Identification Of Linear Time-varying Systems Research Articles

Beurling-Type Density Criteria for System Identification

This paper addresses the problem of identifying a linear time-varying (LTV) system characterized by a (possibly infinite) discrete set of delay-Doppler shifts without a lattice (or other “geometry-discretizing”) constraint on the support set. Concretely, we show that a class of such LTV systems is identifiable whenever the upper uniform Beurling density of the delay-Doppler support sets, measured “uniformly over the class”, is strictly less than 1/2. The proof of this result reveals an interesting relation between LTV system identification and interpolation in the Bargmann-Fock space. Moreover, we show that the density condition we obtain is also necessary for classes of systems invariant under time-frequency shifts and closed under a natural topology on the support sets. We furthermore find that identifiability guarantees robust recovery of the delay-Doppler support set, as well as the weights of the individual delay-Doppler shifts, both in the sense of asymptotically vanishing reconstruction error for vanishing measurement error.

Application of decoupled ARMA model to modal identification of linear time-varying system based on the ICA and assumption of “short-time linearly varying”

A new approach for time-varying (TV) modal parameters identification is proposed in this research. In the identification process, the entire signal is divided into successive short time windows, where the structure response under white noise excitation is transformed into modal coordinates by the Independent Component Analysis (ICA) method. The decoupled time-varying Auto-Regressive Moving-Average (ARMA) model based on the new assumption of “short-time linearly varying” (STLV) is established with the extracted modal coordinates. The TV parameters can be obtained by solving a nonlinear least squares problem. The main motivation of the proposed approach is to simplify the model and reduce the computational difficulty. The effectiveness and accuracy are validated via both the numerical example and experiment.

Modeling and parameter identification of linear time-varying systems based on adaptive chirplet transform under random excitation

In this paper, a time–frequency algorithm based on adaptive chirplet transform for parameter modeling and identification of Linear Time-Varying (LTV) systems under random excitation is presented. It is assumed that the solution of responses of LTV structures is expressed as the sum of multicomponent Linear Frequency Modulated (LFM) signals in a short-time. Then the measured acceleration response is used to perform the adaptive chirplet transform, in which an integral algorithm is employed to reconstruct the velocity and displacement responses. The vibration differential equation with time-varying coefficients is transformed into a simple linear equation. Furthermore, for systems under random excitation, the input–output relation based on correlation function is also derived to estimate the parameters including physicals parameters and instantaneous modal parameters. The full procedure of the method is presented and validated by using simulated responses. The results show that the presented method is accurate and robust for various LTV systems under random excitation.

Robust Adaptive Identification of Linear Time-Varying Systems Under Relaxed Excitation Conditions

An on-line modified least-squares identification algorithm is proposed for linear time-varying systems with bounded disturbances under relaxed excitation conditions. An extra term which enhances the tracking ability for time-varying parameters is added to the covariance’s update law. An indicator of the regressor’s excitation level based on the maximum eigenvalue of the covariance matrix is developed. By combining the maximum eigenvalue with its variation trend shown by the sensitivity of the maximum eigenvalue to change in the covariance matrix, a novel identification law, which is switched between a modified least-squares algorithm and a gradient algorithm based on fixed $\sigma $ -modification, is proposed. The boundedness of the estimation error and the covariance matrix are guaranteed via Lyapunov stability theory. The superiority of the proposed method is verified by simulations.

Identification of Linear Time-Varying Systems Through Waveform Diversity

Linear, time-varying (LTV) systems composed of time shifts, frequency shifts, and complex amplitude scalings are operators that act on continuous finite-energy waveforms. This paper presents a novel, resource-efficient method for identifying the parametric description of such systems, i.e., the time shifts, frequency shifts, and scalings, from the sampled response to linear frequency modulated (LFM) waveforms, with emphasis on the application to radar processing. If the LTV operator is probed with a sufficiently diverse set of LFM waveforms, then the system can be identified with high accuracy. In the case of noiseless measurements, the identification is perfect, while in the case of noisy measurements, the accuracy is inversely proportional to the noise level. The use of parametric estimation techniques with recently proposed denoising algorithms allows the estimation of the parameters with high accuracy.

A Non-Parametric Linear Parameter Varying Approach for Identification of Linear Time-Varying Systems

This paper presents a novel, non-parametric, linear parameter varying (LPV) algorithm for identification of linear time-varying systems. The method estimates the timevarying impulse response function as a LPV Laguerre basis expansion whose coefficients are functions of a scheduling variable (SV). Unlike many parametric LPV identification techniques, the method requires no a priori knowledge of the system order. Monte Carlo simulations of a time-varying, second-order system mimicking variations of reex stiffness dynamics with ankle position demonstrated that the method performs very well. It will be a valuable tool for identification of physiological and engineering systems in conditions where the system order is unknown and its parameters change with a SV.

Modal Parameters Identification of Time-Varying System Based on Hilbert-Huang Transform

Mass distribution of floating structures will change due to such factors as ballast water management during ship sailing or inward and outward port, crane movement during construction, and the loading and unloading of cargoes on the offshore platform structures, which will then make its natural frequency change with time. While traditional modal identification cannot deal with the issues of non-stationary of time-varying system, Hilbert-Huang transform is quite effective for the processing and analysis of non-stationary signal. Therefore, this paper investigates the identification of linear time-varying system using the Hilbert-Huang transform, and conducts a numerical simulation of the two degrees-of-freedom time-varying system. In addition, this paper designs a cantilever beam subjected to a moving mass, and then makes a comparison of the identification results between the present identification method and those from finite element method (FEM) and eigensystemrealization algorithm (ERA), which shows a higher accuracy of the present method for modal identification of time-varying system.

Time-Varying Ankle Mechanical Impedance During Human Locomotion.

In human locomotion, we continuously modulate joint mechanical impedance of the lower limb (hip, knee, and ankle) either voluntarily or reflexively to accommodate environmental changes and maintain stable interaction. Ankle mechanical impedance plays a pivotal role at the interface between the neuro-mechanical system and the physical world. This paper reports, for the first time, a characterization of human ankle mechanical impedance in two degrees-of-freedom simultaneously as it varies with time during walking. Ensemble-based linear time-varying system identification methods implemented with a wearable ankle robot, Anklebot, enabled reliable estimation of ankle mechanical impedance from the pre-swing phase through the entire swing phase to the early-stance phase. This included heel-strike and toe-off, key events in the transition from the swing to stance phase or vice versa. Time-varying ankle mechanical impedance was accurately approximated by a second order model consisting of inertia, viscosity, and stiffness in both inversion-eversion and dorsiflexion-plantarflexion directions, as observed in our previous steady-state dynamic studies. We found that viscosity and stiffness of the ankle significantly decreased at the end of the stance phase before toe-off, remained relatively constant across the swing phase, and increased around heel-strike. Closer investigation around heel-strike revealed that viscosity and stiffness in both planes increased before heel-strike occurred. This finding is important evidence of "pretuning" by the central nervous system. In addition, viscosity and stiffness were greater in the sagittal plane than in the frontal plane across all subgait phases, except the early stance phase. Comparison with previous studies and implications for clinical study of neurologically impaired patients are provided.

Damping Effects Induced by a Mass Moving along a Pendulum

The experimental study of damping in a time-varying inertia pendulum is presented. The system consists of a disk travelling along an oscillating pendulum: large swinging angles are reached, so that its equation of motion is not only time-varying but also nonlinear. Signals are acquired from a rotary sensor, but some remarks are also proposed as regards signals measured by piezoelectric or capacitive accelerometers. Time-varying inertia due to the relative motion of the mass is associated with the Coriolis-type effects appearing in the system, which can reduce and also amplify the oscillations. The analytical model of the pendulum is introduced and an equivalent damping ratio is estimated by applying energy considerations. An accurate model is obtained by updating the viscous damping coefficient in accordance with the experimental data. The system is analysed through the application of a subspace-based technique devoted to the identification of linear time-varying systems: the so-called short-time stochastic subspace identification (ST-SSI). This is a very simple method recently adopted for estimating the instantaneous frequencies of a system. In this paper, the ST-SSI method is demonstrated to be capable of accurately estimating damping ratios, even in the challenging cases when damping may turn to negative due to the Coriolis-type effects, thus causing amplifications of the system response.

Identification of linear time-varying systems using a wavelet-based state-space method

In this paper a new wavelet-based state-space method for the identification of dynamic parameters in linear time-varying systems is presented using free vibration response data and forced vibration response data. For an arbitrarily linear time-varying system, the second-order vibration differential equations are first rewritten as the first-order state equations using the state-space theory. The excitation and response signals are projected using the Daubechies wavelet scaling functions and the first-order state-space equations are transformed into simple linear equations using the orthogonality of the scaling functions. This allows the time-varying equivalent state-space system matrices at each moment to be identified directly via solving the linear equations. The system modal parameters are extracted though eigenvalue decomposition of the state-space system matrices. The stiffness and damping matrices are determined by comparing the identified equivalent system matrices with the physical system matrices. The proposed algorithm is investigated with a two-degree of freedom spring–mass–damper system and a cantilever beam. Numerical results demonstrate that the proposed method is robust and effective with regards to the identification of abruptly, smoothly and periodically time-varying parameters.

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.

A Deconvolution-Based Approach to Structural Dynamics System Identification and Response Prediction

Two general linear time-varying system identification methods for multiple-input multiple-output systems are proposed based on the proper orthogonal decomposition (POD). The method applies the POD to express response data for linear or nonlinear systems as a modal sum of proper orthogonal modes and proper orthogonal coordinates (POCs). Drawing upon mode summation theory, an analytical expression for the POCs is developed, and two deconvolution-based methods are devised for modifying them to predict the response of the system to new loads. The first method accomplishes the identification with a single-load-response data set, but its applicability is limited to lightly damped systems with a mass matrix proportional to the identity matrix. The second method uses multiple-load-response data sets to overcome these limitations. The methods are applied to construct predictive models for linear and nonlinear beam examples without using prior knowledge of a system model. The method is also applied to a linear experiment to demonstrate a potential experimental setup and the method’s feasibility in the presence of noise. The results demonstrate that while the first method only requires a single set of load-response data, it is less accurate than the multiple-load method for most systems. Although the methods are able to reconstruct the original data sets accurately even for nonlinear systems, the results also demonstrate that a linear time-varying method cannot predict nonlinear phenomena that are not present in the original signals.

Subspace-Based Identification of Linear Time-Varying System

A physical parameter identification method for the linear time-varying system is presented in this paper. For an arbitrarily time-varying system, a series of Hankel matrices are established directly from the combined measurements of displacement, velocity, and acceleration together with the input excitations from a single experiment with the system. The time-varying equivalent state-space system matrices of the structure at each time instance are then identified by singular value decomposition using the free or random responses. A time-varying transformation matrix for transforming the identified equivalent system matrices into the physical system matrices is developed to determine the mass, stiffness, and damping matrices. Details of the proposed approach are investigated with a two degrees-of-freedom spring-mass-damper system with three kinds of time-varying parameters. Numerical results show that the proposed method is accurate and effective in identifying time-varying physical parameters.

Experimental verification of an algorithm for identification of linear time-varying systems

Experimental verification of an algorithm for identification of linear time-varying systems

Identification of linear time-varying systems using a modified least-squares algorithm

In this paper a modified version of the standard least-squares algorithm is presented. The aim is to use the proposed modified LS algorithm in linear time-varying systems. The proposed modification involves the addition of extra terms to both the parameter estimates’ and the covariance's update laws. We establish a series of properties on the identification error, the parameter estimates and the covariance matrix. These properties are important in an adaptive control context, because LS algorithms provide an easy way of modifying the parameter estimates in order to avoid singular points in the control scheme.

A robust ensemble data method for identification of human joint mechanical properties during movement.

This paper describes a perturbation method for the identification of linear time-varying systems with an unknown input (voluntary joint input) using ensemble data. The method separates the unknown input and the perturbation through high-pass filtering and recasts the multi-input single-output system identification into single-input single-output system identification. The method is robust to intertrial variation, and can track changes of system dynamics up to 5 Hz. Analysis and simulation are given for the conditions similar to those for the human arm experiments. Experiments show that mechanical properties of the human elbow joint change with voluntary movement speed and that the mean stiffness with voluntary movement is in the range of the posture and is higher than reported before.

Nonparametric Identification of Linear Time-Varying Systems. 1st Report. Basic Study on an Identification Method Based on Nonstationary Spectral Analysis.

This paper describes an approach to the identification of a time-varying transfer function which represents the instantaneous frequency response of a linear time-varying system. It is shown that the identification problem of the time varying transfer function is equivalent to the estimation problem of the nonstationary cross-spectrum between the output process and the appropreately filtered input process. According to this fact, we propose a three step identification procedure which includes two nonstationary spectral estimators. The identification error is asymptotically evaluated and numerical examples are examined to illustrate the performance of the proposed method.

Nonparametric Identification of Linear Time-Varying Systems. 2nd Report, Identification from a Pair of Single Realization of Input/Output Processes.

This paper describes the identification of the time-varying transfer function from a pair of single realization of the input and output processes. We propose an identification method based on the nonstationary spectrum estimator with time-frequency smoothing kernel and derive the optimal kernel to minimize a mean square error measure. Then we show how to construct the optimal estimator approximately using the given data. Simple examples are examined to illustrate the performance of the proposed method.

Identification of Linear Time-Varying Systems Using a Modified Least Squares Algorithm

In the present paper a modified version of the standard LS algorithm is proposed. The aim is to establish a series of convergence properties on the identification error, the parameter estimates and the covariance matrix. Thus the proposed modification renders the LS algorithm suitable for use in LTV systems.This is important in an adaptive control context as LS algorithms provide an easy way of modifying the parameter estimates in order to avoid singular points in the control law (namely by using the “covariance᾿ matrix).

IDENTIFICATION OF LINEAR TIME-VARYING SYSTEMS

This paper is concerned with the identification of linear time-varying systems. The discrete-time state space model of freely vibrating systems is used as an identification model. The focus is placed on identifying successive discrete transition matrices that have the same eigenvalues as the original transition matrices. First a typical subspace-based method is presented to illustrate the extraction of the observability range space using the singular value decomposition (SVD) of a general Hankel matrix. Then, the identification of varying transition matrices is approached by using an ensemble of response sequences. For arbitrarily varying systems, a series of the Hankel matrices are formed by an ensemble set of responses which are obtained through multiple experiments on the system with the same time-varying behaviour. The varying transition matrix at each moment is estimated through the SVD of two successive Hankel matrices. The proposed algorithm is applied to two special cases that require only a single response series, i.e., periodically varying systems and slowly varying systems. The use of the eigenvalues of the transition matrices is discussed and the pseudomodal parameters are defined. Finally, a two-link manipulator subjected to a varying end force is used as an example to illustrate the tracking capability and performance of the proposed algorithm.