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)

System of Systems and Emergence. Part 1: Principles and Framework

The paper is in two parts and in Part (1) attempts to formalise the loose concept of "System of Systems" (SoS) within the context of Systems Theory whilst in Part (2) explores and develops a conceptual framework for emergence that is suitable for further development. We view the notion of SoS as an evolution of the standard notion of systems and provide an abstract and generic definition that is detached from the particular domain. To achieve this we deal first with the abstraction of the fundamental components of the system, describe the different aspects of the structure of a composite system and then embark on the task to explain the difference of the new notion, to the standard notion of Composite Systems. 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 referred to as plays which are usually linked to games. The notion of the play is introduced as an extension of the notion of the system and involves the notion of autonomous agents in place of objects and the notion of scenario in place of interconnection topology. This new definition characterises SoS as a development of the Composite System notion where now the subsystems act as autonomous intelligent agents in a multi-agent system play based on a scenario that possibly involves a game. The notion of emergence is considered within both the framework of Composite and SoS and it is linked to the problem of defining functions on a given system and evaluating their values. The emergence is thus presented as the defining signature of a system including System of Systems

Complexity, Emergence and the Challenges of Assurance: The Need for a Systems Paradigm

The complexity of modern products, systems, and processes makes the task to identify, characterise, and provide sufficient assurance about the desirable properties a major challenge. Stakeholders also demand a degree of enhanced confidence about the absence of undesirable properties with a potential to cause harm or loss. This develops a framework of seven fundamental facets of performance as an ontology for emergent behavioural properties and a separate framework for the emergent structural properties of complex systems. The emergent behavioural aspects are explored and we develop a systems framework for assurance based on an Assessment and Management paradigm each comprising a number of principles and processes. The key argument advanced is that in the face of complexity and incessant change, enhanced confidence in the achievement of desirable and avoidance of undesirable properties requires a systems approach empowered by suitable modelling and relevant diagnostic tools explaining the nature of emergent properties. Our principal focus is on safety, security, and sustainability emergent behavioural (performance) aspects of complex products, systems, and processes

Integration of operations in process systems

The problem of system Integration in the context of an Industrial Enterprise is a multidimensional problem with fundamental dimensions those of: (i) Purpose, goals and objectives, (ii) Overall process operations, (ii) Overall System Design/Redesign, (iii) Information, Data and Software, and (iv) Verification, validation and assurance. Each of the above areas is addressed by respective groups which consider their area as representing the entirety of the problem and they frequently ignore the other important dimensions. The aim of this paper is to consider first the general problem of integration from all its fundamental aspects and focus on the key paradigm of Global Operations. Within this area we consider the issues of system organization and in particular the problems relating to hierarchical organization of the different operational functionalities, the issues of aggregation and disaggregation and the related problems of the "top-down" and "bottom up" approaches. System complexity is considered within this framework and the notion of emergent properties is then considered as a problem of aggregation of behaviours, which may be also seen as projections within the setup of a control and information architecture. Such an architecture is defined for the multi-level hierarchical organisational structure that we consider. We show that systems and control concepts and problems play a central role the development of an overall integration methodology and interpreting the features of the different emergent properties. The subject of modelling emerges with a central role in the effort to develop a methodology for systems integration, as well as quantifying the predictors of relevant emergent properties. The approach introduced here is intimately linked to Multilevel hybrid systems (Hierarchy of Operations), and provides a complementary dimension to issues of System Design (and Re-design) dominated by the theory of Structure Evolving systems [10] (in the total Design and Life-cycle analysis) emerges as the central approach. This paper provides an overview of the subject area and focuses on the development of the general conceptual framework for integration. We additionally develop and propose a systems framework for the evolutionary integration in this paper

Generalised resultants, dynamic polynomial combinants and the minimal design problem

The theory of dynamic polynomial combinants is linked to the linear part of the dynamic determinantal assignment problems (DAP), which provides the unifying description of the dynamic, as well as static pole and zero dynamic assignment problems in linear systems. The assignability of spectrum of polynomial combinants provides necessary conditions for solution of the original DAP. This paper demonstrates the origin of dynamic polynomial combinants from linear systems, examines issues of their representation and the parameterisation of dynamic polynomial combinants according to the notions of order and degree, and examines their 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 of 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 parameterisation 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, which is referred to as the dynamic combinant minimal design (DCMD) problem. An algorithmic approach based on rank tests of Sylvester matrices is given, which produces the minimal order and degree solution in a finite number of steps. Such solutions provide low bounds for the respective dynamic assignment control problems

Symmetries, groups and groupoids for Systems of Systems

In this paper we propose an algebraic model of systems based on the concept of symmetry that can be instrumental in representing Systems of Systems two main characteristics, namely complexity and (hierarchical) emergence

Multi-Agent Systems: A new paradigm for Systems of Systems

We present the notion of Systems of Systems, its drivers, and the challenges we face in conceptualizing, designing, implementing and validating them. In this work in progress we propose Multi-Agent Systems as a new paradigm, taken from Artificial Intelligence, which seems to fit the purpose

Strong stability of internal system descriptions

The paper introduces a new notion of stability for internal autonomous system descriptions that is referred to as “strong stability” and characterizes the case of avoiding overshoots for all initial conditions within a given sphere. This is a stronger notion of stability compared to alternative definitions (asymptotic, Lyapunov), which allows the analysis and design of control systems described by natural coordinates to have no overshooting response for arbitrary initial conditions. For the case of LTI systems necessary and sufficient conditions for strong stability are established in terms of the negative definiteness of the symmetric part of the state matrix. The invariance of strong stability under orthogonal transformations is established and this enables the characterisation of the property in terms of the invariants of the Schur form of the state matrix. Some interesting relations between strong stability and special forms of coordinate frames are examined and links between the skewness of the eigenframe and the violation of strong stability are derived. The characterisation of properties of strong stability given here provides the basis for the development of feedback theory for strong stabilisation

The distance to strong stability

The notion of “strong stability” has been introduced in a recent paper [12]. This notion is relevant for state-space models described by physical variables and prohibits overshooting trajectories in the state-space transient response for arbitrary initial conditions. Thus, “strong stability” is a stronger notion compared to alternative definitions (e.g. stability in the sense of Lyapunov or asymptotic stability). This paper defines two distance measures to strong stability under absolute (additive) and relative (multiplicative) matrix perturbations, formulated in terms of the spectral and the Frobenius norm. Both symmetric and non-symmetric perturbations are considered. Closed-form or algorithmic solutions to these distance problems are derived and interesting connections are established with various areas in matrix theory, such as the field of values of a matrix, the cone of positive semi-definite matrices and the Lyapunov cone of Hurwitz matrices. The results of the paper are illustrated by numerous computational examples

Numerical and Symbolical Methods for the GCD of Several Polynomials

The computation of the Greatest Common Divisor (GCD) of a set of polynomials is an important issue in computational mathematics and it is linked to Control Theory very strong. In this paper we present different matrix-based methods, which are developed for the efficient computation of the GCD of several polynomials. Some of these methods are naturally developed for dealing with numerical inaccuracies in the input data and produce meaningful approximate results. Therefore, we describe and compare numerically and symbolically methods such as the ERES, the Matrix Pencil and other resultant type methods, with respect to their complexity and effectiveness. The combination of numerical and symbolic operations suggests a new approach in software mathematical computations denoted as hybrid computations. This combination offers great advantages, especially when we are interested in finding approximate solutions. Finally the notion of approximate GCD is discussed and a useful criterion estimating the strength of a given approximate GCD is also developed