Orthonormal high-level canonical PWL functions with applications to model reduction

Abstract

This paper is the natural sequel to [1]. An inner product is defined for the linear vector space [S] of all the piecewise linear (PWL) continuous mappings defined over a rectangular compact set S using a simplicial partition This permits us to assure that [S] is a Hilbert space.Then, the notion of orthogonality is introduced and orthonormal bases of PWL functions are formulated. A relevant consequence of the approach is that the problem of function approximation can be translated to the more studied field of approximation in Hilbert spaces of finite dimension. As will be shown, this powerful theoretical framework can be used to generate an optimal scheme for model reduction.

A method for the approximate synthesis of cellular non‐linear networks—Part 1: Circuit definition

“…In our case, this deÿnition-which allows one to consider the linear space PWL[S H ] as an N -dimensional Hilbert space [15]-involves the evaluations of the function f at the N vertices resulting from the simplicial partition of S, and it would turn out to be particularly useful if we were interested in ÿnding an interpolation of the samples {(v j ; f(v j )); j = 1; : : : ; N }.…”

Towards analog implementations of PWL two-dimensional non-linear functions

“…The latter problem has recently been faced [2][3][4] by resorting to the piecewise-linear (PWL) approximation technique proposed by Julià an et al [5,6]. By means of this technique, each constitutive equation y = f(x) for a non-linear resistive element of a circuit (where y is a generic dependent descriptive variable and x is the vector of independent descriptive variables) is replaced with a proper canonical PWL expression.…”

“…The N -dimensional space PWL[S H ] associated with the class of continuous PWL functions f PWL is deÿned by the domain S, its simplicial partition H , and a proper inner product (see [3,5,6] for details). Each function belonging to PWL[S H ] can be represented as the sum of N basis functions (arbitrarily organized into a vector), weighted by an N -length coe cient vector c, which determines the shape of f PWL uniquely.…”

Robust explicit model predictive control via regular piecewise-affine approximation

“…After defining the sets S i , it is possible to introduce the related PWAS function. We choose to define each component of u(x), namely u j (x), j = 1, , m, as the weighted sum of N v linearly independent α-basis functions (Julián et al 2000). Every element of the j-th basis is affine over each simplex and satisfies…”

“…proposed, based on a special class of functions, hereafter referred to as piecewise-affine simplicial (PWAS) functions, proposed by Julián et al (2000). The choice of PWAS functions leads to a regular partition, so that the point-location problem is solved with a negligible effort compared to explicit MPC defined on generic PWA partitions (the reader is also referred to Oliveri et al (2012) for the practical implementation).…”