Analysis Of Linear Systems Research Articles

Design and test of a compact large-stroke dual-drive linear-motion system

This paper presents the design, modelling, analysis, and test of a piezo-actuated dual-drive linear motion system, capable of millimeter stroke with nanometer resolution. The primary actuator is a two-stage bridge-type amplifier with hybrid flexure hinges, which is driven by a piezo-stack actuator and generates output stroke greater than 6 mm. The micro actuation is directly driven by an extra piezo-stack to achieve resolution of 20 nm. A very large amplification ratio of 91 is achieved, which implies not only large stroke but also compactness. An inherent characteristic for the multi-stage compliant displacement amplifier, ‘kick-back’ phenomenon, is first analyzed and eliminated by using the stiffness distribution method. The architectural parameters are optimized by genetic algorithm to find the trade-off among natural frequency, stroke and stiffness ratio. The optimized design was then fabricated and tested. The superiority of the proposed system is remarkable compared with other similar works. The concept and approach outlined are generic and can be extended to this class of actuators.

Shuffle algorithm for fractional descriptor Roesser type continuous-time linear systems

The shuffle algorithm is applied to analysis of the fractional descriptor Roesser type continuous-time linear systems. Using the shuffle algorithm the fractional descriptor linear system is reduced to the equivalent standard system and the system is decomposed into dynamic and static parts. Procedure for computation of the matrices of equivalent standard system and of the dynamical and static parts of the system is proposed.

Reinventing personality from a Volterra nonlinear model of motivation

Multidimensional Laplace Transforms (MLT) is a generalisation of the Laplace transforms widely used in the analysis of linear control systems. The use of MLT allows the response of continuous nonlinear systems to be expressed as a power series of the stimulus. MLT was used in modelling animal motivation. The two distinct motivational mechanisms of priming and homeostasis are combined in a single mathematically rigorous nonlinear dynamic model. Projecting the model to simulate human motivation led to an age-dependent cybernetic personality model. The model comprises four complex dimensions, where sentences are classified to reflect states of motivation. The Age Trend Classification (ATC) is based on empirical evaluation of funniness scores over adult age: a sentence is classified according to changes in its average scores, either as falling, rising, peaking or constant. In contrast, psychometric personality models assume stable human traits; age dependence in adulthood is often considered insignificant. In this respect the age dependent artificial personality is a re-invention of and radically different from current psychological models. This paper explains the nonlinear roots of the motivation model, how the complex motivational dimensions reconcile with psychological traits and suggests a cybernetic hypothesis of human aging. Statistical analysis compares the consistency of Big Five personality with ATC funniness scores, the analysis shows that Cronbach's Alpha coefficients of raw funniness scores are higher in value than Big Five traits, and that the correlation coefficients between raw ATC funniness and Big Five scores show that nearly all ATC correlations with Extroversion and Neuroticism are closely distributed. The high consistency of ATC scores supports the reliability of funniness score data as a source of personality measures, the nearly uniform correlation with two of five personality traits suggests the reliance on such empirical measures as a criterion to select sentences with strong links to personality.

An Optimal H∞ Approach for Local Stability Analysis of Non Linear Digital Systems with Saturation Non Linearity and External Interference

An optimal [Formula: see text] criterion for the stability investigation of non linear digital filters with saturation non linearity and external interference is proposed in this paper. The proposed criterion can be used for overflow oscillation-free realization of the non linear digital system implemented with fixed wordlength machines like digital computers or special purpose embedded systems. The proposed approach ascertains the local asymptotic stability of non linear digital filtering schemes without external interference. Further, by employing [Formula: see text] performance criterion the conditions for local stability and performance measurement are proposed with external interference. In the proposed work, the stability conditions are proposed in linear matrix inequality (LMI) by exploiting energy-based Lyapunov theory, Lipschitz inequality, and saturation arithmetic. The proposed criterion is shown to be less conservative which will improve the operating region and obtain a better estimate of [Formula: see text] performance index compared to the existing local criterion. Moreover, the proposed criterion can be used in selecting the appropriate wordlength of a digital system for tolerable values of interference energy, [Formula: see text] performance level, and fixed-point resolution. This will be an additional advantage of this work compared with existing global criteria. Sufficient examples to show the efficacy of the proposed technique are also provided.

SMT-Based Reachability Analysis of High Dimensional Interval Max-Plus Linear Systems

This paper discusses the reachability analysis (RA) of Interval Max-Plus Linear (IMPL) systems, a subclass of continuous-space, discrete-event systems defined over the max-plus algebra. Unlike standard Max-Plus Linear (MPL) systems, where the transition matrix is fixed at each discrete step, IMPL systems allow for uncertainty on state matrices. Given an initial and a target set, we develop algorithms to verify the existence of IMPL system trajectories that, starting from the initial set, eventually reach the target set. We show that RA can be solved by encoding the IMPL system, as well as initial and target sets, into linear real arithmetic expressions, and then checking the satisfaction of a resulting logical formula via a satisfiability modulo theory (SMT) solver. The performance and scalability of the developed SMT-based algorithms are shown to drastically outperform state-of-the-art RA algorithms applied to IMPL systems, which promises to usher their use in practical, industrial-sized IMPL models.

Introduction to Jan C. Willems’, “Dissipative Dynamical Systems, Part II: Linear Systems With Quadratic Supply Rates”

The two-part article, “Dissipative Dynamical Systems,” by Jan C. Willems, was submitted in January 1972 to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Archive for Rational Mechanics and Analysis</i> and published in the same year (vol. 45, Part I, pp. 321–351, Part II: pp. 352–393). While Part I develops the theory of dissipative systems for general input–state–output systems, Part II confines itself to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linear systems</i> . Moreover, most of Part II is about dissipativity with respect to the supply rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y{\hspace{0.06em}}^{\top}u,$</tex-math></inline-formula> that is, linear <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">passive</i> systems. The main questions addressed in Part II concern the analysis of the set of storage functions of linear passive systems (with the existence of a minimal and maximal element already established in Part I) and the compatibility of storage functions with other external and internal symmetry properties of the system, such as reciprocity and reversibility. (See “The Articles of This Special Issue.”) This links the article closely to the realization theory of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">physical</i> linear systems, in particular, the network synthesis of electrical circuits, the Onsager–Casimir relation of linear thermodynamics, and the input–output analysis of linear viscoelastic systems.

Analysis of Positive Systems With Input Saturation: Invariant Hyperpyramids and Hyperrectangles

This article addresses the problem of estimating the domain of attraction of positive systems under a saturated linear feedback. Compared with the analysis of general saturated linear systems, one remarkable difference is that only the domain of attraction inside the first orthant is of interest. Hence, we tackle this problem by exploiting different shapes of invariant sets confined in the first orthant, including hyperpyramids and hyperrectangles, which are, respectively, the level sets of sum-separable and max-separable Lyapunov functions. For an open-loop positive system, a sufficient condition is given to ensure that a pyramid (or a hyperrectangle) is contractively invariant for the closed loop. This condition is nonconservative for the single-input case. Moreover, for a system without open-loop positivity, we establish conditions such that the closed loop has a hyperrectangular invariant set. Examples show the superiority of the results, in comparison with those existing ones using invariant ellipsoids.

Identifiability analysis of linear ordinary differential equation systems with a single trajectory

Ordinary differential equations (ODEs) are widely used to model dynamical behavior of systems. It is important to perform identifiability analysis prior to estimating unknown parameters in ODEs (a.k.a. inverse problem), because if a system is unidentifiable, the estimation procedure may fail or produce erroneous and misleading results. Although several qualitative identifiability measures have been proposed, much less effort has been given to developing quantitative (continuous) scores that are robust to uncertainties in the data, especially for those cases in which the data are presented as a single trajectory beginning with one initial value. In this paper, we first derived a closed-form representation of linear ODE systems that are not identifiable based on a single trajectory. This representation helps researchers design practical systems and choose the right prior structural information in practice. Next, we proposed several quantitative scores for identifiability analysis in practice. In simulation studies, the proposed measures outperformed the main competing method significantly, especially when noise was presented in the data. We also discussed the asymptotic properties of practical identifiability for high-dimensional ODE systems and conclude that, without additional prior information, many random ODE systems are practically unidentifiable when the dimension approaches infinity.

Consensus error analysis for linear continuous-time multi-agent systems with additive noises

This paper investigates the distributed consensus problem for linear continuous-time multi-agent systems subject to additive process noises. Due to the existence of noises, the asymptotic consensus cannot be achieved. Hence, a distributed consensus protocol is proposed such that all the agents can achieve consensus to some degree. Furthermore, based on Lyapunov theory, we characterize the consensus error under undirected and directed topologies. Numerical examples are presented to verify the effectiveness of two results.

MEDICAL EFFICACY OF A SPONTANEOUS MEDICO-SOCIAL EXPERIMENT (BY THE EXAMPLE OF COVID-19 REGIONAL PREVENTION)

Relevance: studies of the effectiveness of large-scale medico-social experiments (MSE) are quite rare. On the one hand, MSE is, as a rule, a response to force majeure circumstances, which should not be at all in ordinary live. On the other hand, such experiments, as well as their study, are processes characterized by high cost and considerable duration. Nevertheless, the example of COVID-19 regional prevention shows that coordinated interaction of various parts of the social organism can and should be carried out in a critical situation, that the effectiveness of this interaction should be quantified using integrative indicators of sanitary statistics. This interaction illustrates the spontaneous formation of a regional model of preventive healthcare of the future, which is characterized by active socio-economic and information-cognitive support of the main preventive and (or) treatment and diagnostic process aimed at the population of the region using a wide range of activities, including traditional and new educational technologies.&#x0D; Gol: to evaluate the medical effectiveness of the first pre-vaccination year of COVID-19 prevention at the regional level using integrative quantitative indicators of sanitary statistics.&#x0D; Methods: historical, statistical, graphical, linear regression analysis, system analysis and synthesis, organizational and methodological modeling.&#x0D; Results: the slope coefficients in the linear regression formulas corresponding to the annual growth rate (GR) on the graphs of prevalence and incidence in the studied social groups indicated that in the pre-pandemic years, the prevalence in the population was permanently growing - GR was within from +17 to +49 ‰ per year, and the incidence showed relative stability - GR was in the range from +8 to +12 ‰, and even decreased in the children’s population (-17 ‰ annually). Comprehensive preventive measures of the first pandemic year (2020) reversed the general growth trend and accelerated the process significantly. Thus, incidence GR was in the range from -35 to -257 ‰, and prevalence GR - from -179 to -326 ‰.&#x0D; Conclusion: preventive measures against COVID-19, carried out in the form of a large-scale MSE, made it possible to introduce at the regional level a model of systemic integrated preventive healthcare of the future, the medical effectiveness of which is many times greater than the previous organizational and methodological model of “autonomous” healthcare, as far as this can be judged by the emerging trends of change integrative indicators of health (prevalence and incidence).

A delay-variation-dependent stability criterion for discrete-time systems via a bivariate quadratic function negative-determination lemma

This article concerns with stability analysis of discrete linear systems with time-varying delays. Firstly, we extend a quadratic function negative-determination lemma for a single variable to the bivariate case. Secondly, we construct a novel Lyapunov-Krasovskii functional (LKF) with a quadratically delay-dependent matrix to investigate the stability of discrete-time systems with time-varying delays. Based on the proposed lemma, a new delay-variation-dependent stability criterion is derived. Finally, numerical examples are given to illustrate the theoretical result and the proposed criterion is shown to be less conservative than some previous ones.

Finite-Time Stability Analysis of Linear Differential Systems with Pure Delay

Nonhomogeneous systems governed by second-order linear differential equations with pure delay are considered. As an application, the exact solutions of these systems and their delayed matrix functions are used to obtain the finite-time stability results. Our results extend and improve some previous results by removing some restrictive conditions. Finally, an example is provided to illustrate our theoretical results.

Optimal stochastic forcings for sensitivity analysis of linear dynamical systems

Abstract The paper is devoted to the construction of optimal stochastic forcings for studying the sensitivity of linear dynamical systems to external perturbations. The optimal forcings are sought to maximize the Schatten norms of the response. As an example,we consider the problem of constructing the optimal stochastic forcing for the linear dynamical system arising from the analysis of large-scale structures in a stratified turbulent Couette flow.

Towards an objective evaluation of EEG/MEG source estimation methods – The linear approach

The spatial resolution of EEG/MEG source estimates, often described in terms of source leakage in the context of the inverse problem, poses constraints on the inferences that can be drawn from EEG/MEG source estimation results. Software packages for EEG/MEG data analysis offer a large choice of source estimation methods but few tools to experimental researchers for methods evaluation and comparison. Here, we describe a framework and tools for objective and intuitive resolution analysis of EEG/MEG source estimation based on linear systems analysis, and apply those to the most widely used distributed source estimation methods such as L2-minimum-norm estimation (L2-MNE) and linearly constrained minimum variance (LCMV) beamformers. Within this framework it is possible to define resolution metrics that define meaningful aspects of source estimation results (such as localization accuracy in terms of peak localization error, PLE, and spatial extent in terms of spatial deviation, SD) that are relevant to the task at hand and can easily be visualized. At the core of this framework is the resolution matrix, which describes the potential leakage from and into point sources (point-spread and cross-talk functions, or PSFs and CTFs, respectively). Importantly, for linear methods these functions allow generalizations to multiple sources or complex source distributions. This paper provides a tutorial-style introduction into linear EEG/MEG source estimation and resolution analysis aimed at experimental (rather than methods-oriented) researchers. We used this framework to demonstrate how L2-MNE-type as well as LCMV beamforming methods can be evaluated in practice using software tools that have only recently become available for routine use. Our novel methods comparison includes PLE and SD for a larger number of methods than in similar previous studies, such as unweighted, depth-weighted and normalized L2-MNE methods (including dSPM, sLORETA, eLORETA) and two LCMV beamformers. The results demonstrate that some methods can achieve low and even zero PLE for PSFs. However, their SD as well as both PLE and SD for CTFs are far less optimal for all methods, in particular for deep cortical areas. We hope that our paper will encourage EEG/MEG researchers to apply this approach to their own tasks at hand.

Phase space reconstruction and Koopman operator based linearization of nonlinear model for damage detection of nonlinear structures

Vibration responses of structures with inherent nonlinear behaviors can degrade the performance of linear theory based damage detection methods. This paper integrates the phase space reconstruction and Koopman operator to provide a linear representation of strongly nonlinear systems. Similar to the modal analysis of linear systems, the linearized model allows for handling nonlinear vibration responses as a superposition of the discovered nonlinear coordinate basis. This property provides opportunities to identify the structural condition change of structures with initial nonlinearity. The eigen-frequencies extracted from the Koopman operator are served as damage features. The performance of using the eigen-frequencies from dynamic mode decomposition for nonlinear structural damage detection is compared with the natural frequencies obtained from fast Fourier transformation and the time-frequency analysis method to emphasize the superiority of the proposed approach. Two experimental structures exhibiting inherent nonlinearity, namely, a magneto-elastic system and a precast segment beam, are employed to demonstrate the feasibility and effectiveness of using the proposed method for identifying condition change of nonlinear structures. Results demonstrate that the presented nonlinearity linearization framework and the damage feature defined in this study are suitable for reliably identifying the occurrence of structural damage and condition change in structures with inherent nonlinearities.

Neutrino Flavor Pendulum Reloaded: The Case of Fast Pairwise Conversion.

In core-collapse supernovae or compact binary merger remnants, neutrino-neutrino refraction can spawn fast pair conversion of the type ν_{e}ν[over ¯]_{e}↔ν_{x}ν[over ¯]_{x} (with x=μ, τ), governed by the angle-dependent density matrices of flavor lepton number. In a homogeneous and axially symmetric two-flavor system, all angle modes evolve coherently, and we show that the nonlinear equations of motion are formally equivalent to those of a gyroscopic pendulum. Within this analogy, our main innovation is to identify the elusive characteristic of the lepton-number angle distribution that determines the depth of conversion with the "pendulum spin." The latter is given by the real part of the eigenfrequency resulting from the linear normal-mode analysis of the neutrino system. This simple analogy allows one to predict the depth of flavor conversion without solving the nonlinear evolution equations. Our approach provides a novel diagnostic tool to explore the physics of nonlinear systems.

Improved Stability Conditions for Time-Varying Delay Systems via Relaxed Lyapunov Functionals

In this paper, the stability analysis of linear systems with time-varying delays is studied. A novel Lyapunov method is presented, in which positive definiteness of the matrices in common Lyapunov functionals is relaxed by adding what is referred to as a zero-integral functional (ZIF). A general form of auxiliary polynomial-based functionals that contains such ZIF is given. Choosing polynomials of different order as well as exploring double-delay-product (DDP) terms, novel Lyapunov functionals are constructed, which contribute to a set of improved stability conditions expressed in terms of linear matrix inequalities. Finally, numerical examples are provided to corroborate the merits of the proposed method relative to a number of existing methods, and in particular, the effectiveness of the proposed ZIFs and DDP terms in reducing the conservatism of stability conditions.

Quenching of oscillations in a liquid metal via attenuated coupling.

In this work, we report a quenching of oscillations observed upon coupling two chemomechanical oscillators. Each one of these oscillators consists of a drop of liquid metal submerged in an oxidizing solution. These pseudoidentical oscillators have been shown to exhibit both periodic and aperiodic oscillatory behavior. In the experiments performed on these oscillators, we find that coupling two such oscillators via an attenuated resistive coupling leads the coupled system towards an oscillation quenched state. To further comprehend these experimental observations, we numerically explore and verify the presence of similar oscillation quenching in a model of coupled Hindmarsh-Rose (HR) systems. A linear stability analysis of this HR system reveals that attenuated coupling induces a change in eigenvalues of the relevant Jacobian, leading to stable quenched oscillation states. Additionally, the analysis yields a threshold of attenuation for oscillation quenching that is consistent with the value observed in numerics. So this phenomenon, demonstrated through experiments, as well as simulations and analysis of a model system, suggests a powerful natural mechanism that can potentially suppress periodic and aperiodic oscillations in coupled nonlinear systems.

On the Stability of Incommensurate h-Nabla Fractional-Order Difference Systems

This work aims to present a study on the stability analysis of linear and nonlinear incommensurate h-nabla fractional-order difference systems. Several theoretical results are inferred with the help of using some theoretical schemes, such as the Z-transform method, Cauchy–Hadamard theorem, Taylor development approach, final-value theorem and Banach fixed point theorem. These results are verified numerically via two illustrative numerical examples that show the stabilities of the solutions of systems at hand.

Elements of future snowpack modeling – Part 1: A physical instability arising from the nonlinear coupling of transport and phase changes

Abstract. The incorporation of vapor transport has become a key demand for snowpack modeling in which accompanied phase changes give rise to a new, nonlinear coupling in the heat and mass equations. This coupling has an impact on choosing efficient numerical schemes for 1D snowpack models which are naturally not designed to cope with mathematical particularities of arbitrary, nonlinear partial differential equations (PDEs). To explore this coupling we have implemented a stand-alone finite element solution of the coupled heat and mass equations in snow using the computing platform FEniCS. We focus on the nonlinear feedback of the ice phase exchanging mass with a diffusing vapor phase with concurrent heat transport in the absence of settling. We demonstrate that existing continuum-mechanical models derived through homogenization or mixture theory yield similar results for homogeneous snowpacks of constant density. When snow density varies significantly with depth, we show that phase changes in the presence of temperature gradients give rise to nonlinear advection of the ice phase amplifying existing density variations. Eventually, this advection triggers a wave instability in the continuity equations. This is traced back to the density dependence of the effective transport coefficients as revealed by a linear stability analysis of the nonlinear PDE system. The instability is an inherent feature of existing continuum models and predicts, as a side product, the formation of a low-density (mechanical) weak layer on the sublimating side of an ice crust. The wave instability constitutes a key challenge for a faithful treatment of solid–vapor mass conservation between layers, which is discussed in view of the underlying homogenization schemes and their numerical solutions.