A Pseudospectral Approach to High Index DAE Optimal Control Problems

Historically, solving optimal control problems with high index differential algebraic equations (DAEs) has been considered extremely hard. Computational experience with Runge-Kutta (RK) methods confirms the difficulties. High index DAE problems occur quite naturally in many practical engineering applications. Over the last two decades, a vast number of real-world problems have been solved routinely using pseudospectral (PS) optimal control techniques. In view of this, we solve a "provably hard," index-three problem using the PS method implemented in DIDO, a state-of-the-art MATLAB optimal control toolbox. In contrast to RK-type solution techniques, no laborious index-reduction process was used to generate the PS solution. The PS solution is independently verified and validated using standard industry practices. It turns out that proper PS methods can indeed be used to "directly" solve high index DAE optimal control problems. In view of this, it is proposed that a new theory of difficulty for DAEs be put forth.Comment: 14 pages, 9 figure

Efficient reconstructed Legendre algorithm for solving linear-quadratic optimal control problems

AbstractIn this paper, a new numerical approach via reconstructed Legendre orthogonal polynomials (LOPs) is presented to solve the linear quadratic optimal control problems (LQPs). By using the elegant operational properties of orthogonal polynomials, a computationally attractive algorithm is developed for calculating LQP. A numerical example illustrates the techniques and demonstrates the accuracy and efficiency of these controllers

On the Rate of Convergence for the Pseudospectral Optimal Control of Feedback Linearizable Systems

In this paper, we prove a theorem on the rate of convergence for the optimal cost computed using PS methods. It is a first proved convergence rate in the literature of PS optimal control. In addition to the high-order convergence rate, two theorems are proved for the existence and convergence of the approximate solutions. This paper contains several essential differences from existing papers on PS optimal control as well as some other direct computational methods. The proofs do not use necessary conditions of optimal control. Furthermore, we do not make coercivity type of assumptions. As a result, the theory does not require the local uniqueness of optimal solutions. In addition, a restrictive assumption on the cluster points of discrete solutions made in existing convergence theorems are removed.Comment: 28 pages, 3 figures, 1 tabl

Charge-Balanced Minimum-Power Controls for Spiking Neuron Oscillators

In this paper, we study the optimal control of phase models for spiking neuron oscillators. We focus on the design of minimum-power current stimuli that elicit spikes in neurons at desired times. We furthermore take the charge-balanced constraint into account because in practice undesirable side effects may occur due to the accumulation of electric charge resulting from external stimuli. Charge-balanced minimum-power controls are derived for a general phase model using the maximum principle, where the cases with unbounded and bounded control amplitude are examined. The latter is of practical importance since phase models are more accurate for weak forcing. The developed optimal control strategies are then applied to both mathematically ideal and experimentally observed phase models to demonstrate their applicability, including the phase model for the widely studied Hodgkin-Huxley equations.Comment: 24 pages, 12 figure

A new scalable algorithm for computational optimal control under uncertainty

We address the design and synthesis of optimal control strategies for high-dimensional stochastic dynamical systems. Such systems may be deterministic nonlinear systems evolving from random initial states, or systems driven by random parameters or processes. The objective is to provide a validated new computational capability for optimal control which will be achieved more efficiently than current state-of-the-art methods. The new framework utilizes direct single or multi-shooting discretization, and is based on efficient vectorized gradient computation with adaptable memory management. The algorithm is demonstrated to be scalable to high-dimensional nonlinear control systems with random initial condition and unknown parameters.Comment: 23 pages, 17 figure

Two-Stage Path Planning Approach for Designing Multiple Spacecraft Reconfiguration Maneuvers

The paper presents a two-stage approach for designing optimal reconfiguration maneuvers for multiple spacecraft. These maneuvers involve well-coordinated and highly-coupled motions of the entire fleet of spacecraft while satisfying an arbitrary number of constraints. This problem is particularly difficult because of the nonlinearity of the attitude dynamics, the non-convexity of some of the constraints, and the coupling between the positions and attitudes of all spacecraft. As a result, the trajectory design must be solved as a single 6N DOF problem instead of N separate 6 DOF problems. The first stage of the solution approach quickly provides a feasible initial solution by solving a simplified version without differential constraints using a bi-directional Rapidly-exploring Random Tree (RRT) planner. A transition algorithm then augments this guess with feasible dynamics that are propagated from the beginning to the end of the trajectory. The resulting output is a feasible initial guess to the complete optimal control problem that is discretized in the second stage using a Gauss pseudospectral method (GPM) and solved using an off-the-shelf nonlinear solver. This paper also places emphasis on the importance of the initialization step in pseudospectral methods in order to decrease their computation times and enable the solution of a more complex class of problems. Several examples are presented and discussed

Department of Applied Mathematics Academic Program Review, Self Study / June 2010

The Department of Applied Mathematics has a multi-faceted mission to provide an exceptional mathematical education focused on the unique needs of NPS students, to conduct relevant research, and to provide service to the broader community. A strong and vibrant Department of Applied Mathematics is essential to the university's goal of becoming a premiere research university. Because research in mathematics often impacts science and engineering in surprising ways, the department encourages mathematical explorations in a broad range of areas in applied mathematics with specific thrust areas that support the mission of the school