We give a survey of the results in connection with the minimal state space realization problem for linear time-invariant systems. We start with a brief historical overview and a short introduction to linear system theory. Next we present some of the basic algorithms for the reduction of non-minimal state space realizations and for the minimal state space realization of infinite or finite sequen...

This paper examines the realization of acausal weighting patterns with two-point boundary-value descriptor systems (TPBVDSs). Attention is restricted to the subclass of TPBVDSs that are stationary, so that their input-output weighting pattern is shift-invariant, and extendible, i.e., their weighting pattern can be extended outwards indefinitely. Then, given an infinite acausal shift-invariant w...

A standard result of linear system theory states that a SISO rational n-th order transfer function always has a n-th order realization. In some applications one is interested in having a realization with nonnegative entries (i.e. a positive system). In this paper we give a contribution to the discrete-time realization problem by providing explicit necessary and suucient conditions for a third o...

Smoothing algorithms of various kinds have been around for several decades. However, some basic issues regarding the dynamical structure and the minimal dimension of the steady-state algorithm are still poorly understood. It seems fair to say that the subject has not yet reached a definitive form. In this paper, we derive a realization of minimal dimension of the optimal smoother for a signal a...

The minimal realization problem in the max-plus algebra, " Chapter 32 in Open Problems in Mathematical Systems and Control Theory

Given a discrete-time linear system C, a shortest basis for C is a set of linearly independent generators for C with the least possible lengths. A basis B is a shortest basis if and only if it has the predictable span property (i.e., has the predictable delay and degree properties, and is noncatastrophic), or alternatively if and only if it has the subsystem basis property (for any interval J ,...

A solution is presented for the problem of realizing a discrete-time LTI state-space model of minimal McMillan degree such that its :rst N expansion coe;cients in terms of generalized orthonormal basis match a given sequence. The basis considered, also known as the Hambo basis, can be viewed as a generalization of the more familiar Laguerre and two-parameter Kautz constructions, allowing genera...