Nobuko Kosugi

Finite spectrum assignment of systems with general delays

This article presents a new framework for dealing with pure and distributed delays simultaneously and impartially as general delays in algebraic finite spectrum assignment. The framework based on an important property common to general delays gives a sufficient condition for finite spectrum assignability of scalar systems with several general delays and a design method of a controller achieving finite spectrum assignment. Especially in algebraic control theory for delay systems, pure and distributed delays have never been dealt with simultaneously and impartially. The framework is significantly important also for multidimensional system theory because its potential applicability to the discussion on systems with general delays, i.e. the possibility that especially distributed delay operators are used as variables for describing multidimensional systems, is clarified for the first time.

A limit theorem for occupation times of fractional Brownian motion

Recently, N. Kono gave a limit theorem for occupation times of fractional Brownian motion, which result generalizes the well-known Kallianpur-Robbins law for two-dimensional Brownian motion. This paper studies a functional limit theorem for Konos result. It is proved that, under a suitable normalization, the limiting process is the inverse of an extremal process.

New coprimeness over multivariable polynomial matrices and its application to control delay systems

Abstract This paper presents a new notion of coprimeness over multivariable polynomial matrices, where a single variable is given priority over the remaining variables. From a characterization through a set of common zeros of the minors, it is clarified that the presented coprimeness is equivalent to weakly zero coprimeness in the particular variable. An application of the presented coprimeness to control systems with non-commensurate delays and finite spectrum assignment is also presented. Because the presented coprimeness is stronger than minor coprimeness, non-commensurate delays are difficult to deal with in algebraic control theory. The “rational ratio condition” between delays, which can reduce non-commensurate delays to commensurate delays, proves to be both powerful and practical concept in algebraic control theory for delay systems.

Digital redesign of infinite-dimensional controllers based on numerical integration of new representation

Abstract Continuous-time infinite-dimensional controllers that include Laplace transforms of time functions with compact support are indispensable for the advanced control of delay systems. However, no study has yet been conducted on digital redesign for obtaining digital controllers that are used in sampled-data control systems from predesigned continuous-time infinite-dimensional controllers. We introduce a new representational form to describe the continuous-time input–output relation of linear time-invariant systems, called “finite interval integral representation”. Using numerical integration, we approximate the continuous-time input–output relation of a predesigned infinite-dimensional controller in the finite interval integral representation to obtain a digital controller.

Probabilistic safety assessment and management of control laws based on strict Markov analysis

This paper presents a probabilistic safety assessment and management framework based on strict Markov analysis according to the international safety standard, IEC 61508. The framework enables us to take possible common-cause failures into consideration, whose measures are important from a practical viewpoint of system safety. Also the framework establishes the existence of a trade-off between safety and guaranteed control performance quantitatively for the first time ever.

Remarks on Tauberian theorem of exponential type and Fenchel-Legendre transform

Let ( ), ≥ 0 be a nondecreasing right-continuous function such that (0) = 0. The asymptotics of and its Laplace-Stieltjes transform ω( ) = ∫∞ 0 − ( ) are closely linked and results in which we pass from ( ) to ω( ) are called Abelian theorems and ones in converse direction are called Tauberian, and they play a very important role in probability theory. A most well-known result on this subject is Karamata’s theorem (cf. Chapter 1 of [1]). Also the cases when ω( ) and ( ) vary exponentially are treated by many authors (e.g. [2], [3], [4], [8], [9]. See also Chapter 4 of [1]). Among them [2] studied the relationship between the limit of (1/λ) log (1/φ(λ)) as λ → ∞ and that of the Laplace-Stieltjes transform modified as

A new method for solving Bezout equations over 2-D polynomial matrices from delay systems

Abstract In the algebraic system theory of delay systems, it is well known that under spectral controllability or canonicity, a Bezout equation set up with a coprime pair of 2-D polynomial matrices has a solution in polynomial matrices with coefficient belonging to a ring of entire functions. We propose a new method for solving such Bezout equations. The basic concept involves the reduction of a Bezout equation over 2-D polynomial matrices to a simple scalar equation over 1-D polynomials. Due to the basic concept, it can be used to calculate a solution even by hand and is particularly efficient in the absence of modern computer algebra systems.

Bezout equations over bivariate polynomial matrices related by an entire function

This study addresses Bezout equations over bivariate polynomial matrices, where the relationship between two variables is described by a real entire function. This paper proposes a solution method that makes optimal use of minor primeness to reduce such Bezout equations to simple equations over univariate scalar polynomials. The proposed solution method requires only matrix calculations, thus making it very useful, especially in the absence of modern computer algebra systems.

Controller reset strategy for anti-windup based on L 2 gain analysis

When a calculated value of one control input approaches the limitations, the automatic resets of controllers are widely used to prevent windup. However, the automatic reset should only be performed after confirmation that it will not adversely affect the transient performance of the overall control system that may lead to another windup. In this paper, we apply the L2 gain analysis for the entire time to the evaluation of transient responses immediately after a reset, and we propose a new strategy for the automatic resets of controllers, such that windup does not accompany any resets. It succeeds in establishing the applicability of controller resets to anti-windup measures.

Finite spectrum assignment of multi-input systems with non-commensurate delays

We present a new framework for finite spectrum assignment for multi-input systems with non-commensurate delays using an algebraic approach over multidimensional polynomial matrices. By focusing on the solvability of a Bezout equation over multidimensional polynomial matrices, we derive a necessary and sufficient condition for finite spectrum assignability under which a finite number of spectra can be assigned by a control law using a ring of entire functions, i.e. Laplace transforms of all exponential time functions with compact support. Furthermore, using a solution to the Bezout equation, we present a design method for a controller that achieves finite spectrum assignment.