Minimal Realization Research Articles

Versal deformation of realisable Markov parameters

ABSTRACTLet be the set of sequences (L1,… , Ln), , admitting a minimal partial realisation of order d. To each , we associate two sequences of integers with r1 ≥ r2 ≥ … ≥ rβ > 0 = rβ+1 = … = rn and with s1 ≥ s2 ≥ … ≥ sα > 0 = sα+1 = … = sn called the partial Brunovsky column and row indices of L, respectively. Let be the subset of formed by the sequences L for which α + β ≤ n. Let Σco be the set of matrix triples with (F, G) controllable and (H, F) observable. We denote by Σco ≤ the subset of Σco formed by the triples which are minimal partial realisations of the sequences . For every ξ ∈ Σco ≤, we obtain a versal deformation of ξ corresponding to the action of the group , we show a method for obtaining a minimal partial realisation ξ of , and we derive a versal deformation of L from the obtained versal deformation of ξ.

A categorical approach to minimal realization for a fuzzy language

In this study, we provide a general theory of minimal realization for a given fuzzy language in a category theory setting. We introduce categorical characterizations of a run map, reachability map, and observability map for a crisp deterministic fuzzy automaton. Finally, we show that all minimal realizations for a given fuzzy language are isomorphic to the Nerode realization for the same case.

Metric Duality Between Positive Definite Kernels and Boundary Processes

We study representations of positive definite kernels K in a general setting, but with view to applications to harmonic analysis, to metric geometry, and to realizations of certain stochastic processes. Our initial results are stated for the most general given positive definite kernel, but are then subsequently specialized to the above mentioned applications. Given a positive definite kernel K on $$S\times S$$ where S is a fixed set, we first study families of factorizations of K. By a factorization (or representation) we mean a probability space $$\left( B,\mu \right) $$ and an associated stochastic process indexed by S which has K as its covariance kernel. For each realization we identify a co-isometric transform from $$L^{2}\left( \mu \right) $$ onto $$\mathscr {H}\left( K\right) $$ , where $$\mathscr {H}\left( K\right) $$ denotes the reproducing kernel Hilbert space of K. In some cases, this entails a certain renormalization of K. Our emphasis is on such realizations which are minimal in a sense we make precise. By minimal we mean roughly that B may be realized as a certain K-boundary of the given set S. We prove existence of minimal realizations in a general setting.

Exploring the quark flavor puzzle within the three-Higgs doublet model

We extend the standard model with two extra Higgs doublets. Making use of a symmetry principle, we present flavor symmetries based on cycle groups ${Z}_{N}$ that oblige each Higgs doublet to contribute to the mass of only one generation. The Higgs doublets couple to the fermions with different strengths and in this way accommodate the quark mass hierarchy. We systematically search for all charge configurations that naturally lead to the alignment in flavor space of the quark sectors, resulting in a Cabibbo-Kobayashi-Maskawa matrix near to the identity, determined by the quark mass hierarchy, and with the correct overall phenomenological features. The minimal realization is by the group ${Z}_{7}$. We show that only a limited number of solutions exist and that any accidental global symmetry that may occur together with the discrete symmetry is necessarily anomalous. A phenomenological study of each class of solutions concerning predictions to the flavor-changing neutral current phenomena is also performed; for some solutions, it is possible to obtain realistic quark masses and mixing, while the flavor-violating neutral Higgs are light enough to be accessible at the LHC.

Study of texture zeros of fermion mass matrices in minimal extended seesaw mechanism and symmetry realization

In our work, we study the texture zeros of [Formula: see text] in the minimal extended type-I seesaw (MES) with incorporating one extra gauge singlet field “[Formula: see text]”. The MES models deal with the Dirac neutrino mass matrix [Formula: see text], the right-handed Majorana mass matrix [Formula: see text] and the sterile neutrino mass matrix [Formula: see text]. We carry out the mapping of all possible zero textures of [Formula: see text], [Formula: see text] and [Formula: see text] for phenomenologically predictive cases having total eight zeros of [Formula: see text] and [Formula: see text] studied in the literature. With this motivation, we consider (a) [Formula: see text] scheme, (b) [Formula: see text] scheme and (c) [Formula: see text] scheme, where the digits of a pair represent the number of zeros of [Formula: see text] and [Formula: see text], respectively along with the one zero and two zero textures of [Formula: see text]. There are a large number of possibilities of zeros of fermion mass matrices but the implementation of [Formula: see text] transformations reduces it to a very minimum number of basic structures. Interestingly out of four allowed one zero textures of [Formula: see text] without sterile neutrino, only three cases ([Formula: see text], [Formula: see text] and [Formula: see text]) are allowed in MES mechanism for the [Formula: see text] and [Formula: see text] schemes. We find some correlations for different combination of [Formula: see text], [Formula: see text] and [Formula: see text] on enforcement of zeros. We examined all the correlations under the recent neutrino oscillation data and find that only [Formula: see text] survives while both [Formula: see text] and [Formula: see text] are ruled out. Interestingly the one zero textures inherently represent the inverted hierarchy of the mass ordering of light neutrinos. No two zero textures of [Formula: see text] survive in MES although there are a number of allowed structures phenomenologically. The allowed texture zeros are finally realized using a discrete Abelian flavor symmetry group [Formula: see text] with the extension of standard model to include some scalar fields.

Solvability conditions for the positive real lemma equations in the discrete time

Passivity theory and the positive real lemma have been recognised as two of the cornerstones of modern systems and control theory. As digital control is pervasive in virtually all control applications, developing a general theory on the discrete-time positive real lemma appears to be an important issue. While for minimal realisations the relations between passivity, positive-realness and existence of solutions of the positive real lemma equations is very well understood, it seems fair to say that this is not the case in the discrete-time case, especially when the realisation is non-minimal and no conditions are assumed on left- and/or right-invertibility of the transfer function. The purpose of this study is to present a necessary and sufficient condition for existence of solutions of the positive real equations under the only assumption that the state matrix A is asymptotically stable.

Parameter-robust linear quadratic Gaussian technique for multi-agent slung load transportation

This paper copes with parameter-robust controller design for transportation system by multiple unmanned aerial vehicles. The transportation is designed in the form of string connection. Minimal state-space realization of slung-load dynamics is obtained by Newtonian approach with spherical coordinates. Linear quadratic Gaussian / loop transfer recovery (LQG/LTR) is implemented to control the position and attitude of all the vehicles and payloads. The controller's robustness against variation of payload mass is improved using parameter-robust linear quadratic Gaussian (PRLQG) method. Numerical simulations are conducted with several transportation cases. The result verifies that LQG/LTR shows fast performance while PRLQG has its strong point in robustness against system variation.

Degree of Dieudonné determinant defines the order of nonlinear system

ABSTRACTThe paper proves that the order of the minimal state-space realisation of a set of irreducible implicit input--output (i/o) difference equations is equal to the degree of the Dieudonné determinant of a certain polynomial matrix. This matrix is defined by the generic linearisation of the i/o equations, and the map that incorporates the knowledge of the i/o equations in the polynomial coefficients. A simple algorithm to compute the degree of the Dieudonné determinant is given. Several examples demonstrate the theoretical result.

A minimal realization for affine control systems on connected Lie groups

In this work, we study minimal realization problem for an affine control system [Formula: see text] on a connected Lie group [Formula: see text]. We construct a minimal realization by using a canonical projection and by characterizing indistinguishable points of the system.

Transfer Function Estimation in System Identification Toolbox via Vector Fitting

This paper considers black- and grey-box continuous-time transfer function estimation from frequency response measurements. The first contribution is a bilinear mapping of the original problem from the imaginary axis onto the unit disk. This improves the numerics of the underlying Sanathanan-Koerner iterations and the more recent instrumental-variable iterations. Orthonormal rational basis functions on the unit disk are utilized. Each iteration step necessitates a minimal state-space realization with these basis functions. One such derivation is the second contribution. System identification with these basis functions yield zero-pole-gain models. The third contribution is an efficient method to express transfer function coefficient constraints in terms of the orthonormal rational basis functions. This allows for estimating transfer function models with arbitrary relative degrees (including improper models), along with other fixed and bounded parameter values. The algorithm is implemented in the tfest function in System Identification Toolbox (Release 2016b, for use with MATLAB) for frequency domain data. Two examples are presented to demonstrate the algorithm performance.

Displaced photon signal from a possible light scalar in minimal left-right seesaw model

We point out that in the minimal left-right realization of TeV scale seesaw for neutrino masses, the neutral scalar from the right-handed $SU(2{)}_{R}$ breaking sector could be much lighter than the right-handed scale. We discuss for the first time the constraints on this particle from low-energy flavor observables and find that the light scalar is necessarily long-lived. We show that it can be searched for at the LHC via displaced signals of a collimated photon jet and can also be tested in current and future high-intensity experiments. In contrast to the unique diphoton signal (and associated jets) in the left-right case, a generic beyond-standard-model light scalar decays mostly to leptons or jets. Thus, the diphoton channel proposed here provides a new avenue to test the left-right framework and reveal the underlying neutrino mass generation mechanism.

Two Cases of Digraph Structures Corresponding to Minimal Positive Realisation of Fractional Continuous-Time Linear Systems of Commensurate Order

The positive and minimal realisation problem for fractional continuous-time linear single-input and single-output (SISO) systems is formulated. Method based on the one-dimensional digraph for finding a positive and minimal realisation of a given proper transfer function is proposed. Two special cases of the digraph structure are given. Sufficient conditions for the existence of a positive minimal realisation of a given proper transfer function systems are established. The algorithm for computation of a positive minimal realisation is proposed and illustrated with a numerical example. The algorithm is based on a parallel computing method to gain needed speed and computational power for such a solution.

A Minimal Realization Technique for the Dynamical Structure Function of a Class of LTI Systems

The dynamical structure function of a linear time invariant (LTI) system reveals causal dependencies among manifest variables without specifying any particular relationships among the unmeasured states of the system. As such, it is a useful representation for complex networks where a coarse description of global system structure is desired without detailing the intricacies of a full state realization. In this paper, we consider the problem of finding a minimal state realization for a given dynamical structure function. Interestingly, some dynamical structure functions require uncontrollable modes in their state realizations to deliver the desired input-output behavior while respecting a specified system structure. As a result, the minimal order necessary to realize a particular dynamical structure function may be greater than that necessary to realize its associated transfer function. Although finding a minimal realization for a given dynamical structure function is difficult in general, we present a straightforward procedure here that works for a simplified class of systems.

Higgs boson decays to dark photons through the vectorized lepton portal

Vector-like fermions charged under both the Standard Model and a new dark gauge group arise in many theories of new physics. If these fermions include an electroweak doublet and singlet with equal dark charges, they can potentially connect to the Higgs field through a Yukawa coupling in analogy to the standard neutrino portal. With such a coupling, fermion loops generate exotic decays of the Higgs boson to one or more dark vector bosons. In this work we study a minimal realization of this scenario with an Abelian dark group. We investigate the potential new Higgs decays modes, we compute their rates, and we study the prospects for observing them at the Large Hadron Collider and beyond given the other experimental constraints on the theory. We also discuss extensions of the theory to non-Abelian dark groups.

An observer-based control scheme using negative-imaginary theory

This paper presents a full-order state observer-based scheme that transforms a causal, LTI, minimally realized system into a strongly strict negative-imaginary system by defining an auxiliary output based on the observed states. The auxiliary output is used for closed-loop control with positive feedback invoking the internal stability condition of negative-imaginary theory. A set of LMI conditions is derived that determines the value of a design parameter μ required for the proposed scheme to transform a given system with a minimal state-space realization into a strongly strict negative-imaginary system. The proposed scheme is applicable for both stable/unstable and square/non-square systems. In this paper, the framework is further explored for robustness analysis of uncertain systems and numerical examples are given to elucidate effectiveness of the proposed results.

Remembering Prof. R.E. Kalman and His Center for Mathematical System Theory [Historical Perspectives

Prof. Rudolf E. Kalman's name was already a brand as a founder of modern control theory before I joined his center, the Center for Mathematical System Theory (CMST) at the University of Florida, as a student in 1975. By that time, he had published his seminal works on optimal control, minimal realization, linear system structure, Lyapunov stability, and Kalman filters (some of them coauthored) [1]. In the control-theory textbook by Prof. Furuta, I saw a picture of Kalman sitting at his desk, taken in his office at CMST; for some reason I remember this picture as clearly as I remember him.

Review of MIMO Minimal Realization Techniques and Case Study on SCARA Robot Manipulator

This paper is concerned with the review of the main minimal realization techniques for linear time-invariant MIMO systems. The main objective of this paper is to evaluate and compare these techniques, including the coprime fraction approach, in terms of realization configuration, space and time complexities using a case study: a SCARA robot manipulator. Results obtained are valid for both discrete and continuous-time systems.

Articles in Natural Languages and Artificial Languages

The aim of this paper is to investigate the usage of articles (i.e., definiteness and specificity) in natural (i.e., English, Samoan, and Lillooet Salish) and artificial (i.e., Esperanto, Unish, and Sambahsa) languages and then to suggest an optimal pattern of articles in artificial languages. The observation in this paper is supported by the typology of articles in natural languages, language acquisition, markedness of articles, and historical developments of articles. Then, it proposes that articles are a recent and uncommon grammatical realization and show great variation across languages. Finally, this paper proposes the ‘Minimal Realization Principle’ for article uses in artificial languages.

Lattice Structure Realization for The Design of 2-D Digital Allpass Filters With General Causality

This paper presents a lattice structure for the realization of two-dimensional (2-D) recursive digital allpass filters (DAFs) with general causality. We employ four basic lattice sections to realize 2-D recursive DAFs with wedge-shaped coefficient support region like a nonsymmetric half-plane (NSHP) support region. The theory and transfer functions of the realized 2-D lattice DAFs are derived. Some variations of the 2-D lattice structure are also presented. We use the Roesser state space model to verify the minimal realization of the proposed 2-D recursive lattice DAF. We present a least-squares design technique and a minimax design technique to solve the nonlinear optimization problems of the proposed 2-D lattice DAF structure. The novelty of the presented lattice structure is that it not only inherits the desirable attributes of 1-D Gray-Markel lattice allpass structure but also possesses the advantage of better performance over the existing 2-D lattice allpass structures. Finally, several design examples are provided for conducting illustration and comparison.

Two-channel Kondo physics in a Majorana island coupled to a Josephson junction

We study a Majorana island coupled to a bulk superconductor via a Josephson junction and to multiple external normal leads. In the absence of the Josephson coupling, the system displays a topological Kondo state, which had been largely studied recently. However, we find that this state is unstable even to small Josephson coupling, which instead leads at low temperature $T$ to a new fixed point. Most interesting is the case of three external leads, forming a minimal electronic realization of the long sought two-channel Kondo effect. While the $T=0$ conductance corresponds to simple resonant Andreev reflection, the leading $T$ dependence forms an experimental fingerprint for non-Fermi liquid properties.