Linear Time-varying System Theory Research Articles

Static and dynamic eigenvalues in unified stability studies

A framework for unified analysis of small-signal and large-signal power system stability based on static and dynamic eigenvalues is proposed in this paper. The presented implementation is based on Gear's method, which is a two-step integration method for numerical simulation with self-adaptive time-step. Furthermore, it can be easily configured for providing the state matrix as basis for calculating the system eigenvalues during simulation. Thus, the presented framework allows for eigenvalue-based analysis of small-signal dynamics and stability margin at any steady-state operating point during a time-domain simulation. Furthermore, Linear Time-Varying system theory is utilized for modal analysis during large-signal transients. For this purpose, dynamic eigenvalues and eigenvectors are calculated by solving a Riccati equation to generalize the modal analysis during transient conditions. The stability is evaluated by calculating the Lyapunov exponent of the mode-vector of the system. The results from numerical analysis of three case studies are presented to evaluate and illustrate the characteristics of the presented approach for unified small-signal and transient stability analysis.

Computing Natural Frequencies and Mode Shapes of an Axially Moving Nonuniform Beam

Abstract The partial differential equation of motion of an axially moving beam with spatially varying geometric, mass, and material properties has been derived. Using the theory of linear time-varying systems and numerical optimization, a general algorithm has been developed to compute complex eigenvalues/natural frequencies, mode shapes, and the critical speed for stability. Numerical results from the new method are presented for beams with spatially varying rectangular cross sections with sinusoidal variation in thickness and sine-squared variation in width. They are also compared to those from the Galerkin method. It has been found that critical speed of the beam can be significantly reduced by nonuniformity in a beam's cross section.

A New Approach to Compute Natural Frequencies and Mode Shapes of One-Dimensional Continuous Structures With Arbitrary Nonuniformities

AbstractOne-dimensional continuous structures include longitudinal vibration of bars, torsional vibration of circular shafts, and transverse vibration of beams. Using the linear time-varying system theory, algorithms are developed in this paper to compute natural frequencies and mode shapes of these structures with nonuniform spatial parameters (mass distributions, material properties and cross-sectional areas) which can have jump discontinuities. A general numerical approach has been presented to include Dirac-delta functions and their spatial derivatives due to jump discontinuities. Numerical results are presented to illustrate the application of these techniques to the solution of different types of spatial variations of parameters and boundary conditions.

Analytic Expressions for Radar Sea Clutter WSSUS Scattering Functions

Bello’s stochastic linear time-varying system theory has been widely used in the wireless communications literature to characterize multipath fading channel statistics. In the context of radar backscatter, this formulation allows for statistical characterization of distributed radar targets in range and Doppler using wide-sense stationary uncorrelated scattering (WSSUS) models. WSSUS models separate the channel from the effect of the waveform and receive filter, making it an ideal formulation for waveform design problems. Of particular interest in the radar waveform design community is the ability to suppress unwanted backscatter from the earth’s surface, known as clutter. Various methods for estimating WSSUS system functions have been studied in the literature, but to date no analytic expressions for radar surface clutter range-Doppler scattering functions exist. In this work we derive a frequency-selective generalization of the Jakes Doppler spectrum model, which is widely used in the wireless communications literature, adapt it for use in radar problems, and show how the maximum entropy method can be used to extend this model to account for internal clutter motion. Validation of the spectral and stationarity properties of the proposed model against a subset of the Australian Ingara sea clutter database is performed, and good agreement is shown.

A Unified Method for the Analysis of Phase and Amplitude Noise in Electrical Oscillators

This paper describes a unified method to quantify and compare amplitude and phase noise induced by white and colored noise sources in electrical oscillators. The proposed approach relies on the Floquet theory of linear time-varying systems and on the application of averaging techniques to simplify the stochastic equations. Closed-form expressions for the phase and amplitude power spectrum are provided. These results are validated via comparisons with SpectreRF simulations and measurements.

The Swing Control of Cranes with Variable Rope Length Based on Linear Time Varying System Theory

In this paper, following the state-feedback stabilization for time-varying systems proposed by Wolovich, a controller is designed for the overhead cranes with a linearized parameter-varying model. The resulting closed-loop system is equivalent, via a Lyapunov transformation, to a stable time-invariant system of assigned eigenvalues. The simulation results show the validity of this method.

Feedback Control of T-S Fuzzy Systems Based on LTV System Theory

Takage-Sugeno (T-S) fuzzy systems are equivalent to linear time-varying (LTV) systems; the theory of LTV systems can therefore be used to design global feedback controllers to stabilise T-S fuzzy systems. In this paper, a T-S fuzzy system is first transformed into its canonical form in terms of a Lyapunov transform. A global feedback controller can then be designed to assign all parallel D-spectrum eigenvalues (PD eigenvalues) of the closed-loop T-S fuzzy system to the desired locations with negative extended means. The advantage of the proposed scheme is that it allows us to design a single feedback controller to regulate the closed-loop T-S fuzzy system in the whole state-space, and guarantee that the closed-loop system is globally asymptotically stable. A simulation example is given in support of the proposed scheme.

On stationary linear time-varying systems

A comprehensive review of representations of linear timevarying systems is given, both in the time and in frequency domains. Subsequently a definition is given of a stationary deterministic signal. Based on this definition the notion of stationary systems is introduced. These systems have the useful property that the spectral relation between input and output has a simpler form than the corresponding relation for arbitrary time-varying systems. It is shown that causality puts rather severe constraints on the frequency mappings that can be realized by stationary linear systems. An extension of the theory of linear time-varying systems to the case of discrete-time and hybrid systems (analog input, digital output, or vice versa) is discussed. Examples of stationary systems are given, such as a decimator, a periodic sampler, and a bilinear A/D converter. Also, a recently proposed generalized sampling method is analyzed by means of the concepts discussed in this paper.

Algebraic Theory of Linear Time-Varying Systems

An algebraic theory of linear time-varying discrete-time systems is developed in terms of a module structure defined over a noncommutative polynomial ring. The module setup is induced from a semilinear transformation that is derived from the given system. Various structural properties of the module framework are explored including the concepts of cyclicity and n-cyclicity. The module theory is then applied to the study of reachability and state feedback. Results on the construction of feedback controllers are obtained that resemble pole or coefficient assignability in the theory of time-invariant systems.