October 28, 2008
Final Work Presentation
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October 28, 2008
Final Work Presentation
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October 28, 2008
Final Work Presentation
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Introduction
Application

ESP
•
Electro Static Precipitator or Electrostatic air cleaner

ESP is used to filter and remove particles from a flowing gas

ESP contains rows of thin wires(electrodes) situated between large metal
plates(collection electrodes)

High DC voltage between wires and plates

The particles are ionized by high voltage around the wires(cathode) and
then attracted by the plates(anode)

The load in the ESP application shows capacitive behaviour
October 28, 2008
Final Work Presentation
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Introduction
DC/DC Power Converters
•
The need to convert a direct current(DC) voltage source from
one voltage level to another
•
Linear conversion in compare to Switched Mode conversion
•
A DC/DC switching power converter stores the input energy
temporarily and then releases that energy to the output at a
different voltage level (this function motivates the switching
property of the converter)
•
The storage may be in magnetic components, like inductors
and transformers, or in capacitors
•
Different topologies has been introduced to fulfill our
expectations, like Buck, Boost, Buck

Boost, Resonant power
converter and etc.
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Introduction
ESP application and DC/DC power converters
•
The proper high load voltage is
provided by a combination of a
DC/DC power converter and a
transformer
•
The resonant converter topology is
chosen for the power converter
•
The switching is done by the four
transistors
(Transistors can be used as switches
if they are saturated or cut

off)
•
Resonance circuit , switched system
and rectifier
•
Switching frequency and resonance
frequency
•
The load is connected in series to the
resonance circuit
Rectifier
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Final Work Presentation
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Introduction
Controller and Observer
•
Controller and Observer in a resonant converter
•
Resonant phase plane controller
•
Control Signal:
•
States:
Switched System(rectifier)
Resonant circuit
W

1

neg. conducting
W
0

not conducting
W
1

pos. conducting
Resonant current
Resonant voltage
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Final Work Presentation
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Modeling of dynamic systems
•
Why modelling is needed? Analysis of system properties, controller design
and observer design is based on the
dynamic model
of the system
•
The traditional differential equation form,
ABCD form
, A set of equations
derived from physical law governing the system
•
The ABCD form is not global for
non

linear systems
and it eliminates the
algebraic equations
(some information of the system are lost)
•
Hamiltonian modelling
is based on the energy properties of the system
•
Hamiltonian modeling DE

form and DAE

form
•
Power converter is a non

linear system due to existence of the switching
transistors and rectifier, this motivates using Hamiltonian modelling
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Final Work Presentation
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Hamiltonian Modelling
is the Hamiltonian states
is the energy storage function (Hamiltonian function)
is a skew

symmetric matrix models how energy flows within the system,
is a positive semi

definite symmetric dissipation matrix
is a skew

symmetric control matrix
*This model structure is used for modelling the converter*
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Final Work Presentation
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Hamiltonian Modelling
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Hamiltonian states, :
charge on the capacitors, , and the magnetic
flow in the inductors,
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The energy function:
•
where and
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The physical interpretation of
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Final Work Presentation
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Hamiltonian modelling of the converter in DE

form
•
Decreasing the number of the states to four. magnetising inductance , , is
high compared to the resonance inductance, making the magnetising
current comparably small and possible to neglect.
•
Hamiltonian modelling in DE

form
•
Hamiltonian modelling is strongly related to the
Graph Theory
•
Different block matrices in Hamiltonian DE

form must be defined
according to graph theory
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A Hamiltonian observer can be defined based on 4th order Hamiltonian
model of the precess
October 28, 2008
Final Work Presentation
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4th order Hamiltonian model of the converter
•
Power converter is a switching system due
to existence of the transistors and rectifier
•
The controller produce the proper function
to switch the transistors
•
The rectifier has 3 possible conducting
states: Positive conducting, Non

conducting and Negative conducting
•
Rectifier is acting in which subspase!?
Ω1, positive conduction:
Ω0, no conduction:
Ω

1, negative conduction:
October 28, 2008
Final Work Presentation
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Modelling for subspace
Ω
0
•
The graph
in the subspase
Ω
0
•
Spanning tree, tree branches and links
•
Component matrices, intermediate block matrices
and the system block matrices are defined according
to the graph theory
The intermediate block matrices
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Final Work Presentation
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Modelling for subspace
Ω
0
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The system block matrices
The system equation:
= +
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Modelling for the subspaces
Ω
1 and
Ω

1
In the subspaces
Ω
1 and
Ω

1 the converter can be modelled by the graphs
Different matrices must be defined for the upper configurations, the final result:
where:
October 28, 2008
Final Work Presentation
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4th order global model
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A global model is:
•
The 4th order global model of the converter system
where
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An Hamiltonian observer can be introduced based on the 4th order global model
October 28, 2008
Final Work Presentation
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Hamiltonian Observer
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The structure of an Hamiltonian observer is given by
•
From the process model and the observer structure, the estimation error
is given as , satisfies
The error can be regarded as a
. Hamiltonian system!!
October 28, 2008
Final Work Presentation
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Hamiltonian Observer
•
Error in the observer can be regarded as a Hamiltonian process with the
Hamiltonian function(energy function):
it is shown that (Hultgren, lenells 2004) : if
•
Note: The process is modelled by a 4th order Hamiltonian model and the
observer is based on the 4th order Hamiltonian model of the process
October 28, 2008
Final Work Presentation
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Discretization of the observer
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Controller creates control signal from the external signal (reference)
and (estimation of the states)
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Implementation of the controller: Analog or Digital
•
In the converter a phase plane feedback controller is used, this controller
uses Resonance voltage and resonance current as feedbacks
•
The only measurment in this application is the resonance current
•
In discretization of the Hamiltonian observer, the sampling interval is very
important (the error balance must be always negative)
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Final Work Presentation
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Implementation in Matlab and Simulink
Observer
Process
October 28, 2008
Final Work Presentation
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Implementation in Matlab and Simulink
System Model
Hamiltonian Observer
Observer Model
Logic
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Working Plane
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Normalization of voltages and currents!?
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Where is the operating point practically?
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Does the observer works
the same in all regions?
•
Where maximum error in
state estimation happens?
•
Does the observer get
unstable anywhere?
•
Which state has the most error?
•
How we can improve state
estimation in the observer?
•
Different modelling? Optimal Sampling frequency?
Result of simulation will answer the questions, hopefully correct!
iLN
vLN
High load current and
low load voltage
Low load current and
high load voltage
High Power
October 28, 2008
Final Work Presentation
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High load current and Low load voltage, 10MHz sampling frequency
Spikes! High error!
High load current
Square Wave
Different subspaces!
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Final Work Presentation
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High load current and Low load voltage, 30MHz sampling frequency
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Final Work Presentation
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High load current and Low load voltage, 3MHz sampling frequency
Error does not
converge to zero!
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Hamiltonian observer gain
•
How we can improve state estimation? Applying obser gain!? Changing the model!?
•
Applying an observer gain may improve state estimation but not always!
•
Lets have a look at trajectories:
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Low load current and high load voltage, 30MHz sampling frequency
Rectifier is operating mostly
In the non

conducting mode
Spikes are much lower
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Final Work Presentation
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High load current and high load voltage (operating point), 10MHz sampling
frequency
•
K=0
•
K
>0
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High load current and high load voltage (operating point),
30MHz sampling frequency
•
K=0
•
K
>0
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3
rd
order model of the process
•
If we have a lower order model of the system, implementation will be
simpler and we have less calculation in the observer
•
By less calculation in the observer, we can discretize the observer with
higher sampling frequency
•
Which state is candidate to be eliminated
?
•
The load capacitance voltage is only slowly varying due to the high value
of . During one conduction period of the rectifier the load voltage can
be viewed as a constant voltage source, , in series with the voltage
•
The influence of the parallel capacitance, , is only significant when the
rectifier is not conducting. In the case when the rectifier is conducting the
parallel capacitance is connected in parallel with much larger load
capacitance .During the periods when the rectifier is not conducting,
the voltage of the parallel capacitance normally is commuting from to

or vice versa. The length of the non

conducting time periods is only
significant when the load current is small.
October 28, 2008
Final Work Presentation
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3
rd
order model of the process
•
subspace
Ω
0
•
subspaces
Ω
1 and
Ω

1
System equation
Global system equation
October 28, 2008
Final Work Presentation
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High load current and Low load voltage, 10MHz sampling frequency
K=0
K
>0
In this region, the rectifier is in the non

conducting mode for a short time
October 28, 2008
Final Work Presentation
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Low load current and High load voltage, 10MHz sampling frequency
In this region, the rectifier is in the non

conducting mode for most of the time
K=0
K
>0
October 28, 2008
Final Work Presentation
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High load current and high load voltage (operating point),
30MHz sampling frequency
K=0
K
>0
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Final Work Presentation
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Modelling of the system by considering resistance across the load capacitance
The load in this industrial application shows
obvious capacitive behaviour, but due to
existence of corona current around the wires,
the load shows resistivity behaviour too
The modelling takes the same steps as we did for
previous cases. First we define different subspaces
, then we achieve the system equation according to
Hamiltonian modelling and graph theory
October 28, 2008
Final Work Presentation
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Modelling of the system by considering resistance across the load capacitance
subspace Ω0
=
+
subspaces Ω1 and Ω

1
where
October 28, 2008
Final Work Presentation
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Thank you for your attention!
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