4 Time-Varying Linear Dynamic System Model

A high order multiplication perturbation method for linear time-varying dynamic system is presented. Firstly the time domain is dispersed with small time interval and the coefficient matrix of the dynamic system is decomposed into a “large amount” and a “small amount” in each time interval. One order perturbation system is then presented from the original dynamic system by a variable transformation which is also a linear time-varying dynamic system. The coefficient matrix of the new system is also decomposed into a “large amount” and a “small amount” and then higher order perturbation system can be obtained by another variable transformation. The final perturbation system can be solved exactly after abandoning the “small amount” of the coefficient matrix in the final system, and then the answer to the original problem can be determined through a series of inverse transform. Since the transfer matrix is the product of a series of exponential matrix which can be calculated accurately by the precise time integration method, so the proposed method has fine accuracy, efficiency and stability. The proposed method actually is a high order symplectic conservative perturbation method for the Hamiltonian system. The examples show that the proposed method can also give good results even though a large time step selected, and with the increase of the perturbation number, the perturbation solutions can tend to exact solutions rapidly.

Climate-change-based global sea-level rise is of concern because it contributes to significant loss of coastal wetlands and mangroves and to increasing damage from coastal flooding in many regions of the world. Physical mechanisms that describe the dynamic global climate systems and the effect of this system behavior on sea-level rise are inherently complex. In this study, conducted using systematic analysis of historic data on temperature change and sea-level rise, a linear dynamic system model is proposed to predict global sea-level rise and mean surface temperatures. Unlike the semiempirical approaches proposed in the recent literature, this model incorporates the inherent interaction between temperature and sea-level rise into the model. The resulting model, recognized from the historic data, shows that the rate of sea-level rise is proportional to temperature, and this rise is also a function of the temporal state of the sea level. Similarly, the rate of temperature change is a function of the temporal state of the temperature and is also affected by the sea-level rise. The proposed model is also used to predict the sea-level rise during the 21st century. DOI: 10.1061/(ASCE)HE.1943-5584.0000447. © 2012 American Society of Civil Engineers. CE Database subject headings: Climate change; Sea level; Temperature effects; Dynamic models; Predictions. Author keywords: Climate change; Complex systems; Sea-level rise; Temperature change; Dynamic model.

Chang transformation was introduced for decoupling the slow and fast dynamics of a singularly perturbed Linear Time Invariant (LTI) system, and it was subsequently extended to Linear Time-Varying (LTV) systems under the slowly-varying assumption by way of frozen-time eigenvalues. This paper extends Chang transformation from slowly-varying LTV systems to LTV systems, when the singularly perturbed system has a semi-proper coefficient matrix A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">22</sub> (t) and LTV system ẇ(t) = A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">22</sub> (t)w(t) is exponentially stable. Instead of using frozen-time eigenvalues based on slowly time-varying conditions, PD-eigenvalues for LTV systems are employed to characterize the exponential stability, thereby circumventing the slowly-varying constraint. Our results provide a larger bound on epsilon, which is a gauge on the validity of the Chang transformation, in some situations than using previous techniques. We have also recast the original Chang transformation so as to provide more insight into the decoupled subsystems. The insight will be useful in subsequent investigation on estimate of the Singular Perturbation Margin (SPM) for LTV systems. An equivalence relationship has recently been established for LTI system between the SPM and the Phase Margin (PM). The new results in this paper will facilitate the development of a PM-type stability margin metric for LTV systems, and for nonlinear, time-varying systems in the future.

The paper deals with the construction of generalized frozen-time (FT) models-for linear time-varying continuous-time systems. The generalized FT models are higher-order models generated from a given n-dimensional state model by applying a stable differential operator to the state equation. This results in FT models whose coefficients depend on the derivatives of the time-varying system coefficient matrices. The accuracy of the higher-order FT models over local time intervals is investigated in terms of truncated Taylor series. In the last part of the paper, the application of the higher-order FT models to state feedback control is considered.

In this paper, an effective method is proposed called the “Dynamic Routh’s stability Criterion (DRSC)”, which is developed using the Dynamic Pole Motion (DPM) approach (DPMA). This innovative technique extends the classical Routh’s stability criterion—traditionally limited to linear time-invariant (LTI) systems—to encompass a broader range of systems, including linear time-varying and nonlinear dynamic systems. By incorporating the behavior of pole trajectories over time, the proposed method offers a more comprehensive framework for analyzing system stability in more complex and realistic scenarios. DPM approach describes the notion of dynamic poles (DP) in a three dimensional ‘g-plane’, which is an extension to the two dimensional ‘s-plane’. The s-domain approach is based upon Laplace transform and has been used for linear time-invariant (LTI) system only. This novel g-plane framework is versatile enough to be applied to both linear and nonlinear dynamic systems. For the stability of dynamic systems, the DP must present in the left-hand side (LHS) of the g-plane. For nonlinear systems, the locations of DP are the function of system states; and these systems states are the function of initial conditions with amplitude and frequency (AnF) of input signals. Therefore, in nonlinear systems, stability is influenced not only by the initial conditions but also by the AnF of the input signals. For example, for given amplitude of the input signal, system may be unstable at low frequency; however, it may become stable at high frequency or vice-versa. These stability conditions are illustrated by several examples.

The purpose of this paper is to derive necessary and sufficient conditions for target path controllability (TPC) of time-varying dynamical systems, and to relate the results to those of Brockett and Mesarović in 1965. It is shown that conditions for TPC in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W^{p}_{1}</tex> by means of L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> -controls are easy to verify, but are very strong. TPC implies that the number of target variables is less than or equal to the number of control variables.

Recently, a class of adaptive schemes has been developed for rejecting sinusoidal output disturbances with unknown frequencies, phases, and amplitudes. These adaptive schemes require an accurate model of the affected system's dynamics to function, and the controller performance degrades as the modeling error increases. While linear time-invariant (LTI) dynamics can adequately model the dynamics of some systems, a time-varying model of the system dynamics is more appropriate in certain scenarios. This paper proposes a robust adaptive scheme for rejecting disturbances that affect the output of a system described by a known linear time-varying model. The disturbances are described by an unknown number of sinusoidal terms with unknown time-varying frequencies. Analysis is done in ideal case where the disturbance frequencies are perfectly known to provide conditions guaranteeing the existence of a controller and a corresponding set of parameters which reject the disturbance. Analysis is then done in the more realistic adaptive scenario where the disturbance frequencies are unknown, and a robust adaptive scheme for selecting controller parameters which reject the disturbance is proposed. Performance of the controller with respect to the speed of variation in the plant model parameters is assessed via simulation.

A technique for determining the dynamic response characteristics of time-varying linear systems is presented. A model composed of a parallel connection of filters whose impulse responses are orthogonalized exponential functions is used. The input to the system is fed to each of the filters. The coefficients of linear regression of the system output on each of the filter outputs are determined. Time-varying characteristics are measured by determining successively the coefficients from short samples of the input and output signals of the system. Many of the statistical properties of regression coefficients are known and can be used to estimate the length of sample of input and output signals required to determine the coefficients with given variance. The measurement technique has been implemented on a high speed digital computer. Results obtained by applying the technique to measurement of a variety of digitally simulated filters and to measurement of human pilot control characteristics are presented.

A novel predefined-time noise-tolerant zeroing neural network for solving time-varying generalized linear matrix equations

ABSTRACTIn this paper, we describe a constructive test to decide whether a given linear time-varying (LTV) differential system admits a stabilising compensator for the control tasks of tracking, disturbance rejection or model matching and construct and parametrise all of them if at least one exists. In analogy to the linear time-invariant (LTI) case, the ring of stable rational functions, noncommutative in the LTV situation, and the Kučera–Youla parametrisation play prominent parts in the theory. We transfer Blumthaler's thesis from the LTI to the LTV case and sharpen, complete and simplify the corresponding results in the book ‘Linear Time-Varying Systems’ by Bourlès and Marinescu.

Chronic SSRI administration in Lsamp deficient mice

Virtual sensing for dynamic industrial process based on localized linear dynamical system models with time-delay optimization

A mixed-level switching dynamic system for continuous speech recognition

Since multivariate time series (MTS), which lie in a non-Euclidean space, exhibit temporal evolution and correlation characteristics, its classification is considered a non-trivial task. To mitigate the impact of the time-varying characteristics and thus enhance the classification accuracy, in this paper, we propose to model MTS data using a time-varying linear dynamical system followed by a neural network-based classification on the Grassmannian manifold. Our experiments on publicly available MTS datasets show promising classification results.

Traffic Signal Control Algorithms of a Traffic Network