Linear Time-varying Equations Research Articles

An error analysis of iterated defect correction methods for linear differential-algebraic equations

Asymptotic expansions of the global error of iterated defect correction (IDeC) techniques based on the implicit Euler method for linear differential-algebraic equations (dae's) of arbitrary index are analyzed. The dependence of the maximum attainable convergence order on the degree of the interpolating polynomial, number of defect correction steps, and on the index of the differential-algebraic system is given. The efficiency of IDeC method and extrapolation is compared on the basis of numerical experiments and comparing computational cost for both methods. Linear time-varying differential-algebraic equations are investigated by presenting numerical results and extending theoretical results for constant coefficient to these problems.

Dichotomies for Linear Evolutionary Equations in Banach Spaces

In this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a Banach space. The theory we present here applies to general time-varying linear equations in Banach spaces. As a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set. In the case of dissipative systems, our theory applies to the study of the flow in the vicinity of the global attractor. The theory formulated here holds for linear evolutionary systems which are uniformly α-contracting and applies to the study of the linearization of nonlinear equations of the following type: (a) parabolic PDEs, including systems of reaction diffusion equations and the Navier-Stokes equations; (b) hyperbolic PDEs, including the nonlinear wave equation and the nonlinear Schrodinger equation with dissipation; (c) retarded differential equations; and (d) certain neutral differential delay equations.

Index of an implicit time-varying linear differential equation: a noncommutative linear algebraic approach

An intrinsic definition of the index for implicit linear time-varying differential systems is proposed. This definition relies on linear algebraic techniques over the noncommutative principal ideal ring of nonconstant linear differential operators (noncommutative transfer function or state variable representation). Accordance with the existing definition for constant coefficient linear systems is demonstrated.

A parametrically driven PLL lock detector

A novel parametrically driven phase locked loop (PLL) lock detector is presented. The lock detector is described by a time-varying linear differential equation derived from the PLL's nonlinear dynamics. The time-varying nature of this detector equation results from the fact that it is parametrically driven by the output of the PLL's quadrature detector. A lock decision is made by comparing a detector equation state to a user-specified threshold. Also, the detector produces a D.C. voltage which may be capable of driving a phase locked receiver's coherent AGC system. Finally a laboratory experiment is described which demonstrates feasibility of the detector concept. Use of the new lock detector eliminates a class of problems associated with the simple phase quadrature lock detector. The problems result from the fact that the quadrature detector can produce a D.C. component under false lock conditions. This D.C. component may actually operate a phase locked receiver's coherent AGC system. It is possible that the AGC system could be fooled into adjusting receiver gain to produce a nominal (above lock detector threshold) value for the unwanted D.C. component. A positive lock indication would result, and the receiver would appear to be operating properly when, in fact, this was not the case.

A fast and exact time-domain simulation of switched-mode power regulators

A new approach for the time-domain simulation of switched-mode power regulators is developed. The concept of this approach is to model each switch element by a linear time-varying nodal equation, thus formulating a modified nodal equation system that describes the global dynamic behavior of the power regulator. By using the symbolic Laplace transform inversion, the time-domain expressions can be obtained in a closed and continuous form, which allows a fast and exact simulation of the system. The time efficiency of the proposed approach is shown by comparing it with a standard numerical integration method. Various transient responses of a practical switched-mode power regulator due to the start-up, the step change in input voltage, and the step change in reference voltage have been simulated. The simulations include the overcurrent protection limitation, the duty-cycle limitation control, the discontinuous conduction mode, and the continuous conduction mode of operation. >

Bounded-input/bounded-output stability of linear multidimensional time-varying systems

The present study of the stability of systems governed by a linear multidimensional time-varying equation, which are encountered in spacecraft dynamics, economics, demographics, and biological systems, gives attention the lemma dealing with L(inf) stability of an integral equation that results from the differential equation of the system under consideration. Using the proof of this lemma, the main result on L(inf) stability is derived according; a corollary of the theorem deals with constant coefficient systems perturbed by small periodic terms. (O.C.)

A generalized expression for the impulse responses of recursive linear time-variant digital filters

Digital filters described by linear time-variant (LTV) difference equations, especially for the general case of time-variant order, are considered. Three notions, i.e. the linearly dependent length for a group of sequences, the lowest time-variant order and the standard form for LTV difference equations, are presented. It is shown that the lowest time-variant order of an LTV difference equation is equal to the linearly dependent length of its linearly independent homogeneous solutions. The properties of an LTV homogeneous difference equation and its homogeneous solutions are investigated. A method to determine one from the other is given. Based on this, LTV homogeneous difference equations have been studied and the general expression for the impulse responses of all the forward recursive LTV digital filters with a finite time-variant order has been determined. >

Performance of adaptive estimation algorithms in dependent random environments

We consider the convergence properties of certain algorithms arising in stochastic, discrete-time, adaptive estimation problems and operating in random environments of engineering significance. We demonstrate that the algorithms operating under ideal conditions are describable by homogeneous time-varying linear difference equations with dependent random coefficients, while in practical use, these equations are altered only through the addition of a driving term, accounting for time variation of system parameters, measurement noise, and system undermodeling. We present the concept of almost sure exponential convergence of the homogeneous difference equations as an a priori testable robustness property guaranteeing satisfactory performance in practice. For the three particular algorithms discussed, we present very mild conditions for the satisfaction of this property, and thus explain much of their observed behavior.

Exponential stability of linear equations arising in adaptive identification

The stability, properties are examined of various time-varying linear differential equations which arise in model reference adaptive identification schemes. Necessary and sufficient conditions for exponential stability are presented.

Lyapunov Functions for Linear Time-varying Systems

A systematic method of generating explicit Lyapunov functions for linear time-varying differential equations is presented in this paper. The linear time-varying system equation is decomposed into a fixed part and a time-varying part. A simple closed form expression of the Lyapunov function for the fixed part has been derived. This has been suitably augmented and applied to the time-varying system in the determination of stability results. Only first-order time-derivatives on the time-varying parameters of the system are involved in the stability results.

A system theory approach to unified electrical machine analysis†

In this paper the linear transformation which is used in the unified analysis of electrical machines without & commutator is considered. It is shown that, a very straightforward derivation of the so-called commutator transformation is obtained if the underlying mathematical problem is well defined and linear system theory techniques are used to reduce a set of time-varying linear differential equations to a set of time-invariant ones. This leads to a generalization of Park's transformation to an arbitrary polyphase machine, and also yields a better insight into why the usual assumptions of unified machine theory are necessary