Minimal Realization Research Articles

Identification of Structural Matrices of Shear Buildings Using Ambient Vibration Tests with Incomplete Measurements

System identification methods (SIMs) are powerful tools for structural health monitoring. SIMs are used either for identification of modal parameters or for extraction of structural characteristic matrices. The current study aims to identify the real matrices of a structure using structural dynamics theory and stochastic subspace identification based on realization theory. The study focuses on extraction of the condensed mass and stiffness matrices of shear buildings by means of ambient excitation responses for limited structural degrees of freedom. Realization theory states that there are minimal realizations for appropriate identification of the main real-system matrices. The present study shows that, by using values smaller than the minimal realization, condensed structural matrices can be correctly identified. This is accomplished by accurate estimation of the full real-system order. It is shown that successive repetitions of this approach can lead to complete structural matrices. The practical application of this procedure is examined using an experimental model and an analytical six-story shear model. Analysis revealed that, even in the presence of noise, this method can accurately identify structural matrices in all cases with high precision.

Analyzing Controllability of Bilinear Systems on Symmetric Groups: Mapping Lie Brackets to Permutations

Bilinear systems emerge in a wide variety of fields as natural models for dynamical systems ranging from robotics to quantum dots. Analyzing controllability of such systems is of fundamental and practical importance, for example, for the design of optimal control laws, stabilization of unstable systems, and minimal realization of input-output relations. Tools from Lie theory have been adopted to establish controllability conditions for bilinear systems, and the most notable development was the Lie algebra rank condition (LARC). However, the application of the LARC may be computationally expensive for high-dimensional systems. In this paper, we present an alternative and effective algebraic approach to investigate controllability of bilinear systems. The central idea is to map Lie bracket operations of the vector fields governing the system dynamics to permutation multiplications on a symmetric group, so that controllability and controllable submanifolds can be characterized by permutation cycles. The method is further applicable to characterize controllability of systems defined on undirected graphs, such as multi-agent systems with controlled couplings between agents and Markov chains with tunable transition rates between states, which in turn reveals a graph representation of controllability through the graph connectivity.

Robust and Computationally Efficient Digital IIR Filter Synthesis and Stability Analysis Under Finite Precision Implementations

In this paper, an analytical synthesis method for obtaining an optimal infinite impulse response (IIR) state-space realization, say minimal pole-zero and pole-L2 sensitivity realizations, based on minimizing zero/L2 sensitivity measures subject to sparse normal-form state transition matrix is proposed. The proposed nth order realization possesses strong robustness and at most only 4n + 1 multiplications per output sample to prevent instability and output distortion, as well as to keep computational efficiency under finite word-length (FWL) effects. This is important for the IIR filter implemented in fixed-point and portable digital devices. In this paper, the proposed pole-L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> sensitivity minimization method, an alternative approach of pole-zero sensitivity minimization, may solve the issue of zero multiplicity. A normal-form realization has been proven that it can achieve a global minimum of un-weighted pole sensitivity measure as well as zero-input limit-cycle free property. The sparse normal-form realization can be synthesized from an arbitrary initial realization with distinct poles by using an analytical similarity transformation. Based on the derived fixed-point arithmetic model in state-space realizations, the Bellman-Gronwall Lemma, and normal-form transformation, a new word-length estimation to guarantee stability is derived. Finally, numerical simulations are performed to verify the correctness and the effectiveness of the theoretical results.

Identification of MISO systems in Minimal Realization Form

The paper is concerned with identifying transfer functions of individual input channels in minimal realization form of a Multi-Input Single Output (MISO) from the input-output data corrupted by the error in all the variables. Such a framework is commonly referred to as error-in-variables (EIV). A common approach in the existing methods for identification of MISO systems is to estimate a non-minimal order transfer function under a subset of simplistic assumptions like homoskedastic error variances, known order, and delay. In this work, we deal with the challenging problem of identifying order, delay in each input of minimal realization form separately while estimating the transfer functions. We also estimate the heteroskedastic noise variances in each of the multiple inputs and output variables. An automated approach for the identification of MISO systems of minimal realization form in the EIV framework is proposed. Numerical case studies are presented to illustrate the efficacy of the proposed algorithm in identifying the transfer function along with the order, delay, and noise variances.

Масштабная инвариантность строгой иерархии Кадомцева–Петвиашвили

In this paper we show first of all that for solutions of the strict KP hierarchy it is sufficient to work in a standard setting. Further we discuss a minimal realization of the hierarchy and present the scaling invariance of the Lax equations of the hierarchy.

A novel constructive procedure to low-order Fornasini–Marchesini model realization

In this paper, a novel constructive procedure is proposed for the realization of a low-order Fornasini–Marchesini state-space model for a given multidimensional system. In particular, essential properties for the constructions of a backward shift space and a resolvent invariant space associated with the Gleason’s problem from a given transfer function matrix are investigated and some new sufficient conditions for such constructions are developed. Then, based on these conditions, a systematic constructive procedure is proposed for the multidimensional Fornasini–Marchesini model realization. It turns out that the new constructive procedure can generate Fornasini–Marchesini models with much lower orders and even the minimal realizations for a much larger class of multidimensional systems than the existing methods. Nontrivial examples are also provided to illustrate the basic idea as well as the effectiveness of the proposed procedure.

A lower bound on the dimension of minimal positive realizations for discrete time systems

In some applications, one is interested in having a state–space realization with nonnegative matrices (positive realization) of a given transfer function and it is known that such a realization may have a dimension strictly larger than the order of the transfer function itself. The aim of this letter is to provide a lower bound on the minimum dimension of a positive realization taking into account some spectral properties of nonnegative matrices.

Minimal dirac neutrino mass models from hbox {U}(1)_{mathrm{R}} gauge symmetry and left\u2013right asymmetry at colliders

In this work, we propose minimal realizations for generating Dirac neutrino masses in the context of a right-handed abelian gauge extension of the Standard Model. Utilizing only U(1)_R symmetry, we address and analyze the possibilities of Dirac neutrino mass generation via (a) tree-level seesaw and (b) radiative correction at the one-loop level. One of the presented radiative models implements the attractive scotogenic model that links neutrino mass with Dark Matter (DM), where the stability of the DM is guaranteed from a residual discrete symmetry emerging from U(1)_R. Since only the right-handed fermions carry non-zero charges under the U(1)_R, this framework leads to sizable and distinctive Left–Right asymmetry as well as Forward–Backward asymmetry discriminating from U(1)_{B-L} models and can be tested at the colliders. We analyze the current experimental bounds and present the discovery reach limits for the new heavy gauge boson Z^{prime } at the LHC and ILC. Furthermore, we also study the associated charged lepton flavor violating processes, dark matter phenomenology and cosmological constraints of these models.

Minimal Strongly Eventually Positive Realization for a Class of Externally Positive Systems

The minimal positive realization of externally positive systems is still an open problem due to various restrictions caused by the nonnegativity constraint on the state-space matrices. In this paper, the minimal strongly eventually positive realization, which relaxes the nonnegativity constraint on the state-space model and only requires that the state trajectory is nonnegative after a certain number of steps, is proposed for an externally positive system. It is shown that a discrete-time transfer function with a simple strictly dominant root is externally positive if and only if it has a minimal strongly eventually positive realization. A constructive proof is provided. The continuous-time counterpart is also addressed by transforming it into a corresponding discrete-time case.

Realization of 2D (2,2)–Periodic Encoders by Means of 2D Periodic Separable Roesser Models

Abstract It is well known that convolutional codes are linear systems when they are defined over a finite field. A fundamental issue in the implementation of convolutional codes is to obtain a minimal state representation of the code. Compared with the literature on one-dimensional (1D) time-invariant convolutional codes, there exist relatively few results on the realization problem for time-varying 1D convolutional codes and even fewer if the convolutional codes are two-dimensional (2D). In this paper we consider 2D periodic convolutional codes and address the minimal state space realization problem for this class of codes. This is, in general, a highly nontrivial problem. Here, we focus on separable Roesser models and show that in this case it is possible to derive, under weak conditions, concrete formulas for obtaining a 2D Roesser state space representation. Moreover, we study minimality and present necessary conditions for these representations to be minimal. Our results immediately lead to constructive algorithms to build these representations.

Theory of coupled parametric oscillators beyond coupled Ising spins

Periodically driven parametric oscillators offer a convenient way to simulate classical Ising spins. When many parametric oscillators are coupled dissipatively, they can be analogous to networks of Ising spins, forming an effective coherent Ising machine (CIM) that efficiently solves computationally hard optimization problems. In the companion paper, we studied experimentally the minimal realization of a CIM, i.e. two coupled parametric oscillators [L. Bello, M. Calvanese Strinati, E. G. Dalla Torre, and A. Pe'er, Phys. Rev. Lett. 123, 083901 (2019)]. We found that the presence of an energy-conserving coupling between the oscillators can dramatically change the dynamics, leading to everlasting beats, which transcend the Ising description. Here, we analyze this effect theoretically by solving numerically and, when possible, analytically the equations of motion of two parametric oscillators. Our main tools include: (i) a Floquet analysis of the linear equations, (ii) a multi-scale analysis based on a separation of time scales between the parametric oscillations and the beats, and (iii) the numerical identification of limit cycles and attractors. Using these tools, we fully determine the phase boundaries and critical exponents of the model, as a function of the intensity and the phase of the coupling and of the pump. Our study highlights the universal character of the phase diagram and its independence on the specific type of nonlinearity present in the system. Furthermore, we identify new phases of the model with more than two attractors, possibly describing a larger spin algebra.

Minimal positive continuous-time realizations of positive response maps

Positive nonlinear continuous-time local realizations of positive response maps are studied. The realizations are minimal if they are positively reachable and positively observable. Necessary and sufficient conditions for a response map to have a minimal positive local realization are provided. It is also shown that a minimal positive local realization has the minimal dimension among all positive local realizations of the given response map and that it is unique up to an isomorphism of systems.

A construction by deformation of unitary irreducible representations of SU(1,n) and SU(n + 1)

We recover the holomorphic discrete series representations of [Formula: see text] as well as some unitary irreducible representations of [Formula: see text] by deformation of a minimal realization of [Formula: see text].

The absolutely minimal realizations for first-degree nD SISO systems

Using partial fraction decomposition, limit methods and algebraic techniques, finding the absolutely minimal realization matrices for first-degree (multi-linear) nD SISO systems is transformed into solving a system of equations consisting of the all principal minors of an unknown matrix. Without using the symbolic approach by Grobner basis, the necessary and sufficient conditions and construction of the absolutely minimal realizations for the degenerate and non-degenerate of transfer functions are derived. Five examples for first-degree 3D, 4D, and 5D SISO systems are presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure, and computer programs written in Mathematica are also given.

Transforming Regions of Conflict through Trade Preferential Agreements: Israel, Jordan and the QIZ Initiative

AbstractThe Qualified Industrial Zone (QIZ) model of border-located, duty and quota-free industrial regions is a policy tool designed to capitalize on cross-border cooperation to facilitate post-conflict normalization. Using mixed methods, this study presents and evaluates the implementation of the 1996 US initiated Israel-Jordan QIZ agreement and its social, economic and political impact, contributing to broader debates regarding the potential and limitations of post-conflict normalization through trade preference agreements. The Israel-Jordan QIZ framework no longer functions due to multiple circumstances. Yet there are lessons to be gained from the mechanisms of the QIZ implementation which contributed to the only minimal realization of its anticipated potential. We found that economic brokerage fostering trade and industry cross-border cooperation can serve one-sided economic interests while minimizing spillover to political or social realms, contrary to original intentions. The QIZ experience demonstrates the importance of self-motivation of each of the parties regarding specific cooperative ventures in realizing the potential of cross-border cooperation.

Passive controller realization of a bicubic admittance containing a pole at [formula omitted] with no more than five elements for inerter-based mechanical control

This paper is concerned with a passive network synthesis problem about the damper-spring-inerter realization of a bicubic admittance containing a pole at s=0 with at most five elements, where the least number of elements for the possible realizations is four. The admittances of many passive mechanisms (controllers) in inerter-based control systems are in this form. A specific four-element realizability that is a parallel connection of a spring and a three-element subnetwork is first solved. By utilizing the realizability constraints based on graph theory, it is proved that only one configuration can cover all the other four-element cases through the discussions of other possible network graphs. By deriving its realizability condition, a necessary and sufficient condition for the four-element realizations can be combined. More generally, the five-element realizability can be investigated. The specific five-element realizability that is the parallel connection of a spring and a four-element subnetwork is solved. By making use of realizability constraints described by network graphs and eliminating impossible configurations, a set of four configurations is found out to cover all the other five-element cases. By investigating their realizability conditions, a necessary and sufficient condition for the five-element realizations can be combined. The results of this paper can reduce the realizability redundancy compared with classical synthesis approaches like Bott–Duffin procedure, which provide a first critical step towards solving the minimal realization problem of such admittances. The results can be applied to the design and optimization of mechanical control systems based on inerters, and in the long term can also contribute to the development of other areas of circuits and systems.

Regularity of the Kronecker stratification of realisable Markov parameters

ABSTRACT Let be the set of sequences of real matrices , , , admitting a minimal partial realisation of order d and such that the sum of their largest partial row and column Kronecker indices is smaller than or equal to n. This set is a differentiable manifold which can be stratified according to the partial row and column Kronecker indices of the sequences L. Let be the set of triples such that is controllable and is observable. This set can also be stratified according to the Brunovsky indices of controllability and observability of and , respectively. In this paper we show that this stratification is Whitney regular and from it the regularity of the considered stratification of will be proved. The study of generic families of elements in or in is also included.

A Generalized State-Space Aeroservoelastic Model Based on Tangential Interpolation

In this work, a new approach for the generation of a generalized state-space aeroservoelastic model based on tangential interpolation is presented. The resulting system of differential algebraic equations (DAE) is reduced to a set of ordinary differential equations (ODE) by residualization of the non-proper part of the transfer function matrix. The generalized state-space is of minimal order and allows for the application of the force summation method (FSM) for the aircraft loads recovery. Compared to the classical rational function approximation (RFA) approach, the presented method provides a minimal order realization with exact interpolation of the unsteady aerodynamic forces in tangential directions, avoiding any selection of poles (lag states). The new approach is applied first for the generation of an aerodynamic model for the bidimensional unsteady incompressible flow in the time domain. Next, an application on the generation of an aeroservoelastic model for loads evaluation of the flutter reduced order assessment (FERMAT) model under atmospheric disturbances is done, showing an excellent agreement with the reference model in the frequency domain. The proposed aeroservoelastic model of minimal order is suited for loads analysis and multivariable control design, and an application to a gust loads alleviation (GLA) strategy is shown.

A nonlinear time-delay realization for gastroparesis in patients with diabetes

The realization problem is fundamental in control systems theory and is well understood for linear systems. The situation is quite different for nonlinear systems with or without time delays. New results are given herein either as an abstract characterization of the class of delay–free nonlinear systems which are known to admit a realization, or by characterizing the existence of a realization for a wider class of retarded type or neutral type time delay nonlinear systems. The problem is motivated by the very practical modelling and realization problem of diabetes with or without gastroparesis. Up to 4% to 12% patients with diabetes are affected by the so-called gastroparesis which delays the digestion process. Several concurrent state space mathematical models do exist for diabetes and require a re-evaluation based on physiologic facts. These are reviewed herein to derive a (minimal) realization.

Minimal Realizations of Autonomous Chaotic Oscillators Based on Trans-Immittance Filters

This review paper describes a design process toward fully analog realizations of chaotic dynamics that can be considered canonical (minimum number of the circuit elements), robust (exhibit structurally stable strange attractors), and novel. Each autonomous chaotic lumped circuit proposed in this paper can be understood as a looped system, where linear trans-immittance frequency filter interacts with an active nonlinear two-port. The existence of chaos is demonstrated via well-established numerical algorithms that represent the current standard in the field of nonlinear dynamics, i.e., by calculation of the largest Lyapunov exponent and high-resolution 1-D bifurcation diagrams. The achieved numerical results are put into the context of experimental measurement; observed state trajectories prove a one-to-one correspondence between theoretical expectations and practical outputs, i.e., prescribed strange attractors do not represent the chaotic transients. Finally, short term unpredictability of the chaotic flow is demonstrated via calculation of Kaplan–Yorke dimension that is high, i.e., generated waveforms can find interesting applications in the fields of chaotic masking, modulation, or chaos-based cryptography.