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Research Papers

Riccati-Based Discretization for Nonlinear Continuous-Time Systems

[+] Author and Article Information
Triet Nguyen-Van

The Presidential Endowed Chair for Electric
Power Network Innovation by Digital Grid,
The University of Tokyo,
7-3-1 Hongo,
Bunkyo, Tokyo 113-8656, Japan;
Digital Control Laboratory,
Graduate School of Systems 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 Systems 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 Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 8, 2015; final manuscript received December 7, 2015; published online February 3, 2016. Assoc. Editor: Zdravko Terze.

J. Comput. Nonlinear Dynam 11(5), 051003 (Feb 03, 2016) (11 pages) Paper No: CND-15-1124; doi: 10.1115/1.4032382 History: Received May 08, 2015; Revised December 07, 2015

A discretization method is proposed for a rather general class of nonlinear continuous-time systems, which can have a piecewise-constant input, such as one under digital control via a zero-order-hold device. The resulting discrete-time model is expressed as a product of the integration-gain and the system function that governs the dynamics of the original continuous-time system. This is made possible with the use of the delta or Euler operator and makes comparisons of discrete and continuous time systems quite simple, since the difference between the two forms is concentrated into the integration-gain. This gain is determined in the paper by using the Riccati approximation of a certain gain condition that is imposed on the discretized system to be an exact model. The method is shown to produce a smaller error norm than one uses the linear approximation. Simulations are carried out for a Lotka–Volterra and an averaged van der Pol nonlinear systems to show the superior performance of the proposed model to ones known to be online computable, such as the forward-difference, Kahan's, and Mickens' methods. Insights obtained should be useful for developing digital control laws for nonlinear continuous-time systems, which is currently limited to the simplest forward-difference model.

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References

Figures

Grahic Jump Location
Fig. 1

Lotka–Volterra: State x1 of continuous-time, Mickens, Kahan, former, and proposed models for T=0.1  s and T=0.5 s. (a) Mickens model, (b) Kahan model, (c) former model, and (d) proposed model.

Grahic Jump Location
Fig. 2

Lotka–Volterra: State response x2 of continuous-time, Mickens, Kahan, former, and proposed models for T=0.1  s and T=0.5 s. (a) Mickens model, (b) Kahan model, (c) former model, and (d) proposed model.

Grahic Jump Location
Fig. 3

Lotka–Volterra: State response x1 of continuous-time, Mickens, Kahan, former, and proposed models for T=1.0  s and T=2.0  s. (a) Mickens model, (b) Kahan model, (c) former model, and (d) proposed model.

Grahic Jump Location
Fig. 4

Lotka–Volterra: State response x2 of continuous-time, Mickens, Kahan, former, and proposed models for T=1.0  s and T=2.0  s. (a) Mickens model, (b) Kahan model, (c) former model, and (d) proposed model.

Grahic Jump Location
Fig. 5

van der Pol: State response x1 of continuous-time, Mickens, forward-difference, former, and proposed models for T=0.2  s and T=2.0  s. (a) Mickens model, (b) forward-difference model, (c) former model, and (d) proposed model.

Grahic Jump Location
Fig. 6

van der Pol: State response x2 of continuous-time, Mickens, forward-difference, former, and proposed models for T=0.2  s and T=2.0  s. (a) Mickens model, (b) forward-difference model, (c) former model, and (d) proposed model.

Grahic Jump Location
Fig. 7

van der Pol: State-responses x1 and x2 of continuous-time, former, and proposed models for T=5.0  s. (a1) Former model x1, (a2) former model x2, (b1) proposed model x1, and (b2) proposed model x2.

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