Piecewise-Linear Identification of Nonlinear Dynamical Systems in View of Their Circuit Implementations

Abstract

We address here an aspect of the problem concerning circuit implementations of nonlinear dynamical systems that depend on control parameters. In particular, the problem of the identification of such systems is addressed in two steps. The first step uses the state-space reconstruction (through time delay reconstruction associated with principal component analysis) on the basis of scalar time series measured in the systems to be identified. The second step deals with the approximation of the flow in the reconstructed space (through a piecewise-linear approximation technique). The proposed method is first validated with two examples concerning known systems and then applied successfully to two realistic cases.

Piecewise affine direct virtual sensors with Reduced Complexity

“…The distribution of the data in the time interval [0, T ], is also important, since all the main dynamic properties of the system (including transient responses) must lie within this time window (see, e.g., [17]). …”

Digital architectures implementing piecewise-affine functions: An overview

“…The need for circuits realizing a PWA input/output relationship arises in many engineering problems, like adaptive [2] and fuzzy control [15] and nonlinear dynamical systems implementation [16]. Indeed, this kind of circuits may have many practical applications, mainly in the field of real-time embedded systems or whenever it is not feasible or convenient resorting to computers or other general-purpose devices, such as DSP boards.…”

Towards Accurate PWL Approximations of Parameter-Dependent Nonlinear Dynamical Systems With Equilibria and Limit Cycles

“…To attain this objective, we define the following functional: (8) where is a positive constant coefficient.…”

“…Generally speaking, from a modeling point of view (i.e., in terms of number of approximation parameters required to obtain a reasonably accurate model), the method is not particularly efficient, as compared with other approximation methods like splines, neural networks or other kernel-based methods, but its main advantage lies in its quite direct circuit implementation [6], [7], which can be particularly interesting whenever we aim to emulate the behaviors of dynamical systems made up of a large number of elementary units, e.g., neurons [1], [8] or we need dedicated hardware for real-time, smallsize and/or low-power applications (e.g., in smart dust or microcontrol systems). Another advantage of such an approach, not shared, for instance, by the wavelets and prewavelets PWL multi-grid approximations [9], [10], is the simplicity of its theoretical formulation, which allows an easy implementation of a multi-grid resolution approach to functions defined over domains of any (at least in principle) dimensionality [11].…”