Linear Parameter Systems Research Articles

Seismic design of inelastic structures using equivalent linear system parameters: part 2 – application and verification

In this second part of the two-part article, the incorporation of a set of equivalent linear system parameters derived in the preceding article in a direct displacement-based design (DDBD) method is presented. By using the proposed design procedures for reinforced concrete and steel, 16 structures modelled as single degree-of-freedom systems are designed. Their responses are then compared with those obtained from a dynamic inelastic time-history analysis. The results show that the use of the proposed equations to calculate ζeq and T eq and the incorporation of the proposed DDBD procedures to proportion structures will produce reasonably good results with relatively simple computation effort.

Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition

We address the problem of model order reduction (MOR) of parametrized dynamical systems. Motivated by reduced basis (RB) methods for partial differential equations, we show that some characteristic components can be transferred to model reduction of parametrized linear dynamical systems. We assume an affine parameter dependence of the system components, which allows an offline/online decomposition and is the basis for efficient reduced simulation. Additionally, error control is possible by a posteriori error estimators for the state vector and output vector, based on residual analysis and primal-dual techniques. Experiments demonstrate the applicability of the reduced parametrized systems, the reliability of the error estimators and the runtime gain by the reduction technique. The a posteriori error estimation technique can straightforwardly be applied to all traditional projection-based reduction techniques of non-parametric and parametric linear systems, such as model reduction, balanced truncation, moment matching, proper orthogonal decomposition (POD) and so on.

Adaptive recursive M-robust system parameter identification using the QQ-plot approach

A new adaptive algorithm for the robust estimation of parameters of linear dynamic discrete-time systems in the presence of non-Gaussian impulsive noise within a measurement sequence is proposed in this study. Starting from the theory of robust estimation, a simple, adaptive, practically applicable robust approximate maximum likelihood algorithm is derived that, in the cases of contaminated normal distribution of measurement noise, demonstrates a high level of efficiency. The QQ-plot technique, combined with data cleaning based on the robustified winsorisation technique, is used as a framework for the classification of sorted data into the class of regular normally distributed data and the class of irregular data belonging to the contaminating distribution with a variance that is much greater than nominal. The link between the QQ-plot technique and a specific linear regression is established, so that the estimation of statistical parameters of the contaminated measurement distribution is performed using the least-squares technique. Then, the suboptimal maximum likelihood criterion is defined, and the system parameter estimation problem is solved robustly, using the proposed recursive robust parameter estimation scheme. Simulation results illustrate the discussion and show the efficiency of the proposed adaptive recursive parameter estimation algorithm in the presence of glint spikes or outliers.

Calculation and Analysis on the Key Parameters of High-Speed Linear Motor Feed System

High-speed machining is playing a more important role in modern manufacturing technology. The feed system of high-speed machine tool is one of the most important components. The linear motor has been widely used in high-speed machine tool. In this paper，a kind of machine tool feed system directly driven by a linear motor is introduced，and its technical parameters are analyzed and calculated. The main parameters’ influence factors and best scope are given out.These results can offer theoretical basis for manufacturing machining technology.

UNIFORM DISSIPATIVENESS, ROBUST SYNCHRONIZATION AND LOCATION OF THE ATTRACTOR OF PARAMETRIZED NONAUTONOMOUS DISCRETE SYSTEMS

In this series of papers, we study issues related to the synchronization of two coupled chaotic discrete systems arising from secured communication. The first part deals with uniform dissipativeness with respect to parameter variation via the Liapunov direct method. We obtain uniform estimates of the global attractor for a general discrete nonautonomous system, that yields a uniform invariance principle in the autonomous case. The Liapunov function is allowed to have positive derivative along solutions of the system inside a bounded set, and this reduces substantially the difficulty of constructing a Liapunov function for a given system. In particular, we develop an approach that incorporates the classical Lagrange multiplier into the Liapunov function method to naturally extend those Liapunov functions from continuous dynamical system to their discretizations, so that the corresponding uniform dispativeness results are valid when the step size of the discretization is small. Applications to the discretized Lorenz system and the discretization of a time-periodic chaotic system are given to illustrate the general results. We also show how to obtain uniform estimation of attractors for parametrized linear stable systems with nonlinear perturbation.

A New Preconditioner with Two Variable Relaxation Parameters for Saddle Point Linear Systems with Highly Singular(1,1) Blocks

In this paper, we provide new preconditioner for saddle point linear systems with (1,1) blocks that have a high nullity. The preconditioner is block triangular diagonal with two variable relaxation paremeters and it is extension of results in [1] and [2]. Theoretical analysis shows that all eigenvalues of preconditioned matrix is strongly clustered. Finally, numerical tests confirm our analysis.

Experimental study on dynamic characteristics of linear parametrically excited system

An electromagnetic device, acting like a spring with alternating stiffness, has been designed to parametrically excite the cantilever beam transversely in the laboratory. It showed good potential for the experimental investigation of linear parametric system. Therefore, experiments for the natural frequency, the response (both free and forced) spectra and frequency response characteristics of a cantilever beam under electromagnetic excitation, are, respectively, conducted to study the dynamic characteristics of linear parametrically excited system. The vibrational model of the cantilever beam system is established using the assumed mode method. Theoretical results, related to the natural frequency, response spectra and external resonant condition, are presented to verify the experimental results. It is shown that the dynamic characteristics of linear time-periodic system do have some typical features, which differ distinctly from that of the linear time-invariant system.

Identification of Parametric Underspread Linear Systems and Super-Resolution Radar

Identification of time-varying linear systems, which introduce both time-shifts (delays) and frequency-shifts (Doppler-shifts), is a central task in many engineering applications. This paper studies the problem of identification of underspread linear systems (ULSs), whose responses lie within a unit-area region in the delay-Doppler space, by probing them with a known input signal. It is shown that sufficiently-underspread parametric linear systems, described by a finite set of delays and Doppler-shifts, are identifiable from a single observation as long as the time-bandwidth product of the input signal is proportional to the square of the total number of delay-Doppler pairs in the system. In addition, an algorithm is developed that enables identification of parametric ULSs from an input train of pulses in polynomial time by exploiting recent results on sub-Nyquist sampling for time delay estimation and classical results on recovery of frequencies from a sum of complex exponentials. Finally, application of these results to super-resolution target detection using radar is discussed. Specifically, it is shown that the proposed procedure allows to distinguish between multiple targets with very close proximity in the delay-Doppler space, resulting in a resolution that substantially exceeds that of standard matched-filtering based techniques without introducing leakage effects inherent in recently proposed compressed sensing-based radar methods.

Continuous wavelet based linear time-varying system identification

A systematic framework is developed to address the parametric linear time-varying system identification problem, using the continuous wavelet transform (CWT). The system is modeled by a differential equation with unknown parameters and identified via the time–frequency representation, the ratio of the CWTs of the output and the input. The efficient execution of this system identification algorithm requires selecting appropriate scales from the wavelet transform. Scale selection can be formulated as an optimization problem: find a set of scales that contains the most information about the system’s dynamics and has high signal to noise ratio. In addition, selecting scales that have a minimum amount of redundant information is desirable. Three candidate selection metrics are presented that address these criteria and are based on an analytic investigation of the wavelet transform’s probabilistic structure. Finally, a non-linear least squares algorithm, coupled with a scale selection algorithm, is presented to identify the system model. Simulations and experiments verify this algorithm’s capability of tracking different types of model variation.

Towards a new type of energy trap: Classical analog of quantum Landau-Zener tunneling

We present a novel type of an energy trap providing targeted energy transfer in a system of weakly coupled pendulums. Our approach is based on the analogy we revealed between the behavior of two weakly coupled classical parametric pendulums (in linear approximation) and the nonadiabatic Landau-Zener tunneling in a two-state quantum system. This analogy leads us to the prediction of an efficient irreversible transfer of vibration energy from one subsystem to another, when the eigenfrequency of at least one of them changes in time, so that the coupled subsystems pass through an internal resonance. The existence of such a phenomenon is not restricted to coupled pendulums, but is inherent to a wide class of both linear and nonlinear parametric oscillatory systems. This opens up the possibility of designing new types of energy traps and absorbers for the dynamic protection of various mechanical systems.

Abstract only

The discussion of parametric polynomial systems is an old and important problem in many technical and mathematical applications. To recall its importance it suffices to remember the well-known discussions of parametric linear systems and how useful they are for applications. However, many practical problems are associated to non-linear systems, in which the study of the nature of the solutions (i.e. no solution, finite number of solutions, degree of freedom of the solutions, etc) is very important depending on the values of the parameters. Since Gröbner bases were introduced, various approaches have been developed for this problem. Many authors have studied the subject during the last 15 years, among others we can highlight D. Lazard, D. Duval, P. Gianni, M. Kalkbrenner, V. Weispfenning, T. Mora, M. Moreno-Maza, L.Gonzalez-Vega, D. Kapur, D. Wang, Y. Sato, A. Suzuki, for the relevance of their contributions. The development of the subject of this thesis was initiated in 2000 with the first Montes' algorithm dispgb for discussing parametric polynomial systems using Gröbner bases, published in 2002 in the Journal of Symbolic Computation. Montes' algorithm has progressed from that moment, until the final obtention of the Minimal Canonical Comprehensive Gröbner System (MCCGS) of a parametric ideal. During the period 2003-2007 I had the great opportunity and pleasure to participate in this challenging project. My personal contribution to the project consisted of the implementation and improvement of a high number of algorithms that have conformed the evolution up to the current algorithm MCCGS, and updating the theory and algorithms that were initially disperse and partially incomplete. Given a polynomial system with parameters and a term order for the variables, the latest implementation of the algorithm MCCGS performs a discussion on the cancellation of some polynomials in the parameters. It obtains a set of ordered pairs consisting of a segment (constructible subset of the parameter space) and a basis (finite set of polynomials) such that the set of all segments forms a partition of the parameter space; for each pair, the substitution of the parameters in the basis by any value inside the corresponding segment, produces always the reduced Gröbner basis, with respect to the given term order, of the ideal generated by the substitution of the same values in the given polynomials; the segments are uniquely determined by the given system and term order and moreover they are described in a canonical form; the number of pairs in the obtained set is the minimum possible with these properties. All implementations have been done in Maple, and the last release 7.0 is currently available on the web http://www-ma2.upc.edu/~montes/, which integrates the improvements made on the original Montes' algorithm and the final MCCGS algorithm.

On guaranteed parameter estimation of a multiparameter linear regression process

This paper presents a sequential estimation procedure for the unknown parameters of a continuous-time stochastic linear regression process. As an example, the sequential estimation problem of two dynamic parameters in stochastic linear systems with memory and in autoregressive processes is solved. The estimation procedure is based on the least squares method with weights and yields estimators with guaranteed accuracy in the sense of the L q -norm for fixed q ≥ 2 . The proposed procedure works in the mentioned examples for all possible values of unknown dynamic parameters on the plane R 2 for the autoregressive processes and on the plane R 2 with the exception of some lines for the linear stochastic delay equations. The asymptotic behaviour of the duration of observations is determined. The general estimation procedure is designed for two or more parametric models. It is shown that the proposed procedure can be applied to the sequential parameter estimation problem of affine stochastic delay differential equations and autoregressive processes of an arbitrary order.

On solving trust-region and other regularised subproblems in optimization

The solution of trust-region and regularisation subproblems that arise in unconstrained optimization is considered. Building on the pioneering work of Gay, Moré and Sorensen, methods that obtain the solution of a sequence of parametrized linear systems by factorization are used. Enhancements using high-order polynomial approximation and inverse iteration ensure that the resulting method is both globally and asymptotically at least superlinearly convergent in all cases, including the notorious hard case. Numerical experiments validate the effectiveness of our approach. The resulting software is available as packages \({\tt TRS}\) and \({\tt RQS}\) as part of the GALAHAD optimization library, and is especially designed for large-scale problems.

A framework for analyzing nonlinear eigenproblems and parametrized linear systems

Associated with an n × n matrix polynomial of degree ℓ , P ( λ ) = ∑ j = 0 ℓ λ j A j , are the eigenvalue problem P ( λ ) x = 0 and the linear system problem P ( ω ) x = b , where in the latter case x is to be computed for many values of the parameter ω . Both problems can be solved by conversion to an equivalent problem L ( λ ) z = 0 or L ( ω ) z = c that is linear in the parameter λ or ω . This linearization process has received much attention in recent years for the eigenvalue problem, but it is less well understood for the linear system problem. We develop a framework in which more general versions of both problems can be analyzed, based on one-sided factorizations connecting a general nonlinear matrix function N ( λ ) to a simpler function M ( λ ) , typically a polynomial of degree 1 or 2. Our analysis relates the solutions of the original and lower degree problems and in the linear system case indicates how to choose the right-hand side c and recover the solution x from z . For the eigenvalue problem this framework includes many special cases studied in the literature, including the vector spaces of pencils L 1 ( P ) and L 2 ( P ) recently introduced by Mackey, Mackey, Mehl, and Mehrmann and a class of rational problems. We use the framework to investigate the conditioning and stability of the parametrized linear system P ( ω ) x = b and thereby study the effect of scaling, both of the original polynomial and of the pencil L . Our results identify situations in which scaling can potentially greatly improve the conditioning and stability and our numerical results show that dramatic improvements can be achieved in practice.

Nonlinear Model Based Estimation of Rigid-Body Motion Via an Indirect Measurement of an Elastic Appendage

In this paper, we develop and implement a nonlinear model based procedure for the estimation of rigid-body motion via an indirect measurement of an elastic appendage. We demonstrate the procedure by motion analysis of a compound planar pendulum from indirect optoelectronic measurements of markers attached to an elastic appendage that is constrained to slide along the rigid-body axis. We implement a Lagrangian approach to derive a theoretical nonlinear model that consistently incorporates several generalized forces acting on the system. Identification of the governing linear and nonlinear system parameters is obtained by analysis of frequency and damping backbone curves from controlled experiments of the decoupled system elements. The accuracy of the proposed model based procedures is evaluated and its results are compared with those of a previously reported point cluster estimation procedure. Two cases are investigated to yield 1.7% and 3.4% errors between measured motion and its model based estimation for experimental configurations, with a slider mass to pendulum frequency ratios of 12.8 and 2.5, respectively. Motion analysis of system dynamics with the point cluster method reveals a noisy signal with a maximal error of 3.9%. Thus, the proposed model based estimation procedure enables accurate evaluation of linear and nonlinear system parameters that are not directly measured.

Prediction of Modal Parameters of Linear Time-Varying Systems

Many engineering structures, such as cranes, traffic-excited bridges, flexible mechanisms and robotic devices exhibit characteristics that vary with time and are referred to as time-varying or nonstationary. In particular, linear time-varying (LTV) systems have been often dealt with on a case-by-case basis. Many concepts and analytic methods of linear time-invariant (LTI) systems cannot be applied to LTV systems, as for example the conventional definition of modal parameters. In fact, LTV systems violate one of the assumptions of the conventional modal analysis, which is stationarity.

Prediction of Modal Parameters of Linear Time-Varying Systems

Many engineering structures, such as cranes, traffic-excited bridges, flexible mechanisms and robotic devices exhibit characteristics that vary with time and are referred to as time-varying or nonstationary. In particular, linear time-varying (LTV) systems have been often dealt with on a case-by-case basis. Many concepts and analytic methods of linear time-invariant (LTI) systems cannot be applied to LTV systems, as for example the conventional definition of modal parameters. In fact, LTV systems violate one of the assumptions of the conventional modal analysis, which is stationarity.Subspace-based identification methods, proposed in the 1970s, have been attracting much attention due to their affinity to the modern control theory, which is based on the state space model. These methods are now successfully applied to many industrial cases and may be considered reference methods for identifying LTI systems.In this paper the use of a subspace-based method for identifying LTV systems is discussed and applied to both numerical and experimental systems. More precisely a modified version of the SSI method, referred to here as ST-SSI (Short Time Stochastic Subspace Identification) is introduced as well as a method for predicting time-varying stochastic systems using the angle variation between the subspaces; the latter is able to predict the system parameter in the “near” future.

Jitter in sampling and restoration of gaussian random processes at the output of linear parametric system

On a basis of conditional mean method there is represented description of jitter in sampling and optimal reconstruction of nonstationery process at the output of the first order parametric RC system under the influence of white noise. We consider two kinds of system parameters modification: linear and sinusoidal ones. Reconstructing functions are defined, and reconstruction errors are estimated for both cases. Jitter effect is described by random value with beta-distribution.

TH‐C‐210A‐04: New High Resolution Dynamic Detectors and Flow Modifying Stents for Neuro‐Endovascular Image Guided Interventions

As the field of minimally invasive endovascular image‐guided interventions (EIGI) advances, there has been progress in the development of new endovascular devices such as the asymmetric vascular stent for treatment of aneurysms as well as progress in the image guidance systems needed to improve both the diagnoses and interventions and the methods used to evaluate the improved imaging performance. High resolution imaging systems with MTFs extending past 8 Lp/mm and with ability to visualize not only radioopaque markers but the detailed structure of devices such as stents have motivated the development of new asymmetric devices such as the blood flow modifying asymmetric vascular stent (AVS). The AVS has now come through a number of generations from balloon expandable stainless steel strutted structures with laser micro‐welded mesh flow diverters to new super‐elastic nitinol closedcell self‐expanding stents with organic material flow diverters. The methods of laser machining, surface finishing, and deployment will be described with progress in animal models reported. In parallel with advances in EIGI devices has come new high resolution detector development. The detectors have a unique combination of features such as far superior spatial resolution compared with conventional dynamic flat panel detectors, large dynamic range of sensitivity with negligible lag and ghosting to enable both fluoroscopy and angiography, and low noise to enable quantum limited performance over the full useful range including during lowdose fluoroscopy. The Micro‐Angiographic Fluoroscope (MAF) consists of an x‐ray converter phosphor sensed by a micro‐channel plate light image intensifier which is in turn coupled to a high performance CCD camera using a fiber optic taper. The MAF is a region of interest (ROI) imager with 4 cm field of view centered at the interventional site and may be moved in front of a larger conventional detector when improved resolution is needed. The Solid State X‐ray Image Intensifier (SSXII) while having much of the benefits of the MAF in superior imaging capability achieves its great sensitivity using only electron multiplying CCD sensors. An array design is being developed so that the imaging FOV may be expanded by adding modules each with its own EMCCD‐based detector. To more fully characterize detectors, new evaluation methods are being explored. For example, the accurate determination of MTF from measurements of noise only, without the need for a slit or edge will be reported. Also from a careful analysis of noise, the exposure range for detector quantum limited performance can be well demarcated by the instrumentation noise equivalent exposure (INEE). Finally, more realistic linear system parameters that include focal‐spot size, geometry, and scatter provide generalized MTFs and DQEs or GMTFs and GDQEs. All told, there is much happening in EIGI.Learning Objectives1. Appreciate the progress being made in improved EIGI devices and in particular flow modifiers such as the asymmetric vascular2. stent (AVS) for aneurysm treatment.3. Understand the operation of new high‐resolution micro‐angiographic systems including the MAF and SSXII.4. Understand new objective image detector evaluations including INEE, GMTF, GDQE, and determination of MTF from noise5. measurements alone.(Supported in part by NIH Grants R01EB002873, R01 NS43924, R01EB008425, and the Toshiba Medical Systems Corp.)

Brake squeal analysis by coupling spectral linearization and modal identification methods

Brake squeal is induced by self-excited vibrations, consequences of local nonlinearities at the contact interface. This paper deals with a new way to analyze the brake squeal behavior. The proposed method is based on a spectral linearization of the brake nonlinear dynamic response with unilateral contact and friction conditions. The approach enables to identify modal parameters of an equivalent linear system by a combination of the random decrement technique and the Ibrahim time domain method. It is applied to the analysis of a pad/beam squealing contact. The obtained results are compared to the classical complex eigenvalues analysis and nonlinear temporal dynamic finite element analysis ones.