WIND DISTURBANCE CANCELLATION FOR SMALLER ALT-AZIMUTH TELESCOPES

This study focuses on eliminating unknown amplitude wind disturbance for 2-DOF alt-azimuth configuration small telescopes. An adaptive controller is designed to overcome wind disturbance as a set and forget system. The mathematical model is derived based on 2-DOF alt-azimuth configuration. The wind disturbance is modeled as a sum of sinusoidal with unknown amplitude, frequency and phase by using Wind-Gust model. The controller aims to cancel the effect of the disturbance on the altitude and azimuth angles of the telescope while positioning or staying static on a dedicated configuration. The asymptotic stability is proven with the Lyapunov approach. The numerical study is illustrated to success of the proposed controller.

FLUID INHOMOGENEITY INFLUENCING WAVE MOTION

This article deals with a problem that describes the propagation of surface waves in a layer of an inhomogeneous fluid. The authors present a mathematical model that describes wave motions on the surface of an ideal exponentially stratified fluid. In the equations and boundary conditions, the transition to dimensionless variables and quantities has been completed. Next, a linear version of the problem follows, the solution of which is in the form of progressive waves of a steady-state form with unknown amplitude coefficients. This type of solution is substituted into the equations and boundary conditions of the linear problem, which makes it possible to reduce the determination of unknown quantities to the problem of solving a system of ordinary differential equations. Solving the system has allowed identifying two areas of physical parameters with different nature of wave motion. Expressions are obtained for the unknown components of the fluid velocity, pressure, free surface shape, and wave frequency. This article contains the analysis of the influence of various parameters of the problem on the wave motion: graphs of the dependence of the phase velocity of the wave on the stratification parameter are constructed for different layer depths and wavelengths. For a better understanding of the nature of wave motion, the expressions for the trajectories of liquid particles are determined. This has required writing the equations of motion of particles using the obtained expressions for the components of the velocity vector; these equations are solved with the method of asymptotic approximations. A graphical analysis of the effect of the stratification parameter value on the particle trajectory shape is carried out. The results have revealed that an increase in stratification leads to a compression of the trajectory in the vertical direction.

Simulating the complexity of the dark matter sheet I: numerical algorithms

ABSTRACT At early times, dark matter has a thermal velocity dispersion of unknown amplitude which, for warm dark matter (WDM) models, can influence the formation of non-linear structure on observable scales. We propose a new scheme to simulate cosmologies with a small-scale suppression of perturbations that combines two previous methods in a way that avoids the numerical artefacts which have so far prevented either from producing fully reliable results. At low densities and throughout most of the cosmological volume, we represent the dark matter phase sheet directly using high-accuracy interpolation, thereby avoiding the artificial fragmentation which afflicts particle-based methods in this regime. Such phase-sheet methods are, however, unable to follow the rapidly increasing complexity of the denser regions of dark matter haloes, so for these we switch to an N-body scheme which uses the geodesic deviation equation to track phase-sheet properties local to each particle. In addition, we present a novel high-resolution force calculation scheme based on an oct-tree of cubic force resolution elements which is well suited to approximate the force field of our combined sheet+particle distribution. Our hybrid simulation scheme enables the first reliable simulations of the internal structure of low-mass haloes in a WDM cosmology.

Design of a Low-Order Harmonic Disturbance Observer with Application to a DC Motor Position Control

Among various tools implemented to counteract undesired effects of time-varying uncertainties, disturbance observer (DOB)-based controller has gained wide popularity as a result of its flexibility and efficacy. In this paper, a low-order DOB that is capable of compensating for the effects of a biased harmonic disturbance, as well as plant uncertainties is presented. The proposed low-order DOB can asymptotically estimate a harmonic disturbance of known frequency but unknown amplitude and phase, by using measurable output variables. An analysis carried out by using the singular perturbation theory shows that the nominal performance of the system can be recovered from a real uncertain system when the observer gain is sufficiently large. The observer gains that result in the performance recovery of the real uncertain system are obtained from the stability condition of the boundary-layer system. To test the performance of the proposed observer, computer simulations with a numerical example and laboratory experiments using a DC motor system have been carried out. The experimental results show that the proposed low-order DOB-based control scheme can provide enhanced performance.

A Collocation-Based Algorithm for Analyzing Bifurcations in Phase Locked Loops with Tanlock and Sawtooth Phase Detectors

Analysis of bifurcation of second-order analog phase locked loop (PLL) with tanlock and sawtooth phase detectors is investigated. Both qualitative and quantitative analyses are carried out. Qualitatively, the basin boundaries of the attractors were constructed by plotting the stable and the unstable manifolds of the system. The basin boundaries show that the PLL under consideration for certain loop parameters has a separatrix cycle which terminates the limit cycle (out-of-lock state) and the loop pulls-in. This behavior is known in literature as homoclinic bifurcation and the value of the bifurcation parameter where this process occurs is called the pull-in range. Quantitatively, we propose a collocation-based algorithm to compute the separatrix cycle and the pull-in range. The separatrix cycle is approximated by a finite set of harmonics N with unknown amplitudes and by utilizing the fact that this limit cycle bifurcates from a separatrix cycle, a system of nonlinear algebraic equations is derived. For given values of filter parameters and gain, the algorithm numerically solves for the unknown amplitude of the harmonics and the value of the pull-in range simultaneously by evaluating the system at the collocation points. Results demonstrate that phase locked loop with sawtooth phase detector characteristics has the wider pull-in range followed by tanlock and sinusoidal, respectively.