Analysis Of Linear Time-varying Systems Research Articles

Time-Varying Respiratory Mechanics as a Novel Mechanism Behind Frequency Dependence of Impedance: A Modeling Approach

Frequency dependence of respiratory mechanics is a well-established behavior of the respiratory system and is known to be an indicator of severity of obstructive disease, attributed to both tissue viscoelasticity and heterogeneity of airflow in the lung. Despite the fact that respiratory parameters are known to vary in time, often amplified in disease, all analysis methods assume stationarity or short-time stationarity in the parameters used to describe the respiratory system, and the effects of this assumption have not yet been examined in any detail. Here, using a generalized approach, we developed a theory for time-varying respiratory mechanics in time-frequency domain for analysis of linear time-varying systems, then, we analyzed the same respiratory system model with time-varying parameters in the time domain. Both time-frequency domain and time-domain derivations revealed a striking correlation between time-varying behavior of the respiratory system and frequency dependence of resistance. Remarkably, this phenomenon arose from the amplitude of time variations of the elastance. This links two mechanisms that are known to increase in obstructive disease: apparent low frequency increases in resistance and the time variations of reactance.

Dynamic Stability Analysis of Linear Time-varying Systems via an Extended Modal Identification Approach

The problem of linear time-varying(LTV) system modal analysis is considered based on time-dependent state space representations, as classical modal analysis of linear time-invariant systems and current LTV system modal analysis under the “frozen-time” assumption are not able to determine the dynamic stability of LTV systems. Time-dependent state space representations of LTV systems are first introduced, and the corresponding modal analysis theories are subsequently presented via a stability-preserving state transformation. The time-varying modes of LTV systems are extended in terms of uniqueness, and are further interpreted to determine the system’s stability. An extended modal identification is proposed to estimate the time-varying modes, consisting of the estimation of the state transition matrix via a subspace-based method and the extraction of the time-varying modes by the QR decomposition. The proposed approach is numerically validated by three numerical cases, and is experimentally validated by a coupled moving-mass simply supported beam experimental case. The proposed approach is capable of accurately estimating the time-varying modes, and provides a new way to determine the dynamic stability of LTV systems by using the estimated time-varying modes.

On the Perturbation Methods for Vibration Analysis of Linear Time-Varying Systems

Some perturbation methods in the studying vibrations of the linear time-varying (LTV) system are discussed. Three classical perturbation methods, namely, averaging method, harmonic balance method, and multiple scales method with linear scales, have been used from a new perspective based on analytical approximations to the corresponding LTV ordinary differential equations. The deploying beam model has been taken as an example to validate the explicit approximate solutions obtained by these perturbation methods. It is demonstrated that such approximate solutions have good agreement with numerical and exact solutions, excluding the vicinity of the turning point.

Controllability analysis of linear time-varying systems with multiple time delays and impulsive effects

In this paper, the issue of controllability for linear time-varying systems with multiple time delays in the control and impulsive effects is addressed. The solution of such systems based on the variation of parameters is derived. Several sufficient and necessary algebraic conditions for two kinds of controllability, i.e., controllability to the origin and controllability, are derived. The relation among these conditions are established. A numerical example is provided to illustrate the effectiveness of the proposed methods.

Stability analysis of linear time-varying systems: Improving conditions by adding more information about parameter variation

In this paper a new Lyapunov function is proposed for stability analysis of linear time-varying systems. This new function carries more information regarding parameter variation leading to less conservative conditions. Using Finsler’s lemma and a suitable form to describe the high-order time-derivatives of the parameters, finite sets of LMIs are obtained which are progressively less conservative as a pair of parameters grow. Previous results can be seen as a special case and numerical examples are carried out for the sake of illustration.

Stability Analysis for Linear Systems With Singular Second-Order Vector Differential Equations

This technical note is concerned with exponential stability analysis of linear time-varying systems represented by the second-order vector differential equations with the singular leading coefficient matrix. Using bounding techniques on the trajectories of the linear time-varying systems, the stability problem of the time-varying systems is transformed to that of the time-invariant systems and a new sufficient condition for exponential stability is derived. The obtained condition can be applied to the nonsingular case and it is proven that the condition for the nonsingular case is superior to a test presented in the recent literature. Analogously, the proposed method is also extended to the stability analysis for a class of linear time-varying systems with delay. By illustrative examples, better results are obtained by the proposed criteria as compared with some results in the literature.

Stability Analysis of Nonlinear Circuits Driven With Modulated Signals

Existing methods for the large-signal stability analysis of microwave circuits assume a periodic excitation, usually consisting of a single tone. However, practical circuits such as power amplifiers will generally be driven by modulated signals or multiple input tones. This paper presents a procedure for the stability analysis of linear time-varying systems, applicable to circuits excited with nonperiodic input signals. The procedure is based on the determination of the time-varying poles associated to an input-output representation of the system. Under modulated signals, this representation is obtained by linearizing the envelope-transient system that rules the circuit behavior. The formulation and methodology have been applied to a simple nonlinear circuit with arbitrary excitation and a practical power-combined amplifier at 3 GHz with very good agreement in comparison with independent simulations and experimental results.

Formation control of underwater mobile sensing networks

Formation control is essential for an underwater mobile sensing network (UMSN), and an ad hoc network which wirelessly connects underwater vehicles of sensing and/or observing types via acoustic communications, to fulfill mobile sensing tasks. The problem of formation control for a UMSN with varying topology is studied in this paper. The methodology of synthesizing distributed formation controller which stabilizes a UMSN with varying topology is proposed on the basis of the stability analysis of linear time-varying systems.

Nonlinear Control Systems: Analysis and Design

Introduction. 1.1 Linear Time-Invariant Systems. 1.2 Nonlinear Systems. 1.3 Equilibrium Points. 1.4 First-Order Autonomous Nonlinear Systems. 1.5 Second-Order Systems: Phase-Plane Analysis. 1.6 Phase-Plane Analysis of Linear Time-Invariant Systems. 1.7 Phase-Plane Analysis of Nonlinear Systems. 1.8 Higher-Order Systems. 1.9 Examples of Nonlinear Systems. 1.10 Exercises. Mathematical Preliminaries. 2.1 Sets. 2.2 Metric Spaces. 2.3 Vector Spaces. 2.4 Matrices. 2.5 Basic Topology. 2.6 Sequences. 2.7 Functions. 2.8 Differentiability. 2.9 Lipschitz Continuity. 2.10 Contraction Mapping. 2.11 Solution of Differential Equations. 2.12 Exercises. Lyapunov Stability I: Autonomous Systems. 3.1 Definitions. 3.2 Positive Definite Functions. 3.3 Stability Theorems. 3.4 Examples. 3.5 Asymptotic Stability in the Large. 3.6 Positive Definite Functions Revisited. 3.7 Construction of Lyapunov Functions. 3.8 The Invariance Principle. 3.9 Region of Attraction. 3.10 Analysis of Linear Time-Invariant Systems. 3.11 Instability. 3.12 Exercises. Lyapunov Stability II: Nonautonomous Systems. 4.1 Definitions. 4.2 Positive Definite Functions. 4.3 Stability Theorems. 4.4 Proof of the Stability Theorems. 4.5 Analysis of Linear Time-Varying Systems. 4.6 Perturbation Analysis. 4.7 Converse Theorems. 4.8 Discrete-Time Systems. 4.9 Discretization. 4.10 Stability of Discrete-Time Systems. 4.11 Exercises. Feedback Systems. 5.1 Basic Feedback Stabilization. 5.2 Integrator Backstepping. 5.3 Backstepping: More General Cases. 5.4 Examples. 5.5 Exercises. Input-Output Stability. 6.1 Function Spaces. 6.2 Input-Output Stability. 6.3 Linear Time-Invariant Systems. 6.4 Lp Gains for LTI Systems. 6.5 Closed Loop Input-Output Stability. 6.6 The Small Gain Theorem. 6.7 Loop Transformations. 6.8 The Circle Criterion. 6.9 Exercises. Input-to-State Stability. 7.1 Motivation. 7.2 Definitions. 7.3 Input-to-State Stability (ISS) Theorems. 7.4 Input-to-State Stability Revisited. 7.5 Cascade Connected Systems. 7.6 Exercises. Passivity. 8.1 Power and Energy: Passive Systems. 8.2 Definitions. 8.3 Interconnections of Passivity Systems. 8.4 Stability of Feedback Interconnections. 8.5 Passivity of Linear Time-Invariant Systems. 8.6 Strictly Positive Real Rational Functions. Exercises. Dissipativity. 9.1 Dissipative Systems. 9.2 Differentiable Storage Functions. 9.3 QSR Dissipativity. 9.4 Examples. 9.5 Available Storage. 9.6 Algebraic Condition for Dissipativity. 9.7 Stability of Dissipative Systems. 9.8 Feedback Interconnections. 9.9 Nonlinear L2 Gain. 9.10 Some Remarks about Control Design. 9.11 Nonlinear L2-Gain Control. 9.12 Exercises. Feedback Linearization. 10.1 Mathematical Tools. 10.2 Input-State Linearization. 10.3 Examples. 10.4 Conditions for Input-State Linearization. 10.5 Input-Output Linearization. 10.6 The Zero Dynamics. 10.7 Conditions for Input-Output Linearization. 10.8 Exercises. Nonlinear Observers. 11.1 Observers for Linear Time-Invariant Systems. 11.2 Nonlinear Observability. 11.3 Observers with Linear Error Dynamics. 11.4 Lipschitz Systems. 11.5 Nonlinear Separation Principle. Proofs. Bibliography. List of Figures. Index.

Error-bounds for balanced model-reduction of linear time-varying systems

Error-bounds are developed for balanced truncation of linear time-varying systems, leading to an extension of the twice the sum of the tail formulas, well known in the time-invariant case. The approach relies on an operator-theoretic framework for analysis of linear time-varying systems. This provides a multivariable notion of frequency for such systems, which are thus characterized by rational functions of many complex variables, allowing the problem to be formulated in the linear-fractional framework. Using a time-varying version of standard necessary conditions for reduced-order modeling, based on convex operator inequalities, we show that these error-bounds for balanced truncation are related to the closest possible reduced-order modeling error in a sense which parallels the time-invariant case.

LOWER BOUNDS FOR THE STABILITY DEGREE OF PERIODIC SOLUTIONS IN FORCED NONLINEAR SYSTEMS

In this paper the problem of local exponential stability of periodic orbits in a general class of forced nonlinear systems is considered. Some lower bounds for the degree of local exponential stability of a given periodic solution are provided by mixing results concerning the analysis of linear time-varying systems and the real parametric stability margin of uncertain linear time-invariant systems. Although conservative with respect to the degree of stability obtainable via the Floquet-based approach, such lower bounds can be efficiently computed also in cases where the periodic solution is not exactly known and the design of a controller ensuring a satisfactory transient behavior is the main concern. The main features of the developed approach are illustrated via two application examples.

Analysis of linear time-varying systems via Hartley series

A method for finding approximated solutions of linear time-varying systems via Hartley series is proposed. The most relevant properties of the Hartley series are presented first. The operational matrix of integration together with the product and coefficient matrices are developed. They are used to transform dynamical equations of linear, time varying systems to a set of linear algebraic equations. Practical applications relating to the operation of electric networks are given.

Uniform decoupling by restricted static state feedback for linear time-varying analytic systems

In this paper the uniform decoupling problem of nonsquare linear time-varying analytic systems via restricted static state feedback is investigated, for the first time. The problem is treated on the basis of an algebraic approach, whose main feature is that it reduces the problem to that of solving a homogeneous algebraic system of equations. On the basis of this system of equations, necessary and sufficient conditions of simple algebraic nature for the existence of a restricted static state feedback controller that decouples the closed-loop system are established and expressions for a special admissible controller pair are analytically derived.

Uniform disturbance localization with simultaneous uniform decoupling for linear time-varying analytic systems

The problem of uniform disturbance localization with simultaneous uniform decoupling for linear time-varying analytic systems via static state feedback is investigated, for the first time. The approach presented has the main feature that it reduces the problem to that of solving a homogeneous algebraic system of equations. On the basis of this basic system of equations, the following aspects of the problem are resolved: the necessary and sufficient solvability conditions, the general analytical expressions of all admissible controllers and the morphological characteristics of the closed-loop system

State feedback uniform decoupling problem of linear time-varying analytic systems

A new approach to the input-output uniform decoupling problem of linear time-varying analytic systems via proportional state feedback is presented. A major feature of the proposed approach is that it reduces the solution of the uniform decoupling problem to that of solving a linear algebraic system of equations. This system of equations greatly facilitates the solution of the three major aspects of the decoupling problem: the necessary and sufficient conditions, the general analytical expressions for the controller matrices, and the structure of the uniformly decoupled closed-loop system.

Uniform disturbance localization with simultaneous uniform exact model matching for linear time-varying analytic systems

The problem of uniform disturbance localization with simultaneous uniform exact model matching for linear time-varying analytic systems via static state feedback is investigated, for the first time. The primary feature of the approach used is that it reduces the problem to that of solving a nonhomogeneous algebraic system of equations. From this system is obtained a set of necessary and sufficient conditions for the solvability of the problem, and a general analytical expression for all admissible controllers.

Output feedback uniform decoupling problem of linear time-varying analytic systems

A new approach to the input-output uniform decoupling problem of linear time-varying analytic systems via proportional (static) output feedback is introduced. A major feature of the proposed approach is that it reduces the solution of the uniform decoupling problem to that of solving a linear algebraic system of equations, This system of equations greatly facilitates the solution of the three major aspects of the decoupling problem: the necessary and sufficient conditions; the general analytical expressions for the controller matrices; and the structure of the uniformly decoupled closed-loop system.

Uniform exact model matching for a class of linear time-varying analytic systems

The problem of uniform exact model matching is studied via state and output feedback, for a class of linear time-varying analytic systems. The following two major issues are resolved: The nessecary and sufficient conditions for the problems to have a solution and the general analytical expressions for the controller matrices. A major feature of the proposed approach, is that it reduces the uniform exact model matching problem to that of solving a linear nonhomogeneous algebraic matrix equation.

Determinant and statistical analysis of linear time-varying systems via block-pulse functions

The analysis of time-varying linear systems using block-pulse expansion is discussed. Formulae for statistical estimates of the output signal by random input influence are derived. A priori and a posteriori error estimates are established.

Analysis of linear time-varying systems and bilinear systems via Fourier series

A method for finding an approximate solution of linear time-varying systems and bilinear systems via Fourier series is proposed. Properties of Fourier series are first briefly presented and the operational matrix for integration together with the product operational matrix are utilized to reduce the dynamical equation of a linear time-varying system (or bilinear system) to a set of simultaneous linear algebraic equations. Illustrative examples are also given.