Abstract
ABSTRACTA lower bound error for free-run simulation of the polynomial NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous input) is introduced. The ultimate goal of the polynomial NARMAX is to predict an arbitrary number of steps ahead. Free-run simulation is also used to validate the model. Although free-run simulation of the polynomial NARMAX is essential, little attention has been given to the error propagation to round off in digital computers. Our procedure is based on the comparison of two pseudo-orbits produced from two mathematical equivalent models, but different from the point of view of floating point representation. We apply successfully our technique for three identified models of the systems: sine map, Chua's circuit and Duffing–Ueda oscillator. This technique may be used to reject a simulation, if a required precision is greater than the lower bound error, increasing the numerical reliability in free-run simulation of the polynomial NARMAX.
Highlights
The polynomial NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous input) is widely applied to represent all sorts of systems (Billings, 2013; Chen & Billings, 1989)
The ultimate goal of the polynomial NARMAX is to predict an arbitrary number of steps ahead
Free-run simulation of the polynomial NARMAX is essential, little attention has been given to the error propagation to round off in digital computers
Summary
The polynomial NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous input) is widely applied to represent all sorts of systems (Billings, 2013; Chen & Billings, 1989). There are many works based on deterministic or stochastic tools that provide some confidence in simulation of recursive functions Many of those techniques are not practical for nonlinear recursive functions with many terms, such as the polynomial NARMAX, or for non-expert users, who like to measure or at least estimate the error. To deal with the numerical questions in nonlinear dynamic systems, many researchers have used the shadowing property (Chow & Palmer, 1992, 1991; Hammel et al, 1987; Sauer, Grebogi, & Yorke, 1997) This property ensures that a pseudo-orbit (a sequence of points calculated from a recursive function map) is a homeomorphism that remains close to the real orbit (Faranda et al, 2012). It is clear that if we take x ∈ Rn, F of Equation (1) can be seen as a specific case of f in Equation (2)
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