Minimal Realization Research Articles

Linearizations for Rational Matrix Functions and Rosenbrock System Polynomials

Our aim in this paper is twofold: First, for solving rational eigenproblems we introduce linearizations of rational matrix functions and propose a framework for their constructions via minimal realizations. We also introduce Fiedler-like pencils for rational matrix functions and show that the Fiedler-like pencils are linearizations of the rational matrix functions. Second, for computing zeros of a linear time-invariant system $\Sigma$ in state-space form, we introduce a trimmed structured linearization, which we refer to as a Rosenbrock linearization, of the Rosenbrock system polynomial $\mathcal{S}(\lambda)$ associated with $\Sigma.$ Also we introduce Fiedler-like pencils for $\mathcal{S}(\lambda),$ describe an algorithm for their constructions, and show that the Fiedler-like pencils are Rosenbrock linearizations of the system polynomial $\mathcal{S}(\lambda).$

Order reduction of MIMO ARX systems using Laguerre bases

In this article, we explore the use of the Laguerre functions for the estimation of multivariable autoregressive with exogenous (ARX) input model. Each polynomial function of the MIMO ARX model associated to the inputs and to the outputs is expanded on independent Laguerre orthonormal bases. The resulting model is entitled MIMO ARX-Laguerre model. The optimal approximation of which is ensured once the poles characterising each Laguerre orthonormal basis are set to their optimal values. In this paper, a minimal recursive representation realisations for MIMO discrete-time linear systems are derived using Laguerre functions. Further we propose, from input/output measurements, an iterative optimisation algorithm for the free parameters (Laguerre poles). Simulation results show the effectiveness of the proposed optimal modelling method.

Minimal realizations of generalized Nevanlinna functions

Minimal realizations of generalized Nevanlinna functions that carry the information on their generalized poles of nonpositive type in an explicit form are established. These realizations are based on a modification of the basic canonical factorization of generalized Nevanlinna functions whereby the non-minimality problems in realizations that are based directly on the canonical factorization are circumvented.

Contractive determinantal representations of stable polynomials on a matrix polyball

We show that a polynomial p with no zeros on the closure of a matrix unit polyball, a.k.a. a cartesian product of Cartan domains of type I, and such that \(p(0)=1\), admits a strictly contractive determinantal representation, i.e., \(p=\det (I-KZ_n)\), where \(n=(n_1,\ldots ,n_k)\) is a k-tuple of nonnegative integers, \(Z_n=\bigoplus _{r=1}^k(Z^{(r)}\otimes I_{n_r})\), \(Z^{(r)}=[z^{(r)}_{ij}]\) are complex matrices, p is a polynomial in the matrix entries \(z^{(r)}_{ij}\), and K is a strictly contractive matrix. This result is obtained via a noncommutative lifting and a theorem on the singularities of minimal noncommutative structured system realizations.

On the matrix feedback shift register synthesis for matrix sequences

in this paper, a generalization of the linear feedback shift register synthesis problem is presented for synthesizing minimum-length matrix feedback shift registers (MFSRs for short) to generate prescribed matrix sequences and so a new complexity measure, that is, matrix complexity, is introduced. This problem is closely related to the minimal partial realization in linear systems and so can be solved through any minimal partial realization algorithm. All minimum-length MFSRs capable of generating a given matrix sequence with finite length are characterized and a necessary and sufficient condition for the uniqueness issue is obtained. furthermore, the asymptotic behavior of the matrix complexity profile of random vector sequences is determined.

Digraph-Based Algorithm for Finding Minimal Positive Realisation of Two-Dimensional Linear System with Delays

In this paper, the new method of the positive minimal realisation two-dimensional linear system with delays described by general model using multi-dimensional digraphs theory D(n) has been presented. For the proposed method, a digraph-based algorithm was constructed. The algorithm is based on a parallel computing method to gain needed speed and computational power for such a solution. The proposed solution allows minimal digraphs construction for any positive two-dimensional linear system with delays. The proposed method was discussed and illustrated with some numerical examples.

Minimal state space realisation of continuous-time linear time-variant input–output models

ABSTRACTIn the linear time-invariant (LTI) framework, the transformation from an input–output equation into state space representation is well understood. Several canonical forms exist that realise the same dynamic behaviour. If the coefficients become time-varying however, the LTI transformation no longer holds. We prove by induction that there exists a closed-form expression for the observability canonical state space model, using binomial coefficients.

Twin Higgs WIMP dark matter

Dark matter (DM) without a matter asymmetry is studied in the context of twin Higgs (TH) theories in which the LHC naturalness problem is addressed. These possess a twin sector related to the Standard Model (SM) by a (broken) ${\mathbb{Z}}_{2}$ symmetry, and interacting with the SM via a specific Higgs portal. We focus on the minimal realization of the TH mechanism, the fraternal twin Higgs, with only a single generation of twin quarks and leptons, and the $SU(3{)}^{\ensuremath{'}}\ifmmode\times\else\texttimes\fi{}SU(2{)}^{\ensuremath{'}}$ gauge group. We show that a variety of natural twin-WIMP DM candidates are present (directly linked to the weak scale by naturalness), the simplest and most attractive being the ${\ensuremath{\tau}}^{\ensuremath{'}}$ lepton with a mass ${m}_{{\ensuremath{\tau}}^{\ensuremath{'}}}>{m}_{\text{Higgs}}/2$, although spin-1 ${W}^{\ensuremath{'}\ifmmode\pm\else\textpm\fi{}}$ DM and multicomponent DM are also possible (twin baryons are strongly disfavored by tuning). We consider in detail the dynamics of the possibly (meta)stable glueballs in the twin sector, the nature of the twin QCD phase transition, and possible new contributions to the number of relativistic degrees of freedom, $\mathrm{\ensuremath{\Delta}}{N}_{\mathrm{eff}}$. Direct detection signals are below current bounds but accessible in near-future experiments. Indirect detection phenomenology is rich and requires detailed studies of twin hadronization and fragmentation to twin glueballs and quarkonia and their subsequent decay to SM, and possible light twin sector states.

The running coupling of the minimal sextet composite Higgs model

We compute the renormalized running coupling of SU(3) gauge theory coupled to N f = 2 flavors of massless Dirac fermions in the 2-index-symmetric (sextet) representation. This model is of particular interest as a minimal realization of the strongly interacting composite Higgs scenario. A recently proposed finite volume gradient flow scheme is used. The calculations are performed at several lattice spacings with two different implementations of the gradient flow allowing for a controlled continuum extrapolation and particular attention is paid to estimating the systematic uncertainties. For small values of the renormalized coupling our results for the β-function agree with perturbation theory. For moderate couplings we observe a downward deviation relative to the 2-loop β-function but in the coupling range where the continuum extrapolation is fully under control we do not observe an infrared fixed point. The explored range includes the locations of the zero of the 3-loop and the 4-loop β-functions in the $$ \overline{\mathrm{MS}} $$ scheme. The absence of a non-trivial zero in the β-function in the explored range of the coupling is consistent with our earlier findings based on hadronic observables, the chiral condensate and the GMOR relation. The present work is the first to report continuum non-perturbative results for the sextet model.

On fuzzy multiset automata

The objective of this paper is to study fuzzy multiset automata, deterministic fuzzy multiset automata and fuzzy multiset languages. We show that a fuzzy multiset language is accepted by a fuzzy multiset finite automaton iff it is accepted by a deterministic fuzzy multiset finite automaton. We also introduce and study minimal realizations (minimal deterministic fuzzy multiset automata) for a given fuzzy multiset language.

On minimal realization for a fuzzy language and Brzozowski’s algorithm

The purpose of the present work is to provide a simpler solution of a well-known characterization of fuzzy regular languages and a minimal realization for given fuzzy language by using Brzozowski’s algorithm. In between, we associate two crisp deterministic fuzzy automata with a given crisp deterministic fuzzy automaton and show that the reachability and observability maps of the given crisp deterministic fuzzy automaton turn out to be morphisms of category of all crisp deterministic fuzzy

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).

Lepton flavor violating decays of vector quarkonia and of theZboson

We address the impact of sterile fermions on the lepton flavor violating decays of quarkonia as well as of the $Z$ boson. We compute the relevant Wilson coefficients and show that the ${\rm B}(V\to\ell_\alpha\ell_\beta)$, where $V=\phi,\psi^{(n)}$, $\Upsilon^{(n)},Z$ can be significantly enhanced in the case of large sterile fermion masses and a non-negligible active-sterile mixing. We illustrate that feature in a specific minimal realization of the inverse seesaw mechanism, known as $(2,3)$-ISS, and in an effective model in which the presence of non-standard sterile fermions is parameterized by means of one heavy sterile (Majorana) neutrino.

Dynamically induced Planck scale and inflation

Theories where the Planck scale is dynamically generated from dimensionless interactions provide predictive inflationary potentials and super-Planckian field variations. We first study the minimal single field realisation in the low-energy effective field theory limit, finding the predictions n s ≈ 0.96 for the spectral index and r ≈ 0.13 for the tensor-to-scalar ratio, which can be reduced down to ≈ 0.04 in presence of large couplings. Next we consider agravity as a dimensionless quantum gravity theory finding a multifield inflation that converges towards an attractor trajectory and predicts n s ≈ 0.96 and 0.003 < r < 0.13, interpolating between the quadratic and Starobinsky inflation. These theories relate the smallness of the weak scale to the smallness of inflationary perturbations: both arise naturally because of small couplings, implying a reheating temperature of 107−9 GeV. A measurement of r by Keck/Bicep3 would give us information on quantum gravity in the dimensionless scenario.

A naturally light Higgs without light top partners

We demonstrate that the inclusion of a realistic lepton sector can relax significantly the upper bound on top partner masses in minimal composite Higgs models, induced by the lightness of the Higgs boson. To that extend, we present a comprehensive survey of the impact of different realizations of the fermion sectors on the Higgs potential, with a special emphasis on the role of the leptons. The non-negligible compositeness of the $\tau_R$ in a general class of models that address the flavor structure of the lepton sector and the smallness of the corresponding FCNCs, can have a significant effect on the potential. We find that, with the $\tau_R$ in the symmetric representation of $SO(5)$, an increase in the maximally allowed mass of the lightest top partner of $\gtrsim 1$ TeV is possible for minimal quark setups like the MCHM$_{5,10}$, without increasing the tuning. A light Higgs boson $m_H \sim(100-200)$ GeV is a natural prediction of such models, which thus provide a new setup that can evade ultra-light top partners without ad-hoc tuning in the Higgs mass. Moreover, we advocate a more minimal realization of the lepton sector than generally used in the literature, which still can avoid light partners due to its contributions to the Higgs mass in a different and very natural way, triggered by the seesaw mechanism. This allows to construct the most economical $SO(5)/SO(4)$ composite Higgs models possible. Using both a transparent 4D approach, as well as presenting numerical results in the 5D holographic description, we demonstrate that, including leptons, minimality and naturalness do not imply light partners. Leptonic effects, not considered before, could hence be crucial for the viability of composite models.

Minimal Realizations of Three-Port Resistive Networks

This paper is concerned with the minimal realization problem of a third-order real symmetric matrix as the admittance of three-port resistive networks. First, a necessary and sufficient condition is derived for a real symmetric matrix to be realizable as the admittance of three-port resistive networks with four terminals and at most k elements, where k ∈ {1,2,...,5}. Since it is well-known that the matrix must be paramount, necessary and sufficient conditions are obtained for any paramount matrix to be realizable as the admittance of three-port resistive networks with at most k elements, where k ∈ {1,2,3,4}. Moreover, a necessary and sufficient condition is derived for a paramount matrix that cannot be realized with less than five elements to be realizable as the admittance of three-port resistive networks with five elements. Finally, some numerical examples are presented to illustrate the results. The results of this paper can contribute to solving minimal realization problems of one-port and multi-port transformerless networks with more than one kind of elements.

The dark side of electroweak naturalness beyond the MSSM

Weak scale supersymmetry (SUSY) remains a prime explanation for the radiative stability of the Higgs field. A natural account of the Higgs boson mass, however, strongly favors extensions of the Minimal Supersymmetric Standard Model (MSSM). A plausible option is to introduce a new supersymmetric sector coupled to the MSSM Higgs fields, whose associated states resolve the little hierarchy problem between the third generation squark masses and the weak scale. SUSY also accomodates a weakly interacting cold dark matter (DM) candidate in the form of a stable neutralino. In minimal realizations, the thus-far null results of direct DM searches, along with the DM relic abundance constraint, introduce a level of fine-tuning as severe as the one due to the SUSY little hierarchy problem. We analyse the generic implications of new SUSY sectors parametrically heavier than the minimal SUSY spectrum, devised to increase the Higgs boson mass, on this little neutralino DM problem. We focus on the SUSY operator of smallest scaling dimension in an effective field theory description, which modifies the Higgs and DM sectors in a correlated manner. Within this framework, we show that recent null results from the LUX experiment imply a tree-level fine-tuning for gaugino DM which is parametrically at least a few times larger than that of the MSSM. Higgsino DM whose relic abundance is generated through a thermal freeze-out mechanism remains also severely fine-tuned, unless the DM lies below the weak boson pair-production threshold. As in the MSSM, well-tempered gaugino-Higgsino DM is strongly disfavored by present direct detection results.

On minimal realisations of dynamical structure functions

Motivated by the fact that transfer functions do not contain structural information about networks (dependency of state variables), dynamical structure functions were introduced to capture causal relationships between measured nodes in networks. From the dynamical structure functions, (a) we show that the actual number of hidden states can be larger than the number of hidden states estimated from the corresponding transfer function; (b) we can obtain partial information about the true state-space equation, which cannot in general be obtained from the transfer function. Based on these properties, this paper proposes algorithms to find minimal realisations for a given dynamical structure function. This helps to estimate the minimal number of hidden states, to better understand the complexity of the network, and to identify potential targets for new measurements.

${\mathscr H}_\infty$ Problem with Nonstrict Inequality and All Solutions: Interpolation Approach

For given rational matrices ${\boldsymbol V}_a$, ${\boldsymbol U}_a$, ${\boldsymbol V}_b$, ${\boldsymbol U}_b$, we find necessary and sufficient conditions for existence of a stable rational matrix ${\boldsymbol\Phi}$ satisfying $\Vert{\boldsymbol\Phi}\Vert_\infty \leq 1$, ${\boldsymbol V}_a{\boldsymbol\Phi} = {\boldsymbol U}_a$, and ${\boldsymbol\Phi} {\boldsymbol V}_b={\boldsymbol U}_b$. A condition is the positive semidefiniteness of a matrix, denoted by ${\cal R}$. We present a parametrization of all problem solutions. A property of the proposed algorithm is, as a first step, to reduce the problem to a minimal realization, by an orthogonal transformation. Another property is the ability to transform the problem into one with constant matrices, by another orthogonal transformation. A problem motivation is the optimal ${\mathscr H}_\infty$ control problem of descriptor systems. We show by an example that the existing numerical ${\mathscr H}_\infty$ control optimization algorithms, which solve the proble...

Negative Imaginary Lemmas for Descriptor Systems

This technical note studies the negative imaginary properties of descriptor linear systems based on state-space realizations. Under the assumption of a minimal realization, necessary and sufficient conditions are established to characterize the negative imaginary properties of descriptor systems in terms of linear matrix inequalities with equality constraints. In particular, a negative imaginary lemma, a strict negative imaginary lemma and a lossless negative imaginary lemma are developed. A multiple-input and multiple-output RLC circuit network is used as an illustrative example to validate the developed theory.