Alan M. Whitman

Closed-Loop Dynamic Analysis of a Rotary Inverted Pendulum for Control Design

This paper presents an approximate analytical solution for the weakly nonlinear closed-loop dynamics of the sliding phase of a sliding mode controlled rotary inverted pendulum based on the multiple scale method. A locally stable nonlinear sliding mode control law with starting configurations above the horizontal line is presented for the rotary inverted pendulum. The analytical expressions derived from the nonlinear solution of the reduced-order closed-loop dynamics provide both qualitative and quantitative insight into the closed-loop response leading to proper selection of parameters that guarantee stabilization and improve controller performance. The approximate analytical solution is verified through comparison with the exact numerical solution. The control performance predicted by the analytical solution is experimentally demo.

A phenomenological model of EEG based on the dynamics of a stochastic Duffing-van der Pol oscillator network

Abstract In this work, we propose a novel phenomenological model of the EEG signal based on the dynamics of a coupled Duffing-van der Pol oscillator network. An optimization scheme is adopted to match data generated from the model with clinically obtained EEG data from subjects under resting eyes-open (EO) and eyes-closed (EC) conditions. It is shown that a coupled system of two Duffing-van der Pol oscillators with optimized parameters yields signals with characteristics that match those of the EEG in both the EO and EC cases. The results, which are reinforced using statistical analysis, show that the EEG recordings under EC and EO resting conditions are clearly distinct realizations of the same underlying model occurring due to parameter variations with qualitatively different nonlinear dynamic characteristics. In addition, the interplay between noise and nonlinearity is addressed and it is shown that, for appropriately chosen values of noise intensity in the model, very good agreement exists between the model output and the EEG in terms of the power spectrum as well as Shannon entropy. In summary, the results establish that an appropriately tuned stochastic coupled nonlinear oscillator network such as the Duffing-van der Pol system could provide a useful framework for modeling and analysis of the EEG signal. In turn, design of algorithms based on the framework has the potential to positively impact the development of novel diagnostic strategies for brain injuries and disorders.

Propagation of radiation in time-dependent three-dimensional random media

In Ref. [1] (Appendix A) we derived equations governing the frequency and spatial spectrum of radiation propagating in three-dimensional time-dependent random media with randomly varying sound speed c ( x , t). From the spectral equations we determine equations for the energy flux in both the forward and backward directions. We consider media that are spatially homogeneous and isotropic and stationary in time. In order to allow an independence assumption the analysis is restricted to fluctuations that satisfy the conditions τμ ≫ L z /c 0 and τμ ≪ L FS/c 0 where τμ is the characteristic time of the fluctuations, k 0 is the mean radiation wavenumber, L z is the characteristic correlation length of the random fluctuations in the mean propagation direction and L FS is a mean scattering length. We consider various values of γ = (k 0 L z )2/2. When γ ≪ 1 we find the usual radiation transfer equations. When γ ≫ 1, but back-scattering can be neglected, we find the forward-scattering equations. We also consider γ ≫ 1, when back-scattering cannot be neglected. We consider as initial boundary conditions a plane wave and an infinite incoherent source. We present numerical solutions for γ ≪ 1, γ = O (1) and γ ≫ 1 using a simple Gaussian form for the fluctuation correlation function.

The derivation and solution of the modal flux equations for backscattering in a duct in the presence of random space and time fluctuations

Abstract In this paper we first derive the equations governing the energy fluxes propagating in each of the modes of a duct. In each mode there is a forward and backward component and the equations are intended to treat ducts in which backscattering plays a major role. The modal fluxes are coupled since there is transfer of energy between the modes that occurs as a result of random time and space sound-speed fluctuations in the medium in the duct. Since the fluctuations are both space and time dependent the governing equations are radiation transport equations. This is not the case if the fluctuations depend only on space. The basic method is to develop a coupled set of equations for the energy spectra in the modes and then to integrate over the frequency to obtain the fluxes. In the second section of this paper the modal flux equations are solved. A numerical result is presented to show how energy is transferred between modes. It is also shown how the reflected energy varies as a function of duct length.

Effect Of The Turbulence Inner Scale On Scintillation In The Atmosphere

Scintillation indices as high as four to six have been measured in experiments in the atmosphere. The inner scale of turbulence has a dramatic effect on the value of the scintillation index when the index is greater than unity. Here we review theory which shows that, in order to obtain such high values, it is necessary to modify the Kolmogorov form by the addition of an inner scale. We present results for plane wave and point source initial conditions and show that, without an inner scale, the maximum scintillation index that may be obtained is much below experimental results. Moreover, we show that, even with the inner scale included, we must have a point-source-type initial condition rather than a plane-wave-type initial condition. We also discuss the effect of the particular modification of the Kolmogorov form that is used to introduce the inner scale.

Analytical Stability Analysis of Surface Vessel Trajectories for a Control-Oriented Model

An analytical stability analysis of the steady trajectory for a surface vessel with various damping models is presented in this work. The analysis is based on a control-oriented, three degrees-of-freedom model that considers vessel motion only in the horizontal plane. The goal of this study is to understand the vessel trajectories predicted by this reduced order model for model-based control design. Straight line and circular motion stability conditions for each trajectory are derived and presented for the various damping models. The results of this analysis show that either a straight line or a circular steady trajectory is possible, depending on the magnitude of the surge force and the form of the damping model used to represent viscous drag, vortex shedding, and losses due to the surface wake generated by the vessel motion. However, the straight line motion is much less likely for the vessel considered in this work.

Pulse propagation of acoustic waves scattered in a channel

Abstract When acoustic waves are scattered by random sound-speed fluctuations in a two-dimensional channel the energy is continually transferred between the propagating modes. In the multiple- scattering region the energy flux assumes an asymptotic form in which there is equal energy flux propagating in each mode. Here we shall make use of this well known result to show how to obtain an asymptotic form for a pulse of acoustic energy propagating in the channel. In the multiple-scattering region the speed of the acoustic waves in the pulse continually changes as the energy is transferred between the modes. The process is basically a diffusion process around the mean speed of propagation. We shall first show, using physical arguments, that the diffusion coefficient is proportional to the square root of the propagation distance times the mean free path of scattering. The theory governing the acoustic propagation in the channel is formulated in terms of modal coherence equations and we shall next give a brief ...

Design Optimization of Variable Stroke Compressors to Minimize Vibration by Using Asymptotic Analysis

A design optimization of a variable displacement compressor that minimizes its crankshaft vibra tion is presented. A vibration mount installed between the flywheel and crankshaft is used as the optimizing element; the stiffness and damping values of the mount are selected so that the shaft vibration level is min imized over the cruise speed range, while the maximum twist angle acceleration amplitude and transmitted torque are kept within acceptable limits for the entire operating speed range. An asymptotic analysis is used to obtain the nonlinear vibrational response. This analysis provides analytic expressions for the motion and forces in terms of the operating speed and other geometric, kinetic, and dynamic parameters. Consequently, the calculation of the objective and constraint functions is simple to do. A result of the analysis is that the optimal design is insensitive to changes in the magnitude and shape of the cylinder pressure excitation.

Effect of random velocity fluctuations on underwater scattering

There is a well‐developed theory to account for the effect of random fluctuations in the sound speed on the underwater scattering of acoustic radiation. Here it is shown that the theory may be readily extended to include the effect of random fluctuations in fluid velocity. It is assumed that the characteristic time and length scales associated with the velocity correlation functions are long when compared to the corresponding scales (period and wavelength) of the acoustic radiation. The application of the theory to ocean studies is discussed with emphasis on the utility of using an inverse method to find the lateral correlation function.

Pulse propagation of acoustic waves scattered in a three-dimensional channel

Abstract In a previous paper (Whitman et al 1999 Waves Random Media 9 1–11) we discussed the scattering of acoustic waves by random sound-speed fluctuations in a two-dimensional channel and presented an asymptotic form for an acoustic pulse propagating in the channel. Here we include the three-dimensional effect of transverse scattering. We find an asymptotic solution in which initially the two-dimensional mode-transfer effect is more important than the transverse scattering effect. However, for large enough propagation distances the transverse scattering effect dominates the pulse spread. In this paper we shall show the form of the pulse shape in both propagation ranges as well as in the transition regime. We shall begin with a discussion of the physics of the problem and then present a mathematical discussion.