Abstract

This letter focuses on two topics in engineering analysis, which are (1) degree-of-freedom (DOF) in modeling of dynamical systems and (2) simultaneous time and frequency localization of signals. These issues are explained from the perspectives of decision and control by making use of concepts from applied mathematics and theoretical physics. Specifically, a new definition is proposed to clarify the notion of “DOF,” which is consistent with the dimension of the state space of the dynamical system model. Relevant examples are presented on (finite-dimensional) vector spaces over the real field R and/or the complex field C⁠.

Issue Section:
Research Papers
Keywords:
dynamical systems modeling, degree-of-freedom, time-frequency analysis of signals
Topics:
Dynamic systems, Modeling, Signals, Dimensions, Degrees of freedom

References

1.
Naylor
,
A.
, and
Sell
,
G.
,
1982
,
Linear Operator Theory in Engineering and Science
, 2nd ed.,
Springer-Verlag
,
New York
.
Google Scholar
Crossref
Search ADS
 
2.
Cohen
,
L.
,
1995
,
Time-Frequency Analysis
,
Prentice-Hall PTR
,
Upper Saddle River, NJ
.
3.
Joshi
,
S.
,
1991
, “
Design of Optimal Second-Order State Estimators
,”
J. Guid. Control
,
14
(
2
), pp.
466
–
468
.
Google Scholar
Crossref
Search ADS
 
4.
Rajagopalan
,
V.
, and
Ray
,
A.
,
2006
, “
Symbolic Time Series Analysis Via Wavelet-Based Partitioning
,”
Signal Process.
,
86
(
11
), pp.
3309
–
3320
.
Google Scholar
Crossref
Search ADS
 
5.
Subbu
,
A.
, and
Ray
,
A.
,
2008
, “
Space Partitioning Via Hilbert Transform for Symbolic Time Series Analysis
,”
Appl. Phys. Lett.
,
92
(
8
), p.
084107
(3 pages).
Google Scholar
Crossref
Search ADS
 
6.
Mukherjee
,
K.
, and
Ray
,
A.
,
2014
, “
State Splitting and Merging in Probabilistic Finite State Automata for Signal Representation and Analysis
,”
Signal Process.
,
104
, pp.
105
–
119
.
Google Scholar
Crossref
Search ADS
 
7.
Goldstein
,
H.
,
Poole
,
C.
, and
Safko
,
J.
,
2002
,
Classical Mechanics
, 3rd ed.,
Addison Wesley
,
San Francisco, CA
.
8.
Sinha
,
S.
,
2009
,
Classical Mechanics
,
Alpha Science
,
Oxford
.
9.
Rao
,
S.
,
1990
,
Mechanical Vibrations
, 2nd ed.,
Addison-Wesley
,
Reading, MA
.
10.
Kelly
,
S.
,
2012
,
Mechanical Vibrations: Theory and Applications
,
Cengage Learning
,
Stamford, CT
.
11.
Mori
,
Y.
,
2017
,
Mechanical Vibrations: Applications to Equipment
,
Wiley
,
London
.
Google Scholar
Crossref
Search ADS
 
12.
Hoffman
,
K.
, and
Kunze
,
R.
,
1971
,
Linear Algebra
, 2nd ed.,
Springer-Verlag
,
Englewood Cliffs, NJ
.
13.
Gill
,
A.
,
1976
,
Applied Algebra for the Computer Sciences
,
Prentice-Hall PTR
,
Upper Saddle River, NJ
.
14.
Khalil
,
H.
,
2002
,
Nonlinear Systems
, 3rd ed., Chapter 11,
Prentice-Hall
,
Upper Saddle River, NJ
.
15.
Griffiths
,
D.
,
2005
,
Introduction to Quantum Mechanics
, 2nd ed.,
Pearson Prentice-Hall PTR
,
Upper Saddle River, NJ
.
16.
Griffiths
,
R.
,
2002
,
Consistent Quantum Theory
,
Cambridge University Press
,
Cambridge
.
17.
Bishop
,
C.
,
2007
,
Pattern Recognition and Machine Learning
,
Springer
,
New York
.
18.
Murphy
,
K.
,
2012
,
Machine Learning: A Probabilistic Perspective
,
The MIT Press
,
Cambridge, MA
.
Copyright © 2021 by ASME
You do not currently have access to this content.