State space construction for continuous time transfer function matrices via Nerode equivalence

State space realizations and monomial equivalence for convolutional codes

A matrix Euclidean algorithm induced by state space realization

Minimal state-space realization in linear system theory: an overview

On the realization of periodic functions

A simple technique is presented for the minimal state space realisation of product factorable two dimensional systems. The application of this method yields the matrices A, b, cT and the scalar d of the two dimensional Givone-Roesser state space model, having minimal dimension.

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 non-catastrophic), or alternatively if and only if it has the subsystem basis property (for any interval J, the generators in B whose span is in J is a basis for the subsystem CJ). The dimensions of the minimal state spaces and minimal transition spaces of C are simply the numbers of generators in a shortest basis B that are active at any given state or symbol time, respectively. A minimal linear realization for C in controller canonical form follows directly from a shortest basis for C, and a minimal linear realization for C in observer canonical form follows directly from a shortest basis for the orthogonal system C⊥. This approach seems conceptually simpler than that of classical minimal realization theory.

The problem of minimal state space realization of two-dimensional systems is considered. The approach followed is, initially, to derive a circuit realization of the given transfer function involving a minimum number of delay elements. To facilitate this realization, the transfer function is expanded into a continued fraction. Using this circuit realization, an algorithm is proposed which readily provides the matrices of the state space model. The simplicity in deriving this model is due to a novel state space model introduced, which is of cyclic structure. Several examples are presented to illustrate ihe proposed algorithm.

Minimal realizations play a key role in system analysis and synthesis. Among a variety of realizations they are characterised by a minimal dimension, which guarantees that no pole zero cancellations occur, a very important feature for the stability analysis, and are also of importance from the numerical point of view. This paper provides a simple method for minimal realization construction for multi-linear odd rational functions an important class from the practical point of view due to strong links to so-called reactance functions.

A directional partial realization problem

In this paper it is shown that a natural representation of a state space is given by the predictor space, the linear space spanned by the predictors when the system is driven by a Gaussian white noise input with unit covariance matrix. A minimal realization corresponds to a selection of a basis of this predictor space. Based on this interpretation, a unifying view of hitherto proposed algorithmically defined minimal realizations is developed. A natural minimal partial realization is also obtained with the aid of this interpretation.

An algorithm for inverting rational matrices

Let $A$ be a rational function. For any decomposition of $A$ into a composition of rational functions $A=U\circ V$ the rational function $\widetilde A=V\circ U$ is called an elementary transformation of $A$, and rational functions $A$ and $B$ are called equivalent if there exists a chain of elementary transformations between $A$ and $B$. This equivalence relation naturally appears in the complex dynamics as a part of the problem of describing of semiconjugate rational functions. In this paper we show that for a rational function $A$ its equivalence class $[A]$ contains infinitely many conjugacy classes if and only if $A$ is a flexible Latt\`es map. For flexible Latt\`es maps $L=L_j$ induced by the multiplication by 2 on elliptic curves with given $j$-invariant we provide a very precise description of $[ L]$. Namely, we show that any rational function equivalent to $ L_j$ necessarily has the form $ L_{j'}$ for some $j'\in \mathbb C$, and that the set of $j'\in \mathbb C$ such that $ L_{j'}\sim L_{j}$ coincides with the orbit of $j$ under the correspondence associated with the classical modular equation $\Phi_2(x,y)=0$.

<p>This paper demonstrates the pontential of verification based on state spaces reduced by equivalence relations. The basic observation is that quite often, some states of a system are similar, i.e., they induce similar behaviours. Similarity can be formally expressed by defining equivalence relations on the set of states and the set of actions of a system under consideration. A state space can be constructed, in which the nodes correspond to equivalence classes of states and the arcs correspond to equivalence classes of actions. Such a state space is often much smaller than the ordinary full state space, but does allow derivation of many verification results.</p><p>Other researches have taken advantage of the symmetries of systems, which induce a certain kind of equivalence. The contribution fo this paper is to show that a more general notion of equivalence is useful. As example, a communication protocol modelled in the formalism of Coloured Petri Nets is verified. Aided by a computer tool supporting state spaces with equivalence classes, significant reduction of state spaces are exhibited.</p><p><strong>Topics</strong>. State space reduction methods, equivalence vs. symmetry, High-level Petri Nets, communication protocols.</p>

Minimal realizations for discrete-time linear periodic systems

Hankel matrices generated by the Markov parameters of rational functions