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.

References

References
1.
Hartley
,
T. T.
, and
Beale
,
G. O.
,
1994
,
Chicatelli SP. Digital Simulation of Dynamic Systems—A Control Theory Approach
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
2.
Mickens
,
R. E.
,
2000
,
Applications of Nonstandard Finite Difference Schemes
,
World Scientific
,
Singapore
.
3.
Yuz
,
J.
, and
Goodwin
,
G. C.
,
2014
,
Sampled-Data Models for Linear and Nonlinear Systems
,
Springer
,
London
.
4.
Nesic
,
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
(
7
), pp.
1103
1122
.
5.
Triet
,
N. V.
, and
Hori
,
N.
,
2013
, “
New Class of Discrete-Time Models for Non-Linear Systems Through Discretisation of Integration Gains
,”
IET Control Theory Appl.
,
7
(1), pp.
80
89
.
6.
Middleton
,
R. H.
, and
Goodwin
,
G. C.
,
1990
,
Digital Control and Estimation—A Unified Approach
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
7.
Folland
,
G. B.
,
1999
,
Real Analysis: Modern Techniques and Their Applications
,
Wiley
,
New York
.
8.
Borwein
,
P.
,
Erdelyi
,
T.
,
1995
,
Polynomials and Polynomial Inequalities
(Graduate Texts in Mathematics),
Springer
,
New York
.
9.
Kittipeerachon
,
K.
,
Hori
,
N.
, and
Tomita
,
Y.
,
2009
, “
Exact Discretization of a Matrix Differential Riccati Equation With Constant Coefficients
,”
IEEE Trans. Autom. Control
,
54
(
5
), pp.
1065
1068
.
10.
Murray
,
J. D.
,
2002
,
Mathematical Biology I: An Introduction
,
Springer
,
New York
.
11.
Kahan
,
W.
,
1993
, “
Unconventional Numerical Methods for Trajectory Calculations (Lecture Notes)
,” CS Division, Department of EECS, University of California at Berkeley, Berkeley, CA.
12.
Guckenheimer
,
J.
, and
Holmes
,
P.
,
1983
,
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
,
Springer
,
New York
.
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