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1

Transformations of the matrices of linear systems to their canonical form with desired eigenvalues

Kaczorek Tadeusz

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2023

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Vol. 71, nr 6

art. no. e147342

EN

A new approach to the transformations of the matrices of linear continuous-time systems to their canonical forms with desired eigenvalues is proposed. Conditions for the existence of solutions to the problems were given and illustrated by simple numerical examples.

2

Some analysis problems of the linear systems

Kaczorek Tadeusz

Journal of Automation, Electronics and Electrical Engineering

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2022

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Vol. 4, No. 2

7--12

EN

New approaches to the transformations of the uncontrollable and unobservable matrices of linear systems to their canonical forms are proposed. It is shown that the uncontrollable pair (A,B) and unobservable pair (A,C) of linear systems can be transform to their controllable (𝐴̅ , 𝐵̅ ) and observable (𝐴̅ , 𝐶̅ ) canonical forms by suitable choice of nonsingular matrix M satisfying the condition 𝑀[𝐴 𝐵] = [𝐴̅ 𝐵̅ ] and [𝐴 𝐵]𝑀 = [𝐴̂ 𝐵̂ ], respectively. It is also shown that by suitable choice of the gain matrix K of the feedbacks of the derivative of the state vector it is possible to reduce the descriptor system to the standard one.

PL

Zaproponowano nowe podejścia do transformacji niesterowalnych i nieobserwowalnych macierzy układów liniowych do ich postaci kanonicznych. Wykazano, że niesterowalna para (A,B) i nieobserwowalna para (A,C) układów liniowych może być przekształcona do ich postaci kanonicznych sterowalnych i obserwowalnych prze odpowiedni dobór nieosobliwej macierzy M spełniającej warunki 𝑀[𝐴 𝐵] = [𝐴̅ 𝐵̅ ] i [𝐴 𝐵]𝑀 = [𝐴̂ 𝐵̂ ]. Pokazano,żeprzezodpowiednidobórmacierzyKsprzężeniazwrotnego od pochodnej wektora stanu jest możliwa redukcja układu deskryptorowegodoukładustandardowego.

3

Gamma-Reduction in Type Theory of Acyclic Recursion

Loukanova Roussanka

Fundamenta Informaticae

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2019

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Vol. 170, nr 4

367--411

EN

In this article, we introduce Moschovakis higher-order type theory of acyclic recursion Lλar. We present the potentials of Lλar for incorporating different reduction systems in Lλar, with corresponding reduction calculi. At first, we introduce the original reduction calculus of Lλar, which reduces Lλar-terms to their canonical forms. This reduction calculus determines the relation of referential, i.e., algorithmic, synonymy between Lλar-terms with respect to a chosen semantic structure. Our contribution is the definition of a (γ) rule and extending the reduction calculus of Lλar and its referential synonymy to γ-reduction and γ-synonymy, respectively. The γ-reduction is very useful for simplification of terms in canonical forms, by reducing subterms having superfluous λ-abstraction and corresponding functional applications. Typically, such extra λ abstractions can be introduced by the λ-rule of the reduction calculus of Lλar.

4

The Algebra of Expansion

Carlier S., Wells J. B.

Fundamenta Informaticae

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2012

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Vol. 121, nr 1/4

43-82

EN

Expansion is an operation on typings (pairs of type environments and result types) in type systems for the λ-calculus. Expansion was originally introduced for calculating possible typings of a l-term in systems with intersection types. This paper aims to clarify expansion and make it more accessible to a wider community by isolating the pure notion of expansion on its own, independent of type systems and types, and thereby make it easier for non-specialists to obtain type inference with flexible precision by making use of theory and techniques that were originally developed for intersection types. We redefine expansion as a simple algebra on terms with variables, substitutions, composition, and miscellaneous constructors such that the algebra satisfies 8 simple axioms and axiom schemas: the 3 standard axioms of a monoid, 4 standard axioms or axiom schemas of substitutions (including one that corresponds to the usual "substitution lemma" that composes substitutions), and 1 very simple axiom schema for expansion itself. Many of the results on the algebra have also been formally checked with the Coq proof assistant. We then take System E, a λ-calculus type system with intersection types and expansion variables, and redefine it using the expansion algebra, thereby demonstrating how a single uniform notion of expansion can operate on both typings and proofs. Because we present a simplified version of System E omitting many features, this may be independently useful for those seeking an easier-tocomprehend version.

5

Observer design using a partial nonlinear observer canonical form

Röbenack K., Lynch A. F.

International Journal of Applied Mathematics and Computer Science

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2006

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Vol. 16, no 3

333-343

EN

This paper proposes two methods for nonlinear observer design which are based on a partial nonlinear observer canonical form (POCF). Observability and integrability existence conditions for the new POCF are weaker than the well-established nonlinear observer canonical form (OCF), which achieves exact error linearization. The proposed observers provide the global asymptotic stability of error dynamics assuming that a global Lipschitz and detectability-like condition holds. Examples illustrate the advantages of the approach relative to the existing nonlinear observer design methods. The advantages of the proposed method include a relatively simple design procedure which can be broadly applied.

6

A General Framework for Mining Frequent Subgraphs from Labeled Graphs

Inokuchi A., Washio T., Motoda H.

Fundamenta Informaticae

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2005

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Vol. 66, nr 1,2

53--82

EN

The derivation of frequent subgraphs from a dataset of labeled graphs has high computational complexity because the hard problems of isomorphism and subgraph isomorphism have to be solved as part of this derivation. To deal with this computational complexity, all previous approaches have focused on one particular kind of graph. In this paper, we propose an approach to conduct a complete search for various classes of frequent subgraphs in a massive dataset of labeled graphs within a practical time. The power of our approach comes from the algebraic representation of graphs, its associated operations and well-organized bias constraints to limit the search space efficiently. The performance has been evaluated using real world datasets, and the high scalability and flexibility of our approach have been confirmed with respect to the amount of data and the computation time.

7

Canonical Forms of Singular 1 D and 2 D Linear Systems

Kaczorek T.

International Journal of Applied Mathematics and Computer Science

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2003

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Vol. 13, no 1

61-72

EN

The paper consists of two parts. In the first part, new canonical forms are defined for singular 1D linear systems and a procedure to determine nonsingular matrices transforming matrices of singular systems to their canonical forms is derived. In the second part new canonical forms of matrices of the singular 2D Roesser model are defined and a procedure for determining realisations in canonical forms for a given 2D transfer function is presented. Necessary and sufficient conditions for the existence of a pair of nonsingular block diagonal matrices transforming the matrices of the singular 2D Roesser model to their canonical forms are established. A procedure for computing the pair of nonsingular matrices is presented.

8

Realisation Problem for Weakly Positive Linear Systems

Kaczorek T.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2000

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Vol. 48, nr 4

595-606

EN

New canonical forms weakly positive linear systems are introduced. Sufficient conditions for the existence of the positive realisations in canonical forms for weakly positive linear are established. Expilcit formulae for the positive realisations in canonical forms are given and illustrated by two numerical examples.