Realization Theory Research Articles

Developing the explanatory dimensions of part–whole realization

I use Carl Gillett’s much heralded dimensioned theory of realization as a platform to develop a plausible part–whole theory. I begin with some basic desiderata for a theory of realization that its key terms should be defined and that it should be explanatory. I then argue that Gillett’s original theory violates these conditions because its explanatory force rests upon an unspecified “in virtue of” relation. I then examine Gillett’s later version that appeals instead to theoretical terms tied to “mechanisms.” Yet I argue that it too violates the desiderata, since it defines realization for mechanisms in terms of two undefined ideas whose explanatory credentials have not been established—“implementation” and “grounds.” Thus I drop those ideas in favor of an explicit constraint that the parts and properties provide a mechanistic explanation. I also distinguish a special mechanistic theory from a preferred general theory that incorporates other kinds of part–whole explanations that target causal powers or capacities. The result is a theory that has the explanatory virtues of mechanistic theories as well as a broader scope desired by Gillett. I also compare the result to a similar idea from Robert Cummins that has been neglected in recent discussions of realization, namely, his general property analysis rather than his functional analysis. Finally, I defend the preferred general theory against possible objections that attempt to show a conflict between metaphysical demands on a theory of realization versus facts about good scientific explanation.

Quantum learning of classical stochastic processes: The completely positive realization problem

Among several tasks in Machine Learning, a specially important one is the problem of inferring the latent variables of a system and their causal relations with the observed behavior. A paradigmatic instance of this is the task of inferring the hidden Markov model underlying a given stochastic process. This is known as the positive realization problem (PRP), [L. Benvenuti and L. Farina, IEEE Trans. Autom. Control 49(5), 651–664 (2004)] and constitutes a central problem in machine learning. The PRP and its solutions have far-reaching consequences in many areas of systems and control theory, and is nowadays an important piece in the broad field of positive systems theory. We consider the scenario where the latent variables are quantum (i.e., quantum states of a finite-dimensional system) and the system dynamics is constrained only by physical transformations on the quantum system. The observable dynamics is then described by a quantum instrument, and the task is to determine which quantum instrument — if any — yields the process at hand by iterative application. We take as a starting point the theory of quasi-realizations, whence a description of the dynamics of the process is given in terms of linear maps on state vectors and probabilities are given by linear functionals on the state vectors. This description, despite its remarkable resemblance with the hidden Markov model, or the iterated quantum instrument, is however devoid of any stochastic or quantum mechanical interpretation, as said maps fail to satisfy any positivity conditions. The completely positive realization problem then consists in determining whether an equivalent quantum mechanical description of the same process exists. We generalize some key results of stochastic realization theory, and show that the problem has deep connections with operator systems theory, giving possible insight to the lifting problem in quotient operator systems. Our results have potential applications in quantum machine learning, device-independent characterization and reverse-engineering of stochastic processes and quantum processors, and more generally, of dynamical processes with quantum memory [M. Guţă, Phys. Rev. A 83(6), 062324 (2011); M. Guţă and N. Yamamoto, e-print arXiv:1303.3771(2013)].

Stability analysis of Volterra functional equations by realization theory

The method of realization of input–output operators in the form of abstract discrete-time control systems and the frequency-domain method are used for the stability/instability analysis of a class of nonlinear Volterra functional equations. To this end, we construct an associated time-invariant abstract discrete-time control system in some weighted function spaces.

On realization theory of quantum linear systems

The purpose of this paper is to study the realization theory of quantum linear systems. It is shown that for a general quantum linear system its controllability and observability are equivalent and they can be checked by means of a simple matrix rank condition. Based on controllability and observability a specific realization is proposed for general quantum linear systems in which an uncontrollable and unobservable subspace is identified. When restricted to the passive case, it is found that a realization is minimal if and only if it is Hurwitz stable. Computational methods are proposed to find the cardinality of minimal realizations of a quantum linear passive system. It is found that the transfer function G(s) of a quantum linear passive system can be written as a fractional form in terms of a matrix function Σ(s); moreover, G(s) is lossless bounded real if and only if Σ(s) is lossless positive real. A type of realization for multi-input–multi-output quantum linear passive systems is derived, which is related to its controllability and observability decomposition. Two realizations, namely the independent-oscillator realization and the chain-mode realization, are proposed for single-input–single-output quantum linear passive systems, and it is shown that under the assumption of minimal realization, the independent-oscillator realization is unique, and these two realizations are related to the lossless positive real matrix function Σ(s).

Identification of open quantum systems from observable time traces

Estimating the parameters that dictate the dynamics of a quantum system is an important task for quantum information processing and quantum metrology, as well as fundamental physics. In this paper we develop a method for parameter estimation for Markovian open quantum systems using a temporal record of measurements on the system. The method is based on system realization theory and is a generalization of our previous work on identification of Hamiltonian parameters [Phys. Rev. Lett. 113, 080401 (2014)].

Realization Relations in Metaphysics

Realization is a technical term that is used by metaphysicians, philosophers of mind, and philosophers of science to denote some dependence relation that is thought to obtain between higher-level properties and lower-level properties. It is said that mental properties are realized by physical properties; functional and computational properties are realized by first-order properties that occupy certain causal/functional roles; dispositional properties are realized by categorical properties; so on and so forth. Given this wide usage of the term it would be right to think that there might be different dependence relations that this term denotes in different cases. Any relation that is aptly picked out by this term can be taken to be a realization relation. The aim of this state-of-the-field article is to introduce the central questions about the concept of realization, and provide formulations of a number of realization relations. In doing so, I identify some theoretical roles realization relations should play, and discuss some theories of realization in relation to these theoretical roles.

IJARCCE - Computer and Communication Engineering

The theme of this paper is to analyze the effect of a non-linearity (second order constitutive relationship between voltage and current in resistor, inductor and capacitor)) on phase delay and group delay for a unit cell of a conventional transmission line model and then by analyzing the phase and group velocities in a Composite Right Hand Left Hand Meta-material (CRLH-MTM) structure which is being used very extensively in the Transmission line (TL) theory for negative refractive index material realization. The complex propagation constant is determined for the non- linear transmission line model and it is observed that the non-linearity in the unit cell is producing the negative resistance which can be used in different electronic circuits, if the same non-linear devices are possible to be modelled from existing devices. Along with this, it is also observed that in order to remove the guard band that exists in the CRLH-MTM for supporting forward and backward waves, there has to be some non-linearity in the transmission line model which is a new finding after the study.

Cartan geometry of spacetimes with a nonconstant cosmological functionΛ

We present the geometry of spacetimes that are tangentially approximated by de Sitter spaces whose cosmological constants vary over spacetime. Cartan geometry provides one with the tools to describe manifolds that reduce to a homogeneous Klein space at the infinitesimal level. We consider a Cartan geometry in which the underlying Klein space is at each point a de Sitter space, for which the combined set of pseudo-radii forms a nonconstant function on spacetime. We show that the torsion of such a geometry receives a contribution that is not present for a cosmological constant. The structure group of the obtained de Sitter-Cartan geometry is by construction the Lorentz group $SO(1,3)$. Invoking the theory of nonlinear realizations, we extend the class of symmetries to the enclosing de Sitter group $SO(1,4)$, and compute the corresponding spin connection, vierbein, curvature, and torsion.

Codes on Graphs: Fundamentals

This paper develops a fundamental theory of realizations of linear and group codes on general graphs using elementary group theory, including basic group duality theory. Principal new and extended results include: normal realization duality; analysis of systems-theoretic properties of fragments of realizations and their connections; minimal = trim and proper theorem for cycle-free codes; results showing that all constraint codes except interface nodes may be assumed to be trim and proper, and that the interesting part of a cyclic realization is its 2-core; notions of observability and controllability for fragments, and related tests; relations between state-trimness and controllability, and dual state-trimness and observability.

New Two-Mode Canonical Operator Realization of Angular Momentum and Generalized Fractional Fourier Transformation Obtained via Entangled State Representation

By virtue of two mutually conjugate bipartite entangled state representations |η〉 and |ξ〉 (Fan and Klauder, Phys Rev A 49:704, 1994) we find new canonical operator realization (COR) of angular momentum. As its two applications, we point out that the COR helps to develop Schwinger bosonic realization theory, and can generalize the concept of FrFT, i.e., constructing new 2-dimensioanl fractional Fourier transformation with additivity.

Geometrically minimal realizations of Boolean controlled systems

For the theory of realization over semirings, consideration was given to a new problem of generation, that of description of various systems with a given external behavior. A notion was introduced of geometric reduction and geometrically minimal realization based on transformations of arbitrary realizations of the given pulse response into each other, rather than on introducing some invariant of the dimensionality type. This notion was considered for the case of systems over the Boolean semiring: the geometrical reductions were classified, and various properties of the geometrically minimal realizations were studied. Some examples were discussed, as well as one class of non-geometric reductions which enabled us to explain the structure of some sets of geometrically minimal realizations.

On time-reversibility of linear stochastic models

Reversal of the time direction in stochastic systems driven by white noise has been of central importance throughout the development of stochastic realization theory, filtering and smoothing. Similarly, in connection with certain problems in the theory of moments, the duality induced by time reversal was the key to deriving a compact parametrization and an interpretation of solutions. By combining ideas from these two lines of development we present herein a general view and a simplified account of time-reversal in stochastic models.

Higher Order Mixed-Moment Approximations for the Fokker--Planck Equation in One Space Dimension

We study mixed-moment models (full zeroth moment, half higher moments) for a Fokker--Planck equation in one space dimension. Mixed-moment minimum-entropy models are known to overcome the zero net-flux problem of full-moment minimum-entropy M${}_n$ models. A realizability theory for these mixed moments of arbitrary order is derived, as well as a new closure, which we refer to as Kershaw closure. They provide nonnegative distribution functions combined with an analytical closure. Numerical tests are performed with standard first-order finite volume schemes and compared with a finite difference Fokker--Planck scheme.

Comparison of LPV and nonlinear system theory: A realization problem

The paper explores the utilization of the nonlinear realization theory to address the problem of transforming linear parameter-varying input–output (LPV-IO) equations into a state-space form with static dependence on the so-called scheduling parameter. The necessary and sufficient solvability conditions are given, and three additional subclasses of LPV-IO equations are suggested that are guaranteed to have a realization of the considered type.

Realization theory of discrete-time linear switched systems

The paper presents realization theory of discrete-time linear switched systems. We present necessary and sufficient conditions for an input–output map to admit a discrete-time linear switched system realization. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observability. Further, we prove that minimal realizations are unique up to isomorphism. We also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state-space representation from input–output data. The paper uses the theory of rational formal power series in non-commutative variables.

Basic Electrical Parameters Model of Mine Communication Cable Based on Physically Realizable Theory

Based on the physically realizable theory, the basic electrical parameters model of the MHYV 7/0.37 communication cable is established by using physical dimensions and material characteristics of the cable. Then ABCD model and the cable attenuation models are derived. Comparing the calculation attenuation and test data, the basic electrical parameters model is consistent with the actual state of the cable, thereby proving the method of building the model is correct and applicable.

Controllability of dynamical systems. A survey

Abstract The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

Interpolatory [formula omitted] model reduction

We introduce an approach to H∞ model reduction that is founded on ideas originating in realization theory, interpolatory H2-optimal model reduction, and complex Chebyshev approximation. Within this new framework, we are able to formulate a method that remains effective in large-scale settings with the main cost dominated by sparse linear solves. By employing Loewner “data-driven” partial realizations within each optimization cycle, computationally demanding H∞ norm calculations can be completely avoided. Several numerical examples illustrate that our approach will produce high fidelity reduced models consistently exhibiting better H∞ performance than those produced by balanced truncation; these models often are as good as (and occasionally better than) those models produced by optimal Hankel norm approximation. In all cases, reduced models are produced at far lower cost than is possible either with balanced truncation or with optimal Hankel norm approximation.

A new approach to physics-based reduction of power system dynamic models

This paper proposes a new approach to physics-based reduction of large-scale power system dynamic models, described by a set of differential and algebraic equations (DAEs). The proposed technique, based on the balanced realization theory (BRT), replaces the original model by one of reduced order, while preserving the physical meaning of retained state variables. Input and output variables of the model are selected in accordance with the goal of a particular study. State variables removed from the model are those that do not dominantly affect the nature of the phenomenon to be analyzed, as evidenced by the magnitude of the frequency response. The proposed technique provides insight into steps of the mathematical description of reduction for the considered phenomenon. In a given study, the number of state variables that can be eliminated depends on the power system model itself and on the required accuracy. Described approach is applied to two (benchmark and real-world) power systems.

Birkhoffian representation of non-homogenous hamiltonian systems

The relation between Hamiltonian system and Birkhoffian system is discussed in this article. Simultaneously, the theory significance and the practical value of Birkhoffian dynamical systems are also investigated. Furthermore, the Birkhoffian realization theory and methods for constructing Birkhoff’s equation are also studied. Then the primary difficulty and the important investigative directions of Birkhoffian dynamics are pointed out in this article. Finally, the formulations and the significance of generalized Birkhoffian dynamics are given. At the same time the almost-generalized Birkhoff’s equations and its applications are also discussed in briefly.