Minimal Realization Theory Research Articles

A General Categorical Framework of Minimal Realization Theory for a Fuzzy Multiset Language

This paper is to study the minimal realization theory for a fuzzy multiset language in the framework of category theory, which has already provided the tools and techniques for the advancement of several features of theoretical computer science. Specifically, by using the well-known categorical concepts, it is shown herein that there is a minimal realization (called the Nerode realization) for each fuzzy multiset language, and all minimal realizations for a given fuzzy multiset language are isomorphic to it.

A categorical approach to minimal realization for a fuzzy language

In this study, we provide a general theory of minimal realization for a given fuzzy language in a category theory setting. We introduce categorical characterizations of a run map, reachability map, and observability map for a crisp deterministic fuzzy automaton. Finally, we show that all minimal realizations for a given fuzzy language are isomorphic to the Nerode realization for the same case.

Minimal Realizations of Linear Systems: The “Shortest Basis” Approach

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.

A notion of possible controllability for uncertain linear systems with structured uncertainty

This paper introduces a notion of possible controllability for a class of uncertain linear systems with structured uncertainty described by averaged integral quadratic constraints. This notion relates to the question of when a state is controllable for some possible value of the uncertainty. The notion of possible controllability is motivated by a desire to extend the theory of minimal realization for linear time invariant systems to the case of uncertain systems with structured uncertainty.

On the Applicability of the Ho-Kalman Minimal Realization Theory

The reduced-order model of a time-invariant linear dynamical system, excited by a force of an impulsive type, may be readily obtained using the Ho-Kalman minimal-realization algorithm [1]. The method is based upon a particular factorization of the Hankel matrix in the Markovian representation of the discrete-time process. For stochastic systems, the applicability of the theory has been demonstrated by Akaike [2] on the assumption that the excitation is a zero-mean white noise of a gaussian type. Some of the most widely known output-only identification methods, such as Eigensystem Realization Algorithm (ERA), Canonical Variate Analysis (CVA), and Balanced Realization (BR)) are based upon the above-mentioned work, with the aid of a robust factorization technique, such as Singular-Value Decomposition (SVD). Notwithstanding the growing popularity of the above methods, some aspects of their applicability are not yet understood. Two points are of particular interest: the first regards the applicability of the theory in highly damped systems; and the second regards its applicability to systems driven by excitations different from the one hypothesized. The aim of the present work is to define a reliable test on the hypotheses. Some numerical and experimental results are presented.

Geometric framework and minimal realizations of nonlinear systems on fibre bundle

The definition of nonlinear control systems on fibre bundles proposed by Brockett and Willems is incomplete from the mathematical view. A new geometric framework is proposed and a minimal realization theory is developed for nonlinear control systems on fibre bundles which is elaborated as a natural generalization of Sussmann's theory and differs essentially from Van der Schaft's approach. Limitations of realization theory given by Van der Schaft are also discussed.

A remark on bilinear systems and moduli spaces of instantons

Explicit equations are given for the moduli space of framed instantons as a quasi-affine variety, based on the representation theory of noncommutative power series, or equivalently, the minimal realization theory of bilinear systems.

ON THE STRUCTURE OF DIRECTED GRAPHS WITH APPLICATIONS: A Rapprochement with Systems Theory Part II

In Part I, certain fundamental structural properties of directed graphs (digraphs) are explored using a unified systemtheoretic approach. In the spirit of general systems theory, a digraph is modelled as a dynamic system with multiple inputs and outputs. The concept of controllability, observability, and the elegant theory of minimal realization, duality, and Kalman's decomposition in linear systems have all found their positions in digraphs. New computation algorithms are derived. In Part II (Vol, 8, No. 4 of this journal) of this paper, the structure of composite digraphs, and subgraph interaction/decomposition are studied. Applications to computer systems, Markov chains, and to dynamic systems and control are presented.

Principal component analysis in linear systems: Controllability, observability, and model reduction

Kalman's minimal realization theory involves geometric objects (controllable, unobservable subspaces) which are subject to structural instability. Specifically, arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces. This situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization. Structural instability associated with geometric theories is not unique to control; it arises in the theory of linear equations as well. In this setting, the computational problems have been studied for decades and excellent tools have been developed for coping with the situation. One of the main goals of this paper is to call attention to principal component analysis (Hotelling, 1933), and an algorithm (Golub and Reinsch, 1970) for computing the singular value decompositon of a matrix. Together they form a powerful tool for coping with structural instability in dynamic systems. As developed in this paper, principal component analysis is a technique for analyzing signals. (Singular value decomposition provides the computational machinery.) For this reason, Kalman's minimal realization theory is recast in terms of responses to injected signals. Application of the signal analysis to controllability and observability leads to a coordinate system in which the "internally balanced" model has special properties. For asymptotically stable systems, this yields working approximations of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X_{c}, X_{\bar{o}}</tex> , the controllable and unobservable subspaces. It is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces.

Generalized Hankel Matrices and System Realization

We define the Hankel matrix of an adjoint system. Adjoint systems include linear and bilinear systems, automata, and group systems in both the time-varying and time-invariant cases. Our definition of the Hankel matrix unifies the familiar $H_i^j = CA^{i + j} B$ of linear system theory (e.g. R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical Systems Theory, McGraw-Hill, New York, 1969) with the bilinear Hankel matrix of A. Isidori (Direct construction of minimal bilinear realizations from nonlinear input-output maps, IEEE Trans. Automatic Control, AC-18 (1973), pp. 626–631), T. J. Tarn and S. Nonoyama (Realization of discrete-time internally bilinear systems, Proc. IEEE Conf. Decision and Control, 76CH 1150-2CS (1976), pp. 125–133) and the Hankel matrix of M. Fliers (Matrices de Hankel, J. Math. Pure Appl., 53 (1974), pp. 197–224). The time-varying case is subsumed by regarding a time-varying system as a time-invariant system in a sequence category as in M. A. Arbib and E. G. Manes (Time-varying systems, SIAM J. Control, 13 (1975), pp. 1252–1270). For minimal realization theory and duality theory in the framework of this paper see B. D. O. Anderson, M. A. Arbib and E. G. Manes (Foundations of system theory: Finitary and infinitary conditions, Lecture Notes in Economics and Mathematical Systems, 115, Springer-Verlag, New York, 1976), M. A. Arbib and E. G. Manes (Adjoint machines, state-behavior machines and duality, J. Pure Appl. Algebra, 6 (1975), pp. 313–344) and S. J. Hegner (Duality theory fordiscrete-time linear systems, J. Comp. System Sci., 17 (1978), pp. 116-143). However, we lean much less heavily on category theory than in our earlier works on realization. We introduce “adjoint correspondences” as the key algebraic ingredient in generalizing familiar linear system results’ to the nonlinear case. For example, the linear realizability criterion $H_{i + 1}^j = H_i^{i + 1} $ does not make sense in the nonlinear setting; the precise condition needed is that “$H_{i + 1}^j $ and $H_j^{i + 1} $ correspond under adjointness.” We provide a realizability theorem characterizing when a matrix $H_i^j $ can be the Hankel matrix of a system, and offer partial realization and canonical realization theorems which associate systems with finite blocks of a Hankel matrix. We provide a general theory of “dimension in a category,” and relate it to system realization via a simple recursion principle.

Nonlinear controllability and observability

The properties of controllability, observability, and the theory of minimal realization for linear systems are well-understood and have been very useful in analyzing such systems. This paper deals with analogous questions for nonlinear systems.

A minimal canonical form realization for a multivariable econometric system

A minimal realization theory for a multivariable system is developed based on the Hankel matrices, which is closely related to the method given by Silverman (1971). The procedure provided here yields a canonical form dynamical equation. The results obtained are applied to a multivariable econometric system. The proposition of realizability and existence of a minimal realization is demonstrated, which contains both proper and strictly proper systems. Thus the correspondence between minimal realization and canonical form is clarified.

Minimal realizations and canonical forms for bilinear systems

The minimal realization theory for input–output maps that arise from finite- dimensional, continuous time, bilinear systems is discussed. It is shown that an observed bilinear system (i.e. a bilinear system together with an observation functional, but without a fixed initial state) is completely determined by its input–output correspondence, i.e. its multivalued input–output map. A precise formulation and proof of the result are given that continuous canonical forms for bilinear systems do not exist. This is done by adapting to the bilinear case an idea due to Hazewinkel and Kalman for the case of linear systems. In addition, the proof presented here has the advantage of not involving any algebraic geometry, which makes it considerably simpler than the original proof of the linear systems result.

Adjoint machines, state-behavior machines, and duality

A process X: K → K is output if Dyn( X)→ K has a right adjoint; state-behavior if Dyn( X)→ X has both left and right adjoints; and adjoint if X has a right adjoint and K has countable coproducts. Output processes provide the proper setting for a general theory of state observability. We give a minimal realization theory using image factorization of a total response map. We give an adjointness theory for state-behavior machines and a duality theory for adjoint machines which clarifies classical linear system duality and yields an improved duality for nondeterministic automata. Adjoint machines (machines with adjoint input processes) provide the first integration of classical sequential machines (the only state-behavior machines in the category, Set, of sets), metric machines, topological machines, linear systems, nondeterministic automata and Boolean machines. There exist state-behavior machines which are not adjoint (but not in Set).

Fuzzy machines in a category

“Fuzzy theories” and “distributive laws” are used to define “fuzzy systems” in an arbitrary category. The resulting minimal realization theory provides new insights even in classical cases (so that, for non-deterministic sequential machines, the minimal realization problem is reformulated in terms of the structure of join-irreducibles in finite lattices). The definition of “fuzzy theory” is of independent interest and meshes well with philosophical aspects of fuzzy set theory.

Discrete-time machines in closed monoidal categories. I

This paper develops a minimal realization theory for discrete-time machines with structure in a suitable closed monoidal category. By specifying the category a number of applications arise, most of them new. Minimal realization is stated as an adjunction between an input-output behavior functor and a realization functor. The very existence of an adjunction yields several new structural results on minimal realization. As preliminaries, certain aspects of categorical algebra are reviewed, and a theory of discrete-time transition systems is developed. The concept of an X -module and an initial object theorem are especially important. A number of examples of suitable categories is given, but discussion of the resulting machine theories is deferred to a subsequent paper.

Stochastic theory of minimal realization

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.