t the data, choose a data set, perform data preprocessing then apply Prony analysis
g 1999).
13
Import Data
Choose Data Set
Data Preprocessing
Calculate Prony Model
Energy oModes
Squared Error
Residues Pole-Zero
Validate Prony Model
OK: Use it
Figure 1.2.2 Prony Analysis System Identifica
14
Pulsed Corona Reactor Voltage
and Current Measurements
Choose Model Order and Criterion of Fit
f
tion
Not OK: Revise
Loop
1.3 Organization of Thesis
This thesis is divided into three major parts: system identification, graphical user
interface and application of the Prony Toolbox to PCR measurements. The organization
of the thesis is shown in Figure 1.3.1 as given below.
• • Chapter 2
Part-I System Identification, Prony Analysis
•
•
Part-II Graphical User Interface (GUI), Prony Toolbox
Part-III Apply Prony Toolbox to PCR Measurements
Figure 1.3.1 Orga
1.4 Synopsis of Thesis
This thesis applies Prony analysis
implements the least squares version of th
thesis also covers design considerations in
selection, model order selection criteria, s
separation, root inspection and assessing r
Chapter 3Chapter 4
•
•
•
Chapter 5Chapter 6nization Chart of Thesis
to characterize PCR mea
e Prony analysis for syst
cluding data preprocess
ignal subspace selection
esiduals. We have desig
15
Prony Method Modified Prony Methods
Prony Toolbox Design Considerations GUI Programmingin MATLAB GUIs in Prony Toolbox
Results and Discussion Conclusions andFuture Research
surements. It
em identification. The
ing, model order
, signal and noise
ned a software tool in
MATLAB, the Prony Toolbox, to perform Prony analysis. The Prony Toolbox provides
flexibility to compare and display analysis results simultaneously for several parameter
variations. Using the Prony Toolbox, we have carried out a wide range of investigations
to characterize PCR measurements. In conclusion, this thesis includes a discussion of
potential future research which is prompted by this work.
16
CHAPTER 2
PRONY ANALYSIS
Prony analysis has been shown to be a viable technique to model a linear sum of
damped complex exponentials to signals that are uniformly sampled. The Prony analysis
was developed by Gaspard Riche, Baron de Prony in 1795 in order to explain the
expansion of various gases (de Prony 1795, Hildebrand 1974 and Marple 1987). In his
original paper, Prony proposed fitting a sum of exponentials to equally spaced data points
and extended the model to interpolate at intermediate points. The Prony analysis is not
only a signal analysis technique but also a system identification method, which is widely
used in the areas of power system electromechanical oscillation, biomedical monitoring,
radioactive decay, radar, sonar, geophysical sensing and speech processing.
2.1 Comparison Between Fourier Series and Prony Analysis
As compared to other oscillatory signal analysis techniques such as those of
Fourier, Prony analysis has the advantage of estimating damping coefficients apart from
frequency, phase and amplitude. In addition, it best fits a reduced-order model to a high-
order system both in time and frequency domains (Marple 1987). Major differences
between Fourier series and Prony analysis are listed in Table 2.1.1.
17
Table 2.1.1 Comparison Between Fourier Series and Prony Analysis
Fourier Series (FS) Prony Analysis (PA)
1. FS is a Non-Parametric method. PA is a Parametric method.
2. FS fits a sum of undamped complex
exponentials.
PA fits a sum of damped complex
exponentials.
3. FS computes amplitude, phase and
frequency of the signal components.
Apart from amplitude, phase and
frequency, PA also computes damping
coefficients of the signal components.
Fourier series has several drawbacks when it is applied to the time-domain signal
which is corrupted by noise. First, experimental time-domain signals are of finite
duration. Fourier transformation of truncated time-domain signal leads to undesirable
frequency-domain “wiggles” (“Gibbs oscillations”) which make it hard to observe a
small peak in the vicinity of a large peak (Marshall, Verdun 1990). Second, Fourier
transforms distributes time-domain noise uniformly throughout the frequency domain
which leads to limitation in the certainty with which peak frequencies, widths,
magnitudes and phases could be computed. Third, discrete sampling of a time-domain
continuous signal causes limitation in obtaining the spectral information content
(Marshall, Verdun 1990).
The Prony analysis (PA) is known to behave poorly when a signal is embedded in
noise (Marple 1987). It yields parameter estimates with a large bias due to its sensitivity
to measurement noise. It does not make a separate estimate of the noise. It also fits
exponentials to any additive noise present in the signal. When PA is applied to a signal
18
embedded in noise, the damping and frequency terms are typically significantly miss-
estimated; they are usually much greater than the actual values (Marple 1987). Besides
poor fit when signal to noise ratio is small, PA is also known to be inconsistent (Kahn et
al 1992).
2.2 Original Prony Analysis
To derive the mathematical formulation for the original Prony analysis, let us
consider a Pulsed Corona Reactor (PCR) as a linear time-invariant (LTI) dynamic system
as shown in Figure 2.2.1.
Pulsed Corona Reactor
(LTI System) ) )
In Figur
y
x
u
The evo
Suppose
some in
there are
u(t
x(t)Figure 2.2.1 Pulsed Corona Reactor LTI System
e 2.2.1, the signals are referred to as follows:
(t): PCR system response,
(t): State of the PCR system,
(t): Input to the PCR system.
lution of the state of the PCR system is expressed by (2.2.1):
)()()( tButAxdt
tdx+= , where A and B are constant matrices.
that the PCR is brought to an “initial state” 0)( xtx = at time t0, by
put pulse (Hauer, Demeure, Scharf 1990). If the input is removed
no subsequent inputs to the system, then (2.2.1) can be rewritten a
(
19
y(t
(2.2.1)
means of
and
s
)0)( =tu
)()( tAxdt
tdx= . (2.2.2)
Here A is a matrix of size whose eigenvalues are λnn× i, right eigenvectors are pi and
left eigenvectors are qi (Kailath 1980). In (2.2.2), system order is represented by n. The
solution to (2.2.2) is expressed as the sum of n components:
)(0
1
)()( ti
n
i
Ti
iepxqtx λ∑=
= . (2.2.3)
As we have assumed the PCR is an LTI system, we express y(t) in the form
) , where C and D are constant matrices. (2.2.4) ()()( tDutCxty +=
If the input is removed , then (2.2.4) simplifies to: )0)(( =tu
) . (2.2.5) ()( tCxty =
The Prony analysis directly estimates the parameters of the eigen structure described in
(2.2.3) by fitting a sum of complex damped sinusoids to evenly spaced sample (in time)
values of the output:
. (2.2.6) )2cos()( )(
1
^
iit
L
ii tfeAty i φπσ += ∑
=
In (2.2.6), we have utilized the following notations:
Ai: Amplitude of component i,
iσ : Damping coefficient of component i,
iφ : Phase of component i,
fi : Frequency of component i,
L: Total number of damped exponential components,
20
)(^
ty : Estimate of observed data for y(t) consisting of N samples y(tk) = y[k],
k=0,1,2,….N-1 that are evenly spaced.
Using Euler’s theorem, )2cos( ii tf φπ + can be represented as a sum of exponentials:
222)2cos(
22)2()2( iiiiiiii jtfjjtfjtfjtfj
iieeeeeetf
φπφπφπφπ
φπ−−+−+
+=+
=+ . (2.2.7)
Inserting (2.2.7) in (2.2.6) and letting t = kT, the samples of are rewritten as )(^
ty
∑=
=L
i
kiiCky
1][ µ (2.2.8)
where
ijii e
AC ϕ
2= (2.2.9)
, which we refer to as the “poles.” (2.2.10) Tfji
iie )2( πσµ +=
In (2.2.10), T is the sampling period.
The original Prony analysis computes Ci and µi in three basic steps (Pierre 2002):
• Solve linear prediction model, which is constructed by the observed data set.
First write (2.2.8) as a linear prediction model,
][......]2[]1[][ 21 Lkyakyakyaky L −++−+−= . (2.2.11)
In (2.2.11), y[k] is computed for 1,...,2,1, −++= NLLLk . For example, y[L] is
computed at : Lk =
]0[......]2[]1[][ 21 yaLyaLyaLy L++−+−= .
We can write y[k] in matrix form for various values of k as
21
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−−−−
+−−−
=
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
+
La
aaa
LNyNyNy
yLyLyyLyLyyLyLy
Ny
LyLy
..]1[....]3[]2[
..........]2[....][]1[]1[....]1[][]0[....]2[]1[
]1[....
]1[][
3
2
1
(2.2.12)
or
Dad = (2.2.13)
where
d = , D = and a = .
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
+
]1[....
]1[][
Ny
LyLy
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−−−−
+−−−
]1[....]3[]2[..........
]2[....][]1[]1[....]1[][]0[....]2[]1[
LNyNyNy
yLyLyyLyLyyLyLy
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
La
aaa
..3
2
1
Assuming the linear prediction coefficients vector a is estimated by solving the
over-determined least square problem, which is computed using (2.2.13)
LN 2>
a = D\d. (2.2.14)
In MATLAB, a computationally robust way to find a is
a=pinv(D)*d ;
where the pinv function computes the pseudo inverse of D.
• Find roots of characteristic polynomial formed from the linear prediction
coefficients.
))......()((......^^
2
^
111
1 LLLLL aaa µµµµµµµµµ −−−=−−−− −− . (2.2.15)
As vector a is known from (2.2.14), the roots of the polynomial (2.2.15) can be readily
computed.
i
^µ
22
In MATLAB, the roots can be computed as ^µ
muhat=roots([1;-a]);
where the vector [1;-a] describes the polynomial to be rooted.
• Solve the original set of linear equations to yield the estimates of the
exponential amplitude and sinusoidal phase.
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−−−
LN
L
NN
L
L
C
CCC
Ny
yy
..
....
..............
....
1....11
]1[....
]1[]0[
3
2
1
1^1
2
^1
1
^
2^2
2
^2
1
^
1^1
2
^1
1
^
µµµ
µµµ
µµµ
(2.2.16)
or
Y = UC (2.2.17)
where
Y = , U = and C = .
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
− ]1[....
]1[]0[
Ny
yy
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−− 1^1
2
^1
1
^
2^2
2
^2
1
^
1^1
2
^1
1
^
....
..............
....
1....11
N
L
NN
L
L
µµµ
µµµ
µµµ
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
LC
CCC
..3
2
1
In MATLAB, the original linear prediction coefficients Ci can be computed by solving
the over-determined set of equations in (2.2.17).
C = U\Y;
As C and are now known, the amplitude, frequency, phase and damping coefficients
are computed using (2.2.9) and (2.2.10).
^µ
23
2.3 Prony Analysis for Time-Domain Design of IIR Filters
In this thesis, we have utilized the MATLAB’s Signal Processing Toolbox built-in
function prony to perform the Prony analysis. The prony function implements the
Prony analysis for time-domain design of IIR filters (Parks, Burrus 1987). It models a
signal using a specified number of poles and zeros (MATLAB Help 2002). The method
uses a variation of the covariance method of autoregressive (AR) modeling to find the
denominator coefficients, the ai, and then finds the numerator coefficients bi for which
the resulting filter's impulse response matches exactly the first (n + 1) samples of the
given data sequence. The filter is not necessarily stable, but this method can potentially
recover the coefficients exactly if the data sequence is truly an autoregressive moving
average (ARMA) process of the correct order (MATLAB Help 2002).
The transfer function of an IIR filter can be written as (Parks, Burrus 1987)
NN
MM
zazazbzbb
zAzBzH −−
−−
++++++
==........1........
)()()( 1
1
110 . (2.3.1)
H(z) is the z-transform of h[n] and is related by the following equation:
∑∞
=
−=0
][)(n
nznhzH .
We can rewrite (2.3.1) as
B(z) = H(z)A(z) (2.3.2)
Equation (2.3.2) is the z-transform of a discrete time convolution, and it can be written as
a matrix multiplication. Using the first K+1 terms of the impulse response, we can write
(Parks, Burrus 1987)
24
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
N
NKk
M
M
a
aa
hh
h
hhhhh
h
b
bbb
.
.
.
1
.....
.
..
...
00...00
0...0
.
.
.
2
1012
01
0
2
1
0
. (2.3.3)
To compute the ai and bi let us partition the matrices as
. (2.3.4) ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−−
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−−−−−−=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−−
aHh
Hb 1
.
.021
1
In (2.3.4), we have used the following notations:
b: Column vector of the M+1 numerator coefficients of (2.3.1),
a: Column vector of the N denominator coefficients of (2.3.1),
h1: Column vector of the last K-M terms of the impulse response,
H1: (M+1) by (N+1) partition of the matrix in (2.3.3),
H2: (K-M) by N partition of the matrix (2.3.3).
The lower K-M equations are written as
aHh 210 +=
or
aHh 21 −= (2.3.5)
25
Equation (2.3.5) suggests the solution for a.
The upper M+1 equations of (2.3.4) are written as below to calculate b:
aHb 1= . (2.3.6)
If K = M+N, H2 is square. If H2 is not singular, (2.3.5) and (2.3.6) can be solved
respectively for a and b. If H2 is singular, (2.3.5) may have many solutions. In that case,
h[n] is most likely generated by a lower-order system. From this fact, it should become
apparent that the assumed model order could have a significant impact on the Prony
results.
Implementation of MATLAB’s prony function requires that numerator order M
and denominator order N of H(z) in (2.3.1) are known. We have used a user-specified
model order as N and M in the prony function. If numerator order of N of H(z) in (2.3.1)
is specified as zero then the Prony analysis for time-domain design of IIR filters
computes the a vector similar to the a vector computed by the original Prony analysis in
(2.2.15). The original Prony analysis is different from the Prony analysis for time-
domain design of IIR filters in the sense that it can’t compute the b vector.
2.4 Modified Prony Analyses
PA is also a numerically intensive algorithm. It involves solution of an over-
determined set of linear equations and rooting of a high-order polynomial, which are
numerically intensive operations.
There are several algorithms suggested as Modified Prony algorithms (Li, Liu,
Razavilar 1998). If the damping factors of signal components are small and the peak
signal to noise ratio (SNR) is high, the backward linear prediction algorithm or
26
Kumaresan-Tufts (KT) algorithm (Kumaresan, Tufts 1987) attains the Crame΄r-Rao
bound. However, the KT algorithm doesn’t estimate the parameters effectively when the
signals are of lower SNR or large damping factor. Some of the other algorithms which
provide better estimation of signal parameters in the presence of noise are the total least
square (TLS) algorithm (Rahman, Yu 1987), the matrix pensil algorithm (Hua, Sarkar
1990) and the maximum likelihood (ML) algorithm (Bresler, Macovski 1986). Modified
Prony analyses involve the use of forward and backward prediction polynomial zeros,
high prediction orders and singular value decomposition (Holt, Antill 1976) to distinguish
signal roots in the presence of additive noise.
While the version of the Prony Toolbox described in this thesis provides access
only to MATLAB’s built-in prony, extension to modified algorithms should be readily
accomplished, thanks to the modular design of the toolbox.
27
CHAPTER 3
DESIGN BASIS OF THE PRONY TOOLBOX
Application of Prony analysis (PA) to characterize Pulsed Corona Reactor (PCR)
measurements is a real world application. Presently, there exist a few software tools (e.g.
MATLAB, Auto Signal, DSI Tools Ringdown) available to perform PA and each of them
has their own limitations: uncommon data input format, limited number of input signals,
rigid software architecture, bugs and limited features. One of the main constraints of the
existing software systems is inability or inflexibility for simultaneous display and
comparison of several Prony analysis sessions. The analysis comparison ability is
believed to be an essential requirement for the PCR signal characterization. Our
hypothesis requires that the roots of the PA for various sets of experiments can be
simultaneously inspected. Hence, we have designed a software tool, the Prony Toolbox
(PTbox), built around MATLAB functions with a user-friendly graphical interface and
containing all the necessary features to perform PA in the application setting of PCR
characterization.
PA is a multi-step procedure which involves performing an autoregressive (AR)
fit, rooting the AR coefficients for complex exponential parameters, sorting the roots,
completing a least squares fit for complex amplitude parameters, and creating the final
model with signal thresholding. All of these steps involve design considerations: data
input format, data preprocessing and filtering, model order selection criteria, signal-noise
separation, assessing residuals and root inspection. Data preprocessing is required
including filtering and removal of the mean. A detailed numeric summary of a PA
28
session results is an essential ingredient of the software tool. The subsequent sections of
this Chapter describe the design requirements in detail.
3.1 Data Input Format
One of the fundamental themes of PTbox is simplicity. In order to attain this
theme, it is important to design the tool in a data input format that is widely accepted in
the research community and industry. It is also required that the file format is cross-
platform-compatible (Hanselman, Littlefield 2001). The different platforms of
MATLAB should not involve any special treatment. The native binary MAT-files fulfill
all of the above requirements. MAT-files also have the ability to store the data with high
precision and to load the data into the MATLAB base workspace with high speed. Due
to the above qualities of the native binary MAT-files, PTbox is designed to import the
data files in the MAT-file format.
3.2 Reduction in Data Length
Prony analysis involves the solution of an over-determined set of linear equations
and rooting of a high-order polynomial, which are highly numerically intensive
operations (Trudnowski 1995). Down-sampling the signal can increase the
computational efficiency and performance of the Prony analysis. Hence, a flexibility to
down-sample the signal is provided in the PTbox.
29
3.3 Data Range
Apart from down-sampling the signal, reduction in the data length can also be
achieved by selecting a specific region of the data. This feature is especially useful when
the signal damps out rapidly; in such a case, the user can specify a data range as the
specific section of the signal where it is judged significant.
3.4 Data Preprocessing and Filtering
A linear trend, a signal mean, or noise in the raw data can cause significant errors
in the PA results. Hence, it is important to preprocess the data before applying the Prony
analysis. Data preprocessing involves filtering the data to isolate the signal components
by frequency, detrending and removing the mean from the signal. Detrend removes any
linear trend from the signal. It computes the least-squares fit of a straight line to the data
and subtracts the resulting function from the data (MATLAB Help 2002).
3.5 Model Order Determination
PA requires some a priori information about the system to be identified; most
obvious is the model order. Usually, in most real systems, the model order is unknown,
however a system specialist may have a rough idea of the expected order. There are
several methods available to estimate model order. The residue amplitude and minimum
energy criterion are two of these sorting methods that presort the residues according to
their amplitude and energy. Several other criteria exist, including specifically the Akaike
information criterion (Trudnowski 1994) and Minimum Descriptive Length (Wax,
30
Kailath 1985). In the literature, they are known to perform poorly for very closely spaced
modes (Trudnowski, Hauer, Rogers 1995).
As there is no straightforward method to compute the model order of an arbitrary
system, an initial estimate of model order can be obtained from the user. Subsequently,
Prony analysis can be performed based on this initial guess, and residues can be presorted
according to their amplitudes and energies.
A good rule of thumb is to initially assume a model order which is approximately
one-third of the sample data length (Trudnowski, Hauer, Rogers 1995). A subset of the
resulting modes can then be selected by iteratively increasing the number of modes in the
model response until the model reconstruction accurately represents the original data. In
order to implement the above rule of thumb, it is necessary to design the PTbox in such a
way that it allows the user to select the specific residues from the presorted residues list.
The feature to select the presorted residues should be implemented in a simple and user-
friendly way.
3.6 Graphic Mode of Prony Analysis Fit
Prony analysis fit can be displayed in two graphic modes: time and frequency. To
analyze the Prony fit, both graphic modes are necessary. The time-domain graphic mode
provides more information in the case of a signal containing low energy residues. The
frequency domain graphic mode emphasizes the high energy residues and the major
frequency components of the Prony fit.
31
3.7 Signal-Noise Separation
Most real signals are corrupted by additive broad-band noise that cannot simply
be filtered out. It is difficult to separate the noise modes from the signal modes in such a
signal. When there is sufficient signal-noise separation, residue amplitudes and residue
energy plots will typically reveal one or more sharp transitions between the signal
subspace and the noise subspace floor (Autosignal 2002). Hence, residue amplitude and
energy plots should be provided in the PTbox to assist the user to distinguish the signal
modes.
3.8 Root Inspection
The inspection of the Prony roots is believed to be an essential requirement to
validate our hypothesis regarding PCR root clustering. The hypothesis validation also
requires the ability to simultaneously display the Prony roots of several Prony analysis
sessions. The root inspection in the Prony model can also assist the user to distinguish
the signal modes.
3.9 Accuracy of Prony Fit
Computations of some of the key statistics should be available. For example,
squared error and mean squared error (MSE), would be helpful to determine the accuracy
of the Prony fit.
32
3.10 Numeric Summary
A list of the key parameters of the Prony analysis should be displayed so that the
user can analyze the Prony fit. There should be features to export and plot the results
associated with each step of the Prony fit.
3.11 Simultaneous Display of Several Prony Analysis Sessions
The ability to display and analyze several Prony analysis sessions is one of the
fundamental and essential design requirements to validate our hypothesis. This is a
unique feature of PTbox which is not present in any of the existing PA software tools
examined by the author.
3.12 Graphical User Interface
PTbox is designed to provide an interface between the user and the application's
underlying code hence the user can apply PA without knowing cumbersome command
line interface commands.
The graphical user interfaces (GUIs) in the PTbox are built using the GUIDE
(Graphical User Interface Development Environment) tool in MATLAB 6.5. The
GUIDE tool provides an easy and efficient way to compose a GUI and add functionality
to it (Marchand 1999). It allows the application programmer to place and modify
MATLAB graphics objects (i.e., uicontrol (user interface control), uimenu (user
interface menu) and axes) in a graphical manner. GUI development using MATLAB
graphics objects can be time consuming and tedious. It requires a lot of experience and
practice to estimate the position attributes of uicontrol and uimenu elements to obtain
33
the desired look for a GUI (Marchand 1999). The GUIDE circumvents this tedious GUI
development process and provides a simple and intuitive graphical approach to GUI
design. By providing a user-friendly way to design GUIs, GUIDE cuts down the
development time. Using GUIDE, the application programmer can focus on functionality
of the GUI; the appearance and form of the GUI is taken care of by the GUIDE tool.
34
CHAPTER 4
IMPLEMENTING THE FEATURES OF THE PRONY TOOLBOX
Prony Toolbox (PTbox) features a graphical user interface (GUI) written in the native
functionality of MATLAB 6.5. It is a user interface built with graphical objects such as
push buttons, text fields, popup menus, frames, axes, check boxes, radio buttons, edit
boxes and menus. The graphical objects are utilized to promote intuitive learning and
application.
GUI programming in MATLAB 6.5 is implemented using various uicontrol
objects and treating GUIs as figure windows. It involves two basic steps:
• Each uicontrol object is assigned a particular position and size in the GUI layout.
• Each uicontrol object is programmed, via so-called callback functions, to perform
a specific task when a specific event is observed.
In addition, functionalities are provided to save and run the GUI. All of these tasks
can be achieved with GUIDE (MATLAB Help 2002). This Chapter is divided into two
parts. The first part describes the GUI development process and introduces the GUIDE,
which was employed by the programmer to construct the GUIs in PTbox. The second
part discusses the features of PTbox and the underlying technical principles.
4.1 GUI Development Process
Development of the GUIs to perform PA has been an iterative process. Figure
4.1.1 shows a road map for creating GUIs (MATLAB Help 2002).
35
Define Task
F
Draw GUI
igure 4.1.1 G
Test Design
UI Development Process Ro
36
Test Code
Write Code
ad Map
STOP
START
Design
Implementation
Figure 4.1.1 shows that the GUI creation process consisted of two phases, a design and an
implementation phase (MATLAB Help 2002). Defining the task then drawing and
testing the GUI by hand are the components of the design phase. Writing and testing the
code are the components of the implementation phase. It is an important fact that the
design phase should be completed before starting the implementation phase. The road
map also shows that arrows flow in both directions; this reflects that the project is likely
to go through the design and implementation phases multiple times (MATLAB Help
2002).
4.2 Introduction to GUIDE
GUIDE is primarily a set of layout tools: Layout Editor, Object Browser, Menu
Editor, Property Inspector and Tab Order Editor. These tools simplify the process of
laying out the graphical objects and programming the GUI. The main layout tool of the
GUIDE is the Layout Editor, which is a control panel of all the GUIDE tools. It is
displayed when GUIDE is opened. It allows the programmer to design the GUI layout
quickly and easily by dragging components from the component palette into the layout
area (MATLAB Help 2002). Figure 4.2.1 shows the Layout Editor.
37
Figure 4.2.1 Layout Editor in GUIDE
The left side of the Layout Editor, which has the user interface objects, is called the
component palette, and the right hand side, where objects are placed, is called the layout
area. Once components are placed inside the layout area, the programmer can align the
objects using the Alignment Tool. It enables the programmer to position objects with
respect to each other and to adjust the spacing between selected objects. Figure 4.2.2
shows the Alignment Tool. It implements two types of alignment operations:
• Align – This option aligns all selected components to a single reference line,
which is determined by the bounding box that encloses the selected objects. It
aligns in both vertical and horizontal directions.
38
• Distribute – The distribute options provide equal spacing between all components
of a selected group.
Figure 4.2.2 Alignment Tool in GUIDE
Properties of components are set by the Property Inspector. Figure 4.2.3 shows the
Property Inspector as applied to an axes object ‘axes_segimage’. It lists all the settable
properties and their current values.
39
Figure 4.2.3 Property Inspector in GUIDE
Sometimes there is a need to see a hierarchical list of all the objects in the GUI. For this
purpose, the Object Browser can be used. Figure 4.2.4 shows a figure object and its
child objects. It shows the parent object, a figure with title Prepare Prony Data GUI,
and its child objects.
40
Figure 4.2.4 Object Browser in GUIDE
The Menu Editor of GUIDE enables the programmer to create two kinds of menus:
• Menu bar – menu is displayed on the top part of the figure.
• Context menu – menu is displayed when the user right-clicks on graphical
objects.
Figure 4.2.5 shows the Menu Editor for the Prepare Prony Data GUI.
41
Figure 4.2.5 Menu Editor in GUIDE
Once objects are layed out and menus are created, the programmer has to save the GUI.
The GUIDE allows the programmer to store GUIs in two file formats:
• FIG-file – the file has an extension .fig that contains a complete description of the
GUI figure layout and the components of the GUI. The changes made by the
user to the GUI layout in the Layout Editor are automatically saved in the FIG-
file (MATLAB Help 2002).
• M-file – the file has an extension .m that contains the code that controls the GUI,
including the callback functions for its components. By default, GUIDE saves
the GUI objects in a FIG-file and writes blank stubs for each of the callback
42
functions referenced by the objects into this M-file. The programmer has to
program the callbacks according to the application needs.
The next section discusses callback programming.
4.2.1 Callback Programming
The term ‘callback’ refers to the action associated with a specific graphic object.
When a user performs an action, for example a left-click, MATLAB executes the code
defined in the callback function of the object which received the left-click. The
callback’s name is derived from the object’s tag property. For example, the callback
name of a push button whose tag is push_plot is push_plot_Callback.
To add code to the callback of the push_plot push button, the programmer has to open
the GUI M-file in the M-file editor. Figure 4.2.1.1 shows an M-file in which a
push_plot push button is defined. The following code is automatically generated by
GUIDE for this button:
function varargout = push_plot_Callback(hObject, eventdata, handles, varargin) % hObject handle to push_plot (see GCBO) % eventdata reserved - to be defined in a future version % of MATLAB % handles structure with handles and user data (see % GUIDATA)
The programmer has to add the following code in order to activate the plotting associated
with the push_plot push button.
axes(handles.axes_data); [x,y,x_val,y_val] = get_var_names(handles); plot(x_val,y_val,'m'); xlabel('Time'),ylabel('Signal'), title('Original Data Plot');
43
Figure 4.2.1.1 Example of Callback Programming
For other types of callbacks such as radio buttons, edit boxes and popup menus, the get
function is typically employed to get the current value of the component property and
set is employed to specify the value for that particular property. For example, the
following code obtains the user entered value in an edit box:
Dfactor=round(str2num(get(handles.edit_sample,'String')));
The following code enables a push button:
set(handles.push_pronyanalysis,'Enable','on');
44
4.3 Installation of PTbox
The PTbox software consists of a single directory or folder of M-files, FIG-files
and MAT-files. To install the PTbox, the user has to copy the folder to anywhere in the
path of MATLAB. A new folder can be added to the MATLAB path by using either the
setpath function or Set Path… option in the MATLAB command window ‘File’ menu.
PTbox is started when the user enters pronytool at the command window prompt.
4.4 Graphical User Interfaces in PTbox
PTbox consists of five GUI’s:
• Splash GUI
• Prepare Prony Data GUI
• Perform PA GUI
• Compare PA Sessions GUI
• Export Data GUI.
Each of the GUIs is designed to implement a particular task. For example, the Prepare
Prony Data GUI performs data preprocessing tasks and prepares the data for PA. The
data flow in the PTbox is summarized in Figure 4.4.1. First data is obtained and
preprocessed in the Prepare Prony Data GUI and then PA is performed. In the last GUI,
PA sessions are compared. The Export Data GUI allows the export of data associated
with each GUI.
45
I
Each
Splash GU
Prepare Prony Data GUI
Compare PA Sessions GUI
Perform PA GUI
Figure 4
of the above GUIs is describ
Export Data GUI
.4.1 Data Flow Path in PTbox
ed in detail in subsequent sections.
46
4.5 PTbox Splash GUI
Figure 4.5.1 PTbox Splash GUI
The PTbox Splash GUI appears when the user enters pronytool at the MATLAB
command window prompt. This GUI is only an informational GUI. It introduces the
PTbox to the user.
47
4.6 Prepare Prony Data GUI
In this GUI, first data is imported then data preprocessing is performed on the
imported data. Figure 4.6.1 shows this GUI.
Figure 4.6.1 Prepare Prony Data GUI
The main features of this GUI are as follows:
Import Data: A file open dialog window appears when the user presses this push button.
The user has to choose a data file from the traditional file open dialog. Figure 4.6.2
shows the file open dialog window. The data file should be in MATLAB MAT-file form.
Once the file is selected and successfully read, the variables are shown in two list boxes.
48
Figure 4.6.2 File Open Dialog Window
PTbox requires that the data should be stored along with variable names in the MAT-file;
the read fails if the file does not contain variable names. PTbox uses the variable names
for display in the selection list boxes. Presently, PTbox does not provide the flexibility of
modifying the data names. However, this should not be a limitation as the names in the
MAT-file can be modified directly by the user via the load and save commands of
MATLAB.
Plot: This push button plots the variables selected by the user from the dependent and
independent variable list boxes. It is a plot of the original data with no preprocessing.
Decimation: This option allows the user to specify a particular data range and reduce the
data length by decimating the signal. The extent of data length reduction is typically
described by the down sampling factor. Formally, decimation converts a discrete time
signal x[n] to another discrete time signal y[n], which consists of subsamples of x[n]
(Porat 1997). Thus M-fold decimation or down-sampling is described by
49
][][ nMxny = , where n and M are integers. (4.6.1)
Decimation increases the sampling interval of the decimated signal by a factor of M. In
the equation (4.6.1), if the sampling interval of x[n] is T, then that of y[n] is MT (Porat
1997). PTbox obtains the down-sampling factor M from the user and performs M-fold
decimation. For example: if the user has specified a down-sampling factor of two then
the toolbox selects alternate samples of the data, hence the decimated signal samples
appear at a rate slower than that of the original signal by a factor of two.
The down-sampling process can cause aliasing in the frequency domain. Hence, an
antialiasing filter with cut off frequency of Mπ should be used before down sampling.
Presently PTbox does not provide the flexibility of filtering the decimated signal. PTbox
has ‘decimation off’ by default, and consequently the user has to activate the decimation
option.
Data Range: This option allows the user to specify a specific section of the decimated
or undecimated signal. This option is particularly useful when the signal damps out
rapidly. Data range is specified by a left mouse click at two different locations on the
plotted range.
Data Preprocessing: PTbox can preprocess the original data range or specific data range
signal. Several data preprocessing options are available:
• Remove Mean: It subtracts the signal sample mean from each sample. It also
changes the time range of signals. The time range of negative values is discarded
and positive values are picked up during the Remove Mean operation. It scales
50
the PCR signals; amplitude of all other signal is unchanged. The scaling of PCR
signals is done according to the given measured transducer relations to obtain true
voltage and ampere units. It also facilitates the correct computation of energy and
power transferred to the pulsed corona reactor in a pulse. The pulsed voltage
signals and the pulsed current signals are scaled by a factor of 242000 and 710
respectively.
• Detrend: It removes the linear trend from the signal. It computes the least-
squares fit of a straight line (or composite line for piecewise linear trends) to the
data and subtracts the resulting function from the data (MATLAB Help 2002).
In MATLAB, the detrend of the data x is computed as
y=detrend(x);
where the data y is obtained after detrending x.
• No change: It results in no preprocessing of the signal.
By pressing the Go push button, the selected data preprocessing is implemented. After
data preprocessing is performed, the data is ready to send to the Perform PA GUI. This is
done by pressing the Perform Prony Analysis push button.
4.7 Perform PA GUI
In this GUI, the Prony analysis is performed. PTbox uses MATLAB’s Signal
Processing Toolbox built-in prony function to perform Prony analysis. The input to this
GUI is the data from the Prepare Prony Data GUI. Figure 4.7.1 shows the Perform PA
GUI.
51
Figure 4.7.1 Perform PA GUI
The main features of this GUI are as follows:
Model Order: The user has to specify a model order in the editable text box. The GUI
uses the user-specified value to perform the PA.
Graphic Mode: This refers to the graphic display of the PA fit in time or frequency
domain. The frequency domain description of the Prony fit is obtained by computing the
Fast Fourier Transform (FFT) of the Prony estimated signal in the sample or time-
domain.
52
Number of Residues: The number of residues to be retained for the results display must
be specified by the user in an editable box. This number is used to choose only specific
residues according to the selection criteria. The total number of residues specified cannot
exceed the model order previously given. If it exceeds, then a dialog box appears and
prompts the user to specify a number of residues less than or equal to the model order.
For a specific number of residues, the GUI also checks that the conjugate of the last
residue is always included in the PA if it indeed exists. The GUI has been programmed
such that it automatically increments the number of residues by one and picks up the
conjugate mode, if the user missed selecting it.
Mode Sorting Criteria: The GUI sorts the Prony residues according to two criteria:
• Amplitude: Sorts the Prony residues according to their amplitude.
• Energy: Sorts the Prony residues according to their energy level. The energy of a
residue is computed by (4.7.1)
2
1
)( ji
Qj
jii RE µ∑
=
=
= . (4.7.1)
In (4.7.1), we have used the following notations:
Ei: Energy of residue i,
Ri: Amplitude of residue i,
iµ : Pole of residue i,
N: Model order of the system,
Q: Data length of the signal,
i: Index, . Ni ...3,2,1=
53
Mode Selection Options: This feature provides the flexibility of choosing the residues
according to the following options:
• All Modes: All the PA result modes are considered.
• Selected Modes Only: In this option, only the user selected modes are considered.
The user can select multiple modes by pressing CTRL+ left mouse button.
• All But Selected Modes: This is a complement option to the Selected modes only
option. In this case, all the modes are considered except the selected modes.
Results: The GUI shows the PA results in a list box. It displays amplitude, frequency,
damping coefficient and energy of the modes according to the mode sorting criteria and
mode selection options.
Plots: This feature provides the following types of plots to assist the user in validating
the Prony fit:
• Squared Error: In this option, the square of the error between the original signal
and Prony fit is plotted with respect to time. The squared error is computed by
(4.7.2)
2^
⎟⎠⎞
⎜⎝⎛ −= iii yye . (4.7.2)
In (4.7.2), we have used the following notations:
ei: Squared error of signal sample i,
yi: Original value of signal sample i,
54
iy^
: Prony fitted value of signal sample i.
This plot is a quick indication of the performance of the Prony fit. A large
squared error is an indication of incorrect model order or number of residues or missing
signal modes during the mode selection process.
• Poles: This shows the poles of the Prony model.
• Residues: This shows the selected as well as all residues. This plot helps the user
to estimate the correct model order.
• Energy: This shows the energy of the Prony modes.
Mean squared error (MSE):
MSE is the mean of the squared error over the sample data length. MSE is an important
statistic that provides information about the performance of the Prony fit. The main
advantage of MSE over the squared error is that it is a single number. It is computed by
(4.7.3):
Q
eMSE
Qi
ii∑
=
== 1 . (4.7.3)
In (4.7.3), we have utilized the following notations:
MSE: Mean squared error of the signal,
ei: Squared error of sample i, computed using (4.7.2),
Q: Data length of the signal.
55
Normalized Mean squared error (NMSE):
NMSE is a single number which is obtained from normalization of the mean
squared error. NMSE can be computed by (4.7.4)
∑
∑=
=
=
== Qi
ii
Qi
ii
y
eNMSE
1
2
1 . (4.7.4)
In (4.7.4), we have utilized the following notations:
NMSE: Normalized mean squared error of the signal,
yi: original value of signal sample i,
ei: Squared error of sample i, computed using (4.7.2),
Q: Data length of the signal.
Save PA session: The current PA session can be saved by selecting Save from the
Session menu. The GUI saves the sessions in the MATLAB workspace, and an unlimited
number of sessions can be saved. The Session menu also has an option of saving the
current session as a file. It saves the session as a CMP-file, which can be opened in the
Compare PA Sessions GUI. When the user saves a PA session, the Compare Sessions
push button is enabled.
4.8 Compare PA Sessions GUI
This GUI compares several PA sessions simultaneously. Figure 4.8.1 shows the
Compare PA Sessions GUI. Its features include the following:
Compare Set Menu: To compare sessions, first the user has to specify the data to be
compared by either loading the data from the workspace or importing the data from a
56
CMP-file. This menu has options to load the data from the workspace or open the CMP-
file.
Figure 4.8.1 Compare PA Sessions GUI
Sessions List: The GUI displays all the specified saved sessions in a list box. For each
session it displays the data file name, data set, data preprocessing option, decimation
option, decimation factor, data range option, model order, number of modes, mode
sorting criteria and mode selection option.
57
Plots: The GUI plots the poles, squared error, energy and residues for the selected PA
sessions.
4.9 Export Data GUI
PTbox provides the flexibility of saving, analyzing and plotting the PA results
according to the user needs, by exporting the results associated with the Prepare Prony
Data GUI, Perform PA GUI and Compare PA Sessions GUI. Figure 4.9.1 shows the
Export Data GUI. The GUI consists of several check boxes and edit boxes. Each check
box refers to particular axis data from the other GUIs. The edit box provides the
flexibility for the user to specify the data structure name. The data can be exported to the
MATLAB workspace as well as to a file. When the user saves the data in a file, PTbox
stores the data in a single structure named Exported_Data in the user specified file
name. The data structure can be expanded in the workspace using the following
MATLAB commands:
load filename;% Loads the data in the base workspace of MATLAB. mmv2struct(Exported_Data); % Unpacks the Exported_Data Structure.
58
Figure 4.9.1 Export Data GUI
The user can view the variables stored in the workspace by entering the command
workspace or whos at the MATLAB command window prompt.
59
4.10 Common Features
This section describes the features that are present in all of the GUIs except the
Splash GUI.
Toolbar: Except the Splash GUI, all of the GUIs have a toolbar. Figure 4.10.1 shows the
toolbar.
Figure 4.10.1 Toolbar in PTbox
The toolbar is placed underneath the main menu at the top of each GUI. It contains icon
buttons that provide quick and easy access to some of the most used functions: printing
the GUI, annotation, drawing lines on the axis plots and zooming the axis plots.
Context Menu: A context menu is designed for each axis in PTbox. Figure 4.10.2 shows
a typical context menu for an axis in the Perform PA GUI.
Figure 4.10.2 Context Menu for an Axis in PTbox
60
The context menu provides the ability to draw the axis plot in a new figure. It is
activated by right-clicking the mouse on the axis. It provides the flexibility to edit the
axis properties and customize it according to the user’s requirements.
Status Text and Navigation Features: The GUIs in PTbox are provided with a status text
message, which guides the user through subsequent actions. Figure 4.10.3 shows a
typical status text message sequence in the Prepare Prony Data GUI. The status text in
the first message informs the user to import the data, and in the second message, it asks
the user to select the variables to plot the data.
Figure 4.10.3 Status Text Message Examples in PTbox
Navigation within a GUI is designed to be smooth and intuitive by disabling/enabling
push buttons for specific actions. For example, the Plot push button is disabled until the
variable names are loaded in the list box. Figure 4.10.4 shows the enable/disable Plot
push button.
Figure 4.10.4 Navigation Feature: Disable/Enable of Plot Push Button in PTbox
61
Help Menu: A help menu is provided in the main menu of all the GUIs. When the user
left-clicks on any options from the help menu, help is displayed on the default browser.
Figure 4.10.5 shows the help menu.
Figure 4.10.5 Help Menu in PTbox
The help menu provides information about the various features provided in the PTbox. It
also provides access to some demos and their data files.
4.11 The Source Code
PTbox consists of a group of MAT-files, M-files and FIG-files. We have listed
the FIG-files and M-files of the PTbox in Appendix D. The source code of the PTbox is
enclosed in a CD provided with the archival copy of this thesis and available from the
author.
62
CHAPTER 5
RESULTS AND DISCUSSION
5.1 The Measured Data
In this Chapter, Prony analysis (PA) is applied to the Pulsed Corona Reactor
(PCR) measurements obtained from experiments carried out in the Chemical Engineering
Department at the University of Wyoming. PCR measurements were recorded by
feeding Argon or Helium or Nitrogen as an inlet gas at a flow rate of 60 or 100 standard
cubic feet per hour (SCFH). Each inlet gas was subjected to pulsed corona frequencies of
50, 100, 200, 400, 500, 600, 700, 800, 1000, 1200 and 1500 Hz. At each of the listed
pulsed corona frequencies, PCR measurements were collected at a sampling rate of 500
MHz and the sample size of each measurement was 1250. High sampling rate is required
to capture the pulsed voltage, which is applied only for a few microseconds. In order to
complete our analysis systematically, we have categorized the entire experiment set in six
different cases. Each case involves ten PA analyses. The cases are listed in Table 5.1.1a
and Table 5.1.1b.
Table 5.1.1a Case List for Pulsed Current
Pulsed Current CASE 1 CASE 2 CASE 3 Argon Helium Nitrogen
Flow Rate (SCFH) 60 100 60 100 60 100 100 100 100 100 100 100 400 400 400 400 400 400 700 700 700 700 700 700 1000 1000 1000 1000 1000 1000
Pulsed Corona Frequency (Hz)
1500 1500 1500 1500 1500 1500
63
Table 5.1.1b Case List for Pulsed Voltage
Pulsed Voltage CASE 4 CASE 5 CASE 6 Argon Helium Nitrogen
Flow Rate (SCFH) 60 100 60 100 60 100 100 100 100 100 100 100 400 400 400 400 400 400 700 700 700 700 700 700 1000 1000 1000 1000 1000 1000
Pulsed Corona Frequency (Hz)
1500 1500 1500 1500 1500 1500
In this Chapter, we present only the detailed analysis steps for CASE 1. The results for
the other cases are summarized in Appendices A, B, and C.
5.2 File and Variable Nomenclature
Several acronyms are utilized to assign variable and file names. Table 5.2.1
summarizes the acronym list.
64
Table 5.2.1 Acronym Glossary for Variable and File Nomenclature
Acronym Glossary
MSE Mean squared error
NMSE Normalized mean squared error
MO Model order
Ar Argon N Nitrogen He Helium t Time I, i Pulsed current V, v Pulsed voltage
SCFH Standard cubic feet per hour
FW Flow rate of gas (60 or 100) in SCFH G Gas type (Ar or He or N) S Signal type (i or v) f Frequency of PCR signal (100 or 400 or 700 or
1000 or 1500) in Hz
var Variable type (t or i or v)
For each case, the following convention is used to assign file names and variable names.
File names are assigned according to (5.1):
FWG_Sf . (5.1)
For example, the data file of the pulsed current signal of Argon gas at 60 SCFH and at
100 Hz has the following file name:
60Ar_i100 .
Variable names are assigned according to (5.2):
var_FWGf . (5.2)
65
For example, the time and pulsed current data of Argon gas at 60 SCFH and at 100 Hz
are assigned the following variable names:
t_60Ar100 and i_60Ar100.
5.3 Data Preprocessing with PTbox
The actual model order of the Pulsed Corona Reactor system is unknown. To get
an initial estimate of model order we computed the mean squared error (MSE) for an
assumed model order. To carry out this task, first the data set was imported in the
Prepare Prony Data GUI. Decimation and Data Range options were kept in their default
Off states. The mean of the dataset was subtracted from each sample of the dataset by
applying the Remove Mean option during Data Preprocessing. Figure 5.3.1 shows the
application settings in the Prepare Prony Data GUI for the data set of pulsed current for
Argon gas at 60 SCFH and at 100 Hz.
66
Figure 5.3.1 Application Settings in Prepare Prony Data GUI for MO vs. MSE
Analysis
The Remove Mean operation changes the time range. The time range of negative values
is discarded during the Remove Mean operation. It also scales PCR signals; the
magnitude of other signals is unchanged. The scaling of PCR signals is done according
to the given measured transducer relations to obtain true voltage and ampere units. It also
facilitates the correct computation of energy and power transferred to the pulsed corona
reactor in a pulse. The pulsed voltage signals and the pulsed current signals are scaled by
a factor of 242000 and 710 respectively.
67
5.4 Prony Analysis with PTbox
The preprocessed signal was analyzed by the Prony algorithm at an assumed
model order. The Amplitude criterion was chosen to presort the computed modes. The
option All Modes was selected for the results listing. The computed value of MSE was
then recorded against the model order. The details of MSE are discussed in section 4.7 of
Chapter 4. Figure 5.4.1 shows the application settings in the Perform Prony Analysis
GUI for the data set of pulsed current for Argon gas at 60 SCFH, 100 Hz and at MO of 6.
Figure 5.4.1 Application Settings in Perform PA GUI for MO vs. MSE Analysis
The above procedure was applied at different values of MO. Figure 5.4.2a and 5.4.2b
show the plots relating MO and MSE. Table 5.4.1a and 5.4.1b show the data related to
68
Figures 5.4.2a and 5.4.2b at various model orders. The empty cells of MSE values in
Table 5.4.1a and Table 5.4.1b corresponding to I_60Ar100 and I_100Ar700 pulsed
currents respectively, indicates that MSE at that particular MO was not computed.
69
100
1000
10000
4 8 12 16 20 24 28 32 36 40
MO
MSE
I_60Ar1500I_60Ar1000I_60Ar700I_60Ar400I_60Ar100
Figure 5.4.2a Plot of MO vs. MSE of Pulsed Current for Argon Gas at 60 SCFH
100
1000
10000
4 8 12 16 20 24 28 32 36 40MO
MSE
I_100Ar100
I_100Ar400
I_100Ar700
I_100Ar1000
I_100Ar1500
Figure 5.4.2b Plot of MO vs. MSE of Pulsed Current for Argon Gas at 100 SCFH
70
Table 5.4.1a Data of MO vs. MSE of Pulsed Current for Argon Gas at 60 SCFH
I_60Ar400 I_60Ar700 I_60Ar1000 I_60Ar1500 I_60Ar100 MO MSE MSE MSE MSE MSE
6 5474 6815 6321 5851 5175 8 1593 3392 3238 2150 2429 10 607 2525 2226 1597 1721 12 248 1153 1186 736 654 14 287 819 840 407 337 16 179 416 474 225.2
18 167 375 435 233 229.4 20 179 421 470 247
22 193 401 459 230 245 24 194 389 459 242 232 26 179 610 694 339 254 28 179 403 483 228 236 30 153 422 493 226 210 32 171 312 379 198 207 34 143 340 419 200 228 36 141 344 397 199 205
Table 5.4.1b Data of MO vs. MSE of Pulsed Current for Argon Gas at 100 SCFH
I_100Ar100 I_100Ar400 I_100Ar1000 I_100Ar1500 I_100Ar700MO MSE MSE MSE MSE MSE
6 9315 10418 10512 5661 10460 8 4455 3735 8274 3090 8214 10 4371 2820 8470 2858 8629 12 1135 843 4379 1928 4787 14 611 401 4718 1059 4882 16 235 198 3625 616
18 234 197 2699 560 2613 20 223 173 2133 440 2249 22 246 193 2043 405 2218 24 215 153 2000 380 2146 26 231 171 1892 401 2118 28 223 155 1910 370 2072 30 216 153 1809 393 1924 32 230 153 1743 397 1794 34 238 163 1803 385 1928 36 222 147 1717 400 1880
71
We also investigated the relationship between MO and normalized mean squared error
(NMSE). The details of NMSE are discussed in section 4.7 of Chapter 4. The analyses
were repeated to obtain NMSE values at different values of MO. Figure 5.4.3 and Table
5.4.2 show the NMSE values of the analyses of pulsed current for Argon gas at 60 SCFH.
The empty cells of NMSE values in Table 5.4.2 corresponding to I_60Ar100 pulsed
current indicate that NMSE at that particular MO was not computed.
0.001
0.01
0.1
1
0 10 20 30 40
MO
NM
SE
I_60Ar400I_60Ar700I_60Ar1000I_60Ar1500I_60Ar100
Figure 5.4.3 Plot of MO vs. NMSE of Pulsed Current for Argon Gas at 60 SCFH
72
Table 5.4.2 Data of MO vs. NMSE of Pulsed Current for Argon Gas at 60 SCFH
I_60Ar400 I_60Ar700 I_60Ar1000 I_60Ar1500 I_60Ar100 MO NMSE NMSE NMSE NMSE NMSE
6 0.34861 0.59841 0.55689 0.59377 0.3601 8 0.10149 0.29786 0.28528 0.21815 0.16908 10 0.03868 0.2217 0.19609 0.16206 0.11979 12 0.01578 0.10126 0.10447 0.07472 0.04553 14 0.01827 0.07188 0.07404 0.04132 0.0235 16 0.01141 0.03652 0.0418 0.02285
18 0.01063 0.03294 0.03833 0.0237 0.01597 20 0.01293 0.03694 0.04141 0.02507
22 0.01232 0.03525 0.04041 0.02337 0.01707 24 0.01236 0.03413 0.04042 0.02457 0.01619 26 0.01137 0.05353 0.06116 0.03445 0.01764 28 0.01141 0.03538 0.04251 0.02316 0.01643 30 0.00977 0.03704 0.04343 0.023 0.01465 32 0.01091 0.02742 0.03339 0.02016 0.01442 34 0.0091 0.02986 0.03693 0.02036 0.01586 36 0.00901 0.03018 0.03502 0.02025 0.01429
Figures 5.4.2a, 5.4.2b and 5.4.3 display the following characteristics:
• Irrespective of the type of gas and flow rate, the MSE curves tend to decrease
with increase in model order. MSE curves of Argon gas at 60 and 100 SCFH
have a major dip or minima in MSE values at MO of 16. We have named this dip
or minima as the “MO/MSE saddle point.” The location of the MO/MSE saddle
point was determined by visual inspection of MO/MSE curves. This is duplicated
at different pulsed frequencies.
• For a particular gas at a particular flow rate, the MSE curve follows the same
decay trend at different pulsed frequencies.
73
• MSE and NMSE follow the same trend with respect to the model order. The only
obvious difference is the scale; MSE values are on a higher scale as compared to
NMSE values.
5.5 Refining the Model Order in PTbox
The observations made in section 5.4 inspired us to investigate the PCR system
behavior by applying Prony analysis in the vicinity of the MO/MSE saddle point. We
selected MO of 12, 14, 16, 18 and 22, which are in the vicinity of 16 (the MO/MSE
saddle point). We performed ten Prony analysis sessions (corresponding to two flow
rates of Argon gas at five pulsed corona frequencies) for each of the above orders. Each
of these sessions was saved in a comparison file, allowing for comparison with the
Compare Sessions GUI. Table 5.5.1 shows a list of Prony analysis sessions performed
for pulsed current with Argon gas at MO of 12. Each of the sessions in which the
number of modes was the same as the specified model order were selected and compared.
Figure 5.5.1 shows the application settings in the Compare Sessions GUI for the data set
of pulsed current for Argon gas at 60 SCFH, 100 Hz and at MO of 12. Figure 5.5.2,
5.5.3, 5.5.4, 5.5.5 and 5.5.6 show the pole diagrams of the compared sessions for MO of
12, 14, 16, 18 and 22 respectively.
74
Figure 5.5.1 Application Settings in Compare Sessions GUI
75
Table 5.5.1 List of PA Sessions of Pulsed Current for Argon Gas at MO 12
Session Number File Name Data Set Data Preprocess
Option Decimation
Option Data Range
Option
Model Order (MO)
MO Sorting Criteria
Number of Residues
MO Selection Criteria
Session 1 60Ar_i100.mat t_60Ar100 & i_60Ar100 Remove Mean No No 12 Amplitude 12 All Modes
Session 2 60Ar_i400.mat t_60Ar400 & i_60Ar400 Remove Mean No No 12 Amplitude 12 All Modes
Session 3 60Ar_i700.mat t_60Ar700 & i_60Ar700 Remove Mean No No 12 Amplitude 12 All Modes
Session 4 60Ar_i1000.mat t_60Ar1000 & i_60Ar1000 Remove Mean No No 12 Amplitude 12 All Modes
Session 5 60Ar_i1500.mat t_60Ar1500 & i_60Ar1500 Remove Mean No No 12 Amplitude 12 All Modes
Session 6 100Ar_i100.mat t_100Ar100 & i_100Ar100 Remove Mean No No 12 Amplitude 12 All Modes
Session 7 100Ar_i400.mat t_100Ar400 & i_100Ar400 Remove Mean No No 12 Amplitude 12 All Modes
Session 8 100Ar_i700.mat t_100Ar700 & i_100Ar700 Remove Mean No No 12 Amplitude 12 All Modes
Session 9 100Ar_i1000.mat t_100Ar1000 & i_100Ar1000 Remove Mean No No 12 Amplitude 12 All Modes
Session 10 100Ar_i1500.mat t_100Ar1500 & i_100Ar1500 Remove Mean No No 12 Amplitude 12 All Modes
76
Figure 5.5.2 Poles of Pulsed Current for Argon at MO 12
Figure 5.5.3 Poles of Pulsed Current for Argon at MO 14
77
Figure 5.5.4 Poles of Pulsed Current for Argon at MO 16 (Saddle/Clustering Point)
Figure 5.5.5 Poles of Pulsed Current for Argon at MO 18
78
Figure 5.5.6 Poles of Pulsed Current for Argon at MO 22
Observing the pole diagrams of Figure 5.5.2, 5.5.3, 5.5.4, 5.5.5 and 5.5.6, the following
remarks can be made:
• Poles tend to cluster either for the MOS/MSE saddle point MO or for a model
order near to the MO/MSE saddle point. The MO at which the clustering of poles
was most, we named the “clustering point.” The extent of clustering of poles was
determined by visual inspection of pole diagrams.
• Poles are scattered for model orders less or more than the clustering point.
• Poles are closer to the center of the unit circle for a MO less than the clustering
point but they start moving toward the perimeter when MO is increased above the
clustering point.
79
In order to be certain that the above remarks are true irrespective of type of gas, we
carried out the same analyses for each of the CASES 2 through 6. These cases are
summarized in detail in Appendices A, B and C. Only the pole diagrams at the clustering
point model order of these cases are presented here.
Figure 5.5.7 Poles of Pulsed Current for Helium at MO 16 (Clustering Point)
80
Figure 5.5.8 Poles of Pulsed Current for Nitrogen at MO 16 (Clustering Point)
Figure 5.5.9 Poles of Pulsed Voltage for Argon at MO 17 (Clustering Point)
81
Figure 5.5.10 Poles of Pulsed Voltage for Helium at MO 15 (Clustering Point)
Figure 5.5.11 Poles of Pulsed Voltage for Nitrogen at MO 12 (Clustering Point)
82
All MO/MSE saddle points and clustering points are summarized in Table 5.5.2.
Table 5.5.2 MO/MSE Saddle Points and Clustering Points for All the CASES
Pulsed Current Pulsed Voltage CASE 1 CASE 2 CASE 3 CASE 4 CASE 5 CASE 6 Argon Helium Nitrogen Argon Helium Nitrogen
MO/MSE Saddle Point 16 12 16 18 17 14
Clustering Point 16 16 16 17 15 12
Observing Figure 5.5.7, 5.5.8, 5.5.9, 5.5.10 and 5.5.11, we conclude:
• The clustering of poles occurs at the clustering point MO for Argon, Helium and
Nitrogen gases. The clustering is different for each gas.
• The clustering point MO is near to the MO/MSE saddle point and it can be
obtained by refining the MO/MSE saddle point.
• The clustering of poles occurs for pulsed current and pulsed voltage at all the
pulsed frequencies.
The observations made above are important findings as a specific pole clustering may
describe the presence of a particular gas.
This effect might be viewed similar to chemical structure sensing techniques such
as NMR. These preliminary gas studies are precursors to PCR pollution abatement
applications. Through the characterization of simple noble gas reactions and Nitrogen
(the most populous atmospheric gas), chemical engineers are hoping to formulate a
general mechanism model for PCR effects.
83
CHAPTER 6
CONCLUSIONS AND FUTURE RESEARCH
In this thesis, we have applied the classical signal processing technique known as
“Prony analysis” to Pulsed Corona Reactor (PCR) current and voltage analysis. Our
hypothesis is that the system health and reactor contents of the PCR can be detected by
inspecting the roots of the Prony Model. To validate our hypothesis it is required that the
roots of the Prony analysis (PA) for various sets of experiments can be simultaneously
inspected. Hence, we have built a software tool, the Prony Toolbox (PTbox) in
MATLAB with a user-friendly graphical interface and containing all the necessary
features to perform PA in the application setting of PCR characterization.
6.1 The Features of PTbox
The following features are provided in the PTbox:
• Modular software architecture and add-on code flexibility.
• Widely used and cross-platform-compatible MAT-file as a data input format.
• Data length reduction by specifying a data range and applying decimation.
• Removal of the mean and detrend by applying data preprocessing.
• Presorting of modes according to amplitude and energy levels and ability to apply
reconstruction to selected modes from the presorted modes.
• Ability to view the results in time and frequency graphic modes.
• Features to view the plots of Prony results such as squared error, energy and
amplitude of selected or all residues, poles and zeros.
84
• Computation and display of mean squared error (MSE) of a Prony fit.
• Ability to compare several PA sessions simultaneously.
• Flexibility of exporting the data associated with each graphical user interface
(GUI) of the PTbox in the base workspace or a file.
• Ability to plot results associated with each step of the PA in a new figure.
• General features for printing via the GUI: annotation, drawing lines on the
axis plots and zooming the axis plots.
• Navigation features including disable/enable of graphic objects and guiding
text messages.
• Help to use the PTbox.
6.2 PTbox Applied to the PCR Data
We have applied Prony analysis to pulsed voltage and pulsed current signals
associated with the reactor feed gases Argon, Helium and Nitrogen at two flow rates and
five pulsed corona frequencies. The application settings of the GUIs in the PTbox were
duplicated for each of the PA sessions. Model order (MO) of the system was computed
by analyzing the plots of MSE at different pulsed corona frequencies and for two flow
rates of the reactor feed gas. As discussed in Chapter 5, we observed that the MSE
curves tend to decrease with increase in model order for all the analyzed gases. We also
observed that for a particular gas at two different flow rates the MSE curves have a dip or
minima, which is duplicated at different pulsed frequencies. We named this dip or
minima the MO/MSE saddle point. We found the MO/MSE saddle point for each of the
reactor feed gases.
85
Applying PA in the vicinity of the MO/MSE saddle point allowed us to refine the model
order. Prony roots at different pulsed corona frequencies for each gas were compared
and inspected. We observed that the clustering of poles occurs either for the MO/MSE
saddle point MO or for the MO near to it for each of the reactor feed gases. The MO at
which the clustering of poles was most, we named the “clustering point.” The extent of
clustering of poles was determined by visual inspection of pole diagrams. The clustering
of poles occurs for pulsed current as well as for pulsed voltage. The clustering of poles
was specific for each of the reactor feed gases. This observation is very important as a
specific pole clustering may describe the presence of a particular gas.
6.3 Future Research Areas
There are several areas where future research has been prompted by this thesis.
The following describes some areas of opportunities for future work.
6.3.1 Breadth of Data Analysis
We have applied Prony analysis with specific application settings in PTbox, for
example, Decimation and Data Range options were kept in their default Off states, the
Amplitude criterion was chosen to presort the computed modes, the option All Modes was
selected for the results listing. Changing the application settings in PTbox could provide
an opportunity to analyze the data from an entirely different perspective. There are
several ways by which the application settings in PTbox could be changed, for example,
specify a Data Range for the signal, the Energy criterion could be chosen to presort the
computed modes, the option Selected Modes could be selected to pick specific modes.
86
The different application settings in PTbox open a wide range of opportunity for data
analysis.
6.3.2 Validation of the consistency of experimental process
In the future research work, several data sets could be recorded and analyzed for
an identical experiment. This analysis could validate the consistency of experimental
process. It could also determine noise consistency in the experimental setup.
6.3.3 Geometry of Pulsed Corona Reactor
Another potential area of future research work is the investigation of data sets for
different geometric shapes and chemical compositions of PCR. In this thesis, we have
analyzed the data only for a particular geometric shape of the reactor. Analyzing data
obtained from different shapes of the reactor could verify the degree to which pole
clustering is dependent upon the geometry of the reactor.
6.3.4 Pole Clustering
The phenomenon of pole clustering at a specific MO/MSE saddle point itself is a
wide and challenging topic for future research work. With the help of chemists and
chemical engineers, we could seek the answers to explain the phenomenon of pole
clustering. In this thesis, we determined the location of the MO/MSE saddle point and
extent of clustering of poles by visual inspection. In the future work, algorithms could be
developed to pick the MO/MSE saddle point and to define a clustering factor which can
87
determine the extent of clustering of poles. A model of PCR measurements could be
simulated to study the clustering effect in the presence of different noise levels.
6.3.5 Enhancement Features of PTbox
The modular design of the PTbox opens several other future research areas. The
version of the PTbox described in this thesis implements only MATLAB’s Signal
Processing Toolbox built-in prony. Extension to modified Prony analyses could be
readily accomplished. Some of the limitations of the PTbox can be removed. For
example, the version of Compare Sessions GUI described in this thesis assigns a single
graphic shape to all of the poles compared in it; it only assigns a different color for each
of the compared PA sessions. This limitation could be removed by designing the
Compare Sessions GUI in such a way that it changes the graphic shape of pole display for
each of the compared PA sessions. This feature could provide an opportunity to analyze
the location of poles without requiring a color print of the Poles diagram.
Another limitation of PTbox is that it does not provide the flexibility of filtering
the decimated signal. The down-sampling process can cause aliasing in the frequency
domain. Hence, an antialiasing filter should be used before down-sampling. In the future
work, PTbox can incorporate the features to filter the decimated signal.
In this thesis, we have used MSE values to direct the selection of model order of
the system. The version of Prony Analysis GUI used in this thesis computes the mean
squared error value only for an arbitrary model order. The user has to compute MSE for
each model order, save each of these MSE values, then combine and summarize the
results outside PTbox. This limitation burdens the user with a lot of extra work to
88
perform an analysis between MO and MSE. Future work could be done in designing a
separate GUI that could automate the computation of MSE for a range of MO.
A GUI could be designed which sets the user preferences in PTbox. For example,
the preference GUI could allow the user to change the colors of the lines in the plots.
More features could be added to the toolbar. In summary, the opportunity to enhance the
PTbox features is like a “creeping monster.”
6.4 Conclusions
In this thesis, we combined the knowledge of chemical and electrical engineering
to achieve the thesis objective. This research work is a significant achievement toward
multi-disciplinary research in electrical engineering. We developed PTbox to
characterize PCR measurements using Prony analysis. We validated our hypothesis by
applying PA to pulsed voltage and pulsed current using PTbox. The observations made
from our PA results are encouraging and promising for future research work. This thesis
was successful in accomplishing the objectives.
89
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Consistency of Prony’s Method and Related Algorithms,” Journal of Comput. Graph. Statistics, vol. 1, pp. 329-349, 1992.
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• Marshall, G. Alan and Francis R. Verdun, Fourier Transforms in NMR, Optical and Mass Spectrometry: A User’s Handbook, Elsevier, pp.179-180, 1990.
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• Parks, T. W. and C. S. Burrus, Digital Filter Design, John Wiley & Sons, pp. 226-229, 1987.
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• Pierre, J. W., Digital Signal Processing Lecture Notes, University of Wyoming, Spring 2002.
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• Scharf, L., Statistical Signal Processing: Detection, Estimation and Time Series Analysis, Addition-Wesley Publishing Company, pp. 484-513, 1991.
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91
APPENDIX A
PA RESULTS OF PULSED VOLTAGE
FOR ARGON GAS
As described in Chapter 5, this Appendix summarizes the Prony analysis results
of CASE 4 of the PCR measurement data. CASE 4 involves the data of pulsed voltage of
Argon gas at a flow rate of 60 and 100 SCFH. The pulsed voltage is measured at pulsing
frequencies of 100, 400, 700, 1000 and 1500 Hz. This Appendix follows the file and
variable nomenclature discussed in section 5.2.
Prony results of pulsed voltage for Argon gas show that the MO/MSE saddle
point MO is 18 and the clustering point MO is 17. The following sections describe the
Prony results in detail.
A.1 MO vs. MSE Analysis of Pulsed Voltage
Prony analysis was applied to CASE 4, using PTbox with the application settings
as described in the sections 5.3 and 5.4. Figure A.1.1a and A.1.1b show the plots relating
MO and MSE. Table A.1.1a and A.1.1b show the data related to Figures A.1.1a and
A.1.1b at various model orders. The empty cells of MSE values in Tables A.1.1a and
A.1.1b indicate that MSE at that particular MO was not computed.
92
1.E+05
1.E+06
1.E+07
1.E+08
6 10 14 18 22 26 30 34
MO
MSE
V_60Ar100V_60Ar400V_60Ar700V_60Ar1000V_60Ar1500
Figure A.1.1a Plot of MO vs. MSE of Pulsed Voltage for Argon Gas at 60 SCFH
1.E+06
1.E+07
1.E+08
6 10 14 18 22 26MO
MSE
V_100Ar100V_100Ar400V_100Ar700V_100Ar1000V_100Ar1500
Figure A.1.1b Plot of MO vs. MSE of Pulsed Voltage for Argon Gas at 100 SCFH
93
Table A.1.1a Data of MO vs. MSE of Pulsed Voltage for Argon Gas at 60 SCFH
V_60Ar100 V_60Ar400 V_60Ar700 V_60Ar1000 V_60Ar1500 MO MSE MSE MSE MSE MSE
6 15296094 14656022 6614138 6802852 4159705 8 14979686 12774910 5211715 5089902 3621700 10 12009512 12158791 4781096 4608467 2998763 14 10621729 8890623 3814960 1962916 16 9567212 3356768 18 12826176 7301410 1451203 1370175 20 8856959 7615333 1490471 1624206 2069574 22 3792444 4581385 2138817 1385520 2010454 24 9427679 9391398 1579125 1115303 1937903 26 7160477 4854816 1421742 1238651 1797646 28 3516716 4053176 1329577 727950 1852704 30 4433740 4556624 1075265 590444 1816365
Table A.1.1b Data of MO vs. MSE of Pulsed Voltage for Argon Gas at 100 SCFH
V_100Ar100 V_100Ar400 V_100Ar700 V_100Ar1000 V_100Ar1500 MO MSE MSE MSE MSE MSE
6 21335648 21983548 48605237 11796749 8 17076217 40871802 42581758 7785632 10 13139085 13157805 34239190 36049770 6686156 12 12720502 12017391 37544081 32128814 6895708 13 13347010 14 26385825 23167038 6362893 16 31070763 4447410 18 14755959 9901041 22466760 27342156 6996783 20 22230028 21478910 25785484 3680759 22 9367498 27548745 29054354 2912249 23 69277680 24 20273129 29838838 25 9069614 26 15525339 9335982 1904896 28 5393605
Figure A.1.1a and A.1.1b show that the MO/MSE saddle point is close to a model order
of 18.
94
A.2 Pole Diagrams in the Vicinity of the MO/MSE Saddle Point for Pulsed Voltage
In this section, we have selected MO of 12, 14, 17, 18 and 22, which are in the
vicinity of 18 (the MO/MSE saddle point). We performed ten Prony analysis sessions for
each of the above orders. We have used the application settings in the Compared
Sessions GUI as described in section 5.5. Observing Figure A.2.1, A.2.2, A.2.3, A.2.4
and A.2.5 we concluded that the clustering of poles occurs maximum for the model order
of 17, hence the clustering point MO is 17.
Figure A.2.1 Poles of Pulsed Voltage for Argon at MO 12
95
Figure A.2.2 Poles of Pulsed Voltage for Argon at MO 14
Figure A.2.3 Poles of Pulsed Voltage for Argon at MO 17 (Clustering Point)
96
Figure A.2.4 Poles of Pulsed Voltage for Argon at MO 18 (Saddle Point)
Figure A.2.5 Poles of Pulsed Voltage for Argon at MO 22
97
APPENDIX B
PA RESULTS OF PULSED VOLTAGE AND PULSED CURRENT
FOR HELIUM GAS
As described in Chapter 5, this Appendix summarizes the Prony analysis results
of CASE 2 and CASE 5 of the PCR measurement data. CASE 2 and CASE 5 involve the
data of pulsed current and pulsed voltage of Helium gas respectively, at a flow rate of 60
and 100 SCFH. The pulsed currents and pulsed voltages are measured at pulsing
frequencies of 100, 400, 700, 1000 and 1500 Hz. This Appendix follows the file and
variable nomenclature discussed in section 5.2.
Prony results of pulsed current and pulsed voltage for Helium gas show that the
MO/MSE saddle point MO is 12 and 16 respectively. The clustering point MO of pulsed
current and pulsed voltage is 17 and 15 respectively. The sections B.1 through B.4
describe Prony results of pulsed current and pulsed voltage.
B.1 MO vs. MSE Analysis of Pulsed Current
Prony analysis was applied to CASE 2, using PTbox with the application settings
as described in the sections 5.3 and 5.4. Figure B.1.1a and B.1.1b show the plots relating
MO and MSE. Table B.1.1a and B.1.1b show the data related to Figures B.1.1a and
B.1.1b at various model orders. The empty cells of MSE values in Tables B.1.1a and
B.1.1b indicate that MSE at that particular MO was not computed.
98
0
200
400
600
800
1000
1200
1400
1600
1800
2000
4 8 12 16 20 24 28
MO
MSE
I_60He100I_60He400I_60He700I_60He1000I_60He1500
Figure B.1.1a Plot of MO vs. MSE of Pulsed Current for Helium Gas at 60 SCFH
10
510
1010
1510
2010
2510
3010
3510
4010
4510
4 8 12 16 20 24 28
MO
MSE
I_100He100I_100He400I_100He700I_100He1000I_100He1500
Figure B.1.1b Plot of MO vs. MSE of Pulsed Current for Helium Gas at 100 SCFH
99
Table B.1.1a Data of MO vs. MSE of Pulsed Current for Helium Gas at 60 SCFH
I_60He100 I_60He400 I_60He700 I_60He1000 I_60He1500 MO MSE MSE MSE MSE MSE
6 1185 1588 1761 1759 1331 10 960 1182 936 756 828 12 293 262 203 208 267 14 261 201 167 174 195 16 131 106 128 152 163 18 130 106 125 150 162 20 145 113 125 149 159 22 123 103 125 155 159 24 125 104 123 147 159 26 117 100 142 175 178
Table B.1.1b Data of MO vs. MSE of Pulsed Current for Helium Gas at 100 SCFH
I_100He100 I_100He400 I_100He700 I_100He1000 I_100He1500MO MSE MSE MSE MSE MSE
6 2400 4617 4437 3466 8 876 861 1095 1131 851 10 1232 1008 922 807 695 12 191 245 235 233 232 14 166 167 174 184 208 16 114 167 99 110 134 18 112 97 101 110 132 20 91 75 90 101 121 22 82 66 88 98 124 24 81 64 86 97 115 26 72 64 85 96 111
Figure B.1.1a and B.1.1b show that the MO/MSE saddle point is close to a model order
of 12.
100
B.2 Pole Diagrams in the Vicinity of the MO/MSE Saddle Point for Pulsed Current
In this section, we have selected MO of 10, 12, 16 and 18, which are in the
vicinity of 12 (the MO/MSE saddle point). We performed ten Prony analysis sessions for
each of the above orders. We have used the application settings in the Compared
Sessions GUI as described in section 5.5. Figure B.2.1, B.2.2, B.2.3 and B.2.4 show the
pole diagram of the compared sessions for MOs of 10, 12, 16 and 18 respectively.
Figure B.2.1 Poles of Pulsed Current for Helium at MO 10
101
Figure B.2.2 Poles of Pulsed Current for Helium at MO 12 (Saddle Point)
Figure B.2.3 Poles of Pulsed Current for Helium at MO 16 (Clustering Point)
102
Figure B.2.4 Poles of Pulsed Current for Helium at MO 18
Observing Figure B.2.1, B.2.2, B.2.3 and B.2.4 we concluded that the clustering of poles
occurs maximum for the model order of 16, hence the clustering point MO is 16.
B.3 MO vs. MSE Analysis of Pulsed Voltage
Prony analysis was applied to CASE 5, using PTbox with the application settings
as described in the sections 5.3 and 5.4. Figure B.3.1a and B.3.1b show the plots relating
MO and MSE. Table B.3.1a and B.3.1b show the data related to Figures B.3.1a and
B.3.1b at various model orders. The empty cells of MSE values in Tables B.3.1a and
B.3.1b indicate that MSE at that particular MO was not computed.
103
1.E+06
1.E+07
6 10 14 18 22 26
MO
MSE
V_60He100V_60He400V_60He700V_60He1000V_60He1500
Figure B.3.1a Plot of MO vs. MSE of Pulsed Voltage for Helium Gas at 60 SCFH
1.E+06
1.E+07
6 10 14 18 22 26
MO
NM
SE
V_100He100V_100He400V_100He700V_100He1000V_100He1500
Figure B.3.1a Plot of MO vs. MSE of Pulsed Voltage for Helium Gas at 100 SCFH
104
Table B.3.1a Data of MO vs. MSE of Pulsed Voltage for Helium Gas at 60 SCFH
V_60He100 V_60He400 V_60He700 V_60He1000 V_60He1500 MO MSE MSE MSE MSE MSE
6 4357941 3966221 2988065 2379389 2286022 8 3743759 3281829 2478291 2274276 2021376 10 4021429 3033940 2803785 2496496 1664961 12 3283779 2666819 2261065 2086596 1627916 14 3344972 2234303 1904176 1623255 15 2158341 1741830 1392031 16 2933526 17 2489308 1683485 1905576 1893732 1620435 18 1884188 2305716 2194749 1734738 19 2285775 20 2414261 1844546 2069417 1632451 22 2459830 2006457 1954539 1983187 1640485 23 3112182 2225462 24 4305461 2394466 1674877
Table B.3.1b Data of MO vs. MSE of Pulsed Voltage for Helium Gas at 100 SCFH
V_100He100 V_100He400 V_100He700 V_100He1000 V_100He1500MO MSE MSE MSE MSE MSE
6 9838485 6691275 5579760 4082467 2085584 8 6462333 4923144 4933210 3655925 1844308 10 5966056 4263349 4996132 3990686 2013106 11 4334154 3425010 12 5576613 3829162 1949600 13 3913448 3097131 1911543 14 5407738 3904545 15 4966441 3581235 3300993 2611288 16 1830369 17 3855195 2804228 2844835 2321780 18 7572415 3969827 4388622 1959263 19 6863945 4746158 20 7228773 4857772 1817052 22 4082487 2586457 3267559 2923129 1710098 23 3625756 4251519 1798181 24 8968003 4279463
105
Figure B.3.1a and B.3.1b show that the MO/MSE saddle point is close to a model order
of 17.
B.4 Pole Diagrams in the Vicinity of the MO/MSE Saddle Point for Pulsed Voltage
In this section, we have selected MO of 10, 12, 15 and 17, which are in the
vicinity of 17 (the MO/MSE saddle point). We performed ten Prony analysis sessions for
each of the above orders. We have used the application settings in the Compared
Sessions GUI as described in section 5.5. Figure B.4.1, B.4.2, B.4.3 and B.4.4 show the
pole diagram of the compared sessions for MOs of 10, 12, 15 and 17 respectively.
Figure B.4.1 Poles of Pulsed Voltage for Helium at MO 10
106
Figure B.4.2 Poles of Pulsed Voltage for Helium at MO 12
Figure B.4.3 Poles of Pulsed Voltage for Helium at MO 15 (Clustering Point)
107
Figure B.4.4 Poles of Pulsed Voltage for Helium at MO 17 (Saddle Point)
Observing Figure B.4.1, B.4.2, B.4.3 and B.4.4 we concluded that the clustering of poles
occurs maximum for the model order of 15, hence the clustering point MO is 15.
108
APPENDIX C
PA RESULTS OF PULSED VOLTAGE AND PULSED CURRENT
FOR NITROGEN GAS
As described in Chapter 5, this Appendix summarizes the Prony analysis results
of CASE 3 and CASE 6 of the PCR measurement data. CASE 3 and CASE 6 involve the
data of pulsed current and pulsed voltage of Nitrogen gas respectively, at a flow rate of
60 and 100 SCFH. The pulsed currents and pulsed voltages are measured at pulsing
frequencies of 100, 400, 700, 1000 and 1500 Hz. This Appendix follows the file and
variable nomenclature discussed in section 5.2.
Prony results of pulsed current and pulsed voltage for Helium gas show that the
MO/MSE saddle point MO is 16 and 16 respectively. The clustering point MO of pulsed
current and pulsed voltage is 14 and 12 respectively. The sections C.1 through C.4
describe Prony results of pulsed current and pulsed voltage.
C.1 MO vs. MSE Analysis of Pulsed Current
Prony analysis was applied to CASE 3, using PTbox with the application settings
as described in the sections 5.3 and 5.4. Figure C.1.1a and C.1.1b show the plots relating
MO and MSE. Table C.1.1a and C.1.1b show the data related to Figures C.1.1a and
C.1.1b at various model orders. The empty cells of MSE values in Tables C.1.1a and
C.1.1b indicate that MSE at that particular MO was not computed.
109
10
100
1000
6 10 14 18 22 26 30MO
MSE
I_60N100I_60N400I_60N700I_60N1000I_60N1500
Figure C.1.1a Plot of MO vs. MSE of Pulsed Current for Nitrogen Gas at 60 SCFH
1
11
21
31
41
51
61
71
6 8 10 12 14 16 18 20 22 24 26 28
MO
MSE
I_100N100I_100N400I_100N700I_100N1000I_100N1500
110
Figure C.1.1b Plot of MO vs. MSE of Pulsed Current for Nitrogen Gas at 100 SCFH
Table C.1.1a Data of MO vs. MSE of Pulsed Current for Nitrogen Gas at 60 SCFH
I_60N100 I_60N400 I_60N700 I_60N1000 I_60N1500 MO MSE MSE MSE MSE MSE
8 680 538 158 118 131 10 657 523 95 40 44 12 239 179 47 26 28 14 63 53 25 26 28 16 49 48 21 20 23 18 25 24 21 21 23 20 17 16 18 18 21 22 16 16 16 19 23 24 16 14 17 19 23 26 15 14 15 15 17 28 15 13 15 15 16
Table C.1.1b Data of MO vs. MSE of Pulsed Current for Nitrogen Gas at 100 SCFH
I_100N100 I_100N400 I_100N700 I_100N1500 I_100N1000 MO MSE MSE MSE MSE MSE
8 56 65 57 25.7 39 10 44 41 39 24.2 33 12 26 17 19 21.7 26 14 27 20 20 22.8 16 17 13 14 16.5 18 18 11 10 10 11.5 12 20 12 24 21 6.3 5.4 22 10 20 19 8 5.6 24 10 21 17 9.3 5.7 26 10 15 12 10.7 5.4
Figure C.1.1a and C.1.1b show that the MO/MSE saddle point is close to a model order
of 16.
111
C.2 Pole Diagrams in the Vicinity of the MO/MSE Saddle Point for Pulsed Current
In this section, we have selected MO of 12, 15, 16 and 17, which are in the
vicinity of 16 (the MO/MSE saddle point). We performed ten Prony analysis sessions for
each of the above orders. We have used the application settings in the Compared
Sessions GUI as described in section 5.5. Figure C.2.1, C.2.2, C.2.3 and C.2.4 show the
pole diagram of the compared sessions for MOs of 12, 15, 16 and 17 respectively.
Figure C.2.1 Poles of Pulsed Current for Nitrogen at MO 12
112
Figure C.2.2 Poles of Pulsed Current for Nitrogen at MO 15
Figure C.2.3 Poles of Pulsed Current for Nitrogen at MO 16 (Clustering Point)
113
Figure C.2.4 Poles of Pulsed Current for Nitrogen at MO 17
Observing Figure C.2.1, C.2.2, C.2.3 and C.2.4 we concluded that the clustering of poles
occurs maximum for the model order of 16, hence the clustering point MO is 16.
C.3 MO vs. MSE Analysis of Pulsed Voltage
Prony analysis was applied to CASE 6, using PTbox with the application settings
as described in the sections 5.3 and 5.4. Figure C.3.1a and C.3.1b show the plots relating
MO and MSE. Table C.3.1a and C.3.1b show the data related to Figures C.3.1a and
C.3.1b at various model orders. The empty cells of MSE values in Tables C.3.1a and
C.3.1b indicate that MSE at that particular MO was not computed.
114
1.E+06
1.E+07
1.E+08
6 10 14 18 22 26
MO
MSE
V_60N100V_60N400V_60N700V_60N1000V_60N1500
Figure C.3.1a Plot of MO vs. MSE of Pulsed Voltage for Nitrogen Gas at 60 SCFH
1.E+05
1.E+06
1.E+07
6 10 14 18 22 26
MO
MSE
V_100N100V_100N400V_100N700V_100N1000V_100N1500
Figure C.3.1b Plot of MO vs. MSE of Pulsed Voltage for Nitrogen Gas at 100 SCFH
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Table C.3.1a Data of MO vs. MSE of Pulsed Voltage for Nitrogen Gas at 60 SCFH
V_60N100 V_60N400 V_60N700 V_60N1000 V_60N1500MO MSE MSE MSE MSE MSE
6 17014399 13979122 8117569 4952961 2599980 8 16723211 13638522 7613483 4486063 2293000 10 12864241 10930362 6032033 3960080 2286607 11 14118154 11726748 5911563 3724511 2114066 12 14192280 11139822 3350076 1881946 13 5628572 14 13122515 10236138 3274414 1840010 15 10976417 6422400 16 12968972 5292941 17 7953723 2858690 18 6108625 5588204 9810361 7323523 3604213 20 4493940 22 4076977 3999230 4861020 3204660 2029233 23 5990605 24 11045306 5449915 2758848 26 5773493 7548137 9312592 5301822 2471656
Table C.3.1b Data of MO vs. MSE of Pulsed Voltage for Nitrogen Gas at 100 SCFH
V_100N100 V_100N400 V_100N700 V_100N1000 V_100N1500 MO MSE MSE MSE MSE MSE
6 3545803 2081698 2374353 2834970 2881933 8 2542451 1311290 1654888 1955590 2224250 10 4194742 2449106 2688501 3248662 3498020 12 1787365 807256 851372 1058727 1940731 14 880004 349574 524337 584003 1355057 15 549022 1280341 16 1036064 370342 17 313252 220374 365659 18 649941 217287 19 806295 303358 365008 286853 552864 21 863703 23 697062 232932 322045 225116 307784 24 681516 391363 25 268763 369847 288643 26 761908 245785 419374 27 312022 228466
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Figure C.3.1a and C.3.1b show that the MO/MSE saddle point is close to a model order
of 14.
C.4 Pole Diagrams in the Vicinity of the MO/MSE Saddle Point for Pulsed Voltage
In this section, we have selected MO of 10, 11, 12 and 14, which are in the
vicinity of 14 (the MO/MSE saddle point). We performed ten Prony analysis sessions for
each of the above orders. We have used the application settings in the Compared
Sessions GUI as described in section 5.5. Figure C.4.1, C.4.2, C.4.3 and C.4.4 show the
pole diagram of the compared sessions for MOs of 10, 11, 12 and 14 respectively.
Figure C.4.1 Poles of Pulsed Voltage for Nitrogen at MO 10
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Figure C.4.2 Poles of Pulsed Voltage for Nitrogen at MO 11
Figure C.4.3 Poles of Pulsed Voltage for Nitrogen at MO 12 (Clustering Point)
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Figure C.4.4 Poles of Pulsed Voltage for Nitrogen at MO 14 (Saddle Point)
Observing Figure C.4.1, C.4.2, C.4.3 and C.4.4 we concluded that the clustering of poles
occurs maximum for the model order of 12, hence the clustering point MO is 12.
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APPENDIX D
MATLAB CODE OF PRONY TOOLBOX
The source code of the PTbox is enclosed in a CD of the archival copy of this
thesis and is available from the author. It contains the M-files and FIG-files of the Prony
Toolbox. Table D.1 lists the M-files with a brief description.
Table D.1 List of M-files in PTbox
File Name File Description applyprony Apply the Prony analysis
comparesessions Contains the Compare PA Sessions GUI exportresults Export the Prony results associated with the GUIs in PTbox fft_analysis Implement the Prony fit in frequency graphic mode mmv2struct Expand a structure and load the fields in the base workspace.
perform_resultslist Prepare the results list box data of Perform PA GUI performprony Contains the Perform PA GUI
prepare_comparelist Prepare compare list box data of the Compare PA Sessions GUI
prepare_savedata Prepare the data structure to be saved from the Prepare Prony Data GUI and the Perform PA Analysis GUI.
preparecomparedata Prepare the data for the Compare PA Sessions GUI pronyabout Provide a brief introduction about the Prony Toolbox
pronyanalysistool Contain the Prepare Prony Data GUI Pronytool Contain the PTbox Splash GUI
Pronytoolbar Implement the Prony toolbar in all the GUIs remove_mean Implement Remove Mean in Prepare Prony Data GUI
windowing_data Implement Data Range in Prepare Prony Data GUI
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