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Analysis and Design of Oscillatory Control Systems
"... This paper presents analysis and design results for control systems subject to oscillatory inputs, i.e., inputs of large amplitude and high frequency. The key analysis results are a series expansion characterizing the averaged system and various Liealgebraic conditions that guarantee the series can ..."
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Cited by 11 (4 self)
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This paper presents analysis and design results for control systems subject to oscillatory inputs, i.e., inputs of large amplitude and high frequency. The key analysis results are a series expansion characterizing the averaged system and various Liealgebraic conditions that guarantee the series can be summed. Various example systems provide insight into the results. With regards to design, we recover and extend a variety of point stabilization and trajectory tracking results using oscillatory controls. We present novel developments on stabilization of systems with positive trace and on tracking for second order underactuated systems.
Stability and Nonlinear Dynamics in Thyristor and Diode Circuits
, 2001
"... Contents 6 Stabilityan# n##9118# dyn#118# in thyristoran# diode circuits 1 6.1 Introduc#T9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6.2 Ideal diode and thyristorswitc hing rules . . . . . . . . . . . . . . . . . 3 6.3 Static VAR system example . . . . . . . . . . . . . ..."
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Cited by 1 (0 self)
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Contents 6 Stabilityan# n##9118# dyn#118# in thyristoran# diode circuits 1 6.1 Introduc#T9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6.2 Ideal diode and thyristorswitc hing rules . . . . . . . . . . . . . . . . . 3 6.3 Static VAR system example . . . . . . . . . . . . . . . . . . . . . . . . 4 6.4 Poinc## emap................................ 6 6.5 Jac#T7T# of Poinc## emap ......................... 8 6.5.1 Thyristorc#isto tfunc#v#S and transversality . . . . . . . . . . 9 6.5.2 Relations between on and o# systems . . . . . . . . . . . . . . 10 6.5.3 Derivation ofJac#78#v formula . . . . . . . . . . . . . . . . . . 11 6.5.4Disc#TvTT ofJac#78vq formula . . . . . . . . . . . . . . . . . . 14 6.6Switc hing damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6.6.1 Simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6.6.2Switc hing damping in the SVC example . . . . . . . . . . . . . 17 6.6.3 Variational equation . . . . . . .
Fixed Point Problems  an Introduction
, 1996
"... This paper surveys a number of fundamental results on the existence and uniqueness of fixed points for certain classes of possibly nonlinear operators. I do not try to be exhaustive, but merely to present the results that are more useful in the context of signal and image reconstruction. Some specif ..."
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Cited by 1 (1 self)
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This paper surveys a number of fundamental results on the existence and uniqueness of fixed points for certain classes of possibly nonlinear operators. I do not try to be exhaustive, but merely to present the results that are more useful in the context of signal and image reconstruction. Some specific aspects pertaining to linear operators, and linear operators in finite dimensional spaces, are also discussed. It is shown that the set of fixed points of a nonexpansive operator is either empty or convex. Under rather general conditions this shows that the minimum norm solution of an operator equation of the form x = Ax exists and is unique, provided that A is nonexpansive. This is a new result, which has interesting practical consequences in signal and image reconstruction problems and in other engineering applications.
ANALYSIS OF OSCILLATORY CONTROL SYSTEMS
"... This paper presents analysis results for control systems subject to oscillatory inputs, i.e., inputs of large amplitude and high frequency. The key results are a series expansion characterizing the averaged system and various Liealgebraic conditions that guarantee the series can be summed. Some ill ..."
Abstract

Cited by 1 (1 self)
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This paper presents analysis results for control systems subject to oscillatory inputs, i.e., inputs of large amplitude and high frequency. The key results are a series expansion characterizing the averaged system and various Liealgebraic conditions that guarantee the series can be summed. Some illustrative example systems provide insight into the results; control design applications are discussed in a companion paper.
On Monodromy Matrix Computation
"... We present a study on the critical time step for the numerical integration based on the RungeKutta method of the monodromy matrix (the fundamental matrix solution) associated with a set of n firstorder linear ordinary differential equations with periodic coefficients. By applying the LiapunovSchm ..."
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We present a study on the critical time step for the numerical integration based on the RungeKutta method of the monodromy matrix (the fundamental matrix solution) associated with a set of n firstorder linear ordinary differential equations with periodic coefficients. By applying the LiapunovSchmidt method, for any dimension n and systems which are perturbations of autonomous systems, we give an approximation to the critical time step which involves the autonomous part as well as the periodic perturbation.
Generalized Viscoelastic 1Dof Deterministic Nonlinear Oscillators
, 1999
"... The theory of deterministic generalized viscoelastic linear and nonlinear 1D oscillators is formulated and evaluated. Examples of Du#ng, Mathieu, Rayleigh, Roberts and van der Pol oscillators and pendulum responses are investigated. The e#ects of structural damping on oscillator performance is also ..."
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The theory of deterministic generalized viscoelastic linear and nonlinear 1D oscillators is formulated and evaluated. Examples of Du#ng, Mathieu, Rayleigh, Roberts and van der Pol oscillators and pendulum responses are investigated. The e#ects of structural damping on oscillator performance is also considered. Numerical solution protocols are developed and results are discussed to determined the influence of viscoelastic damping on oscillator motion. The numerical examples show that the inclusion of linear or nonlinear viscoelastic material properties significantly influence oscillator responses as related to amplitudes, phase shifts and energy loses when compared to equivalent elastic ones. Keywords : damping, Du#ng oscillator, integraldi#erential equations, Mathieu oscillator, nonlinear deterministic oscillators, numerical analysis, pendulum, Rayleigh, Roberts and van der Pol oscillators, structural damping, linear and nonlinear viscoelasticity 1 INTRODUCTION The study of 1DOF ...
3STEADYSTATE OSCILLATIONS IN NONLINEAR SYSTEMS
"... The preceding chapter introduced the notion of a sinusoidalinput describing function (DF). Some of the implications of this type of linearization are discussed there. Here we apply the DF to the study of steadystate oscillations. For DF utilization to be meaningful, certain conditions must be fulf ..."
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The preceding chapter introduced the notion of a sinusoidalinput describing function (DF). Some of the implications of this type of linearization are discussed there. Here we apply the DF to the study of steadystate oscillations. For DF utilization to be meaningful, certain conditions must be fulfilled
The Poincare map of randomly perturbed periodic motion
, 1206
"... A system of autonomous differential equations with a stable limit cycle and perturbed by small white noise is analyzed in this work. In the vicinity of the limit cycle of the unperturbed deterministic system, we define, construct, and analyze the Poincare map of the randomly perturbed periodic motio ..."
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A system of autonomous differential equations with a stable limit cycle and perturbed by small white noise is analyzed in this work. In the vicinity of the limit cycle of the unperturbed deterministic system, we define, construct, and analyze the Poincare map of the randomly perturbed periodic motion. We show that the time of the first exit from a small neighborhood of the fixed point of the map, which corresponds to the unperturbed periodic orbit, is well approximated by the geometric distribution. The parameter of the geometric distribution tends zero together with the noise intensity. Therefore, our result can be interpreted as an estimate of stability of periodic motion to random perturbations. In addition, we show that the geometric distribution of the first exit times translates into statistical properties of solutions of important differential equation models in applications. To this end, we demonstrate three examples from mathematical neuroscience featuring complex oscillatory patterns characterized by the geometric distribution. We show that in each of these models the statistical properties of emerging oscillations are fully explained by the general properties of randomly perturbed periodic motions identified in this paper.