Equivalence of AR‐representations in the light of the impulsive‐smooth behaviour

Abstract

SUMMARYThe paper presents a new notion of equivalence of non-regular AR-representations, based on the coincidence of the impulsive-smooth behaviours of the underlying systems. The proposed equivalence is characterized by a special case of the usual unimodular equivalence and a restriction of the matrix transformation of full equivalence (Int.

Impulse Controllability: From Descriptor Systems to Higher Order DAEs

“…2 More precisely, similar to [4, equation (1.3)] and [15,Equation (2)], in (2) above, p i is δ (i) , the LHS R(p)w, is interpreted as convolution of R(p) and w, while in the RHS of (2), for convenience of matrix equations, the "coefficients" of δ (i) are on the right of δ (i) . It is important to note that, when dealing with initial conditions and half-line solutions, R(d/dt)w = 0 is not a "homogeneous" system of differential equations: this is central to this paper.…”

“…The following proposition is about a necessary and sufficient condition for the equality of two impulsive-smooth behaviors termed as the so-called fundamental equivalence in [15]. For a polynomial matrix P (ξ), let δ M (P ) denote the number of poles at infinity of P (ξ).…”

“…The initial condition for the system is given by Z R v and the initial condition space is the image of Z R denoted as Im Z R . We assume rank R[ξ] (R(ξ)) = n and hence the system in ( 2) is solvable for all initial conditions: see [15,Theorem 5 and Corollary 6].…”

“…In order to deal with this, we will consider (1) with impulsive-smooth distributions [7], [29] as the underlying function class: linear combinations of distributions with support at zero and smooth functions on R + . The "impulsive-smooth" behaviors are of interest in switched or multimode systems: see [5], [14], [15], for example. Interconnection of systems has been studied by [24], [28], and conditions for regular interconnection and regular feedback have been derived in the context of smooth behaviors.…”

Equivalence conditions for behaviors and the Kronecker canonical form

“…Of course, there is quite some literature on the classification of singular systems or autoregressive systems that is indirectly related to our paper; see, e.g. Pugh et al [21,22]. However, the classification problems treated by these authors are different to ours and their results will therefore not be discussed here any further.…”