F Rikus Eising

Between controllable and uncontrollable

The distance between a system ( A, B ) and the set of uncontrollable systems is the minimum of the smallest singular value of [λ I − A, B ]with respect to λ.

Realization and stabilization of 2-d systems

During recent years several state-space models concerning discrete 2-D systems (systems with two time parameters) have appeared in the literature. These are used, for example, in image processing. To these models are attached the names of Attasi [1], Fornasini-Marchesini [2], Givone-Roesser [3]. In this paper it is shown that all these models are special cases of a new model which is a straightforward generalization of the 1-D case. Under certain conditions the existence of a stabilizing feedback is shown. In the last part connections with [4] are made.

State-space realization and inversion of 2-D systems

In this paper the state-space realization results of [1] for causal 2-D systems are generalized to a much larger class of 2-D systems. We introduce a generalized notion of a state-space realization for which the state can still be recursively evaluated. The results include a realization method for a class of NSHP filters. In the second part we introduce inverse 2-D systems with inherent delay. Some results concerning existence of an inverse with inherent delay for a 2-D system will be given. It will be shown that, in general, a causal 2-D system cannot have a causal inverse (with inherent delay). Furthermore, it will be shown that a causal 2-D system always has an inverse with inherent delay in the larger class of 2-D systems mentioned above.

Pole assignment for systems over rings

A very simple proof of the pole assignment theorem for systems over a principal ideal domain (and other rings) is given. Furthermore, an algorithm is presented. Extensions are also indicated.

Controllability and observability of 2-D systems

In this note a necessary and sufficient condition for modal controllability (modal observability) of a 2-D system as defined in [3], is obtained in terms of controllability (observability) of a system as is derived in [1]. Furthermore, it is shown that modal controllability (modal observability) is a generic property.

Realization algorithms for systems over a principal ideal domain

In this paper realization algorithms for systems over a principal ideal domain are described. This is done using the Smith form or a modified Hermite form for matrices over a principal ideal domain. It is shown that Hos algorithm and an algorithm due to Zeiger can be generalized to the ring case. Also a recursive realization algorithm, including some results concerning the partial realization problem, is presented. Applications to systems over the integers, delay differential systems and two-dimensional systems are discussed.

Exact model matching of 2-D systems

In this note the 2-D model mathing problem is considered. The purpose of this note is not to present a complete solution to this problem but rather to indicate some possible solutions. By using a generalized dynamic cover (GDC) [5] a first level realization, [1] is constructed for the desired compensator.

Separability of 2-D transfer matrices

In this note it is shown that every proper 2-D transfer matrix is feedback equivalent to a separable proper 2-D transfer matrix.

Pole assignment, a new proof and algorithm

In this paper a new proof of the pole assignment theorem is given. This proof is a very straightforward one. It is not based on canonical forms and also the reduction to the single input case (Heymanns lemma) is not used. Furthermore, an algorithm is given which allows to take into account numerical aspects with respect to the feedback construction for the multi-input case. Furthermore the non-uniqueness of the feedback matrix in the multi-input case may be exploited in order to reduce the gains.

Low-order realizations for 2-D transfer functions

In this note, a realization result for casual 2-D transfer functions with separable numerator or separable denominator is generalized. A possible dimension of the local state space will be determined and system matrices will be given.