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ANALYSIS AND DESIGN OF RESONANT
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ANALYSIS AND DESIGN OF RESONANT FREQUENCY
CONTROL SYSTEMS WITH APPLICATIONS

By

Daniel Smith

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Mechanical Engineering

2007

ABSTRACT

ANALYSIS AND DESIGN OF RESONANT FREQUENCY CONTROL
SYSTEMS WITH APPLICATIONS

BV

J

Daniel Smith

Resonant systems arise in many areas throughout science and engineering. Some
examples of systems which must be excited at resonance for optimal performance
include ultrasonic motors, inductive heating loads, cavity resonators, and cyclotrons.
Due to disturbances such as environmental conditions, load variation and degradation,
the resonant frequency of these systems can shift, resulting in loss of performance.
This necessitates employment of a resonant frequency control system that maintains
lock between the excitation frequency and the resonant frequency.

In this thesis, three resonant frequency control methods are investigated. The first
uses nonlinear feedback with a phase lead compensator to start and sustain oscillation
at the resonant frequency of the system. The second method varies the excitation
frequency to track the resonant frequency of the system by seeking the frequency cor-
responding to a local maximum of the magnitude response. The third method varies
a parameter in the resonant system to tune the resonant frequency to the excitation
frequency by seeking the parameter value corresponding to a local maximum of the
magnitude response. Non-linear models of these control systems are developed and
linearized to obtain tractable models for analysis and design. In addition, design
guidelines are provided and results are illustrated through two applications: a novel
RF plasma ignition system, and a dynamic vibration absorber. Simulation results are
presented for the dynamic vibration absorber and the developed RF plasma ignition
system to illustrate the effectiveness of the proposed control methods. Also, experi-
mental results for the preposed ignition system are presented to further demonstrate

the features of the preposed methods and the viability of the ignition system.

This work is dedicated to my family, who has made it possible for me

to become the person I am today.

iii

ACKNOWLEDGMENTS

I would like to acknowledge my adviser, Dr. Cevat Gekcek, for his generous
investment of expertise and time, and for his dedication to my develepment. His
mentoring has contributed immensely to my personal and professional growth and
will have lifelong impact. Also, I would like to thank Dr. Harold Scheck and his staff
for providing resources which made engine testing of the RF plasma ignition system
possible. In addition, I would like to thank my family and friends for their support

and understanding through this endeavor.

iv

TABLE OF CONTENTS

LIST OF FIGURES ............................. vii
1 Introduction ................................ 1
1.1 Motivation ................................. 2
1.1.1 RF Plasma Ignition System ................... 3

1.1.2 Dynamic Vibration Absorber .................. 7

1.2 Problem Formulation ........................... 8
1.3 Resonant Frequency Control Systems .................. 9
1.3.1 Self-Oscillating Resonance Tracking Control .......... 9

1.3.2 Resonance Seeking Control .................... 10

1.3.3 Resonance Tuning Control .................... 10

1.4 Thesis Overview .............................. 11
1.4.1 Thesis Organization ....................... 11

1.4.2 Original Contributions ...................... 11

2 Modeling .................................. 13
2.1 Resonance ................................. 13
2.2 RF Plasma Ignition System ....................... 14
2.3 Vibration Absorber ............................ 17
2.4 Resonator ................................. 20
2.5 Self-Oscillating Resonance Tracking Control .............. 20
2.6 Resonance Seeking Control ........................ 22
2.7 Resonance Tuning Control ........................ 24

3 Analysis .................................. 27
3.1 Self-Oscillating Resonance Tiacking Control .............. 27
3.2 Resonance Seeking Control ........................ 30
3.3 Resonance Tuning Control ........................ 35

4 Design ................................... 40
4.1 Design Guidelines ............................. 40
4.1.1 Self-Oscillating Resonance Tracking Design ........... 40

4.1.2 Resonance Seeking Control System ............... 42

4.1.3 Resonance Tuning Control System ............... 43

4.2 Simulation ................................. 44
4.2.1 Self-Oscillating Resonance Simulation .............. 44

4.2.2 Resonance Seeking Control Simulation ............. 46

4.2.3 Resonance Tuning Control Simulation ............. 47

4.3 Experimental Results and Discussion .................. 49

5 Conclusions ................................ 55
5.1 Summary ................................. 55
5.2 Future W'erk ................................ 57

BIBLIOGRAPHY .............................. 58

vi

1.1

2.1
2.2
2.3
2.4
2.5
2.6
2.7

3.1
3.2

4.1
4.2
4.3
4.4
4.5
4.6

4.7
4.8
4.9

LIST OF FIGURES

Corona discharge from bench testing. ..................

Proposed RF ignition system. ......................
Model of dynamic vibration absorber ...................
Bede plot of MSD with and without. dynamic vibration absorber.

Model of resonant system. ........................
Self-oscillating resonance tracking system model .............
Model of resonance seeking control system ................

Model of resonance tuning control system. ...............

Simplified linear model of resonance seeking control system. .....

Simplified linear model of resonance tuning control system .......

Amplitude and angular frequency of the capacitor voltage. ......
Angular frequency response for ignition system. ............
Magnitude response of the primary mass .................
Tuned parameter response .........................
Bede plot of MSD with and without dynamic vibration absorber.

Amplitude of voltage across the capacitor V2 for Open-loop (solid) and
closed-loop (dashed) systems. ......................
Corona at the beginning of an ignition event ...............
Flame propagation at the end of a 4 ms ignition event. ........

Burn time with respect to ignition event duration ............

4.10 Burn time with respect to ignition timing. ...............

vii

35
39

45
47
48
49
50

51
52
53
54
54

CHAPTER 1

Introduction

Resonant systems are found commonly throughout the disciplines of science and en-
gineering. In many cases resonance is destructive, and therefore should be avoided.
In some systems, however, resonance is useful, if not critical for proper operation of a
system. hr'faintaining resonance may be difficult since operating conditions, wear, ag-
ing, manufacturing defects and other disturbances often cause the resonant frequency
or excitation frequency of such systems to vary. Resonant frequency control methods
are therefore proposed in this work which compensate for these variations to ensure
oscillation at the resonant frequency of the resonator. Resonant frequency control
is based on two approaches: tracking the resonant frequency of a system by varying
the excitation frequency, referred to as resonance tracking control, and by varying
the resonant frequency of the system to track the excitation frequency, referred to as
resonance tuning control. The aim of this thesis is to develop systematic modeling,
analysis and design methods for resonant frequency control systems and to apply
them to two applications to demonstrate that the methods are effective and useful.
Three resonant frequency control methods are presented; two are based on resonance
seeking control methods, and one on resonance tuning control methods. In addition,
two applications for these methods, an RF plasma ignition system, and a dynamic
vibration absorber are presented.

This section will discuss the motivation and applications for the research on reso-

nance seeking control, as well as briefly discuss the methods that have been developed

to solve this problem.

1 . 1 Motivation

Resonant loads, such as ultrasonic motors, piezoelectric transducers, induction heat—
ing loads, resonant inverter loads, microelectremechanical gyroscopes, cyclotrons,
cavity resonators, microwave heating loads, plasma processing loads and bandpass
wireless comn’mnicatien loads, arise in several areas of science and engineering [1]-
[20]. Such loads must be driven at their resonant frequencies for optimal performance
[II-[20]. However, even if these leads are driven initially at their resonant frequencies,
disturbances, such as temperature change, lead variation, manufacturing variabil-
ity, aging, fatigue damage, microphonics or electromagnetic detuning, can cause their
resonant or excitation frequencies to drift with time and significantly impair their per-
formance. This necessitates employment ef resonant frequency control systems that
excites such systems at their resonant frequencies or tune the resonant frequency of
the system to the excitation frequency in the face of such inherent disturbances.
The lock between the excitation frequency and resonant frequency of a resonant
system can be achieved by tracking the resonant frequency and varying the excitation
frequency accordingly to match the resonant frequency. Alternatively, the resonant
frequency of the resonant system can be tuned to match the excitation frequency.
It should be pointed out that in some applications (ultrasonic motors, piezoelectric
transducers, induction heating loads, resonant inverter loads, microwave heating loads
and plasma processing leads) the former approach is more feasible while in some
other applications (microelectremechanical gyroscopes, cavity resonators, cyclotrons
and bandpass wireless communication leads) the latter approach is more feasible.
The resonance frequency control problem has received significant attention re-

cently in the contexts of the specific applications listed above. When the model of

the resonator is a lightly damped second order system, the phase of its impedance
or admittance function is locally odd about the resonant frequency and its value is
0, 7r/ 2, or —7r / 2 rad at the resonant frequency. Thus, for such resonators, the phase
characteristic of its transfer function measured by a phase detector can be used for
resonance seeking control purposes. Recent resonant frequency control methods with
general formulations for second order systems which take advantage of this character-
istic are as follows. In [21], analysis and design methods are developed for a resonance
tracking control method for resonating an RLC circuit using phase-locked loop. In
[22] resonance tracking is achieved by using proportional feedback around the second
order system and adaptively controlling the feedback gain using the error between the
excitation and resonant frequencies obtained by a phase detector. In [23] a resonance
tuning control method is developed for second order systems, in which a system pa-
rameter is adaptively tuned to change the resonant frequency of the resonator, again
using phase detection. In many practical cases, however, the order of the system is
often inherently higher than two. In these situations, the phase detector based reso-
nance seeking control method is usually inadequate. Further methods suitable for a
broader range of resonant systems are desirable. Therefore, the goal of this thesis is to
introduce and investigate control methods that seek the unknown resonant frequency
of a system and drive it at its resonant frequency, and to apply such tools to two

applications, an RF plasma ignition system and a dynamic vibration absorber.

1.1.1 RF Plasma Ignition System

Modern spark ignited internal combustion engines are required to minimize exhaust
emissions while maximizing fuel efficiency. These requirements call for precise control
of the ignition timing to run the engine at the knock threshold and the ignition
energy to improve the combustion efficiency. The effects of ignition energy and timing

on combustion quality have been widely studied. Increasing ignition energy and

precisely controlling its timing have been shown to reduce combustion delay, rise
time, unwanted emissions and cycle-to-cycle variability while increasing lean limits,
knock limits and efficiency [24]—[26]. Thus the ignition system plays a crucial role in
the optimal Operation of modern engines.

Ignition systems have been incrementally refined for many years. Their basic na—
ture, however, has remained essentially unchanged and their energy transfer efficiency
is still extremely low (less than 1%) [24]. As a result, the spark generated by con-
ventional ignition systems is usually weak, stationary and confined between the two
electrodes of the spark plug. The weak and stationary spark causes slow or incom-
plete burn of the air-fuel mixture resulting in low fuel economy, high emissions, and
reduced power [27]——[30]. With these conventional ignition systems, the achievable
lean limit and exhaust gas recirculation tolerances are also modest [27]—[30]. Further,
the performance of conventional ignition systems is known to deteriorate especially
during cold start or with fouled spark plugs [24].

Competition among automotive manufacturers as well as government regulations
for better performance necessitate development of new ignition systems that may
overcome the shortcomings of conventional ignition systems. Recent efforts toward
this goal have focused on modifications to conventional ignition systems [31]-[34].
All of these efforts, however, are still hindered by poor spark plug longevity and
inherently low efficiency due to high secondary coil resistance. Other non-conventional
efforts include using a high frequency transformer to achieve high voltage for ignition
[35], using an RF resonator for this purpose [36], and utilizing a pulsed high power
laser beam focused through a window into the combustion chamber for ignition [37].
Systems utilizing a high frequency transformer still have a relatively low operating
frequency resulting in performance similar to that of a conventional ignition systems.
RF resonators lack robustness since slight component variations easily detune the

resonator, resulting in inadequate voltage. Laser induced ignition is not feasible in

automotive applications due to its high power and optical accessibility to cylinder
requirements. As a result, none of these ignition systems have found widespread
applications in automotive industry.

An RF plasma torch is a viable and promising ignition system for achieving in-
creased ignition energy and precisely controlled ignition timing [38], [39]. This type of
ignition systems uses a high quality factor RF resonator excited by an RF generator
to start and sustain an RF plasma torch for a prescribed duration inside the com-
bustion chamber. Plasmas can be classified as DC, AC and RF plasmas according to
their excitation frequency f. For a spacing between the electrodes of a few millime-
ters, the classification is roughly as follows. When f < 1 kHz, the generated plasma
is turned off between the excitation cycles: and this type of plasma is called a DC
plasma. When 10 kHz < f < 1 MHz, the generated plasma is on continuously but the
ions cross the sheath in less than one excitation cycle and follow the instantaneous
sheath potential. This type of plasma is called an AC plasma. When f > 10 MHz,
the plasma is on continuously and it takes many excitation cycles to cross the sheath
region. This type of plasma is called a RF plasma. Concerning ignition systems, the
plasma generated by a conventional ignition system is a DC plasma.

The DC plasma requires a large sheath voltage. Its efficiency is low since a consid-
erable amount of energy is dissipated in non-ionizing collisions. For the RF plasma,
the sheath size changes with each RF cycle that moves the charged particles back and
forth through the plasma. This displacement current generates heat dissipation that
is proportional to the square of the frequency. Hence, the efficiency of the RF plasma
is very high. The efficiency of an AC plasma lies between that of a DC plasma and
a RF plasma. It is generally easier to control the plasma density and the peak ion
bombardment energy with RF excitation.

In this work an RF plasma torch is employed to ignite and initiate controlled

combustion. It is believed that RF plasma ignition has several advantages over con-

ventional technologies. The proposed RF ignition system is expected to: (1) accelerate
the initial flame development by lowering the effective ignition temperature and time,
thus decreasing heat losses to the chamber walls, resulting in higher overall thermal
efficiency as well as a faster and more powerful burn; (2) provide a better match
with the ionization and dissociation energies of the molecules present in the com-
bustion chamber, resulting in lowered radiation and conduction heat losses; (3) have
beneficial effects on the fuel decomposition rates prior to combustion; (4) reduce the
troublesome spark advance requirement due to the precise control of ignition timing
and energy; (5) extend the lean limit and enhance the EGR tolerance by precisely
adjusting the plasma power and its duration according to engine operating conditions;
(6) yield reduced electrode wear and spark plug fouling as a result of RF excitation;
and (7) cause low electromagnetic interference.

RF plasma is generated by the ignition system without the need for a ground
electrode, and develops as a large, spider shaped corona discharges emanating from
the regions of highest electromagnetic field strength. Despite all of its potential
benefits, this ignition system requires its RF resonator to be excited precisely at
its resonant frequency to generate the plasma torch. Thus, it is extremely sensitive
to environmental changes and manufacturing tolerances. Hence, for proper practical
operation, it requires a control system to track the resonant frequency of the resonator
and drive it at that frequency despite environmental changes and manufacturing
tolerances.

To demonstrate the event that such a system can generate, Figure 1.1 shows
a corona discharge from the system. This discharge contains a number of plasma
streamers that are caused by ’avalanches’ of ions. When a high electromagnetic
field periodically draws in or pushes away ions, they collide with other molecules
generating more ions that in turn are pushed and pulled away, creating an avalanche

effect. Corona discharges are therefore capable of dispensing energy into a much larger

volume than arc—based ignition systems, where the current flow naturally concentrates
into the path generated at breakdown. Failure of the system to properly track the
resonant frequency will either severely degrade such a discharge, or will prevent the

discharge completely.

 

Figure 1.1. Corona discharge from bench testing.

Such a discharge is inherently more efficient, and has potential to significantly
improve combustion with a resulting increase and power and reduction in emissions.
Due to the simplicity and effectiveness of the system proposed, it offers excellent
potential for automotive ignition systems, as well as many other applications. Thus,
developing resonance frequency control methods suitable for such a system will make
the concept viable, and will potentially provide a significant contribution toward

automotive engine technology.

1.1.2 Dynamic Vibration Absorber

The second example presented is a classical dynamic vibration absorber problem
consisting of a sinusoidally excited mass-spring-damper (MSD) with a tuned absorber
attached. In this configuration, a properly tuned absorber can effectively attenuate

the sinusoidal force exerted on the primary system. However, it is assumed that the

excitation frequency or system parameters are not precisely known. With known
values, properly tuning an absorber is a relatively easy. However, if the variables are
not precisely known, it is necessary to implement a control system that will adjust
the absorber parameters to accommodate for deviations in the parameter values of
either the absorber or the primary system.

For this thesis, a varying parameter in the absorber will be assumed [40]—[44]. This
parameter will be tuned to accomplish the goal of minimizing the magnitude response
of the primary system. For an undamped system, minimization occurs when the
natural frequency of the absorbing mass matches the excitation frequency [45]. At this
condition the absorbing mass resonates, and the resulting forces on the original mass
counter the excitation forces, thereby attenuating the magnitude response. Thus,
designing a control system for seeking the maximum value of amplitude of the isolated
absorbing MSD by varying the stiffness is one proposed solution. Therefore the design
problem is not varying the frequency of excitation, but rather adapting the parameters

of the absorber to attenuate vibration in the primary system.

1 .2 Problem Formulation

Considering the motivation and applications discussed above, the following resonant
frequency control problem is formulated: Given a resonator, develop a model which
includes the resonator and an appropriate resonant frequency control system. Then
determine the performance of the feedback system (stability, robustness, disturbance
rejection, reference tracking, rise time, settling time, maximum overshoot, etc). Fi-
nally, design the parameters of the overall system so that the system achieves certain
desired performance objectives.

This problem is motivated by two classifications of resonant systems: second order
systems and higher order systems. For each type of resonator, this thesis will present

an appropriate resonant frequency control system. The resonant frequency control

systems will be demonstrated with simulation results, and with experimental results

from the RF plasma ignition system.

1.3 Resonant Frequency Control Systems

As mentioned above, two approaches to solving the resonant frequency control prob-
lem are considered in this work. First, resonance tracking control is considered, where
the resonant system is excited at its resonance frequency by varying the excitation
frequency. Second, resonance tuning control is considered, where a structural pa-
rameter in the system is varied to tune the resonant frequency of the resonator to
the excitation frequency. The first method that will be presented is self-oscillating
resonance tracking systems, and is a form of resonance tracking control. This method
primarily well suited to second order resonators. The second method that will be
proposed is an adaptive control method which tracks the resonant frequency of the
resonant system by controlling the frequency of excitation. This method is thus a
form of resonance tracking control, and will be referred to as resonance seeking cen-
tral. The third method is also and adaptive control method, but varies a parameter of
the resonant system to tune the resonant frequency to the excitation frequency. This
method is therefore will be referred to as resonance tuning control. These last two
methods are suitable for higher order resonators, but may also be used for resonant

frequency control of second order systems.

1.3.1 Self-Oscillating Resonance Tracking Control

Self—oscillating resonance tracking control is a method by which resonant systems self-
start and sustain oscillation. Feedback is designed with two elements: a compensator,
which is designed to compensate for phase changes in the loop, and a nonlinearity,
which is designed to provide self-starting ability and an amplitude limitation mecha-

nism. The describing function method [46] will be used to determine the conditions

necessary for resonance, and design guidelines will be given for designing the com-
pensator and nenlinearity. This resonant frequency control method is well-suited for
second order systems, and the main analysis and design methods in this work include
a time delay as found in the RF plasma ignition system. Design of compensators

suitable for higher order systems will also be briefly discussed.

1.3.2 Resonance Seeking Control

Resonance seeking control is an adaptive method based on that found in [47] that
seeks the frequency corresponding to a local maximum in the magnitude response of
a system. The system is excited with a variable frequency oscillator. The frequency
of the sinusoid is controlled through feedback by estimating the derivative of the
magnitude response of the resonant system with respect to the excitation frequency.
From this derivative, the frequency corresponding to the local maximum value of the
magnitude response of the system can easily be sought. The derivative is estimated
by perturbing the excitation frequency of the resonant system with a low amplitude,
low frequency sinusoid, then synchronously demodulating the absolute value of the
output of the system. After being filtered, the signal is fed back to complete the
control loop. The low pass filter used for synchronous demodulation plays the role of

the controller in the system, thus design methods focus primarily on this element.

1.3.3 Resonance Tuning Control

Resonance tuning control is also an adaptive method that seeks a parameter value of
the resonant system corresponding to a local maximum in the magnitude response.
The resonant system is excited with a fixed amplitude, fixed frequency sinusoidal
signal. The variable system parameter is controlled through feedback by estimating
the derivative of the magnitude response of the resonant system with respect to the

adjustable parameter value from which the maximum value can easily be sought. The

10

derivative is estimated by perturbing the adjustable parameter value of the system
with a low amplitude, low frequency sinusoid, then synchronously demodulating the
absolute value of the output of the system. After being filtered, the signal is fed
back to complete the control loop. Again, the low pass filter used for synchronous
demodulation plc ys the role of the controller in the system, thus design methods focus

primarily on this element.

1 .4 Thesis Overview

1.4.1 Thesis Organization

This thesis is organized as follows: Chapter 2 provides the models of the resonant
frequency control systems as well as models for the motivating application. Chapter
3 presents analysis of the resonant frequency control methods. Chapter 4 discusses
the issues surrounding the design of resonant frequency control systems and provides
simulation results of each resonance tuning system as well as experimental results
for the RF plasma ignition system. Finally, Chapter 5 gives conclusions and outlines

some future research directions.

1.4.2 Original Contributions

The primary contribution of this thesis is the development of systematic analysis and
design methods for three resonant frequency control topologies. The first resonant
frequency control method achieves resonant frequency tracking using an integrated
self-oscillating feedback topology. A phase-lead compensator is used to compensate
for the propagation delay in the RF generator and a signum type nonlinearity is used
to achieve, the amplitude limitation for sustained oscillation. The describing function
method is employed for analysis and design of the self-oscillating resonance tracking

system. The second method varies the excitation frequency to track the resonant

11

frequency of the system by seeking the frequency corresponding to a local maximum
of the magnitude response. The third method varies a parameter in the resonant
system to tune the resonant frequency to the excitation frequency by seeking the
parameter value corresponding to a local maximum of the magnitude response.

Nonlinear models that accurately predict the performance of the resonant fre-
quency control systems are developed. These developed models are then linearized to
obtain more tractable linear time-invariant models that can be used for both analysis
and design of the resonant frequency control systems. Based on the developed linear
time-invariant models, specific guidelines for designing the resonant frequency control
systems are also provided.

Two useful applications of these methods are presented. A novel RF plasma ig-
nition system has been developed and tested on engine, and a control system based
on a self-oscillating resonance tracking feedback topology has been implemented and
tested. The proposed self-oscillating resonance tracking control method and the res-
onance seeking control method are applied through simulation of the ignition system
to determine its potential effectiveness. Results from bench and on engine testing
are also presented. These results are promising, and show that the system is a viable
alternative to conventional ignition systems. The second example is based on the
classical problem of tuning a dynamic vibration absorber to match the excitation fre—
quency of a system. It is assumed that the absorber is a simple mass-spring-damper
system with variable stiffness, and that. the excitation frequency of the primary sys-
tem is constant. Thus, the resonance tuning method is employed to tune the stiffness
in the absorber to minimize amplitude of the primary system, and results are verified
with simulation.

This thesis has generated one published work [48], and another paper is in prepa-

ration [49].

12

CHAPTER 2

Modeling

This chapter will present the models for the ignition system and dynamic vibration
absorber, and a generic model for a resonator. Three control topologies that will be
used for resonant frequency control are also proposed. The concept of resonance is

also defined in this chapter for clarification.

2. 1 Resonance

Scientist and engineers in various fields have used many definitions of resonance
[6],[45],[47],[50],[51]. With such a diverse field of applications, it is difficult to reconcile
these various concepts to one common definition of resonant frequency. Therefore, to
prevent confusion the definition of resonant frequency will be fixed through two defi—
nitions. One definition will apply to second order systems, and the second to higher
order resonant systems. Second order systems, such as the RLC circuit found in the
RF plasma ignition system, have a single peak in the magnitude response very near
their natural frequency. This can be seen in the second order system of the common

form

2
can

— 32 + 2Cwns + (442,,

 

H(s) (2.1)

where can is the natural frequency of the system and C is the damping ratio. When

such a system is lightly damped, the magnitude response |H(jw)| has a peak near

13

w 2 run. Thus, the first definition of resonance is given as

Definition 2.1: The resonant frequency, (or, ofa second order system of the form
where 0 < C < 0.707 is defined as wn [50/,[52/.

F or higher order systems with transfer function H (s), the definition of resonance is
not fixed since the properties of systems vary considerably. For example, a resonator
may have more than one peak in its magnitude response. Thus the following definition
of resonant frequency is given.

Definition 2.2: The resonant frequency, wr, of a higher order system is de-
fined as the frequency wr at which the magnitude response, [H ( jwr)|, reaches a local
mari mum [51].

It. should be noted that for second order systems this peak frequency occurs, more
precisely, at to,- = wn\/I_——2C—2. Thus, if C is small, can z can. Therefore, the local
maximum in [H (jw)| occurs approximately at can. Thus, for C sufficiently small,

either definition can be applied to second order systems.

2.2 RF Plasma Ignition System

Prior to discharge, a spark plug can be modeled as a cylindrical capacitor with a
very small capacitance value. Connecting an inductor in series with the spark plug
forms a series RLC circuit, in which the resistor represents the losses in the physical
components. By driving this RLC circuit with a suitable RF generator as shown in
Figure 2.1 exactly at its resonant frequency, a very high RF voltage can be developed
across the spark plug that results in an RF plasma torch. In this figure, R, L and
C are the resistance, inductance and capacitance of the series RLC circuit; T, L1,
Cp, C A and V form the RF generator; and D is the timing and gate driver circuit
of the RF generator. The RF plasma torch generated this way can be used to ignite
air-fuel mixtures in an internal combustion engine with appropriate ignition timing

and duration.

14

V

nil—JET

 

 

 

 

 

 

 

 

 

 

Figure 2.1. Proposed RF ignition system.

The gate driver and the RF generator can be modeled as
A(s) = Ke—ST, (2.2)

where r is the propagation delay of the gate driver and RF generator combination
and K is the gain of the RF generator. The load of the RF generator is a series RLC
circuit with impedance
1
Z = R+ L —. 2.3
(8) 8 + SC ( )
The transfer function from the RLC circuit input voltage to the voltage across the

capacitor is

 

1
H(s>— , ,5? 1 (2 4)
S +18+m
Letting
can —1— (2.5)

be the angular resonant frequency and
g = _ _ (2.6)

be the damping ratio of the RLC circuit, the transfer function H (s) can be expressed

as

wt

(2.7)

 

H(s) —

— 32 + 2Cwns + a2,

'

15

Thus, the resonant frequency of this system can be defined by either method discussed
above since there is only one peak in the magnitude response resulting from a single
set of lightly damped complex conjugate poles.

Assuming that the input of the RLC circuit is of the form
’U1(t) 2 V1 cos(wt), (2.8)

where V1 is the amplitude and U.) is the angular frequency of the input voltage, the

voltage across the capacitor in the steady-state is
'vg(t) = |H(jw)|l/1cos[wt + 1H(jw)]. (2.9)
Thus, the amplitude of the voltage across the capacitor in the steady-state is
V2 = IH(jw)IV1- (2-10)

If the angular frequency of the input is exactly equal to the angular resonant frequency

of the RLC circuit, then the voltage V2 becomes

V2 = QVi, (2.11)
where
1

is the quality factor of the RLC circuit. Thus, the amplitude of the voltage across
the capacitor (spark plug before the breakdown) in the steady-state can be made suf-
ficiently high to start a corona discharge if Q of the RLC circuit is sufficiently high.
A Q factor of approximately 500 can be achieved by properly minimizing the losses
in the RLC circuit. If, on the other hand, the input angular frequency of the RLC
circuit is slightly different (even only 1%) from the resonant frequency of the RLC
circuit, then the voltage across the capacitor will not be sufficient enough to start a
corona discharge. Hence, even if the RLC circuit is driven initially at its resonance,

its resonant frequency or driving frequency may shift in time due to environmental

16

changes, reducing the achievable high voltage across the spark plug drastically and
rendering the ignition system completely inadequate. This mandates the use of a con—
trol system that tracks the resonant frequency of the RLC load connected to the RF
generator and drives it at its resonant frequency despite environmental disturbances.
For control purposes the current through the load is used for feedback since it can
be easily measured. The transfer function C(s) is the transfer function of the RLC
circuit from its voltage input to the current through it and is given by

( ) is
S 2 ,
s2 + 2Cwns + (i272,

 

(2.13)

where ’y = L/C.

The nominal values for the RLC circuit are R = 5.80 52, L = 14.86 x 10’6 H,
C = 10.10 x 10‘12 F. Thus, the nominal values of tag, C and Q are won = 81.63 x 106
rad/s, (n = 2.39 x 10""3 and Qn = 209.13. The gain of the RF generator is K = 20.00
and the propagation delay of the RF generator and its gate driver is r = 43.00 x 10—9
3. Due to the environmental disturbances, L or C values can change within 5% of
their respective nominal values so that the angular resonant frequency of the RLC

circuit can take any value in the interval [77.74 X 106, 85.92 x 106] rad/s.

2.3 Vibration Absorber

The classical tuned dynamic vibration absorber problem is to design an absorber
such that when attached to a primary system, it will minimize its vibration due to
excitation at a particular frequency. This is useful for systems where resonance of the
primary system without an absorber is a problem. The primary system is assumed to
be an MSD system, and the absorber will be another MSD mounted directly on the
mass of the primary system. Figure 2.2 shows a diagram of a standard absorber. In
this diagram, ha and hp are spring constants, ca and cp are damping constants, and

ma and mp are masses of the absorber and primary system, respectively. 270, is the

17

displacement of the absorber relative to the primary mass, and rp is the displacement

of the primary mass relative to its stationary mount, and u(t) represents a force acting

on the primary mass that is assumed to be of the form u(t) = F0 sin wot.

 

 

_Iwa

 

 

a

I::J

 

 

 

Tnp

_pp

 

 

 

 

I-dcp I(

ut)

Figure 2.2. Model of dynamic vibration absorber.

Thus, the goal is to minimize the vibration in the primary mass. This requires

the system be excited at the mode corresponding to the resonant frequency of the

absorbing MSD system. The following state space equation governs the dynamics of

the vibration absorber system,

 

 

r
it} (Ir 1
. ___JL _.:iL
$2 _ mp mp
£173 0 0
$4 .423 s— —(
'P 'P
0
L
m
+ p u t ,
O <)
1
—mp

 

0 0 3:1
12L 3a.
mp mp 172
0 1 $3
1 1 ,. l 1
fiswflh‘im+mhm “
(2.14)

18

where 2:1 = xp and r3 2 ma — rp. It is assumed that the position asp of the original
mass is available for feedback. The forced mass-spring—damper example selection was
chosen to have mass mp = 0.1 kg, spring stiffness kp = 1000 N/ m, and a damping
rate of cp = 0.09 Ns/m, and the absorber was chosen with mass mp = 0.02 kg, a
nominal spring stiffness ha = 200 N /m, and a damping value ca = 0.02 Ns/ m. The
design of the absorber is tailored to the natural frequency of the primary system,
wnl = 100 rad/s. Figure 2.3 shows a bode plot for the primary MSD system with
and without a vibration absorber. With the absorber, it is clear that the magnitude
response of the primary system is minimized at the resonant frequency, which occurs

at about 100 rad/sec.

 

 

 

 

 

 

 

 

 

 

 

0 e . e e .
A —20-
m
E
Q, —40—
“U
3
'a -60
b0
3
Z.
-80..
e
A —45~ —
to
Q)
E
Q) -90“ "
3
i
—135— U -
_180 i i . i . 1 i i LL; 3
10' 10 10

w (rad / see)

Figure 2.3. Bode plot of MSD with and without dynamic vibration absorber.

If the excitation frequency of the primary system or the parameters of the absorber

19

are disturbed, the absorber must be retuned for the magnitude response of the primary
system to be minimized. To accomplish this, the parameter ha is considered to
be a variable stiffness and it is further assumed that this stiffness can be actively
adjusted [40]-[44]. Controlling this value allows the absorber to be dynamically tuned

to account for changes in the resonant frequency or excitation frequency.

2.4 Resonator

Based on the above applications, it is desirable to formulate the resonant frequency
control problems with a generic resonator model. Thus, based on the intended ap-
plications, a resonator model is proposed in Figure 2.4. In this figure, u(t) and y(t)
are the system input and output, respectively, while H (s,§ , It) is the resonant sys-
tem, where 6 = [51 -~§n]T is a adjustable structural parameter of the system, and
Ii 2 [n1 - - - ren]T is a structural parameter of the system which may drift due to distur-
bances discussed above but which cannot be adjusted. These parameters are assumed

to be constant but unknown, with nominally known values.

 

u(t) ————) H(s,€, n) ‘———-) y(t)

 

 

 

Figure 2.4. Model of resonant system.

2.5 Self-Oscillating Resonance Tracking Control

This section presents the topology and modeling of a self-oscillating resonance fre-
quency tracking control system as a solution to resonating second-order systems with
a time delay. The RF plasma ignition system can be modeled as such, with the RLC
circuit being modeled a second-order system, and the inherent time delay in the gate

driver and RF amplifier considered to be significant.

20

l\='Iot.ivated by the previous discussion, the following self-oscillating resonance fre-
quency tracking system shown in Figure 2.5 is considered. In this figure, the transfer

function A(s) represents the driver system and is assumed to be in the form
A(s) = Ke‘ST, (2.15)

where r is the time delay in the driver system and K is the gain of the driving system;
C(s) is the transfer function of the resonant system, given by

H is

s = . 2.16
32 + 2Cwns + “2,2, ( )

 

The transfer function C (s) is a phase-lead type compensator to be designed to com-

pensate for the phase delay in the driving system and is assumed to be in the form
C(s) = 1+ 773, (2.17)

where 77 is a design parameter, and the static nonlinearity <p(u) is a signum type non-
linearity to achieve amplitude limiting mechanism for self-oscillation and is assumed
to be of the form

Wt) = .3 sgn (a), (2.18)
where sgn is the standard signum function and ,8 is a design parameter. Moreover,
17(t) is the input resonant system, y(t) is the output of the resonant system, u(t) is
the input of the signum nonlinearity, u(t) is the output of the signum nonlinearity
and the negative feedback sign is used for the sake of convention.

The parameters of the resonant system are assumed to be constant but unknown
with known nominal values to take into account the effect of environmental distur-
bances on the lead. Moreover, the output of the resonant system is assumed to be
available for feedback purposes.

With the above assumptions, the self-oscillating resonance tracking problem is to
design the compensator C (s) and the signum nonlinearity cp(u) to start and sustain
a self-oscillation with angular frequency sufficiently close to the nominally known

resonant frequency of the resonant system despite inherent disturbances.

21

 

 

 

 

 

 

 

 

 

 

 

 

 

“005) W)

 

 

 

 

 

 

 

 

Figure 2.5. Self-oscillating resonance tracking system model.

2.6 Resonance Seeking Control

In this section, the model for resonance seeking control is presented. This control
method is a form of adaptive control which seeks the excitation frequency corre-
sponding to a local maximum in the magnitude response of a system. Thus, the
proposed control system is designed to excite systems at their resonant frequency,
as given in the first definition above. Unlike the self-oscillating resonance tracking
control topologies, the resonant system is excited by a constant amplitude, variable
frequency sinusoidal signal source. The sinusoidal input signal u(t) to the resonant
system with transfer function H (s) in the steady state results in the sinusoidal out-
put, y(t), the magnitude of which should be maximized. Thus the input is assumed

to take the form

u(t) = Vsin(wst + 9) (2.19)

where V is the amplitude of the input, ws is the angular driving frequency, and 0 is

the phase angle. Thus, at steady state, the output y(t), is
y(t) = V|H(jw3)| sin[wst + 6 + £H(jw3)]. (2.20)

The magnitude M (tag) is defined as the average of the absolute value of y(t), and is
given by
2 .

22

The magnitude M (ws) reaches a local maximum when ids = our by the second defini-

tion of resonant frequency given above. Thus, resonance occurs when
. I
M (wr) = 0 (2.22)

and

M"(w,~) < 0. (2.23)

As discussed previously, ws can be determined nominally, but may deviate from
car causing degradation in performance of the system. Thus, a closed loop system is
used to enforce the condition that (us :3 car.

For the proposed control system, it is assumed that the output of the resonant
system is measurable for feedback purposes. The control system requires access to
M’ (cos), in order to properly regulate tag. This derivative cannot be measured directly,
so a method for estimating it is developed. To accomplish this estimation, a small
sinusoidal perturbation is added to the source frequency ws. The average magnitude
response with respect to this perturbation is then estimated.

To demonstrate how this is accomplished, Figure 2.6 presents the proposed control
system. In this figure, H (s) is the transfer function of the system that shall be excited
at its resonant frequency. The absolute value of the signal y(t) is taken, resulting in
the signal m(t). The average of m(t) thus is M (cos). The multiplication operator and
filter F (s) perform synchronous demodulation to estimate the required derivative
M ’ (eds). The lower section of the system represents the variable frequency oscillator,
with amplitude V and instantaneous angular driving frequency ws(t), as well as the
nominal angular center frequency (.00 and the small perturbation frequency 8 cos(ot).
The signals m(t) and z(t) represent the input and output, respectively, of the filter
F (s) The controlled phase of the variable frequency oscillator is represented by 6(t),

and the kw is the angular frequency feedback gain.

23

cos(ot)

 

 

 

I»!
A

H-
V

u(t) H(s) y(t) | . I in“) 5’3“) F(s)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

V sin(-) 3-1 MSU) 6(t) kw A

 

 

 

 

 

 

 

 

 

 

 

 

wo + E cos(ot)

Figure 2.6. Model of resonance seeking control system.

2.7 Resonance Tuning Control

In this section, the model for resonance tuning control is presented. This control
method is a form of adaptive control which tunes a system parameter value é corre-
sponding to local maxima in the magnitude response of the resonant system. Thus,
the proposed control system is designed to excite systems at their resonance frequency
as defined in Definition 1 by tuning the resonator to the excitation frequency rather
than seeking the resonant frequency of the resonator. The system is excited by a sinu-
soidal signal source u(t), with constant amplitude V, and constant angular frequency
tag. The sinusoidal input signal u(t) to system H (3) results in the sinusoidal output,
y(t), the magnitude of which should be maximized. Further, a structural parameter,
6, of the system is adjusted by a second input signal £3, unlike the resonance seeking
system. This signal {S is assumed to slowly time varying. Since the control system
will tune this parameter to achieve resonant frequency control, the transfer function

for the resonant system H (s) will be denoted H (s, g ) An example of such a system

24

is the dynamic vibration absorber, there the adjustable parameter .5 is the variable

stiffness ha. The output the system at steady state therefore assumes the form
y(t) = VIHIJ'woa €s(t)ll Sinf‘wot + ZHljws, {sftll} (224)

The magnitude Mwofis) is defined as the average of the absolute value of y(t), and
is given by

2
Areas.) = guffaws.» (2.25)

The magnitude Mwo(£s) reaches a local maximum when 53 = Er, where £7. is the
parameter value corresponding the the maximum value of [H (jw0,{ )| Therefore,
considering the second definition of resonant frequency above, but applying it to a

structural parameter rather than a frequency, resonance occurs when

["1th (gr) : 0 (2-26)
and
Agog.) < 0. (2.27)

As discussed previously, £3 can be determined nominally, but may deviate from
67. causing a degradation in performance of the system. Thus, a closed 100p system
is used to enforce the condition that {s 2 .3.

For the proposed control system, it is assumed that the output of the system is
measurable for feedback purposes. The control system requires access to MLOKS),
in order to properly regulate {3. This derivative cannot be measured directly, so
a method for estimating it is developed. To accomplish this measurement, a small
sinusoidal perturbation is added to the source signal {3. The average magnitude
response with respect to this perturbation is then estimated.

To demonstrate how this is accomplished, Figure 2.7 presents the proposed control
system. In this figure, H (s,{ ) is the transfer function of the system that shall be

excited at resonance. The absolute value taken the signal y(t) is taken, yielding

25

the signal m(t). The average of m(t) thus becomes Atfwo(£3), the scaled magnitude
response of the resonant system at steady state. The multiplication operator and filter
F (s) perform synchronous demodulation to estimate the required derivative hfsofig).
The signals m(t) and z(t) represent the input and output, respectively, of the filter
F (s) The lower section of the system provides the parameter value feedback loop
gain kg, perturbation frequency ecos(ot), and addition of initial parameter value {0
to yield the instantaneous parameter value £,(t).

cos(ot)

 

 

 

Vsin wot—+1 H(S, ) y(t) l . [ m(t) m(t) F(s)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

£0 + e cos(ot)

Figure 2.7. Model of resonance tuning control system.

26

CHAPTER 3
Analysis

In this section, the resonant frequency control models developed in Chapter 2 are
considered and analysis methods are developed. To take into account disturbances
that affect the resonant frequency of the systems, the uncontrolled parameters of the
resonant system are assumed to be constant but unknown. Three main resonant fre-
quency control systems are analyzed. First, self-oscillating resonance tracking control
is analyzed for second order systems with time delay. Next, resonance seeking and
resonance tuning control are developed. Nonlinear and linear models for both reso-
nance seeking and resonance tuning control systems are also presented to aid in the

design process.

3.1 Self-Oscillating Resonance Tracking Control

Consider the self-oscillating resonance tracking system shown in Figure 2.5. The
method of describing function is used to facilitate analysis and design of this system.
Since the resonant system is assumed to have sufficiently high quality factor, the
output is practically equal to the first harmonic of the signal in the loop. Thus, the
use of the describing function method is justified.

Assuming that the signal y(t) in the steady-state is essentially of the form

y(t) = Ycos(u2t), (3.1)

27

it follows that the signal u(t) in the steady-state is given by
u(t) = |C(jw)|Ycos[wt + ZC(jw)]. (3.2)

Thus, the signal u(t) in the steady-state can be expressed as
(u(t) = NIIC(J'w)IYIIC(jw)IYcoslwt + Acne] + e(t). (3.3)

where e(t) is the sum of higher order harmonics and N(a) is the describing function

(equivalent gain) of the signum nonlinearity given by

4
N(cr) = —fi. (3.4)
no
Then, the signal .1:(t) in the steady-state can be expressed as
wit) = IAiijNllCUwHYIICUwIIYcoslwt + mow) + mow) + 7r] (3 5)

-a(t) * e(t),
where a(t) is the inverse Laplace transform of A(s). Thus, the signal y(t) in the

steady-state turns out to be

y(t) = IGIJ'w)|IAIJ'wHNIICUWHYI|C(J'w)IY€08th + 400w)

MAW) + 406w) + W] — y(t) * a(t) ... e(t), (36)

where g(t) is the inverse Laplace transform of C(s). Hence, assuming that the term

y(t) a: a(t) * e(t) is negligibly small due to filtering in C(s), it follows that
1+ C(jwMUwWIICUw)IYICUw) = o. (3.7)
which is the (first-order) harmonic balance equation. Equivalently,

IGUWHlAljwllNlICIJ'wHYl[C(J'wfl COSIZGIJ'w) + 114010) + ZCIJ'w)I = -1 (3.8)

and

IGIJ'w)I|A(jw)

 

N [

 

CIJWIIYIICijll siI'llZGIJ'w) + 14010) + 400W] = 0- (3.9)

28

Moreover, since |A(jw)|N[|C(jw)|Y]|C(jw)| 3&4 0, equation (3.9) can be simplified as
[C(jw)| sin[ZC(jw) + ZA(jw) + 4C(jw)] = 0. (3.10)

This equation is independent of Y and thus can be easily solved for to to determine
the estimated angular frequency of oscillation. Once the estimated angular frequency
of oscillation is determined, the estimated amplitude of oscillation can be determined
by solving equation (3.8) for Y.

It follows from equation (3.10) that the estimated angular frequency of oscillation
w is equal to the angular resonant frequency of the resonant system independent of

environmental disturbances provided that
1.4(jw) + (C(jw) E 0, mod 7r. (3.11)

However, it is impossible to satisfy this condition by a finite dimensional transfer
function C(s). Nevertheless, with the assumed simple form of C(s), the phase-lag
introduced by the driving system can be compensated at least for the nominal angular

frequency Lung. Noting that the phase of the transfer function A(jw) is
ZAUw) = —wr, (3.12)

the design parameter 77 in the transfer function C (s) is selected to satisfy

ZCanQ) + ZAUojno) E 0, mod 71'. (3.13)
Thus, it turns out that
1
(”720

With this choice, it follows from equation (3.10) that
|C(jw)| sin[lC(jw)] z 0, (3.15)

provided that
ACUw) + ZA(jw) z 0, mod 7r. (3.16)

29

Hence, the angular oscillation frequency w is approximately equal to the angular
resonant frequency can of the resonant system provided that the above condition is
satisfied. Clearly, equation (3.10) can be solved numerically to determine a better
estimate for the angular oscillation frequency.

Having determined the angular oscillation frequency to, equation (3.8) can be
solved numerically to estimate the amplitude of the oscillation Y. An analytical
estimate for the amplitude of the oscillation can be also obtained by using the ap-

proximation w a: can. With this assumption, it follows from equation (3.8) that

 

 

 

. 201071
N Cyan, Y = - , 3.17
i < 0| 1 7K .—__1+n,w, < >
when wnr E (—7r/2, +7r/2), mod (27r) and
. 2 w
Nilcawnm = + C " (3.18)

“/K\/ 1 + 772w?
when wnr E (+7r/ 2, +3rr/ 2), mod (2n). Hence, the amplitude of the oscillation is

given by
_ 27m
— it run

Y

 

sgn (Nucewnnn). (3.19)
Note further that the parameter 6 turns out to be

nY . . '
5 = Hin sgn (N[|C(]wn)|Y]). (3.20)

 

3.2 Resonance Seeking Control

This section details the development of a nonlinear model that can accurately rep-
resent the resonance seeking control system. For additional use in system design, a
simple linear time-invariant model is also developed.

Consider the system model presented in Figure 2.6. The angular frequency of the

driving signal, w3(t), can be written as
w3(t) = wo -l— 6(t) + ecos(ot), (3.21)

30

where

e(t) = raga). (3.22)
It follows then that. the driving signal can be expressed as
u(t) = Vsin[u10t + ¢V(t)], (3.23)

where
e
eéV(t) = 6(t) + — sin(ot). (3.24)
o
The signal 6(t) is assumed to be slowly time varying, and o is assumed to be

sufficiently small relative to too, resulting in the following approximation:

y(t) = Ysinlwot + a(t)] (3.25)
where
Y = VIHIJ'wsttllI, (326)
and
¢y(t) = 6(t) + 33in(ot) + 2H[jw,(t)]. (3.27)

It follows that the instantaneous magnitude is

7710) = Iy(t)I (328)
which can also be written as
m(t) = y(t)sgnly(t)l
= Y sin[w()t + ¢y(t)]sgn{sin[w0t + ¢y(t)]} (3.29)

Using the following Fourier series expansion,

00

._ _ 4 1 ‘. ,
sgn[s1n(a:1:)] — X Z ;s1n(wn.r), (3.30)
nodd
It can easily be shown that
4 °° 1
sgn[y(t)] = g 2 7f sm[w()nt + n.c_by(t)]. (3.31)
n Odd

31

Therefore, from equation (3.29) it follows that

41’ °° 1
m(t) = 7 sin[w0t + ¢y(t)] 2 ; sin[to()nt + ncby(t)]. (3.32)
nOdd
Applying the trigonometric identity sin2(:r) = %[1 — cos(2:r)] yields

m(t) = gfl — cos[2w0t + 2¢y(t)]} + p(t), (3.33)

where p(t) represents higher order terms. Noting that

M[w3(t)] = g, (3.34)
it follows that
m(t) = .M[a23(t)]{1 — cos[2w0t + 2¢y(t)]} + p(t) (3.35)

Assuming e is sufficiently small, M [ws(t)] is expanded into its Taylor series about

(e(t) = too + e(t). M [w3(t)] is therefore approximated as
M[w3(t)] = M[w(t)] + A/I'[w(t)]ecos(ot) (3.36)
It follows then, from equation (3.35), that
m(t) = {le(t)l + M ’Iw(t)l€ COSI0(t)I} {1 - COSIQwot + Qty/(01} + W)

An expression has now been obtained that contains the derivative of the magnitude
response with respect to frequency. To isolate this term, the signal is multiplied by
cos(ot). So from Figure 2.6,

m(t) = m(t) cos(ot) (3.37)

With some trigonometric manipulation, m(t) can be written as
a(t) -_- git/mean + q(t) (3.33)

where the term q(t) includes higher frequency components around o, 20, 2w0 — o,

2w0 + o, (n — 1)'w0 + o, (n — 1)'w0 + o, (n + 1)'w0 + o, and (n + 1)w0 -— o. Passing

32

the signal r(t) through the low pass filter F (s) attenuates these signals, so assuming

these signals are entirely eliminated,
e(t) = §f(t) * rlr[’[tu(t)], (3.39)

where f (t) is the impulse response of the filter F (s), and * is the convolution Opera-

tion. It follows from (3.22) that
' kw5 . I
0(t) = ——2—f(t) *1"! [m(t)]. (3.40)

Considering that 9 = w(t) — Leo, the equation becomes

a(t) — too = k—‘gg-flt) * A/I’[w(t)], (3.41)
where
(u(t) = ws(t) — ecos(ot). (3.42)

This equation is an approximation of the dynamic behavior of the frequency in the
resonance seeking control system. If the assumptions stated above are satisfied, simu-
lation indicates that this nonlinear equation accurately describes the dynamics of the
proposed system. Based on this equation, and assuming that the closed loop system
is stable, the steady state value toss of m(t) must satisfy

1:

Therefore, if F (3) contains at least one integrator, M ’ [w(t)] will approach a steady
state value of zero. Thus the instantaneous frequency (u(t) asymptotically approaches
the resonant frequency of the resonator wr, or similarly the driving frequency ws(t)
asymptotically approaches wr + 5 cos(ot). Thus, to ensure that w3(t) at steady state
is very close to the resonant frequency (er, 5 should be chosen to be sufficiently small.

For design purposes, however, a simplified linear time-invariant model is desirable.
To accomplish this, the nonlinear equation (3.41) can be linearized about its nominal

frequency too, or for a more useful linearization, about its nominal system parameters

33

(no and so. To linearize about wo, take 6w(t) = w(t) — too, and expand M ’ [wo + 5w(t)]

into its Taylor series. Retain only the first two terms to give
k
6w(t) = 4530:) * (Mao) + M"(w0)6w(t)]. (3.44)

The linearization from this expansion results in a less straightforward final form since
M’ (too) still depends on the uncontrollable system parameter K3. This leads to lin-
earization about the nominal frequency we and nominal resonant system parameter
no. Take 6s = h: — so, and denote M (w) as M (w, Ii) since it is also dependent on the
system parameter It. Then M ' [too + 6w(t), so + 6ft] is expanded into its Taylor series

and higher order terms are ignored, yielding

 

 

+ Err/(s, K.) wwo 6n . (3.45)
85 nzno

Note that the derivatives of M in this equation are all dependant on nominal values

and that M ’ (too, so) = 0. Thus the linearized equations can be written as

 

(5w(t) = kf(t) * [ricer — 6w(t)]. (3.46)
where
kwe (92
.. 2 __ _ / - .47
and

2
5§a3trce,r)

@390
“‘“06K, c348)

wzwo
nzno

 

 

(SLUT = — 62
WMGU, K)

 

where (Star is the deviation of the resonant frequency from the nominal value too.

Lastly, from equation (3.42) it can be seen that

6w3(t) = 6w(t) + ecos(ot). (3.49)

34

To further demonstrate these equations, a block diagram is presented in Figure 3.1
that represents the linearized model in equation (3.46). Thus, the closed 100p reso-
nance seeking control system behavior can be analyzed with standard linear system
techniques. Simulation results indicate that this linear time-invariant model is quite
accurate, given the above assumptions.

5 cos(ot)

 

 

(Star kF(s) + 6W3

 

 

 

 

 

Figure 3.1. Simplified linear model of resonance seeking control system.

Also, it is important to note that F (3) plays a primary role in the resonance seeking
control system. It can be easily seen that the linear system of equation 3.46 is stable
if all the roots of the characteristic equation 1 + kF(s) = 0 have negative real parts.
Thus, the original nonlinear system shown in Figure 2.6 is locally asymptotically

stable if this condition is satisfied.

3.3 Resonance Tuning Control

This section details the development of a nonlinear model that can accurately repre-
sent the resonance tuning control system. As in the last section, for additional use in
system design, a simple linear time-invariant model is also developed.

Consider the system model presented in Figure 2.7. The parameter g of H (s),
is tuned to adapt the resonant frequency of the system to the excitation frequency.

Thus, the input signal {3(t) which controls this parameter can be written as

{3(t) = {0 + {e(t) + ecos(ot), (3.50)

35

where
€C(t) = k£z(t). (3.51)
The driving signal can be expressed as

u(t) = Vsin(w0t), (3.52)

The signal {e(t) is assumed to be slowly time varying, and a is assumed to be

sufliciently small relative to 50, resulting in the following approximation:

y(t) = Y Sin(w0t + (by) (3.53)
where
Y = VIHLiwo. €s(t)]|, (3.54)
and
W = éHljwo,€s(t)] (3.55)

It follows that the instantaneous magnitude is

m(t) = |y(t)l (3.56)

which can also be written as

m(t) = y(t)ssnly(t)l- (3-57)

Recalling equation (3.30), it follows that

00

4 1
sgn[y(t)] = E E h sin(w0nt + n¢y(t)). (3.58)
n=1,3,5,...

Thus, the signal m(t), by equation (3.53), becomes

m(t) = fl;— sin[w0t + ¢y(t)] Z isinhuont + ngby(t)]. (3.59)
nOdd

Applying the trigonometric identity sin2(r) = %[1 — cos(2:r)] yields

m(t) = 3::{1—eos[2eot+2ey(t)]}+p(t) (3.60)

36

where p(t) represents higher order terms. Noting that

2Y
Il/Iw0[§3(t)] = ?, (3.61)
it follows that
m(t) = Maggy-(If )]{1 — cos[2w0t + 2d>y(t )]} + p(t) (3.62)

Assuming e is sufficiently small, Mwo [53(t)] is expanded into its Taylor series about

£(t) = £0 + £C(t). 11le [§,g(t)] is therefore approximated as
tween = Meolfltll + MLOKIIIIECOSIUIUl (3.63)

It then follows that equation (3.62) becomes

m(t) = {MwoKU ])+M’. 0)o[€(t]ecos[ (t )]}{1—cos[2w0t+2d)y(t)]}

+726) (3.64)

An expression has now been obtained that contains the derivative of the magnitude
response with respect to the adjustable parameter. To isolate this term, the signal is

multiplied by cos(ot). So from Figure 2.7,
m(t) = m(t) cos(ot) (3.65)
With some trigonometric manipulation, m(t) can be written as
m(t) =—II’ [at )]+ q(t). (3.66)

where the term q(t) includes higher frequency components around 0, 2o, 2w0 — a,
21420 + o, (n —1)u20 + o, (n -1)uI0 + o, (n +1)w0 + o, and (n +1)w() — o. Passing the
signal m(t) through the low pass filter F (s) attenuates these signals. Assuming these

signals are entirely eliminated, it follows that

2(t)=-:-f(t) II’ [at )1 (3.67)

37

where f (t) is the impulse response of the filter F (s) and III is the convolution Operation.

It follows from 3.51 and 3.67 that

 

e(t) = kggfit) I M1,, [an]. 1368)

Considering that {e(t) = 6 (t) — too, the equation becomes

at) — £0 = 4574) I molten, (3.69)

where

{(t) = {3(t) — eces(ot). (3.70)

This equation is an approximation of the dynamic behavior of the adjustable param-
eter in the resonance seeking control system. Therefore, if the system is stable and
F (3) contains at least one integrator, MLO[€(t)] will approach a steady state value of
zero. Thus the parameter value f (t) asymptotically approaches the parameter value
67-, or similarly the tuning signal {3(t) asymptotically approaches {T +5 cos(ot). Thus,
to ensure that the average value of §3(t) at steady state is very close to the parameter
value corresponding to resonance 67., 8 should be chosen to be sufficiently small.

As in resonance seeking control, the nonlinear equation (3.69) is linearized about
the nominal adjustable parameter {0 and about nominal fixed system parameter so

below. Mw0(§) is denoted NIWO (é , It) since it is also dependent on K3. Thus linearization

 

yields
e(t) = Int) I [It — e(t)]. (3.71)
where
k = £33 (Ba—:2Meok. 4) it“ (3.72)
and 0

 

82
Willow (67 K') {:50
6.5,. = _ a2 “”0674. (3.73)
fillle (63 I"2)

 

 

5:60
5:50

38

Also. 6£(t) = €(t) -— £0 and 6s = fi —— H0. Further, 66+ is the deviation of the tuned

parameter from the nominal value 50. Lastly, from equation 3.70 it can be seen that
(5639?) = 6§(t) + €cos(ot). (3.74)

To further demonstrate these equations, a block diagram is presented in Figure 3.2
that represents the linearized model in equation 3.71. Thus, the closed loop reso-
nance tuning control system behavior can be analyzed with standard linear system
techniques, and similar results and methods from resonance seeking control are ap-

plicable.

e cos(ot)

 

 

657. kF(s) + as,

 

 

 

 

 

Figure 3.2. Simplified linear model of resonance tuning control system.

39

CHAPTER 4
Design

In this chapter, the design problem formulated above is considered and some guide-
lines for designing the resonant frequency control system are provided. Using these
guidelines the actual resonant frequency control systems are simulated with the pre—
viously discussed parameters of the application. These simulation results illustrate
the performance of the resonant frequency control systems and verify the accuracy
of the proposed models. All simulations are carried out in MATLAB/Simulink. In
addition, experimental results for the RF plasma ignition system are presented to

demonstrate its feasibility.

4. 1 Design Guidelines

4.1.1 Self-Oscillating Resonance Tracking Design

For second order systems with time delays, as shown in Figure 2.5 the compensator
parameter 77 can be selected using equation (3.14). For a desired output amplitude

Y, the design parameter 6 can be selected using the nominal values of the resonant

system as
WCIIwTIOY
B = _._ 4.1
+ 2,7111" ( )
when wnor E (—7r/2, +7r/2), mod (2n) and
7r(anOY
: _— 4.2
[3 271W ( )

40

when wnOT E (+7r/ 2, +37r/ 2), mod (277), where Lang, (7,, and cm are the nominal values
of too, C and 7, respectively.

In order to self-start the oscillation, the system shown in Figure ?? with 1,9(u)
replaced by N (a) must be unstable for sufficiently large N (a). When the amplitude
of the output is small, the signum nonlinearity provides the necessary high gain for
the fundamental component to self—start the oscillation. The system then generates
a periodic waveform with a growing envelope until the signum nonlinearity limits the
amplitude of the oscillation to the value estimated by equation (3.8). The transfer
function C(s) filters out all but the fundamental component of this periodic wave-
form yielding an essentially sinusoidal output at the angular frequency estimated by
equation (2.5).

It Should be pointed out that a higher order compensator C (s) can be used to
better compensate for the delay in the loop. Simulation results do, however, indicate
that the selected simple compensator is adequate for the intended ignition application.
In addition, a static saturation type nonlinearity can also be used in place of the
signum nonlinearity for amplitude limiting.

If the system is higher than second order, and a unique solution to the harmonic
balance equations exists with simple phase compensator of the form given in equation
(2.17 ), the simple phase lead compensator solution presented above may still be suit-
able. However, if the disturbances to the resonant system cause large deviations in
the natural frequency of the system, requiring more robustness, or the harmonic bal-
ance equations yield multiple solutions, a higher order compensator may be required.
Notice that the second-order system of equation (2.16) with no time delay and no
compensation is robust to disturbance in the natural frequency, as 400w”) = 0 for
any wn. Thus, using cancelation to convert a system to this form ideally will result

in such robustness. Assuming that the dominant poles in the form of equation (2.16)

41

can be factored out such that

0(8) : Gr(3)Gnr(S)
73
s2 + 2Cwns + log,

 

(4.3)

= Cm(s)

it is easily seen that choosing C(s) to directly cancel Cm(s) will yield the robust
form discussed above. This is a simple method, but it should be noted that it is often
impractical since it may yield improper, unstable, or complicated compensators, or
cancelation of unstable zeros or poles. In addition, Gm- may have terms that could be
considered dominant near the resonant frequency. Thus, careful analysis of the system
should be carried out before designing such a compensator. If such a compensator
is not suitable, yet more robustness to disturbance to the natural frequency is still
needed, using compensator which satisfies the necessary phase condition similar to

that in equation (3.11) and one or more of its derivatives may be a viable alternative.

4.1.2 Resonance Seeking Control System

The primary component of the system requiring careful design for the resonance
seeking control topology is the low-pass filter. This filter plays a critical role in
the control strategy. Ftorn the linearized model, it is clear that the filter acts as
the controller. Thus, the goal of the resonance seeking control system is to find
the ’controller’ F (s) such that the ’output’ w3(t) tracks the ’input’ tor satisfactorily.
Having obtained this linear model, the use of standard linear control design methods
can be used, with the assumption that the linear model is accurate.

The following guidelines should thus be followed when designing such a control
system:

- The parameter 5 should be sufficiently small so that the it is reasonable to assume
the higher-order terms of the Taylor series can be ignored and the perturbation signal

applied to wS(t) is negligible.

42

- The filter F (3) should stabilize the closed loop system shown in Figure 2.6.

- The filter F (9) must contain at least one integrator to achieve near-zero steady-
state tracking error.

- The bandwidth of the filter F (5) must be sufficiently small to reject the high
frequency terms at o and too.

- The bandwidth of the filter F (3) should be sufficiently large to reduce settling
time of the system.

It should also be noted that the resonance seeking control system can be designed
to stabilize about a local minimum in the magnitude response of the resonant system

using similar guidelines.

4.1.3 Resonance ’Ihning Control System

The feedback loop in the resonance tuning control system is very similar to that of
the resonance seeking problem, as found in the analytical section. Thus, some of the
filter design guidelines will not be restated. As above, the primary component of
the system requiring careful design for the resonance tuning control topology is the
low-pass filter, and similar guidelines should be followed when designing it. As found
previously, from the linearized model, it is clear that the filter acts as the controller.
Thus, the goal of the resonance tuning control system is to find the ’controller’ F (s)
such that the ’output’ {e(t) tracks the ’input’ 6r satisfactorily. This approach allows
the use of standard linear control design methods, with the assumption that the
linear model is accurate. Like the resonance seeking control method, resonance tuning
control systems can be designed to stabilize about a local minimum as well as a local
maximum of the magnitude response of the resonant system. The dynamic vibration
absorber example is a case when stabilizing about a local minimum is necessary.

Simulations for this example are presented below.

43

4.2 Simulation

4.2.1 Self-Oscillating Resonance Simulation

Simulation results are presented below to compare the ability of the open—loop and
closed-loop systems to initiate and sustain robust oscillation at the resonant frequency
of the RLC circuit in the RF plasma ignition system. Feasibility of the ignition system
will also be assessed by observing trends in timing and magnitude.

The model used for simulation is defined by equations (2.15)-(2.18). Using the

guidelines developed earlier, 13 is selected as 6 = 5 and C(s) is designed as
C(s) = 1+ 4.73 x 10‘%. (4.4)

With these values, the system shown in Figure 2.5 is simulated by assuming that L
and C undergo 5% increase at t = 50 x 10’6 s. The instantaneous amplitude and
angular frequency of the capacitor voltage for the closed-loop system are shown in
Figure 4.1. For comparison purposes, the same system is also simulated by breaking
the loop at the output of the nonlinearity and letting u(t) = Ssgn [cos(w()t)] (the
nominal driving input). The instantaneous amplitude and angular frequency of the
capacitor voltage for the Open—loop system are also shown in Figure 4.1. The nominal
angular resonant frequency is also plotted in this figure with dashed line for reference.
In both cases, the instantaneous amplitude and angular frequency of the capacitor
voltage are obtained from the output y(t) numerically.

It should be pointed out that results obtained for closed-loop system from simula-
tion agree well with those calculated based on the previous analysis. Using equation
(3.10), the steady-state angular frequency of the output with R, L and C at their
respective nominal values is estimated to be 81.63 x 106 rad/s. Using this value in
equation (3.8), the steady-state amplitude of the output for this case is estimated to
be 21.95 A. The same values are determined from simulation to be 81.62 x 106 rad/s

and 21.95 A, respectively. Similarly, with L and C shifted as described above and R

44

30 l l l l l 7 l I I

 

 

25*- .

closed-loop

-15— -

kV

10 - (— open—loop _

 

 

 

 

90 100

 

to
O
_.
_.
_1
_
.1
.1
_1
_1
_1

 

(— open-loop

00
U1
I

 

w, Mrad/s
55°

\1
U!
I

 

closed-loop

 

 

 

70 1 1 1 1 1 1 1 1 1
0 IO 20 30 40 50 60 70 80 90 100

Figure 4.1. Amplitude and angular frequency of the capacitor voltage.

at its nominal value, the steady-state angular frequency and amplitude of the output
are estimated to be 77.77 X 106 rad/s and Y = 21.66 A, respectively. These values
are determined from simulation to be 77.76 x 106 rad/s and 21.71 A, respectively.
These simulation results demonstrate that the analysis and design methods devel-
oped above are quite accurate. The performance achieved by the resonance tracking
system considered above indicates that the proposed method is a viable solution to

the resonance tracking problem.

45

4.2.2 Resonance Seeking Control Simulation

The RF plasma ignition system is a suitable example for resonance seeking control
due to its very distinctive peak in the magnitude response of the RLC circuit corre-
sponding to the natural frequency. Thus, the system model and nominal parameters
discussed above will be used for the resonator for simulation of resonance seeking
control. The following parameter values were chosen for the system. The loop gain
was chosen as kw = 35 X 1019, the system input amplitude as V = 100 V, the pertur-
bation frequency as o = 3.26 x 105, and the amplitude of the perturbation frequency

as e = 81.62 x 103 rad/s. The filter, F(s) was chosen as

1
3(32 + 2.18 x 1053 +1.18 x 1010)

 

F(s) = (4.5)

Figure 4.2 shows the driving frequency ws for two different values of initial fre-
quency wo, rug 2 0.98am and too = 1.05am, to represent changes in the values of the
RLC circuit that result in a shift in the natural frequency. The system effectively
tracks natural frequency. Smaller disturbances in the nominal or natural frequencies
result in much faster settling. This is due to the high M ’ (to) values that occur locally
around the natural frequency, effectively increasing the loop gain, and causing faster
settling. That is also the reason for the slow response for the larger deviation. M ’ (cos)
is minimal when ws deviates a large amount from too.

Comparing the resonance seeking control system to the self-oscillating resonance
tracking control system reveals that the self-oscillating resonance frequency control
system is a more suitable control system for implementation due to the simplicity of
the control method, as well as the resonance frequency tracking performance. How-
ever, resonance seeking control is useful since the algorithm is not limited to second
order systems and it is not dependant of the phase of the system, but strictly maxi-

mizes the magnitude response of the resonant system.

46

 

 

h-
.—

79 1 l 1 1 1 l i
O l 2 3 4 5 6

t, ms

\l
w)-
0

10

Figure 4.2. Angular frequency response for ignition system.

4.2.3 Resonance Tuning Control Simulation

The classical dynamic vibration absorber problem is ideal for resonance tuning con-
trol. Instead of maximizing the magnitude response of the system, however, the
vibration absorption problem calls for minimization of the magnitude response of the
primary system. For this simulation, the excitation frequency is perturbed from the

value to the absorber is tuned, resulting in the following force on the primary mass
F = 100 sin 95t N. (4.6)

Excitation at this frequency does result in minimal displacement of the primary mass
since the absorber is not excited at the frequency for which it was tuned. Thus, a
resonance tuning control system with the following parameters is implemented. The
loop gain is chosen as kg = —300, the perturbation amplitude E = 0.70 N/m, the
perturbation frequency a = 9.5 rad/sec, and the nominal parameter value 1:20 = 200

N/m. The filter F(s) is chosen to be

1
3(32 + 0.95003 + 0.2256) '

 

F(s):

47

Figure 4.3 shows the displacement of the primary MSD system rp(t) based on the
above control system. Clearly the system accurately tracks the minimal magnitude
response of the system as designed. The steady state amplitude which remains is

due to the slight amount of damping in the system. Figure 4.4 shows the input

 

350 I I a I

300 *

 

 

I -- 1 1

40 60 80 100

 

Figure 4.3. Magnitude response of the primary mass.

{3 applied to the variable stiffness ha. The response shows stable tracking of the
parameter to minimize the damped mass displacement. Figure 4.5 is a bode plot
showing the change in the phase and magnitude characteristics corresponding to the
nominal resonant system and the system with the parameter value after the control
system reaches steady state. From this plot, it is evident that the control system is
effective at tuning the system to the excitation frequency.

Thus, the resonance tuning system is a viable method for tuning the parameters

of a resonator to minimize or maximize the magnitude response.

48

205

 

200m

195 -

has, N/m
p-a
to
o

185 r I

180 I WWII“ IIIIIIIIIIIIIII IIIIIIIII'IIII “IIIIIIIIII'II'IIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII1III1I1I1IIII

 

 

 

0 2e 46 t 66 85 100
, s

Figure 4.4. Tuned parameter response.

4.3 Experimental Results and Discussion

This section will discuss preliminary experimental results of the primary motivating
application, the RF plasma ignition system. To further investigate the feasibility of
the ignition system, bench testing and engine testing were performed. Bench test-
ing consisted of developing an open-loop plasma torch generation system capable of
accurate timing and duration control. The closed-loop system was also implemented
during bench testing after a robust open-loop system had been developed. Engine
testing involved taking high-speed video of ignition and combustion events in an op-
tically accessible engine to determine the capabilities and benefits of the system.
The closed loop control topology selected for the ignition system was self-
oscillating resonance frequency tracking control due to the simple mature of the con-
trol system and the very fast response to parameter perturbations. Figure 4.6 shows
the amplitude of the open-loop and closed-loop systems during resonance buildup

with low supply voltage to prevent breakdown. As expected, the closed-loop sys-

49

 

l
N
CD

I

 

  

|H(jw)| (113
c!»
C)

With resonant tuning control

 

 

1H (J'w) (deg)

 

 

 

 

 

 

 

 

—135~ -
-180 i i i 447 . . 1

102
w (rad/sec)

Figure 4.5. Bode plot of MSD with and without dynamic vibration absorber.

tern response has comparable amplitude and response time to the open-loop system
response when resonance is manually maintained.

Experimental data also shows that resonance buildup before breakdown occurs
in approximately 50 ps depending on the system configuration and breakdown can
easily be timed within a 5 us window. The resonant system clearly has a much
faster and more precise response time compared to a conventional inductive ignition
system, which has a typical rise time of 300-500 V/us. This has the potential to
reduce cycle-to—cycle variability, to more accurately approach the knock threshold,
and to allow more accurate calibration for cylinder-to—cylinder deviations and other
issues requiring consistent and accurate ignition events.

The most recent testing of the system has been on a single cylinder, optically ac-

50

 

25*

. I u \Avavs‘g‘ v .

 

 

 

0 ioo éoo 360 460
t, as

Figure 4.6. Amplitude of voltage across the capacitor V2 for open-loop (solid) and closed-
loop (dashed) systems.

cessible, direct injection, gasoline engine with the ignition system running open-100p.
The engine has a quartz window in the piston to allow an axial View of combus-
tion progression. Initial testing was completed in open-loop mode with no fuel being
injected to verify that ignition events could occur at high pressures. Following suc-
cessful completion of these tests, various ranges for ignition timing, ignition duration,
and fuel duration were tested. All load, timing and fuel points showed repeated and
successful ignition events.

Two images showing the beginning and end of a 4 ms ignition event are shown
below. Figure 4.7 shows the commencement of a typical ignition event approximately
150 ps after the start of plasma formation. At this point in the event, the mixture has
already begun to burn around the individual plasma streamers. This was determined
by analyzing several different combustion videos of similar ignition events. For refer-
ence, the estimated diameter of the corona is about 2 cm. Figure 4.8 shows that the
flame has already propagated through much of the cylinder before the 4 ms ignition

event has even completed.

51

 

Figure 4.7. Corona at the beginning of an ignition event.

The ignition system showed excellent ignition characteristics even in very short
duration events of less than 150 as Lean limits were also tested on the optically
accessible engine, and the system consistently ignited the leanest mixtures tested,
even in cold start conditions. However, with the equipment available and the very
limited number of cycles being run for each test, quantification of limits and potential
benefits was not possible.

Figure 4.9 shows the approximate burn time obtained by observing the high speed
videos obtained from engine testing. Video was taken of the axial view of the com-
bustion chamber through a quartz window in the piston. The following data is from
operation with ignition timing at 12 degrees before top dead center (BTDC) at the
engine running at low load. There is clearly a minimum duration of about 0.5 ms to
achieve faster burn times with this ignition system, although shorter burn times do
effectively and consistently ignite the mixture.

Figure 4.10 shows the approximate burn time obtained by observing the high

52

 

Figure 4.8. Flame propagation at the end of a 4 ms ignition event.

speed videos obtained from engine testing. The affect of ignition timing on the burn
duration are as expected, with the ’hook’ pattern used for timing optimization present.
Comparative data is not currently available for the system, so quantification of the

benefits of the RF plasma ignition system is not yet possible.

53

 

12.5- .

H
I—Ii—‘H
HUN

'

Burn Duration, ms
H
11—1 .0
o 01

to
cs: 01
T

 

9°
01
r
l

 

 

l l l

0 0:5 1 1:5 é 2.15 3 3.5 4
Ignition Event Duration, ms

 

00

Figure 4.9. Burn time with respect to ignition event duration.

 

II--| I—I I—I I—I r—I
O I? 1.; Ce e
.1

Burn Duration, ms

10

 

 

 

 

4 6 53 140 112 1‘4 116 1'8 20
Ignition Timing, Degrees BTDC

Figure 4.10. Burn time with respect to ignition timing.

54

CHAPTER 5

Conclusions

5. 1 Summary

The primary focus of this thesis is the systematic development of analysis and design
methods for three resonant frequency control methods. The first approach to resonant
frequency control is through a self-oscillating resonance tracking feedback topology.
The second approach utilizes variable frequency excitation to seek the resonant fre-
quency of the system. The last approach varies a parameter in the resonant system to
tune the resonant frequency of the resonator to the excitation frequency. Two appli—
cations are presented to illustrate the effectiveness of the resonant frequency control
systems. The first is a novel resonance-based RF plasma ignition system composed of
a high quality factor series RLC circuit driven by an RF generator to start and sus-
tain an RF plasma torch inside the combustion chamber. The second is a dynamic
vibration absorber composed of a primary mass-spring-damper system, and tuned
absorber consisting of a mass-spring-damper where the spring has variable stiffness.

The self-oscillating resonant frequency control method that is presented utilizes a
first-order phase lead compensator to compensate for phase changes in the loop and
a nonlinearity is used to achieve amplitude limitation for sustained oscillation. The
describing function method is employed for analysis and design of the self-oscillating

resonance tracking system, and detailed guidelines are established for the design of

55

the self-oscillating resonance frequency tracking system for second-order systems with
time delays.

The second method of resonant frequency control that is presented is an adaptive
method that seeks a local maximum of the magnitude response of the resonant sys-
tem. This is accomplished by perturbing the driving frequency then synchronously
demodulating the absolute value of the output of the resonator to estimate the deriva-
tive of the magnitude response with respect to the driving frequency. This derivative
is then used for tracking purposes.

The third method of resonant frequency control that is presented is also an adap-
tive method that seeks a local maximum of the magnitude response of the resonant
system, but instead of seeking the maximum frequency, the controller adjusts a pa-
rameter of the resonator to cause the resonant frequency of the resonator to track
a constant excitation frequency. This is accomplished by perturbing the adjustable
parameter and synchronously demodulating the absolute value of the output of the
resonator to estimate the derivative of the magnitude response with respect to the
driving frequency. The derivative is then used for tracking purposes.

Nonlinear models that accurately predict the performance of the resonant fre-
quency control systems are developed. These developed models are then linearized to
obtain more tractable linear time-invariant models that can be used for both analysis
and design of the resonant frequency control systems. Based on the deve10ped linear
time-invariant models, specific guidelines for designing the resonant frequency control
systems are also provided.

Design guidelines are introduced for each resonant frequency control method.
Based on these design methods, results from simulation of the RF plasma ignition sys-
tem and dynamic vibration absorber are presented to illustrate the benefits. Promis-
ing results from bench and engine testing of the RF plasma ignition system which

indicate that it is feasible for practical applications are also shown. Ignition is very

56

consistent under pressure and in cold start conditions with relatively small cycle—to-
cycle variation. The ignition energy and timing of the developed ignition system can
be controlled very precisely. The system generates corona discharges, which are a
far more efficient and effective means of ignition than arc discharges. Spark plugs
without a ground electrode can therefore be used, reducing the quenching effect the
electrodes normally have on the discharge while dispersing the ignition energy over a

large volume.

5.2 Future Work

Future theoretical work will include further development of resonant frequency control
methods stemming from resonance seeking and resonance tuning control. Methods of
combining resonance tracking and resonance tuning control to further enhance reso-
nant frequency control system performance will be researched, as well as improving
the adaptive control methods to achieve faster response time. In addition, general
methods for systematic analysis and design of self-oscillating resonance tracking con-
trol for higher-order systems will be investigated.

Future work also will include investigation of modelling of the ignition system
during and after breakdown to provide more effective compensator designs and other
types of static nonlinearities for more effective compensation of system parameter
changes during operation. This will help optimize the energy transfer to the mixture
after breakdown since the breakdown changes the system configuration and natural
frequency considerably. Future design work for the RF plasma ignition system will
focus on mechanical system development and packaging for on-engine testing so that
the benefits of the system can be quantified. Positive results are expected based
on the research supporting corona discharge ignition and the testing that has been

completed.

57

[ll

[2]

l4]

l5]

[6]

[7]

l8]

l9]

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