Behavior Of Linear Systems Research Articles

Experimental Evaluation of Damping in a Steel Tubular Member of an Offshore Platform Jacket in Air

Abstract Pluck tests were performed on a major diagonal member of an offshore platform jacket in the fabrication yard to develop data for assessment of structural damping. The initial displacements were varied in order to investigate dependence of damping on amplitude. The response histories were typically composed of six or seven vibrational modes having closely spaced frequencies in the 3.4–4.2-Hz range. System identification was accomplished using an output error approach. Response histories reconstructed from the system parameters identified by this method exhibit excellent fidelity to the corresponding recorded data traces. The analysis results support the assumption of linear system behavior and lead to a suggested value of 0.002 for the critical damping ratio.

Frequency domain direct parameter identification for modal analysis: State space formulation

This paper outlines the theoretical development of a modal identification technique for the dynamic behaviour of linear time-invariant systems. Using a state space formulation for the equations of motion for a mechanical structure, an algorithm is developed to identify its state transition matrix from measured multiple input/multiple output relations (frequency response functions). From the identified matrix quantities, it is possible to derive a modal model of natural frequencies, damping values, mode shapes and modal participation factors. Aspects of data reduction via singular value analysis are discussed to extract valid state variables from the entire set of response stations. Emphasis is put on the application of this frequency domain direct parameter identification technique for the analysis of non-uniformly sampled frequency response functions, as generated for example when automated stepped-sine excitation is applied for modal testing of mechanical structures.

More about continuity of dynamical systems

In this note we continue the investigations started in [1]. We give sufficient conditions for the convergence of the external behavior of discrete-time time-invariant finite-dimensional linear (input/state/output) systems. We also state some interesting conjectures.

Asymptotic behaviour and boundedness of linear systems with time varying coefficients

Motivated by developments in spacecraft dynamics, the asymptotic behaviour and boundedness of solution of a special class of time varying systems in which each term appears as the sum of a constant and a time varying part, are analysed in this paper. It is not possible to apply standard textbook results to such systems, which are originally in second order. Some of the existing results are reformulated. Four theorems which explore the relations between the asymptotic behaviour/boundedness of the constant coefficient system, obtained by equating the time varying terms to zero, to the corresponding behaviour of the time varying system, are developed. The results show the behaviour of the two systems to be intimately related, provided the solutions of the constant coefficient system approach zero are bounded for large values of time, and the time varying terms are suitably restrained. Two problems are tackled using these theorems.

A transfer matrix approach to feedback invariants of linear systems: Application to decoupling

The transfer matrix approach allows one to exhibit fairly easily fundamental invariants of multivariable systems under the action of transformation groups. These transformation groups include the classical state feedback, state, and input changes of coordinates. Then the solvability of numerous classical control problems (e.g. decouplability, disturbance rejection, model following) can be expressed directly in terms of these invariants. In this expository paper we present in a unified way three lists of feedback invariants related to the infinite behavior of linear systems. These invariants turn out to be crucial in studying feedback decoupling problems. The main tools for this study are factorization of transfer matrices over the P.I.D. of proper rational functions. When dealing with stability we factor transfer matrices over the P.I.D. of proper rational stable functions. We illustrate the use of these invariants by studying some decoupling problems.

Identification of dynamic Green's functions in structural networks

This paper presents the mathematical and computational basis of a method for experimentally identifying the dynamic behavior of linear structural systems. The method consists of exciting the structure with specific pulses and reconstructing the dynamic Green's functions by deconvolution of the measured response. The reconstruction procedure involves the solution of an ill-posed integral equation in the presence of noise. Inverse Gaussian functions, which have mathematical properties that render the deconvolution problem tractable, are used as probing pulses. Two examples are presented for which dynamic Green's functions are reconstructed from synthetic response data: 1) a nondispersive medium consisting of a mechanical slider with free body motion, and 2) a network that may be representative of a large orbiting structure with rigidly connected torsional members and Timoshenko beams. It is shown that residual effects of noise that may be detectable in the reconstructed Green's functions are effectively smoothed out when these functions are used to predict the response to forces such as may be induced by control systems. Finally, an example is presented of deconvolution implemented in slow motion.

Uniqueness of the asymptotic behaviour of autonomous and non‐autonomous, switched and non‐switched, linear and non‐linear systems of dimension 2

AbstractAn explicit criterion is given which guarantees that the solutions of a system of two first‐order differential equations have a unique asymptotic behaviour as t'→ + ∞. the equations are allowed to be time‐dependent, with isolate discontinuities (switching), and non‐linear. In particular, continuously differentiable and piecewise linear systems, as far as the dependence on the state variables is concerned, are considered. the particular case of a linear system switched back and forth between two 2 × 2 matrices is treated in detail, with an outline of an algorithm to decide on the converence to zero of the solutions.The case of switching between N matrices is a straightforward generalization that is, however, not covered in this paper. It is shown that the problem of unique asymptotic behaviour of piecewise linear systems can always be treated by considering such switched linear systems. Furthermore, an example shows that even some continuously differentiable systems may be reduced to this case. Finally, the connection with the work of Brayton and Tong on the stability of differential equations, also using sets of matrices, is given.

Rotating Gratings Reveal the Temporal Transfer Function of the Observing System

Abstract It is shown that a rotating sinusoidal grating is a useful sweep signal for the analysis of the temporal behaviour of linear imaging systems. For a suitably chosen angular velocity and spatial frequency, the spatial a.c. component of the grating appears modulated in one dimension at the output of the system. The profile of this modulation is the temporal transfer function (TTF) of the system. A quantitative analysis of this effect is presented, and results from experiments with a photographic camera are shown. It is proposed to use this method in the field of vision research since it is presently the only way to demonstrate the complex-valued TTF of the visual system for suprathreshold grating stimuli. The main consequence of the first psycho-physical investigations was the discovery of a phase reversal at the origin of the TTF of the human visual system for gratings of low spatial frequency.

Transient stability of a power system using a non-Luré-type Lyapunov function

A new viewpoint is offered on the construction of a non-Lure-type Lyapunov function for studying the dynamic behaviour of linear systems with a non-linear feedback of the ‘classical’ sector type. The approach makes it possible to show that a particular relation between the parameters defining the Lyapunov function must be satisfied in order to ensure an estimate of the attraction domain closest to the true stability boundary.

Oscillation and nonoscillation for systems of two first-order linear differential equations with delay

Functional differential equations with delay provide a mathematical model for a physical or biology system in which the rate of change of the system depends upon its past history. In the literature, there are several papers [14] concerning the study of oscillatory behavior of solutions of first order differential equations with time delay. But there are few results [S] about the oscillation of linear system of first order equation with delay. In this paper, we study the oscillatory behavior of linear system with delay. First, in Section 2, we consider the case of constant coefficients in details. We show that the delay can cause or destroy the oscillations. We present constructive procedure to generate or destroy oscillations of a system. In section 3 we consider the case of variable coefficients and obtain some results analogous to the result in Section 2.

Structural properties and limit behaviour of linear stochastic systems in Hilbert spaces

Structural properties and limit behaviour of linear stochastic systems in Hilbert spaces

Variation in efficiency of parallel algorithms

This paper presents the investigation of the variation in efficiency of a parallel algorithm as a function of the number of processors. A parallel algortithm was developed which predicted the behavior of a typical linear engineering system by solving a set of linear equilibrium equations. Experimental results of the algorithm were obtained by simulating the parallel process on a uniprocessor. The results showed the parallel method to have certain advantages over certain sequential methods, thus showing promise for this parallel algorithm in being competitive against some sequential algorithms.

On stability of linear time-varying infinite-dimensional discrete-time systems

The asymptotic behaviour of linear time-varying infinite-dimensional discrete-lime systems is considered. The introduced notions are: weak power equistability, power equistability, uniform power equistability, l p -equistability, uniform / l p -equistability and l p(x) -equistability. It is shown that they are identical. A generalization of the concept of spectral radius of a single operator is also proposed. It is proven that any time-varying system is uniformly power equistable if and only if the generalized spectral radius of the sequence of the operators which define the system considered is less than one.

Relaxation phenomena and induced polarization

We deal with the general subject of the linear system behavior of material media that have dispersive electrical properties. In this review we deal with phenomenological theory which begins with a general statement of Ohm's law that relates the macroscopic electric field and the current density at a point in the medium. The complex constant of proportionality is the complex resistivity that forms the basis of the discussion. Various ways to represent the frequency dependence of the complex resistivity are described. The corresponding time domain or transient response characteristics are also considered in the context of the induced polarization properties of the medium. We point out that it is often fruitful to represent the dispersion as a superposition of basic relaxation processes. In fact, if such distributions are sufficiently broad, the effective complex resistivity varies in magnitude in almost a linear manner with the logarithm of frequency. The corresponding phase is approximately a constant that is proportional to the slope of the amplitude curve on log frequency basis. This same slope parameter is also descriptive of the transient decay of the system when the linear response is plotted on a log time base. This general behavior seems to be a general property of such composite media as textured rock, mineralized material and biological suspensions where we can expect that the scale of the dispersive mechanisms is quite broad.

Singularly perturbed systems of diffusion type and feedback control

Asymptotic approximations describing the behaviour of linear systems of diffusion type (convective or non-convective) with a small diffusivity, to which a feedback control of distributed or boundary type based on point sensors is applied, are constructed and proven to be correct. As a consequence one can find a near-optimal feedback control for a cost minimization problem with a quadratic performance index measuring the deviation of the stationary state from an ideal state, under the restriction of a prescribed exponential degree of stability of the stationary state.

Estimate for the root-location of linear systems via the lyapunov matrix equation

It is well known that the Lyapunov matrix equation can be utilized to obtain the estimate for the transient behavior of linear constant systems. This paper shows that the equation gives information not only on the maximum real part of the characteristic roots of system matrix but also on other extremal values to these roots.

Realization theory of infinite-dimensional linear systems. Part II

This paper studies the real-time behavior of constant linear systems. A function space Λ is introduced to give a precise language to discuss the working mode of systems. It is shown that a realization of a constant linear input/output map produces the desired outputs during the application of inputs. A differential equation description is derived for those systems whose weighting patterns are sufficiently smooth. The notion of topological observability in bounded time yields a necessary and sufficient condition under which the canonical realization of a constant linear input/output map has a Banach state space.

Recent developments in the acoustic spectroscopy of violins

Between the oscillation of the bowed violin string and its final acoustic output stands the intricate dynamical system of the violin body and air cavity. The behavior of this (linear) system is under all circumstances completely characterized by its normal modes, or “resonances,” provided they are analyzed not only as to frequency but also as to complex eigenfunctions. By assembling such information we are in a position to compute the coupling of each mode to the driving points at the bridge and to all acoustic-field variables. The approach will be illustrated by data on several violins and by the way the corresponding transfer functions change with change in parameters such as soundpost placement. [Work supported by NSF.]

Cambridge Linear Analysis and Design Programs

This paper contains a brief description of a set of interactive computer programs which can be used to aid in the analysis of the dynamic behaviour of linear multivariable systems. The main emphasis is on frequency response methods including Nyquist and Bode arrays, closed-loop Nyquist and boae arrays, characteristic gain loci and generalised root loci. Several methods of designing control schemes using these analysis methods are provided within the programs. Other analysis aids include the drawing of hiemann surfaces, the calculation of zeros and closed-loop pole positions, the calculation of frequency responses, the calculation of time responses, and a complex matrix manipulation package. Two special features of this set of programs are the flexibility of the form of the system model, and the ability to read files of commands from disk.The programs consist of about 30,000 lines of source code, and are written in Fortran IV except for the more recent additions which are in a language which compiles into Fortran IV. The overlay structure useo allows the programs to run on a PDP11 in less than 64K bytes of core.

The use of frequency transmission concepts in linear multivariate system analysis

Abstract The classical work by Nyquist and Bode and its extensions to the multivariable case are based on a frequency-response approach to feedback systems. Modern state-space techniques, on the other hand, make use of an internal system description as dictated by a set of first-order linear differential equations. Relevant to both approaches is the problem of frequency propagation which in this paper is seen as the means of unifying the external and internal system description. The inextricable association of frequency transmission with the concept of (A, B)-invariant subspaces and controllability subspaces empowers this approach with the ability to effect a penetrating study of a number of aspects of linear system behaviour such as the restriction of the state vector in given subspaces and the characterization of such subspaces, the characterization of multivariate zeros and the categorization of systems into degenerate and non-degenerate. In this paper the study of the transmission of a particular freq...