Properties and Classification of Generalized Resultants and Polynomial Combinants

Polynomial combinants define the linear part of the Dynamic Determinantal Assignment Problems, which provides the unifying description of the frequency assignment problems in Linear Systems. The theory of dynamic polynomial combinants have been recently developed by examining issues of their representation, parameterization of dynamic polynomial combinants according to the notions of order and degree and spectral assignment. Dynamic combinants are linked to the theory of “Generalised Resultants”, which provide the matrix representation of polynomial combinants. We consider coprime set polynomials for which assignability is always feasible and provides a complete characterisation of all assignable combinants with order above and below the Sylvester order. The complete parameterization of combinants and coresponding Generalised Resultants is prerequisite to the characterisation of the minimal degree and order combinant for which spectrum assignability may be achieved

The Euclidean Division as an Iterative ERES-based Process

Considering the Euclidean division of two real polynomials, we present an iterative process based on the ERES method to compute the remainder of the division and we represent it using a simple matrix form

Structured squaring down and zero assignment

The problem of zero assignment by squaring down is considered for a system of p-inputs, n-outputs and n-states (m > p), where not all outputs are free variables for design. We consider the case where a k-subset of outputs is preserved in the new output set, and the rest are recombined to produce a total new set of p-outputs. New invariants for the problem are introduced which include a new class of fixed zeros and the methodology of the global linearization, developed originally for the output feedback pole assignment problem, is applied to this restricted form of the squaring down problem. It is shown that the problem can be solved generically if (p − k)(m − p) > δ*, where k (k < p) is the number of fixed outputs and δ* is a system and compensation scheme invariant, which is defined as the restricted Forney degree

A Symbolic-Numeric Software Package for the Computation of the GCD of Several Polynomials

This survey is intended to present a package of algorithms for the computation of exact or approximate GCDs of sets of several polynomials and the evaluation of the quality of the produced solutions. These algorithms are designed to operate in symbolic-numeric computational environments. The key of their effectiveness is the appropriate selection of the right type of operations (symbolic or numeric) for the individual parts of the algorithms. Symbolic processing is used to improve on the conditioning of the input data and handle an ill-conditioned sub-problem and numeric tools are used in accelerating certain parts of an algorithm. A sort description of the basic algorithms of the package is presented by using the symbolic-numeric programming code of Maple

Structure evolving systems and control in integrated design

Existing methods in Systems and Control deal predominantly with Fixed Systems, that have been designed in the past, and for which the control design has to be performed. The new paradigm of Structure Evolving Systems (SES), expresses a new form of system complexity where the components, interconnection topology, measurement-actuation schemes may not be fixed, the control scheme also may vary within the system-lifecycle and different views of the system of varying complexity may be required by the designer. Such systems emerge in many application domains and in the engineering context in problems such as integrated system design, integrated operations, re-engineering, lifecycle design issues, networks, etc. The paper focuses on the Integrated Engineering Design (IED), which is revealed as a typical structure evolution process that is strongly linked to Control Theory and Design type problems. It is shown, that the formation of the system, which is finally used for control design evolves during the earlier design stages and that process synthesis and overall instrumentation are critical stages of this evolutionary process that shapes the final system structure and thus the potential for control design. The paper aims at revealing the control theory context of the evolutionary mechanism in overall system design by defining a number of generic clusters of system structure evolution problems and by establishing links with existing areas of control theory. Different aspects of model evolution during the overall design are identified which include cases such as: (i) Time-dependent evolution of system models from “early” to “late” stages of design. (ii) Design stage-dependent evolution from conceptualisation to process synthesis and to overall instrumentation. (iii) Redesign of given systems and constrained system evolution. Within each cluster a number of well defined new Control Theory problems are introduced, which may be studied within the structural methodologies framework of Linear Systems. The problems posed have a general systems character, but the emphasis here is on Linear Systems; an overview of relevant results is given and links with existing research topics are established. The paper defines the Structural Control Theoretic context of an important family of complex systems emerging in engineering design and defines a new research agenda for structural methods of Control Theory

Systems of Systems: A Control Theoretic View

This paper considers the notion of SoS as an evolution of the standard notion of systems, provides a clear distinction to the standard notion of composite systems and aims to provide an abstract and generic definition that is detached from the particular domain as well as a classification of the families of SoS. We present a new abstract definition of the notion of System of Systems as an evolution of the notion of Composite Systems, empowered by the concept of autonomy and participation in tasks usually linked to games. Control theoretic concepts and methodologies are used to provide the characterization of the notion of "systems play" that is used as the evolution of the notion of the interconnection topology. In this set up the subsystems in SoS act as autonomous intelligent agents in a multi-agent system that is defined by a central task and possibly a game

The Minimal Design Problem on Dynamic Polynomial Combinants

The theory of dynamic polynomial combinants is linked to the linear part of the Dynamic Determinantal Assignment Problems, which provides the unifying description of the pole and zero dynamic assignment problems in Linear Systems. The fundamentals of the theory of dynamic polynomial combinants have been recently developed by examining issues of their representation, parameterization of dynamic polynomial combinants according to the notions of order and degree and spectral assignment. Central to this study is the link of dynamic combinants to the theory of "Generalised Resultants", which provide the matrix representation of the dynamic combinants. The paper considers the case of coprime set polynomials for which spectral assignability is always feasible and provides a complete characterisation of all assignable combinants with order above and below the Sylvester order. A complete parameterization of combinants and respective Generalised Resultants is given and this leads naturally to the characterisation of the minimal degree and order combinant for which spectrum assignability may be achieved, referred to as the "Dynamic Combinant Minimal Design" (DCMD) problem. Such solutions provide low bounds for the respective Dynamic Assignment control problems

Geometric and algebraic properties of minimal bases of singular systems

For a general singular system with an associated pencil T(S), a complete classification of the right polynomial vector pairs x(s), u(s)), connected with the N{script}r{T(S)}, rational vector space, is given according to the proper-nonproper property, characterising the relationship of the degrees of those two vectors. An integral part of the classification of right pairs is the development of the notions of canonical and normal minimal bases for N{script}r{T(S)} and N{script}r{R(S)} rational vector spaces, where R(s) is the state restriction pencil of Se[E, A, B]. It is shown that the notions of canonical and normal minimal bases are equivalent; the first notion characterises the pure algebraic aspect of the classification, whereas the second is intimately connected to the real geometry properties and the underlying generation mechanism of the proper and nonproper state vectors x(s). The results describe the algebraic and geometric dimensions of the invariant partitioning of the set of reachability indices of singular systems. The classification of all proper and nonproper polynomial vectors x(s) induces a corresponding classification for the reachability spaces to proper-nonproper and results related to the possible dimensions feedback-spectra assignment properties of them are also given. The classification of minimal bases introduces new feedback invariants for singular systems, based on the real geometry of polynomial minimal bases, and provides an extension of the standard theory for proper systems (Warren, M.E., & Eckenberg, A.E. (1975)

Unimodular equivalence and similarity for linear systems

The problem of finding the mapping between unimoduar transformations relating two minimal matrix fraction descriptions (MFDs) of a transfer function, and the similarity transformations relating the respective minimal state-space representations is considered. It is shown that the problem is equivalent to finding the relation of MFDs of the input-state transfer functions of the two systems. This relation turns out to be an equivalence relation involving the unimodular and the similarity matrices relating the MFDs and the state-space systems, respectively. A canonical form for MFDs under this equivalence relation is obtained and it is shown that it leads to a canonical state-space representation, via a realisation procedure

Dynamic Polynomial Combinants and Generalised Resultants

The theory of constant polynomial combinants has been well developed [2] and it is linked to the linear part of the constant Determinantal Assignment problem [1] that provides the unifying description of the pole and zero assignment problems in Linear Systems. Considering the case of dynamic pole, zero assignment problems leads to the emergence of dynamic polynomial combinants. This paper aims to demonstrate the origin of dynamic polynomial combinants from Linear Systems, and develop the fundamentals of the relevant theory by establishing their link to the theory of Generalised Resultants and examining issues of their parameterization according to the notions of order and degree. The paper provides a description of the key spectral assignment problems, derives the conditions for arbitrary assignability of spectrum and introduces a parameterization of combinants according to their order and degree