This paper systematically analyzes linear oscillators, e.g., spring-mass-damper systems or RLC-circuits that are controlled by an extension of a phase-locked loop (PLL). These systems are often used in measurement applications where the stability and dynamics directly influence the measurement quality. Therefore, a description of the control loop in terms of phase signals is sought. However, the classical oscillator turns into a highly nonlinear system when it is formulated in amplitude/phase-variables of its input and output signals. Up to now, there were made either ab-initio assumptions of slowly varying parameters or trial-and-error designs. The novel approach proposed in this paper derives a universally valid description in state space form which enables the use of standard methods of nonlinear system theory. Using this description, the stability of phase controlled oscillators is analyzed by means of Lyapunov functions. A linearization is applied in order to effectively design the controller and optimize the closed-loop dynamics. Simulations with the original nonlinear systems are conducted to justify the linear approach. Thereby, two application scenarios are under consideration: Tracking of the desired target value (target phase shift) and resonance tracking (changes of the system parameters). It is found that including the phase dynamics of the oscillator significantly improves the description of the closed-loop behavior. Finally, the results are validated experimentally for an application measuring the viscosity of fluids.

References

References
1.
Kutin
,
J.
,
Smrečnik
,
A.
, and
Bajsić
,
I.
,
2003
, “
Phase-Locking Control of the Coriolis Meter's Resonance Frequency Based on Virtual Instrumentation
,”
Sens. Actuators, A
,
104
(
1
), pp.
86
93
.
2.
Goodbread
,
J.
,
Sayir
,
M.
,
Häusler
,
K.
, and
Dual
,
J.
,
1998
, “
Method and Device for Measuring the Characteristics of an Oscillating System
,”
U.S. Patent No. 5,837,885
.https://www.google.com/patents/US5837885
3.
Sell
,
J.
,
Niedermayer
,
A.
, and
Jakoby
,
B.
,
2011
, “
A Digital PLL Circuit for Resonator Sensors
,”
Sens. Actuators, A
,
172
(
1
), pp.
69
74
.
4.
Langdon
,
R. M.
,
1985
, “
Resonator Sensors-A Review
,”
J. Phys. E: Sci. Instrum.
,
18
(
2
), p.
103
.
5.
Haueis
,
M.
,
Dual
,
J.
,
Cavalloni
,
C.
,
Gnielka
,
M.
, and
Buser
,
R.
,
2001
, “
A Fully Packaged Single Crystalline Resonant Force Sensor
,”
J. Micromech. Microeng.
,
11
(
5
), p.
514
.
6.
Axelson
,
P.
, and
Johnsson
,
A.
,
1976
, “
Phase-Locked Loop Technique to Record Resonance Frequency of Plant Tissue
,”
Physiol. Plant.
,
36
(
2
), pp.
113
117
.
7.
Valtorta
,
D.
, and
Mazza
,
E.
,
2006
, “
Measurement of Rheological Properties of Soft Biological Tissue With a Novel Torsional Resonator Device
,”
Rheol. Acta
,
45
(
5
), pp.
677
692
.
8.
Park
,
S.
,
Tan
,
C.-W.
,
Kim
,
H.
, and
Hong
,
S. K.
,
2009
, “
Oscillation Control Algorithms for Resonant Sensors With Applications to Vibratory Gyroscopes
,”
Sensors
,
9
(
8
), pp.
5952
5967
.
9.
Smithgall
,
D.
,
1975
, “
A Phase-Locked Loop Motor Control System
,”
IEEE Trans. Ind. Electron. Control Instrum.
,
22
(
4
), pp.
487
490
.
10.
Mizutani
,
Y.
,
Suzuki
,
T.
,
Ikeda
,
H.
,
Yoshida
,
H.
, and
Shinohara
,
S.
,
1998
, “
Frequency Control of MOSFET Full Bridge Power Inverter for Maximizing Output Power to Megasonic Transducer at 3 MHz
,” IEEE Industry Applications Conference, 33rd
IAS
Annual Meeting
, St. Louis, MO, Oct. 12–15, Vol.
3
, pp.
1644
1651
.
11.
Kuang
,
Y.
,
Jin
,
Y.
,
Cochran
,
S.
, and
Huang
,
Z.
,
2014
, “
Resonance Tracking and Vibration Stabilization for High Power Ultrasonic Transducers
,”
Ultrasonics
,
54
(
1
), pp.
187
194
.
12.
Wang
,
C.
,
Yu
,
H.-H.
,
Wu
,
M.
, and
Fang
,
W.
,
2007
, “
Implementation of Phase-Locked Loop Control for MEMS Scanning Mirror Using DSP
,”
Sens. Actuators, A
,
133
(
1
), pp.
243
249
.
13.
Gökcek
,
C.
,
2003
, “
Tracking the Resonance Frequency of a Series RLC Circuit Using a Phase Locked Loop
,” 2003
IEEE
Conference on Control Applications
, June 23–25, pp.
609
613
.
14.
Sun
,
X.
,
Horowitz
,
R.
, and
Komvopoulos
,
K.
,
2002
, “
Stability and Resolution Analysis of a Phase-Locked Loop Natural Frequency Tracking System for MEMS Fatigue Testing
,”
ASME J. Dyn. Syst., Meas., Control
,
124
(
4
), pp.
599
605
.
15.
Munzinger
,
C.
,
Weis
,
M.
,
Seemann
,
W.
,
Rudolf
,
C.
, and
Kern
,
D.
,
2010
, “
Dynamiksteigerung adaptronische strebe zur kompensation geometrischer maschinenfehler
,”
Adaptronik für Werkzeugmaschinen
, Vol.
1
,
Shaker Verlag
,
Aachen, Germany
, pp.
19
45
.
16.
Häusler
,
K.
,
Reinhart
,
W.
,
Schaller
,
P.
,
Dual
,
J.
,
Goodbread
,
J.
, and
Sayir
,
M.
,
1996
, “
A Newly Designed Oscillating Viscometer for Blood Viscosity Measurements
,”
Biorheology
,
33
(
4–5
), pp.
397
404
.
17.
Rüst
,
P.
,
Cereghetti
,
D.
, and
Dual
,
J.
,
2013
, “
A Micro-Liter Viscosity and Density Sensor for the Rheological Characterization of DNA Solutions in the Kilo-Hertz Range
,”
Lab Chip
,
13
(
24
), pp.
4794
4799
.
18.
Kharrat
,
C.
,
Colinet
,
E.
, and
Voda
,
A.
,
2008
, “
H∞ Loop Shaping Control for PLL-Based Mechanical Resonance Tracking in NEMS Resonant Mass Sensors
,”
IEEE
Sensors Conference
, Lecce, Italy, Oct. 26–29, pp.
1135
1138
.
19.
Brack
,
T.
, and
Dual
,
J.
,
2015
, “
Multiple Frequency Tracking of a Torsional Oscillator—Applications in Dynamic Viscometry
,”
International Congress on Sound and Vibration (ICSV)
.
20.
Meirovitch
,
L.
,
2001
,
Fundamentals of Vibrations
,
McGraw-Hill Higher Education
,
New York
.
21.
Bellescize
,
H.
,
1932
, “
Onde Electrique
,”
La Reception Synchrone
, Vol.
11
, Paris.
22.
Hsieh
,
G.
, and
Hung
,
J.
,
1996
, “
Phase-Locked Loop Techniques. A Survey
,”
IEEE Trans. Ind. Electron.
,
43
(
6
), pp.
609
615
.
23.
Gardner
,
F.
,
2005
,
Phaselock Techniques
,
Wiley
,
Hoboken, NJ
.
24.
Best
,
R.
,
2007
,
Phase-Locked Loops: Design, Simulation, and Applications
,
McGraw-Hill Professional
,
New York
.
25.
Kern
,
D.
,
Brack
,
T.
, and
Seemann
,
W.
,
2012
, “
Resonance Tracking of Continua Using Self-Sensing Actuators
,”
ASME J. Dyn. Syst., Meas., Control
,
134
(
5
), p.
051004
.
26.
Kroupa
,
V.
,
2003
,
Phase Lock Loops and Frequency Synthesis
,
Wiley
,
Chichester, UK
.
27.
Dorf
,
R.
, and
Bishop
,
R.
,
2008
,
Modern Control Systems
,
Prentice Hall
,
Upper Saddle River, NJ
.
28.
Strogatz
,
S.
,
2001
,
Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering
,
Perseus Books Group
,
Cambridge, UK
.
29.
Lunze
,
J.
,
2013
,
Regelungstechnik: Systemtheoretische Grundlagen, Analyse und Entwurf einschleifiger Regelungen
,
9th ed.
, Vol.
1
,
Springer Vieweg Verlag
,
Berlin/Heidelberg
.
30.
Dual
,
J.
,
Sayir
,
M.
, and
Goodbread
,
J.
,
1990
, “
Viscometer
,” U.S. Patent No. 4,920,787.
31.
Preumont
,
A.
,
2011
,
Vibration Control of Active Structures
,
Springer
,
The Netherlands
.
32.
Dual
,
J.
, and
O'Reilly
,
O.
,
1993
, “
Resonant Torsional Vibrations: An Application to Dynamic Viscometry
,”
Arch. Appl. Mech.
,
63
(
7
), pp.
437
451
.
33.
Oppenheim
,
A. V.
,
Schafer
,
R. W.
, and
Buck
,
J. R.
,
1999
,
Discrete-Time Signal Processing
,
2nd ed.
,
Prentice-Hall
,
Upper Saddle River, NJ
.
34.
Zhou
,
K.
, and
Doyle
,
J.
,
1998
,
Essentials of Robust Control
(Prentice Hall Modular Series for Engineering),
Prentice Hall
,
Upper Saddle River, NJ
.
35.
Bemporad
,
A.
,
Morari
,
M.
, and
Ricker
,
N.
,
2015
, “
Model Predictive Control Toolbox for Matlab—User's Guide; Version 5.1
,”
The MathWorks
,
Natick, MA
.
36.
Liberzon
,
D.
,
2011
,
Calculus of Variations and Optimal Control Theory: A Concise Introduction
,
Princeton University Press
,
Princeton, NJ
.
37.
García
,
C. E.
,
Prett
,
D. M.
, and
Morari
,
M.
,
1989
, “
Model Predictive Control: Theory and Practice—A Survey
,”
Automatica
,
25
(
3
), pp.
335
348
.
38.
Landau
,
I.
,
Lozano
,
R.
,
M'Saad
,
M.
, and
Karimi
,
A.
,
2011
,
Adaptive Control—Algorithms, Analysis and Applications
,
2nd ed.
,
Springer
,
London
.
You do not currently have access to this content.