Presence Of Small Disturbances Research Articles

A lie group PMP approach for optimal stabilization and tracking control of autonomous underwater vehicles

AbstractIn this research, we explore a finite horizon optimal stabilization and tracking control scheme for the dynamical model of a 6‐DOF Autonomous Underwater Vehicle (AUV). Dynamical equations of the AUV are represented in a Lie group () framework, encompassing both translational and rotational motions. Utilizing a left Lie group action on , we define error function for velocities via a right transport map to effectively address optimal trajectory tracking. The optimal control objective is formulated as a trade‐off problem, aiming to minimize both errors and control effort simultaneously. Left action on yields the left trivialized Hamiltonian function from which the concomitant state and costate dynamical equations are derived using Pontryagin's Minimum Principle (PMP). Consequently, the resulting two‐point boundary value problem is solved to obtain optimal trajectories. We demonstrate the optimality of the resulting solution obtained from the derived control law. For ensuring boundedness in the presence of small disturbances, this study incorporates the effects of internal parametric uncertainties associated with added mass and inertia components, along with the influence of external disturbances induced by ocean currents. Through simulation validations, we confirm the alignment of our results with the theoretical developments, demonstrating that the proposed control law effectively mitigates both parametric uncertainties and ocean current disturbances.

Optimization and stability of the contrast transfer function in aberration-corrected electron microscopy

The Contrast Transfer Function (CTF) describes the manner in which the electron microscope modifies the object exit wave function as a result of objective lens aberrations. For optimum resolution in C₃-corrected microscopes it is well established that a small negative value of C₃, offset by positive values of C₅ and defocus C₁ results in the most optimal instrument resolution, and optimization of the CTF has been the subject of several studies. Here we describe a simple design procedure for the CTF that results in a most even transfer of information below the resolution limit. We address not only the resolution of the instrument, but also the stability of the CTF in the presence of small disturbances in C₁ and C₃. We show that resolution can be traded for stability in a rational and transparent fashion. These topics are discussed quantitatively for both weak-phase and strong-phase (or amplitude) objects. The results apply equally to instruments at high electron energy (TEM) and at very low electron energy (LEEM), as the basic optical properties of the imaging lenses are essentially identical.

On the least-time suboptimal feedback problem in Lp-space

A design procedure developed in this paper leads to a feedback suboptimal, least time control system which is assumed to operate in the presence of small disturbances. The approach is based on the L-problem in functional analysis and the control signal is assumed to be bounded in its norm associated with Lp-space. Computer simulation results are included to demonstrate the effectiveness of the technique.

The Dynamic Behavior of Articulated Pipes Conveying Fluid With Periodic Flow Rate

The plane motion of two rigid, straight articulated pipes conveying fluid is examined. In contrast to previous work, the flow rate is not taken as constant, but is allowed to have small periodic oscillations about a mean value, as would be expected in a pump-driven system. It is shown that in the presence of such disturbances, both parametric and combination resonances are possible. When the system can also admit loss of stabilty by static buckling or by flutter, it is found that the presence of small periodic disturbances constitutes a destabilizing effect. Floquet theory and converging infinite determinant expansions are used to illustrate a basic difference between systems which lose stability by divergence and those that lose stability by flutter. An algebraic criterion is obtained for the minimum amplitude of flow rate oscillation required for the system to be affected by the presence of small disturbances.

Optimization and Control of Nonlinear Systems Using the Second Variation

A feedback control scheme is described that maximizes a terminal quantity while satisfying specified terminal conditions, in the presence of small disturbances. The scheme can also be used in a rapidly converging computation technique to find exact solutions to the nonlinear two-point boundary value problems occurring in the calculus of variations. The scheme is based on a linear perturbation from a nominal optimum path and, as such, involves the second variation of the calculus of variations. A simple analytical example is given for thrust direction control to place a vehicle in orbit. Numerical examples of both the control scheme and the optimization technique are given for a lifting vehicle re-entering the earth’s atmosphere at parabolic speed.