Theory Of Algebraic Systems Research Articles

On behavioral model identification

This paper presents theory and algorithms for system identification suitable for the framework of behavioral system theory. An algorithm based on algebraic system theory and realization theory (Ho-Kalman realization) in a subspace model identification framework is presented. The novel approach taken consists of the formulation of input–output data as an impulse response model and the subsequent application of state-space realization. The method provides an approximation framework for simultaneous model identification and signal reconstruction.

A triangular-norm-based propositional fuzzy logic

In 1997, Wang introduced the formal system L ∗ for a type of fuzzy propositional calculus given by a particular t-norm different from the three famous ones (Lukasiewicz, Gödel, product). Recently, Pei proved the completeness theorem for L ∗ with respect to W ̄ -semantic by developing a theory of algebraic systems. In this paper, we aim to prove it by essentially metamathematical method. Furthermore, we discussed originally the completeness for the schematic extension L n ∗ of L ∗ with respect to W n -semantic.

Application of the theory of algebraic systems for creating a hierarchy of solid structures formed under equilibrium and nonequilibrium conditions

Molecular systems and structures of crystals are described by ‘‘groups’’ as algebraic systems. Molecular systems are described by ‘‘point group symmetries’’ and crystals, by ‘‘space groups.’’ A ‘‘ring’’ algebraic system describes complicated molecular systems with partially ~most often ‘‘hidden’’ ! ordered and almost independent subsystems ~some metalloorganic compounds and complexes, oligomers and polymers, including fullerenes and nanopipes, biological molecules!. A ‘‘field’’ algebraic system generates structures which have all the characteristics of fields ~‘‘sources,’’ ‘‘sinks,’’ ‘‘gradients,’’ and ‘‘vortices’’ !. These systems describe physical systems which can be fractally ordered. These last grow in energy and concentration ~mass! fields and create aperiodic structures such as quasicrystals. Their aperiodicity is caused by the absence of a translation operator in the solid, but they can have rotational symmetry and fractal ordering. A real representation of the growing spatial structures after multiplying by a complex number makes it possible to explain the different types of spatial and energy oscillations controlled by the mass and energy fluxes. The formation of macroscopic periodic systems without a periodicity on an atomic scale has been observed during MOCVD processes.

Abstract and behaviour module specifications

The theory of algebraic module specifications and modular systems was developed initially mainly on the basis of equational algebraic specifications. We show that it is in fact almost independent of what kind of underlying specification framework is chosen. More specifically, we present a formulation where this framework appears as an indexed category or, equivalently, specification frame. The ensuing theory is called the theory of abstract module specifications. We are able to prove main results concerning the correctness and compositionality of abstract module specifications in a purely categorical way, assuming the existence of pushouts of morphisms between abstract specifications that allow model amalgamation, functor extension and/or suitable free constructions. Then, by instantiating the theory of abstract module specifications to the behaviour specification frame in the sense of Nivela and Orejas, we obtain a theory of behaviour module specifications.

On the Sequence of Consecutive Powers of a Matrix in a Boolean Algebra

In this paper we consider the sequence of consecutive powers of a matrix in a Boolean algebra. We characterize the ultimate behavior of this sequence, we study the transient part of the sequence, and we derive upper bounds for the length of this transient part. We also indicate how these results can be used in the analysis of Markov chains and in max-plus algebraic system theory for discrete event systems.

The Max-plus Algebra Approach To RailwayTimetable Design

A scheduled railway system that operates according to a cyclic timetable naturally exhibits a cyclic (periodic) behaviour. In a max-plus algebra setting such a system can be modelled as a linear (discrete event) dynamic system. The computation of a timetable then reduces to solving an eigenvalue problem for which efficient algorithms have been developed. Moreover, the max-plus algebra system theory contains stability analysis and simulation facilities. This paper shows that the max-plus algebra approach offers an efficient interactive timetable design framework which directs attention to the critical components in the railway system.

Fuzzy ideals and congruences of lattices

General theory of algebraic fuzzy systems developed earlier by the authors is applied to extend the Stone's prime ideal theorem to fuzzy ideals of distributive lattices. A correspondence between fuzzy ideals and fuzzy congruences in a distributive lattice is obtained and certain important properties of these are discussed.

Recursiveness of zero structures for system matrices of interconnected linear systems via module theoretic tools

The importance of the zeros of multivariable linear systems is well-known in terms of measure obstructions to the controllability and the observability. In this paper, a recursive decomposition of interconnected systems is outlined by taking into account the sequential structure of the connections. The paper extends the coordinate, module-theoretic studies from the elementary algebraic systems theory to include the case of such linear interconnected systems which need not to be controllable or observable. Also, the properties of controllability and observability, the decoupling zeros and the signal bloking issues are characterized.

An algebraic theory of object-oriented systems

The paper presents an algebraic specification of net objects. By net objects we mean those that are defined in object-oriented paradigms and those defined with nested relations in complex database models. An algebraic specification is set up that involves structures of net objects, accesses of net objects, and some features of object-oriented programming, such as multiple inheritance and polymorphism. Objects and their relationships are then characterized formally in the modeling, which utilizes the hierarchical approach in the algebraic theory of abstract data types, and is further developed by adding mechanisms from existing object systems. Categories of net objects are presented with the properties of accesses among them.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Algebraic fuzzy systems

A general theory of algebraic fuzzy systems is developed by introducing the notion of a fuzzy L -subset, corresponding to a set L of subsets of a nonempty set X assuming truth values in a complete Brouwerian lattice. It is observed that the set FL of all fuzzy L -subsets of X is a closure (algebraic closure) fuzzy set system if and only if L is a closure (algebraic closure) set system on X . A fuzzy L -subset generated by a fuzzy subset is described. Modularity and distributivity of the complete lattices L and FL are discussed. A method to construct fuzzy L -subsets from certain family of members of L is established. Certain applications of the general theory are mentioned.

Intuitionistic theory of algebraic systems and heyting-valid analysis

Intuitionistic theory of algebraic systems and heyting-valid analysis

Algebraic results for linear-equation structures involving polynomial block-matrix partitions. Some implications in decentralized control theory

This paper outlines as first objective a set of previous results in the field of algebraic systems theory for the case of block-partitioned polynomial matrices. The formulation is relevant for obtaining a mathematical outline being available for analysis of fixed modes when transforming a decentralized information, which involves the use of local controllers of which each one uses a part of the output components for feedback, in the centralized one. This centralized controller involves the use of a controller which possesses all the information, being available by feedback. The main theoretical results are presented in the ring of polynomial matrices and the associated ring of quotients of rational matrices. The results are extendable without difficulty to other sets of matrices over rings.

Pole assignment under changes in dynamics of multivariable systems

A formulation is presented, based upon algebraic systems theory, to deal with the problem of robustness under combined parametrical and/or structural changes in the plant. The key design objective is the maintenance of the dominant closed-loop poles close to their nominal values, while prescribing their placement due to the un model led dynamics to the subsets of the complex plane, which have small influence in the system response. The mathematical framework is the ring of polynomial matrices for input-output descriptions and its relationships with the rational transfer matrix fraction descriptions.

Algebraic theory of linear time-invariant feedback systems with two-input two-output plant and compensator

This paper present an algebraic theory for linear time-invariant multi-input multi-output systems with two-input two-output plant and compensator, using the factorization approach. This system configuration considered by Doyle and Nett is most general in that any interconnection of two systems can be represented in terms of this scheme. Other system configurations encountered in compensator design problems are special cases of this system configuration. The analysis and synthesis applies to continuous-time as well as discrete-time systems, and to lumped as well as distributed systems, since the algebraic setting is completely general. The compensator parametrization has four degrees of freedom. The paper is self-contained and is tutorial in the sense that it develops the main results of Nett without introducing the concept of containment.

Pole assignability in polynomial rings, power series rings, and Prüfer domains

The topics treated in this paper have their origins in the area of algebraic systems theory. However, the paper itself should be classified as pure commutative algebra and we shall present it as such in the body of the text. Still, it is appropriate to give a brief paragraph of motivation. If a physical system is governed by a pair (F, G) of matrices, then the stability of the system can be determined by examining the eigenvalues of the matrix F. If the system is unstable, a “feedback” matrix K can sometimes be employed in such a way that the eigenvalues of the matrix F+ GK measure the stability of the (modified) system (F+ GK, G). In this manner, an unstable system can be rendered stable. The pole assignability problem over commutative rings is one method of attacking the problem of finding such matrices K. The paper is divided into four sections. Section 1 is given over almost entirely to defining the properties in which we shall be interested. It concludes with a theorem about residuating and lifting the properties. Section 2 is concerned with the preservation of the properties under polynomial ring and power series ring formation. Section 3 is concerned with “feedback cyclization,” a strong form of pole assignability. Section 4 is concerned with pole assignability over Priifer domains. A complete elaboration of our results must await the introduction of the necessary terminology. For now, we mention the following results in somewhat vague language. If R is a zero-dimensional ring, then the pole assignability problem is solvable in R[X] and in R[ [Xl]. From a ringtheoretic standpoint, almost any class of commutative rings contains members for which the feedback cyclization problem is solvable. On the other hand, if R is a ring with 1 in its stable range, then the feedback cyclization problem is solvable in R if and only if a certain nice matricial property

Remarks on the inversion of Hankel matrices

The paper continues the investigation into the links between algebraic system theory, more specifically the partial realization problem, and the problem of Hankel matrix inversion. The representation of the inverse of a Hankel matrix as a Bezoutiant and the Bezout equation play a principal role.

Stability analysis of electrical networks containing lossless transmission lines – algebraic system theoretical consideration

AbstractThis paper deals with electrical networks containing lossless transmission lines and discusses the stability from the viewpoint of an algebraic system theory. Specifically, we discuss the effect of the initial conditions of the transmission lines on the stability of the system. We point out that the coprimeness of certain matrices determined by the transmission lines is crucially important for stability and derive an equivalent condition to coprimeness in terms of the pole‐zero cancellations, as well as the structure of the interconnections among transmission lines. An easily tractable stability criterion is developed.

Algebraic theory of linear multivariable feedback systems

This paper presents an algebraic theory for analysis and design of linear multivariable feedback systems. The theory is developed in an algebraic setting sufficiently general to include, as special cases, continuous and discrete time systems, both lumped and distributed. Designs are implemented by construction of a controller with two vector inputs and one vector output. Use of controllers of this type is shown to generate convenient stability results, and convenient global parametrizations of all I/O maps and all disturbance-to-output maps achievable, for a given plant, by a stabilizing compensator. These parametrizations are then used to show that any such I/O map and any such disturbance-to-output map may be simultaneously realized by choice of an appropriate controller. In the special case of lumped systems, it is shown that the design theory. can be reduced to manipulations involving polynomial matrices only. The resulting design procedure is thus shown to be more efficient computationally. Finally, the problem of asymptotically tracking a class of input signals is considered in the general algebraic setting. It is shown that the classical results on asymptotic tracking can be generalized to this setting. Additionally, sufficient conditions for robustness of asymptotic tracking, and robustness of stability are developed.

Symmetric linear systems: an application of algebraic systems theory

Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.

New results on the algebraic theory of linear systems: the solution of the cover problems

We give the complete solution of a large class of problems in linear system theory, the so-called cover problems. These problems are formulated and solved both in the state-space and in the input-output frameworks. The key concept, which allows the effective parametrization of all solutions of the cover problems, is that of the partial realizations of a sequence of matrices. It is shown that the solutions of the state-space cover problems can be expressed as state spaces of the partial realizations of appropriately defined sequences of matrices, and the solution of the input-output cover problems can be expressed as a simple function of the functions of the partial realizations of the sequences of mentioned above.