Smooth/impulsive linear systems: Axiomatic description

(Singular) state models and (singular) LTID systems

“…The equivalence theorem (see Theorem 2 in Lomadze (2009) and Corollary 1 in Lomadze & Mahmood (2009)) states that for two AR models (A 1 , B 1 ) and (A 2 , B 2 ) to determine the same behaviour, it is necessary and sufficient that A 2 = UA 1 for some unimodular polynomial matrix U, and B 2 = VB 1 for some biproper rational matrix V.…”

“…Closing the section, we want to formulate an ARrepresentation theorem (see Theorem 2 in Lomadze & Mahmood (2009)), though we shall not use it in the sequel. The theorem states that for a subspace B ⊆ M q to be an LTID system, it is necessary and sufficient that B satisfy the following three conditions:…”

“…Recall that an AR model is called reduced, if its transition matrix is a diagonal matrix with powers of s on the diagonal (see Lomadze & Mahmood (2009)). Every AR model can be brought into a reduced form.…”

“…These rank conditions are weaker than the above-mentioned rank conditions on the scalar matrices G, [ K M ], G H and K. These are models that may have singularity at infinity. By LTID systems, we shall understand systems that together with C ∞ trajectories may contain impulsive trajectories and that were introduced in Lomadze (2009)and Lomadze & Mahmood (2009).…”

Impulse Controllability: From Descriptor Systems to Higher Order DAEs

“…In this paper we extend these results to a larger space of solutions, now allowing impulses and their derivatives too. Regular feedback interconnection and smooth/impulsive behaviors have been studied respectively in [12] and [13] using techniques from homological 1 algebra.…”