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Research Papers

Discretization of Nonautonomous Nonlinear Systems Based on Continualization of an Exact Discrete-Time Model

[+] Author and Article Information
Triet Nguyen-Van

Digital Control Laboratory,
Graduate School of System
and Information Engineering,
University of Tsukuba,
1-1-1 Tennoudai,
Tsukuba 305-8573, Japan
e-mail: nvtriet@digicon-lab.esys.tsukuba.ac.jp

Noriyuki Hori

Digital Control Laboratory,
Graduate School of System
and Information Engineering,
University of Tsukuba,
1-1-1 Tennoudai,
Tsukuba 305-8573, Japan
e-mail: hori@iit.tsukuba.ac.jp

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received April 9, 2013; final manuscript received October 12, 2013; published online November 20, 2013. Assoc. Editor: Hashem Ashrafiuon.

J. Dyn. Sys., Meas., Control 136(2), 021004 (Nov 20, 2013) (9 pages) Paper No: DS-13-1152; doi: 10.1115/1.4025711 History: Received April 09, 2013; Revised October 12, 2013

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.

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References

Moon, F. C., 1992, Chaotic and Fractal Dynamics, Wiley, New York.
Strogatz, S. H., 1994, Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MA.
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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. [CrossRef]
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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.
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Figures

Grahic Jump Location
Fig. 1

(a) Self-sustained oscillation (C1(s)): Phase plane and time response of the continuous-time, the forward-difference, and the proposed models, for x0 = −1, y0 = −1.5, ε = 1.5, A = 1, ω = 2, T = 0.1. (b) Quasi-periodic oscillation (C(1(b)): Phase plane and time response of the continuous-time, the forward-difference, and the proposed models, for x0 = −1, y0 = −1.5, ε = 1.5, A = 3, ω = 2, T = 0.1. (c) Fundamental oscillation (C1(c)): Phase plane and time response of the continuous-time, the forward-difference, and the proposed models, for x0 = −1, y0 = −1.5, ε = 1.5, A = 3, ω = 3, T = 0.1. (d) Harmonic oscillation (C1(d)): Phase plane and time response of the continuous-time, the forward-difference, and the proposed models, for x0 = −1, y0 = −1.5, ε = 1.5, A = 8, ω = 3, T = 0.1.

Grahic Jump Location
Fig. 2

(a) (C2(a)): Phase plane and time response of the continuous-time, the forward-difference, and the proposed models, for x0 = −1, y0 = −1.5, ε = 3, A = 1, ω = 2, T = 0.1. (b) (C2(b)): Phase plane and time response of the continuous-time and the proposed models, for x0 = −1, y0 = −1.5, ε = 4, A = 1, ω = 2, T = 0.1. (c) (C2(c)): Phase plane and time response of the continuous-time, and the proposed models, for x0 = −1, y0 = −1.5, ε = 5, A = 1, ω = 2, T = 0.1.

Grahic Jump Location
Fig. 3

(a) (C3(a)): Phase plane and time response of the continuous-time, the forward-difference, and the proposed models, for x0 = −1, y0 = −1.5, ε = 1.5, A = 1, ω = 2, T = 0.2. (b) (C3(a)): Phase plane and time response of the continuous-time, and the proposed models, for x0 = −1, y0 = −1.5, ε = 1.5, A = 1, ω = 2, T = 0.3.

Grahic Jump Location
Fig. 4

(a) (C3(b)): Phase plane and time response of the continuous-time, 4th order Runge-Kutta, and the proposed models, for x0 = −1, y0 = −1.5, ε = 1.5, A = 1, ω = 2, T = 0.3. (b) (C3(c)): Phase plane and time response of the continuous-time, 4th order Runge-Kutta, and the proposed models, for x0 = −1, y0 = −1.5, ε = 1.5, A = 1, ω = 2, T = 0.5.

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