Time-varying Linear Differential Equations Research Articles

Controllability and stabilizability of linear time-varying distributed hereditary control systems

This paper is concerned with the controllability and stabilizability problem for control systems described by a time-varying linear abstract differential equation with distributed delay in the state variables. An approximate controllability property is established, and for periodic systems, the stabilization problem is studied. Assuming that the semigroup of operators associated with the uncontrolled and non delayed equation is compact, and using the characterization of the asymptotic stability in terms of the spectrum of the monodromy operator of the uncontrolled system, it is shown that the approximate controllability property is a sufficient condition for the existence of a periodic feedback control law that stabilizes the system. The result is extended to include some systems which are asymptotically periodic. Copyright © 2014 John Wiley & Sons, Ltd.

Sufficient conditions of the various stabilities of the linear time-varying delayed differential equations

Due to the appearance and the study of the ornithopter and flexible-wing micro air vehicles, etc., the time-varying systems become more and more important and ubiquitous in the study of the mechanics. In this letter, the sufficient conditions of the uniform asymptotic stability are first presented for the delayed time-varying linear differential equations with any time delay by employing the Dini derivative, Lozinskii measure and the generalized scalar Halanay delayed differential inequality. They are especially based on the estimation of the arbitrary solutions but not the fundamental solution matrix since their solutions' space is infinite-dimensional. Then some sufficient conditions of the stability, asymptotic stability and uniform asymptotic stability of the delayed time-varying linear system with a sufficiently small time delay are reported by employing Taylor expansion and Dini derivative. It implies that these stabilities can be guaranteed by the Lozinskii measure of the matrix composing of the time delay and the coefficient matrices of the system.

Modal analysis of asymmetric rotor system with isotropic stator using modulated coordinates

A new modal analysis method for rotor systems with periodically time-varying parameters is proposed. The essence of the method is to introduce the modulated coordinates such that the periodically time-varying linear differential equations can be effectively transformed to the equivalent time-invariant linear differential equations. This paper presents the modal analysis procedure for the asymmetric rotor system, of which rotating and stationary parts possess asymmetric and isotropic properties, respectively. With a simple asymmetric rotor model, the analytical modal analysis procedure is illustrated and compared with the conventional method based on formulation in the rotating coordinates. A numerical example with a flexible asymmetric rotor model is also provided to demonstrate the effectiveness of the proposed method.

ORIENTATION CONTROL OF PARTICLES AND FIBERS FOR POLYMERIC LIQUIDS AND FIBER REINFORCED MATERIALS

ABSTRACT Molecules in polymeric liquids and fibers in fiber-reinforced materials are modelled using the spheroidal model or the dumbbell model of the Kelvin type. A method is developed to transform the nonlinear governing equation of orientation for dilute suspensions to a system of time-varying linear differential equations. Thereby the finding of the exact solution becomes much easier than before, where the introduced scalar variable z plays a key role both in the finding of the solution and in the formulation of the criterion of strong flow. Moreover, by utilizing the results of the exact solution, the technologically important problem of fiber (or particle) orientation control is further analyzed; hence, the criterion of controllability for a target orientation is established and the allowable set in the space of parameters is identified. The procedures to choose suitable parameters to perform specific orientation alignment are suggested and demonstrated, and guarantee that the most fibers (or particles) can be controlled to the preferred orientation within a finite time with desired precision.

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.

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.

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