Behavior Of Linear Systems Research Articles

Closed-Form Solutions for the Response of Linear Systems to Fully Nonstationary Earthquake Excitation

New explicit closed-form solutions are derived for the evolutionary correlation and power spectral–density (PSD) matrices characterizing the nonstationary response of linear elastic, both classically and nonclassically damped, multi-degree-of-freedom (MDOF) systems subjected to a fully nonstationary earthquake ground motion process. The newly developed earthquake ground motion model considered represents the temporal variation of both the amplitude and the frequency content typical of real earthquake ground motions. To illustrate the analytical results obtained, a three-dimensional unsymmetrical building equipped with viscous bracings is considered with a single-component ground motion acting obliquely with respect to the building principal directions. These new analytical solutions for structural response statistics are very useful in gaining physical insight into the nonstationary response behavior of linear dynamic systems subjected to realistic stochastic earthquake ground motion models. Furthermore, the evolution in time of the cross-modal correlation coefficients is examined and compared with the classical stationary solution for white-noise ground motion excitation. The effects of cross-modal correlations on various mean-square response quantities also are investigated using the analytical solutions obtained.

Eine vollständige Charakterisierung des Verhaltens energetischstabiler linearer zeitdiskreter Systeme

This paper investigates the stability behaviour of time variant linear time discrete systems. Two versions of stability of the energy-norm are compared. The equivalence of both concepts is proved. The BIBOstability of time discrete linear systems is also investigated. It is shown, that a similar result for the BIBOstability holds. The results clarify some relevant concepts in the literature.

First Harmonic Analysis of Linear Control Systems with High-Gain Saturating Feedback

A first harmonic approach (describing function method) is used in this work to study the behavior of linear control systems with a saturating high-gain linear feedback. It is shown that when the eigenvalues of the closed-loop system are located deeper in the left-half complex plane, several unstable periodic orbits shrink to the origin, thus leading to an unstable equilibrium point. This dynamical behavior is interpreted as the vanishing of the region of attraction of the origin when a saturating high-gain feedback is used.

Componentwise stability of linear systems: a non-symmetrical case

In this paper, we present an extension of the concept of componentwise asymptotic (exponential) stability of linear systems in the non-symmetrical case. The main motivation for these results is the need for a more detailed evaluation of the dynamical behaviour of linear systems in electrical engineering and biology. Necessary and sufficient conditions for componentwise asymptotic (exponential) stability in the non-symmetrical case are given. The symmetrical case is obtained as a particular case. © 1997 by John Wiley & Sons, Ltd.

Analysis and redesign of an antilock brake system controller

Antilock braking (ABS) systems are a combination of hydromechanical braking components with switching solenoids and relays which operate the brakes in response to wheel and vehicle speed information. The control law which calculates the braking is a complex set of logical rules designed to allow for variations in road conditions and the nonlinear nature of tyre friction characteristics. The piecewise linear action of the actuator components (solenoids and relays) and the logical control rules make it difficult to analyse the dynamical behaviour of an ABS using conventional methods. The paper describes the analysis and subsequent redesign of an ABS using a new method for studying the dynamical behaviour of piecewise linear systems. The method is applied to a conventionally designed ABS controller. Undesirable performance features, not evident from conventional simulation and testing procedure, are found and the controller is redesigned to give a satisfactory dynamical performance.

A hierarchical stress release model for synthetic seismicity

We construct a stochastic dynamic model for synthetic seismicity involving stochastic stress input, release, and transfer in an environment of heterogeneous strength and interacting segments. The model is not fault‐specific, having a number of adjustable parameters with physical interpretation, namely, stress relaxation, stress transfer, stress dissipation, segment structure, strength, and strength heterogeneity, which affect the seismicity in various ways. Local parameters are chosen to be consistent with large historical events, other parameters to reproduce bulk seismicity statistics for the fault as a whole. The one‐dimensional fault is divided into a number of segments, each comprising a varying number of nodes. Stress input occurs at each node in a simple random process, representing the slow buildup due to tectonic plate movements. Events are initiated, subject to a stochastic hazard function, when the stress on a node exceeds the local strength. An event begins with the transfer of excess stress to neighboring nodes, which may in turn transfer their excess stress to the next neighbor. If the event grows to include the entire segment, then most of the stress on the segment is transferred to neighboring segments (or dissipated) in a characteristic event. These large events may themselves spread to other segments. We use the Middle America Trench to demonstrate that this model, using simple stochastic stress input and triggering mechanisms, can produce behavior consistent with the historical record over five units of magnitude. We also investigate the effects of perturbing various parameters in order to show how the model might be tailored to a specific fault structure. The strength of the model lies in this ability to reproduce the behavior of a general linear fault system through the choice of a relatively small number of parameters. It remains to develop a procedure for estimating the internal state of the model from the historical observations in order to use the model for forward prediction.

Dynamical anti-reset windup method for discrete-time saturating systems

This paper presents a dynamical anti-reset windup method for multivariable discrete-time saturating control systems. With the intention of maintaining the behavior of unsaturated linear systems as far as possible, an additional dynamical compensator is determined explicitly for the saturating system. Due to the additional compensator, the state positioning problem of saturated systems is completely resolved. The proposed method guarantees total stability of resulting systems, and asymptotic stability in the absence of exogeneous inputs can be addressed easily. The resulting dynamics of the compensated controller reflects the linear closed-loop system. The proposed method is applicable to any open-loop stable plants with saturating actuators and linear controllers. Simulation examples are included to illustrate the effectiveness of the proposed method.

An experimental impact oscillator

The present work considers the impacting behavior of a piecewise linear experimental system, a previous version of which has successfully displayed a wide variety of non-linear, including chaotic, behavior in other previous experiments [1–3]; making use of the ‘ball rolling on a hill’ concept, a cart is constrained to roll along an ‘energy well’ track, shaped in the form of a parabola such that the governing equations of motion of the cart are almost linear. In contrast to previous studies, where the impact condition was fixed (usually at equilibrium), the rigid barrier is placed at varying positions along the track in this study to provide a displacement constraint that gives rise to a discontinuity in the restoring force on the cart (provided solely by gravity in the non-impacting region). Varying the position of the impact can significantly change the overall behavior of the system, and this experimental study will concentrate on behavior where the impact point is varied among three different positions along the parabolic track, one chosen at a position less than equilibrium, one chosen at equilibrium, and one chosen greater than equilibrium; wide and varied responses are observed for these cases. Data were collected with the LabVIEW object-oriented, programmable interface, about which a few salient features are also discussed.

The Dynamic Stiffness Method for Linear Rotor-Bearing Systems

The behavior of linear rotor-bearing systems is investigated by using the exact approach of the dynamic stiffness method, which entails the use of continuous rather than lumped models. In particular, the theoretical formulation for rotor systems with anisotropic bearings is developed by utilizing the complex representation of all the involved variables. The proposed formulation eventually leads to the 8 × 8 complex dynamic stiffness matrix of the rotating Timoshenko beam; this matrix proves to be related, by a simple rule, to the 4 × 4 dynamic stiffness matrix, which describes rotor systems with isotropic bearings. The method is first applied to the critical speeds evaluation of a simple rotor system with rigid supports; for this case, the exact results of the dynamic stiffness approach are compared to the usual convergence procedure of the finite element method. Successively, the steady-state unbalance response of two rotor systems with anisotropic supports is analyzed; for these examples, the dynamic stiffness results compare favorably with the results of the finite element and the transfer matrix analysis performed by other authors.

The Existence of a Discrete-Event Representation of Linear Continuous-Variable Systems with Quantised State Measurements

This paper is concerned with the representation of stable linear discrete-time continuous-variable systems with quantised state measurements. It is assumed that the state space IRn is partitioned into disjoint regions Qx and that the number z of the region Qx(z) to which the state x belongs at time k is used as the qualitative value of x(k): [x] = z. For a given z(0) = [x(0)] the system can generate different qualitative trajectories [X] = [x(0)], [x(1)], [x(2)]… The paper presents a necessary and sufficient condition under which the sequence [X] is a Markov chain. This condition is rather restrictive and shows that there is, in general, no representation of the qualitative behaviour of linear systems by some stochastic automaton.

Componentwise stability of continuous-time delay linear systems

This note deals with a special type of asymptotic (exponential) stability, namely componentwise asymptotic (exponential) stability for linear continuous-time delay systems. The main motivation for these results is the need for the evaluation in a more detailed manner of the dynamical behavior of linear systems in electrical engineering and biology. Necessary and sufficient conditions for componentwise asymptotic (exponential) stability are given.

An explicit closed-form solution for linear systems subjected to nonstationary random excitation

Explicit, closed-form solutions are presented for the correlation matrix and evolutionary power spectral density matrix of the response of a linear, classically damped MDOF system subjected to a uniformly modulated random process with the gamma envelope function. The effects of the statistical cross-modal correlations on the evolutionary mean square responses are investigated. A simple MDOF system subjected to ground motion is used as an illustrative example. Through this study, additional insight is gained into the nonstationary behavior of linear dynamic systems.

The decay time of the time-dependent transient stochastic dynamics, in presence of an external force

We characterize the transient behavior of the time-dependent linear stochastic systems in presence of an external force, by means of the integration of the nonlinear relaxation time (NLRT), supported by the quasi-deterministic (QD) theory. The formalism is applied to study the transient dynamics of the switch-on times of a single mode semiconductor laser, in presence of an external optical signal, as a prototype model. The time scale, for this laser model is essentially given in terms of a relevant scaling parameter.

Oscillatory behaviour of controllable single-input single-output positive linear discrete-time systems

Oscillatory behaviour of controllable single-input single-output positive linear discrete-time systems

Stability behavior of linear time-varying systems

Stability behavior of linear time-varying systems

Identification of linear and nonlinear dynamic systems using recurrent neural networks

This paper describes the use of Elman-type recurrent neural networks to identify dynamic systems. Networks as originally designed by Elman ( Cognitive Sci., 1990, 14, 179–211) and also those in which self-connections are made to the context units were employed to identify a variety of linear and nonlinear systems. It was found that the latter networks were more versatile than the basic Elman nets in being able to model the dynamic behaviour of high order linear and nonlinear systems.

The convergence of object dependent resolution in maximum likelihood based tomographic image reconstruction

Study of the maximum likelihood by EM algorithm (ML) with a reconstruction kernel equal to the intrinsic detector resolution and sieve regularization has demonstrated that any image improvements over filtered backprojection (FBP) are a function of image resolution. Comparing different reconstruction algorithms potentially requires measuring and matching the image resolution. Since there are no standard methods for describing the resolution of images from a nonlinear algorithm such as ML, the authors have defined measures of effective local Gaussian resolution (ELGR) and effective global Gaussian resolution (EGGR) and examined their behaviour in FBP images and in ML images using two different measurement techniques. For FBP these two resolution measures are equal and exhibit the standard convolution behaviour of linear systems.

Seismic response spectra of a combined system by Green's functions

A parametric study of the response of a class of linear secondary systems to earthquake excitations is made. The combined dynamical system studied consists of a single-degree-of-freedom oscillator attached to a continuous cantilever beam. The generalized differential equation governing free vibration is solved exactly for the modes of the combined system. Floor response spectra for records from the 1985 Mexico City and the 1940 El Centro earthquakes are presented in several formats. The floor response spectrum concept is an extension of the widely used ground response spectrum and may provide useful insight into the behavior of linear secondary systems. The effects of each of the system parameters are examined. It is shown that the frequency characteristics of the excitation modify the relative importance of the system parameters. The study also demonstrates the effectiveness and limitations of the cascade approximation which is often applied as an approximate solution to the problem.

State Space Decoupling of Non-Minimum Phase Systems

The decoupling of the input - output behaviour of linear multivariable systems by constant state- feedback generally requires the compensation of all invariant zeros, which causes unstability in the case of non - minimum phase systems. The paper presents a method for partial but stable decoupling, ensuring without compensations coupling of only one output with several inputs. The application of a dynamic state-feedback however allows complete and stable decoupling with the disadvantage of an increased number of disagreeable transmission zeros.

Structure-variable Model-following Control for Hydraulic Drives

Mechanical systems can be improved in their efficiency, if electronic signal processing is added. A typical example is the digital control of hydraulic drives. A successful industrial application of hydraulic drives depends on robustness, reliability and low cost. This paper presents a control scheme for hydraulic drives, in which low cost proportional valves are used instead of much more expensive servovalves. In order to reduce the electronic expense, the concept of pulse modulation is proposed. Therefore, digital-analog converters and relatively expensive analog power amplifiers can be saved. An essential disadvantage of the components mentioned above is their rather non-linear behaviour. For this reason, a special model-following control scheme is developed in order to force a linear system behaviour. The intended linearization is achieved in three steps.1.feed forward compensation of the static nonlinearities of the valve.2.partial improvement of the dynamics by a proportional feedback and.3.inbedding of the just mentioned compensated feedback system into a particular structure-variable model-following control scheme. By prescribing the linear model system, the dynamic behaviour of the hydraulic drive can be influenced within some practical limits.The paper describes the nonlinear hydraulic components lations of the proposed control scheme. In particular, investigations are given. and presents the theoretic reresults of practical test-stand.