Abstract
We consider the discretization of continuous-time nonlinear systems described by normal forms. In particular, we consider the case when the input to the system is generated by a B-spline hold device to obtain an approximate discrete-time model. It is shown that the corresponding sampled-data model and its accuracy (measured in terms of the local truncation error) depend on the smoothness of the input and on the applied integration strategy, namely, the truncated Taylor series expansion. Moreover, the sampling zero dynamics of the discrete-time model are asymptotically characterized as the sampling period goes to zero, and it is shown that these zero dynamics converge to the asymptotic sampling zeros of the linear case.
Highlights
The theory of normal forms has been studied for continuous and discrete-time nonlinear systems [1]–[4]
We show that the numerical integration method, namely a truncated Taylor series expansion can exploit the B-spline assumption on the
An approximate sampled-data model for a class of nonlinear systems affine in the input has been proposed based on normal forms, B-spline functions and the truncated Taylor series expansion
Summary
The theory of normal forms has been studied for continuous and discrete-time nonlinear systems [1]–[4]. When discretizing a continuous-time system, the corresponding sampled-data model includes extra zeros (linear case) or zero dynamics (nonlinear case), that depend on the hold device used to generate the system input [12]. In [8], a truncated Taylor series expansion was proposed to discretize nonlinear systems, explicitly characterizing the sampling zero dynamics. Based on the latter result, [7], [11] proposed a more accurate sampleddata model for relative degree two, in order to guarantee the stability of the resulting model. DISCRETE-TIME SYSTEMS In the first part of this section we present important concepts that will be used later in Section V to develop the main result of the paper, namely, an approximate sampled-data model for nonlinear systems. In the subsection, we present the sampled-data model for an n-th order integrator
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