An innovative approach is proposed for generating discrete-time models of a class of continuous-time, nonautonomous, and nonlinear systems. By continualizing a given discrete-time system, sufficient conditions are presented for it to be an exact model of a continuous-time system for any sampling periods. This condition can be solved exactly for linear and certain nonlinear systems, in which case exact discrete-time models can be found. A new model is proposed by approximately solving this condition, which can always be found as long as a Jacobian matrix of the nonlinear system exists. As an example of the proposed method, a van der Pol oscillator driven by a forcing sinusoidal function is discretized and simulated under various conditions, which show that the proposed model tends to retain such key features as limit cycles and space-filling oscillations even for large sampling periods, and out-performs the forward difference model, which is a well-known, widely-used, and on-line computable model.

References

References
1.
Moon
,
F. C.
,
1992
,
Chaotic and Fractal Dynamics
,
Wiley
,
New York
.
2.
Strogatz
,
S. H.
,
1994
,
Nonlinear Dynamics and Chaos
,
Addison-Wesley
,
Reading, MA
.
3.
Lambert
,
J. D.
,
1973
,
Computational Methods in Ordinary Differential Equations
,
John Wiley & Sons
,
New York
.
4.
Hartley
,
T. T.
,
Beale
,
G. O.
, and
Chicatelli
,
S. P.
,
1994
,
Digital Simulation of Dynamic Systems—A Control Theory Approach
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
5.
Slotine
,
J.-J. E.
, and
Li
,
W.
,
1991
,
Applied Nonlinear Control
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
6.
Yuz
,
J. I.
, and
Goodwin
,
G. C.
,
2005
, “
On Sampled-Data Models for Nonlinear Systems
,”
IEEE Trans. Autom. Control
,
50
, pp.
1477
1489
.10.1109/TAC.2005.856640
7.
Nešić
,
D.
, and
Teel
,
A. R.
,
2004
, “
A Framework for Stabilization of Nonlinear Sampled-Data Systems Based on Their Approximate Discrete-Time Models
,”
IEEE Trans. Autom. Control
,
49
, pp.
1103
1122
.10.1109/TAC.2004.831175
8.
Hirota
,
R.
,
2000
,
Lectures on Difference Equations—From Continuous to Discrete Domains
,
Science Publishing
,
Tokyo, in Japanese
.
9.
Mickens
,
R. E.
,
1994
,
Nonstandard Finite Difference Models of Differential Equations
,
World Scientific
,
Singapore
.
10.
Middleton
,
R. H.
, and
Goodwin
,
G. C.
,
1990
,
Digital Control and Estimation—A Unified Approach
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
11.
Lakshmikantham
,
V.
, and
Trigiante
,
D.
,
1988
,
Theory of Difference Equations: Numerical Methods and Applications
,
Academic Press
,
San Diego, CA
.
12.
Mori
,
T.
,
Nikiforuk
,
P. N.
,
Gupta
,
M. M.
, and
Hori
,
N.
,
1989
, “
A Class of Discrete-Time Models for a Continuous-Time System
,”
IEE Proc.-D: Control Theory Appl.
,
136
, pp.
79
83
.10.1049/ip-d.1989.0012
13.
Zwillinger
,
D.
,
1997
,
Handbook of Differential Equations
,
3rd ed.
,
Academic Press
,
San Diego, CA
.
14.
Shiobara
,
H.
, and
Hori
,
N.
,
2011
, “
Exact Time-Discretization of Differential Riccati Equations With Variable Coefficients
,”
Proceedings of 13th IASTED International Conference on Control and Applications
,
C. W.
de Silva
, ed.,
Vancouver
, pp.
90
95
.
15.
Morimoto
,
Y.
,
2000
, “
Frequency Pulling of Quasi-Periodic Oscillation in Forced van der Pol Oscillator
,”
IEICE Trans. Fundamentals
,
E83-A7
, pp.
1479
1482
.
You do not currently have access to this content.