42,476 research outputs found
Further Results on Lyapunov Functions for Slowly Time-Varying Systems
We provide general methods for explicitly constructing strict Lyapunov
functions for fully nonlinear slowly time-varying systems. Our results apply to
cases where the given dynamics and corresponding frozen dynamics are not
necessarily exponentially stable. This complements our previous Lyapunov
function constructions for rapidly time-varying dynamics. We also explicitly
construct input-to-state stable Lyapunov functions for slowly time-varying
control systems. We illustrate our findings by constructing explicit Lyapunov
functions for a pendulum model, an example from identification theory, and a
perturbed friction model.Comment: Accepted for publication in Mathematics of Control, Signals, and
Systems (MCSS) on November 20, 200
A global observer for attitude and gyro biases from vector measurements
We consider the classical problem of estimating the attitude and gyro biases
of a rigid body from vector measurements and a triaxial rate gyro. We propose a
simple "geometry-free" nonlinear observer with guaranteed uniform global
asymptotic convergence and local exponential convergence; the stability
analysis, which relies on a strict Lyapunov function, is rather simple. The
excellent behavior of the observer is illustrated through a detailed numerical
simulation
μ-Dependent model reduction for uncertain discrete-time switched linear systems with average dwell time
In this article, the model reduction problem for a class of discrete-time polytopic uncertain switched linear systems with average dwell time switching is investigated. The stability criterion for general discrete-time switched systems is first explored, and a μ-dependent approach is then introduced for the considered systems to the model reduction solution. A reduced-order model is constructed and its corresponding existence conditions are derived via LMI formulation. The admissible switching signals and the desired reduced model matrices are accordingly obtained from such conditions such that the resulting model error system is robustly exponentially stable and has an exponential H∞ performance. A numerical example is presented to demonstrate the potential and effectiveness of the developed theoretical results
Exact Solutions for Domain Walls in Coupled Complex Ginzburg - Landau Equations
The complex Ginzburg Landau equation (CGLE) is a ubiquitous model for the
evolution of slowly varying wave packets in nonlinear dissipative media. A
front (shock) is a transient layer between a plane-wave state and a zero
background. We report exact solutions for domain walls, i.e., pairs of fronts
with opposite polarities, in a system of two coupled CGLEs, which describe
transient layers between semi-infinite domains occupied by each component in
the absence of the other one. For this purpose, a modified Hirota bilinear
operator, first proposed by Bekki and Nozaki, is employed. A novel
factorization procedure is applied to reduce the intermediate calculations
considerably. The ensuing system of equations for the amplitudes and
frequencies is solved by means of computer-assisted algebra. Exact solutions
for mutually-locked front pairs of opposite polarities, with one or several
free parameters, are thus generated. The signs of the cubic gain/loss, linear
amplification/attenuation, and velocity of the coupled-front complex can be
adjusted in a variety of configurations. Numerical simulations are performed to
study the stability properties of such fronts.Comment: Journal of the Physical Society of Japan, in pres
Adiabatic stability under semi-strong interactions: The weakly damped regime
We rigorously derive multi-pulse interaction laws for the semi-strong
interactions in a family of singularly-perturbed and weakly-damped
reaction-diffusion systems in one space dimension. Most significantly, we show
the existence of a manifold of quasi-steady N-pulse solutions and identify a
"normal-hyperbolicity" condition which balances the asymptotic weakness of the
linear damping against the algebraic evolution rate of the multi-pulses. Our
main result is the adiabatic stability of the manifolds subject to this normal
hyperbolicity condition. More specifically, the spectrum of the linearization
about a fixed N-pulse configuration contains essential spectrum that is
asymptotically close to the origin as well as semi-strong eigenvalues which
move at leading order as the pulse positions evolve. We characterize the
semi-strong eigenvalues in terms of the spectrum of an explicit N by N matrix,
and rigorously bound the error between the N-pulse manifold and the evolution
of the full system, in a polynomially weighted space, so long as the
semi-strong spectrum remains strictly in the left-half complex plane, and the
essential spectrum is not too close to the origin
Complex Dynamics and Synchronization of Delayed-Feedback Nonlinear Oscillators
We describe a flexible and modular delayed-feedback nonlinear oscillator that
is capable of generating a wide range of dynamical behaviours, from periodic
oscillations to high-dimensional chaos. The oscillator uses electrooptic
modulation and fibre-optic transmission, with feedback and filtering
implemented through real-time digital-signal processing. We consider two such
oscillators that are coupled to one another, and we identify the conditions
under which they will synchronize. By examining the rates of divergence or
convergence between two coupled oscillators, we quantify the maximum Lyapunov
exponents or transverse Lyapunov exponents of the system, and we present an
experimental method to determine these rates that does not require a
mathematical model of the system. Finally, we demonstrate a new adaptive
control method that keeps two oscillators synchronized even when the coupling
between them is changing unpredictably.Comment: 24 pages, 13 figures. To appear in Phil. Trans. R. Soc. A (special
theme issue to accompany 2009 International Workshop on Delayed Complex
Systems
A Comparison of LPV Gain Scheduling and Control Contraction Metrics for Nonlinear Control
Gain-scheduled control based on linear parameter-varying (LPV) models derived
from local linearizations is a widespread nonlinear technique for tracking
time-varying setpoints. Recently, a nonlinear control scheme based on Control
Contraction Metrics (CCMs) has been developed to track arbitrary admissible
trajectories. This paper presents a comparison study of these two approaches.
We show that the CCM based approach is an extended gain-scheduled control
scheme which achieves global reference-independent stability and performance
through an exact control realization which integrates a series of local LPV
controllers on a particular path between the current and reference states.Comment: IFAC LPVS 201
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