Behavior Of Linear Systems Research Articles

Limits of linearity in squeeze film behavior of a single degree of freedom microsystem

This paper presents a theoretical investigation of squeeze film flow in systems employing microplates parallel to a substrate and undergoing large amplitude normal vibration. Most previous models of squeeze film damping assume small oscillation amplitude with linear system behavior, but it is often unclear how small the vibrations must be to actually elicit this response. In addition, fluid inertia effects are usually overlooked. This study provides a compact nonlinear solution for the incompressible hydrodynamic forces with specific terms describing fluid inertia and viscous damping. Numerical analysis (the explicit Runge-Kutta method) is applied to solve the nonlinear governing equation. The effects of frequency, oscillation amplitude, aspect ratio (of gap to length), and Reynolds number on the dynamic response of the system are investigated. The overall system response depends strongly on the actuation frequency and system properties. It is found that a simple criterion of validity for the linear system assumption is not possible. Near resonance, the vibration input amplitude (relative to the initial gap) must be very small indeed for linearity (∼0.001), while in other cases the relative amplitude can be greater than one.

Linear recruitment leads to allocation and flexibility in collective foraging by ants

Positive feedback in communication is central to the emergence of collective behaviour in animal groups. The nonlinearity hypothesis predicts that group behaviour will be defined in large part by the degree of cooperative interaction in this feedback. Strong interactions mean that the effectiveness of communication grows nonlinearly with signal quantity, leading to robust spatial patterning and consensus decision making. These predictions have been supported in many experimental systems, but the corresponding behaviour of linear systems, where signal effectiveness is proportional to signal quantity, has not been well explored. We examined the consequences of tandem running, a linear method of recruitment, for collective decision making by foraging colonies of the ant Temnothorax rugatulus. We found that colonies collectively chose the better of two unequal feeders, but they allocated foragers equally when feeders were identical. This result accords with theoretical predictions and contrasts with symmetry breaking seen in species with highly nonlinear trail pheromone recruitment. Colonies were also able to reallocate foragers when the quality of two unequal feeders was switched, again in accord with theoretical expectations. We built a model based upon the behaviour of individual ants and found that colony-level qualities of choice, allocation and flexibility can be predicted by two behaviours: quality-dependent linear recruitment and quality-dependent attrition. These experiments are the first thorough investigation of the consequences of linear recruitment for collective animal behaviour.

A methodology for the prediction of structure level costs based on element condition states

The determination of work programmes for bridges that maximise benefit to all stakeholders requires consideration of not only the condition of the elements of which a bridge is comprised, but also the performance of the structure as a whole. This is required because although some costs can be related to the condition of the elements, others, such as the costs of travelling over the bridge, cannot. One possible methodology to do this involves linking relevant costs to structure performance states (SPS) that are determined from the element condition states (CSs). However, even a bridge with a moderate number of elements and element CSs results in a large number of possible SPSs and an unwieldy amount of required work to estimate the costs associated with each. This work can be drastically reduced, however, by exploiting the almost linear system behaviour of the bridge that can occur between many combinations of element CSs. This article presents a methodology for relating SPSs to element CSs for the purpose of determining structure level costs. This methodology is demonstrated in several examples, wherein the effects of exploiting the linearity of the system behaviour to reduce the computational effort are also explored.

Teaching and Learning the Interplay Between Chance and Determinism in Nonlinear Systems

That the interplay of random and deterministic processes may result in both the limited predictability of nonlinear systems and the formation of structures seems to be a most valuable general insight into the nature of science. This study investigates the possibility of teaching and learning the interplay of chance and determinism in nonlinear systems in school science instruction. An analysis of the relevant scientific literature and an explorative learning process study with 30 11th-grade students are included. Three experiments displaying the behaviour of nonlinear systems (deterministic chaos, self-organization and fractals) and one experiment demonstrating the behaviour of linear systems were discussed using a teaching experiment design. The findings show that although the majority of students initially considered chance and determinism as contradictory conceptions, they finally developed sound explanations concerning the interplay of chance and determinism in the investigated nonlinear systems.

Multi-model identification of a fractional non linear system

Takagi-Sugeno fuzzy models have proved to be a powerful tool for non-linear system identification. Due to their simplicity, linear integer sub-models are mainly used in such structure. In this paper multi-model approach, based on Takagi-Sugeno form is used to perform identification of fractional non-linear systems, applying fractional local models. In this purpose, an output error method is used, and the algorithm is extended to the fractional case. Numerical simulations are performed in order to analyze the two sub-models: integer and fractional, fitting ability to approximate the non linear system behaviour. The validation of the method is carried out on real data of the heat diffusion.

Distance between Behaviors and Rational Representations

In this paper we study notions of distance between behaviors of linear differential systems. We introduce four metrics on the space of all controllable behaviors which generalize existing metrics on the space of input-output systems represented by transfer matrices. Three of these are defined in terms of gaps between closed subspaces of the Hilbert space $\mathcal{L}_2(\mathbb{R})$. In particular we generalize the “classical” gap metric. We express these metrics in terms of rational representations of behaviors. In order to do so, we establish a precise relation between rational representations of behaviors and multiplication operators on $\mathcal{L}_2(\mathbb{R})$. We introduce a fourth behavioral metric as a generalization of the well-known $\nu$-metric. As in the input-output framework, this definition is given in terms of rational representations. For this metric, however, we establish a representation-free, behavioral characterization as well. We make a comparison between the four metrics and compar...

Lateral resolution and transfer characteristics of vertical scanning white-light interferometers

White-light interferometers are widely used for high-accuracy topography measurement in industrial and scientific applications. A common way to characterize a white-light interferometer is to assume small surface amplitudes resulting in linear transfer characteristics described by the instrument transfer function (ITF). However, the well-known batwing effect gives rise to systematic errors, causing extra nonlinearity to the ITF. In this paper a model to simulate an interference pattern in the image plane as it is obtained by a vertical scanning white-light interferometer is introduced in order to overcome the limitation of small surface amplitudes. Repeating the simulation procedure for different height positions of the object results in an image stack that can be analyzed by the same algorithms as real measurement data. The simulation results agree with experimental observations: the batwing effect occurs in certain situations and the correct amplitude of a rectangular grating structure can be obtained as long as the structure is optically resolved. Both simulation, as well as experimental results, provide transfer characteristics of broader bandwidth than predicted by theoretical approaches based on linear system behavior.

Longitudinal Aircraft Dynamics and the Instantaneous Acceleration Center of Rotation

Nonminimum-phase zeros, that is, closed-right-half-plane (CRHP) zeros, affect both the open-and closed-loop behavior of continuous-time linear systems in undesirable ways [1]. For example, an asymptotically stable linear system with an odd number of positive zeros experiences initial undershoot to a step input (see “Initial Undershoot”). Moreover, under the rules of root locus, zeros in the open-right-half plane (ORHP) attract closed-loop poles, which limits the controller gain and thus the performance of the closed-loop system.

A framework and model system to investigate linear system behavior in Escherichia coli

BackgroundThe ability to compose biological systems from smaller elements that act independently of the other upon assembly may help make the forward engineering of biological systems practical. Engineering biology in this manner is made difficult by the inherent nonlinear response of organisms to genetic devices. Devices are inevitably coupled to one another in the cell because they share the same transcriptional machinery for expression. Thus, new properties can emerge when devices that had been characterized in isolation are expressed concurrently. We show in this report that, similar to physical systems, the Escherichia coli (E. coli) transcriptional system can exhibit linear behavior under "small" perturbation conditions. This, in turn, allows devices to be treated as independent modules.ResultsWe developed a framework and model system consisting of three devices to investigate linear system behavior in E. coli. Our framework employed the transfer curve concept to determine the amount of nonlinearity elicited by the E. coli transcriptional system in response to the devices. To this effect, the model system was quantitatively characterized using real-time quantitative PCR to produce device transfer curves (DTCs). Two of the devices encoded the bacterial neomycin phosphotransferase II (nptII) and chloramphenicol acetyl transferase (cat), while the third encoded the jellyfish-originating green fluorescent protein (gfp). The gfp device was the most nonlinear in our system, with nptII and cat devices eliciting linear responses. Superposition experiments verified these findings, with independence among the three devices having been lost when gfp was present at copy numbers above the lowest one used.ConclusionsWe show that linear system behavior is possible in E. coli. Elucidation of the mechanism underlying the nonlinearity observed in gfp may lead to design rules that ensure linear system behavior, enabling the accurate prediction of the quantitative behavior of a system assembled from individually characterized devices. Our work suggests that biological systems follow principles similar to physical ones, and that concepts borrowed from the latter (such as DTCs) may be of use in the characterization and design of biological systems.

Application of Extrinsic Information Transfer Charts to Anticipate Turbo Code Behavior

This paper examines turbo codes that are currently introduced in many international standards, including the UMTS standard for third generation personal communications and the ETSI DVB-T standard for Terrestrial Digital Video Broadcasting. The convergence properties of the iterative decoding process associated with a given turbo-coding scheme are estimated using the analysis technique based on so-called extrinsic information transfer (EXIT) chart. This approach provides a possibility to anticipate the bit error rate (BER) of a turbo code system using only the EXIT chart. It is shown that EXIT charts are powerful tools to analyze and optimize the convergence behavior of iterative systems utilizing the turbo principle. The idea is to consider the associated SISO stages as information processors that map input a priori LLR’s onto output extrinsic LLR’s, the information content being obviously assumed to increase from input to output, and introduce them to the design of turbo systems without the reliance on extensive simulation. Compared with the other methods for generating EXIT functions, the suggested approach provides insight into the iterative behavior of linear turbo systems with substantial reduction in numerical complexity.

Equivalence of rational representations of behaviors

This article deals with the equivalence of representations of behaviors of linear differential systems. In general, the behavior of a given linear differential system has many different representations. In this paper we restrict ourselves to kernel and image representations. Two kernel representations are called equivalent if they represent one and the same behavior. For kernel representations defined by polynomial matrices, necessary and sufficient conditions for equivalence are well known. In this paper, we deal with the equivalence of rational representations, i. e. kernel and image representations that are defined in terms of rational matrices. As the first main result of this paper, we will derive a new condition for the equivalence of rational kernel representations of possibly noncontrollable behaviors. Secondly we will derive conditions for the equivalence of rational representations of a given behavior in terms of the polynomial modules generated by the rows of the rational matrices. We will also establish conditions for the equivalence of rational image representations. Finally, we will derive conditions under which a given rational kernel representation is equivalent to a given rational image representation.

Periodic orbits for perturbations of piecewise linear systems

We consider the existence of periodic orbits in a class of three-dimensional piecewise linear systems. Firstly, we describe the dynamical behavior of a non-generic piecewise linear system which has two equilibria and one two-dimensional invariant manifold foliated by periodic orbits. The aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system. In order to analyze this situation, we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation. By using this function, we state some results of existence and stability of limit cycles in the perturbed system, as well as results of bifurcations of limit cycles. The techniques presented are similar to the Melnikov theory for smooth systems and the method of averaging.

Phase Behavior of Linear Mixtures in the Context of Equation of State Models

Models of the equation of state (EOS) type, based on the van der Waals (vdW) EOS, are able to deal with phase equilibrium properties of asymmetric systems over wide ranges of pressure, composition, and temperature. EOS type models are of great practical interest. In general, EOSs always give a nonideal behavior for mixtures, regardless the form of the set mixing rules. In other words, EOSs have no access to the “ideal solution” state. However, EOS type models do have access to the “linear system” (LS) state. A LS is a mixture whose EOS parameters depend linearly on composition. In a LS, the partial molar parameter of a given component within a multicomponent system equals the pure compound parameter. This could be seen as an alternative form of “ideality”. In this work, we study the phase equilibria of linear systems. We evaluate the phase behavior of binary linear (and also nonlinear) systems by generating different types of charts including pressure−temperature projections of phase equilibrium univarian...

Approximating distributional behaviour of linear systems using Gaussian function and its derivatives

The paper is concerned with defining families of smooth functions that can be used for the approximation of impulsive types of solutions for linear systems. We review the different types of approximations of distributions in terms of smooth functions and explains their significance in the characterization of system properties where impulses were used for their characterisation. For controllable systems, we establish an interesting relation between the time t and sigma (volatility) in the approximation of distributional solutions. An algorithm is then proposed for the calculation of the coefficients of the input required to minimize the distance of our desired target state before and after approximation is proposed. The optimal choice of sigma is derived for a pre-determined time t for the state transition.

Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results

The present state-of-the-art article is devoted to the analysis of new trends and recent results carried out during the last 10years in the field of fractional calculus application to dynamic problems of solid mechanics. This review involves the papers dealing with study of dynamic behavior of linear and nonlinear 1DOF systems, systems with two and more DOFs, as well as linear and nonlinear systems with an infinite number of degrees of freedom: vibrations of rods, beams, plates, shells, suspension combined systems, and multilayered systems. Impact response of viscoelastic rods and plates is considered as well. The results obtained in the field are critically estimated in the light of the present view of the place and role of the fractional calculus in engineering problems and practice. This articles reviews 337 papers and involves 27 figures.

DESIGN OF RECONFIGURABLE MULTIPLE ELEMENTS MICROSTRIP RECTANGULAR LINEAR ARRAY ANTENNA

This paper presents a reconfigurable multiple element microstrip rectangular linear array antenna integrated with radio frequency (RF) switches. The corporate feed design concept is used to excite the linear array antenna that consists of 8 elements of rectangular patches at 5.8 GHz. Two PIN diode switches were deployed at the feeding line to activate the two arrays of patches that are located on the left and right sides of the antenna structure. The behavior of the reconfigurable multiple element linear antenna array system has been investigated with respect to the beam shaping characteristic. The comparisons of the performance between two structures, with and without Wilkinson Power Divider (WPD) are discussed in this paper. Two different beam patterns were achieved through the reconfigurable antenna at different number of elements design that incorporates with PIN diode switches and modified WPD concept. The simulation and the measurement results for 4 and 8 elements array antenna structure are presented.

Static and dynamic scaling of semiflexible polymer chains—a molecular dynamics simulation study of single chains and melts

A molecular dynamics study of the effect of increasing molecular chain stiffness on the dynamic properties of single linear chains and polymer melts is presented. The chain stiffness is controlled by a bending potential. By means of a normal mode analysis the systematic crossover behavior of coarse-grained linear polymer systems from Rouse modes to bending modes with increasing mode number p is presented systematically for the first time via simulation data. The magnitude and the onset of the region where crossover behavior occurs is investigated in dependence of a systematic variation of the chains’ persistence lengths. For long wavelength modes the well-known p−2 behavior is observed which represents the Rouse scaling, whereas for the bending modes one observes a distinct p−4 scaling of the mean square mode amplitudes and the relaxation times. Additionally, the effect of this crossover behavior on the monomer dynamics given by their mean square displacements is investigated. The findings are contrasted with previous simulation studies of semiflexible chain behavior and it is shown that those studies—due to too small persistence lengths Lp—were actually limited to the crossover regime where both, Rouse and bending modes contribute to the chain dynamics, and no distinct p−4 scaling can be observed. Using an expansion of the monomer position vector, a simple theory of the observed scaling behavior is proposed which allows for deriving analytic expressions that describe the presented simulation data very well.

Automated behavioral modeling and analytical model-order reduction by application of symbolic circuit analysis for multi-physical systems

The aim of symbolic analysis that has its origin in the design of analog circuits is the extraction of dominant system behavior by automated derivation of approximated symbolic formulas. Since exact symbolic analysis will yield exceptionally complex expressions even for rather small systems a class of symbolic approximation techniques have been developed that allow a reduction of the complexity of symbolic equations and their later solution by means of mixed symbolic and numerical strategies. Hence, it becomes possible to reduce the underlying nonlinear differential-algebraic systems of equations (DAE systems) of component-based networks and systems to a behavioral description of a predefined accuracy. It is a major advantage of the approach that the model simplification is performed by an automatic error control and that the simplified model is physically interpretable again. The contribution will give an overview of the symbolic tool Analog Insydes algorithms for extraction of dominant behavior of linear systems, e.g. formulas for poles and zeros as well as algorithms for generating behavioral models from nonlinear DAEs. Moreover, the underlying methodology has been extended to the application of analysis and modeling of gas-pipeline nets and mixed electrical and mechanical systems. For the latter a library was developed in cooperation with the Fraunhofer IIS/EAS for symbolic models of micro-mechanical elements that can be connected to networks, even together with electrical components.

Projective re-normalization for improving the behavior of a homogeneous conic linear system

In this paper we study the homogeneous conic system \(F: Ax = 0, x \in C\setminus \{0\}\) . We choose a point \(\bar s \in {\rm int}C^*\) that serves as a normalizer and consider computational properties of the normalized system \(F_{\bar s} : Ax = 0, \bar s^T x = 1, x \in C\) . We show that the computational complexity of solving F via an interior-point method depends only on the complexity value \(\vartheta\) of the barrier for C and on the symmetry of the origin in the image set \(H_{\bar s} := \{Ax {\bar s}^Tx = 1, x \in C\}\) , where the symmetry of 0 in \(H_{\bar s}\) is$$ {\rm sym}(0, H_{\bar s}) := {\rm max}\{\alpha : y \in H_{\bar s} \Rightarrow -\alpha y \in H_{\bar s} \} .$$We show that a solution of F can be computed in \(O(\sqrt{\vartheta}{\rm ln}(\vartheta/{\rm sym}(0, H_{\bar s}))\) interior-point iterations. In order to improve the theoretical and practical computation of a solution of F, we next present a general theory for projective re-normalization of the feasible region \(F_{\bar s}\) and the image set \(H_{\bar s}\) and prove the existence of a normalizer \({\bar s}\) such that \({\rm sym}(0,H_{\bar s}) \ge 1/m\) provided that F has an interior solution. We develop a methodology for constructing a normalizer \({\bar s}\) such that \({\rm sym}(0, H_{\bar s}) \ge 1/m\) with high probability, based on sampling on a geometric random walk with associated probabilistic complexity analysis. While such a normalizer is not itself computable in strongly-polynomial-time, the normalizer will yield a conic system that is solvable in \(O(\sqrt{\vartheta}{\rm ln}(m\vartheta))\) iterations, which is strongly-polynomial-time. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective re-normalization methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46% decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1,000 × 5,000.

Oscillatory behavior of linear matrix Hamiltonian systems

AbstractWe establish some new oscillation criteria for the matrix linear Hamiltonian system X ′ = A (t)X + B (t)Y, Y ′ = C (t)X –A *(t)Y by using a new function class X and monotone functionals on a suitable matrix space. In doing so, many existing results are generalized and improved. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)