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

Properties of double (or two-dimensional) Laplace transform (2DLT) for analysis of linear time-varying (LTV) systems are explored. The one-dimensional to two-dimensional expansion formulas, as well as the two-dimensional to one-dimensional reduction formulas, are outlined. The merit of the proposed technique for frequency analysis of LTV systems is demonstrated.

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.

This paper addresses the stability analysis of linear continuous systems under interval uncertainties. An interval generalization of the known Raus criterion is suggested to estimate the stability of the system considered. It is based on obtaining the interval extensions of the elements of the Raus matrix which are nonlinear functions of independent system parameters. The case when these elements are independent intervals is considered. The interval extensions are also determined by using modified affine arithmetic. Two sufficient conditions on stability and instability of the linear system considered are obtained. A numerical example illustrating the applicability of the method suggested is solved at the end of the paper.

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.

This paper deals with singular value decomposition (SVD) for a class of linear time-varying systems. The class considered here describes switched linear systems with periodic switching. Based on an appropriate input-output description, the calculation method of singular values and singular vectors is derived. The SVD enables us to characterize the dominant input-output signals using singular vectors, which form orthogonal systems in input and output spaces. Then SVD is applied to switched linear systems to improve the transient response. A numerical example is provided to demonstrate the proposed method.

The stability of a particular class of shaping filters, namely, the periodic shaping filters, is considered in this paper. The main objective of the present paper is to show that by considering the output covariance function of this periodic shaping filter it is possible to determine the stability of this class of shaping filters in the bounded input-bounded output sense. Furthermore, it is demonstrated that if γ(t, τ) is separable in t and τ, and the components of the separating vectors that result in the scalar covariant function are jointly linearly independent over one period of the time interval, then the resulting shaping filter is uniformly asymptotically stable. The results of this paper are presented in a series of theorems that, in effect, enable the determination of stability properties of such systems by using their structural properties only. The approach taken in this paper is thus parallel to that of Lyapunov's second method in stability analysis of linear time-varying systems.

Solution of multipoint boundary value problems via block-pulse functions

A new approach has been developed in this paper to solve time-independent Schrödinger wave equation for any arbitrary potential and space varying mass as well. The method is based on the state transition matrix used in the analysis of linear time-varying systems, and can determine both bound states and reflection and transmission coefficients associated with scattering problems. Numerical examples for the computation of eigenvalues and eigenmodes associated with bound states are presented for quadratic potential, quartic potential, constant potential well and arbitrary potential well with both constant and space-varying or position-dependent masses. Similarly, transmission coefficients for scattering problems without any infinite potential, and time delays for scattering problems with an infinite potential are computed for arbitrary potential wells.

The direct method presented in this paper for the calculation of the mean square ensemble average of nonstationary functions can be of great use in the analysis of linear time-varying systems with periodical random phase inputs. The results can be obtained either by a closed-form solution or by a single computer run instead of numerous Monte Carlo simulations. The method can be equally applied for adjoint analysis.

This technical note addresses the stability analysis of linear systems governed by continuous-time difference equations with multiple delays. We propose new sufficient conditions for L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -exponential stability and exponential stability within the Lyapunov-Krasovskii framework. These conditions are delayindependent, and provide constructive exponential decay rate estimates of the system response. It is also shown that these conditions are less conservative than previous results in the literature.

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

In this paper, Beveral sovable classes of linear time-varying systems are identified and explicit methods for finding their solutions are given. It is also shown that any linear time-varying system can be transformed into an algebraically equivalent commutative linear time-varying system or a time-invariant system, provided that an appropriate non-singular (time-varying) algebraic transformation can be found. Consequently it is concluded that the eommutativity of a linear time-varying system is not an inherent property of a dynamic system, but rather is a representation, property.

Some sufficient conditions for exponential stability of linear neutral functional differential equations

Youla-like parametrizations are derived for a class of linear time-varying systems. In addition these parametrizations are used within a predictive control scheme for LTV-systems with guaranteed stability.

The relationship between two gap metrics is investigated for a class of time-varying linear systems. Specifically, systems are considered to be causal linear maps between finite-energy continuous-time signals of nonuniformly lower-bounded support, and by definition, the corresponding input-output graphs admit normalized so-called strong-right and strong-left representations. First, a time-varying generalization of Vinnicombe's $\nu$-gap is revisited to rectify an omission in the original developments and to verify compliance with the axioms of a metric. Subsequently, it is shown that this generalized $\nu$-gap metric induces the same topology as an apposite adaptation of Feintuch's time-varying gap metric for discrete-time systems defined over finite-energy signals of uniformly lower-bounded support. The two generalized metrics are therefore qualitatively equivalent in robust stability analysis for linear time-varying feedback interconnections. Quantitatively, the $\nu$-gap between a given pair of systems is never larger in value than the Feintuch gap.