Fast Mesh Refinement in Pseudospectral Optimal Control

Mesh refinement in pseudospectral (PS) optimal control is embarrassingly easy --- simply increase the order $N$ of the Lagrange interpolating polynomial and the mathematics of convergence automates the distribution of the grid points. Unfortunately, as $N$ increases, the condition number of the resulting linear algebra increases as $N^2$; hence, spectral efficiency and accuracy are lost in practice. In this paper, we advance Birkhoff interpolation concepts over an arbitrary grid to generate well-conditioned PS optimal control discretizations. We show that the condition number increases only as $\sqrt{N}$ in general, but is independent of $N$ for the special case of one of the boundary points being fixed. Hence, spectral accuracy and efficiency are maintained as $N$ increases. The effectiveness of the resulting fast mesh refinement strategy is demonstrated by using \underline{polynomials of over a thousandth order} to solve a low-thrust, long-duration orbit transfer problem.Comment: 27 pages, 12 figures, JGCD April 201

A Gauss pseudospectral transcription for optimal control

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2005.Includes bibliographical references (p. 237-243).A pseudospectral method for solving nonlinear optimal control problems is proposed in this thesis. The method is a direct transcription that transcribes the continuous optimal control problem into a discrete nonlinear programming problem (NLP), which can be solved by well-developed algorithms. The method is based on using global polynomial approximations to the dynamic equations at a set of Gauss collocation points. The optimality conditions of the NLP have been found to be equivalent to the discretized optimality conditions of the continuous control problem, which is not true of other pseudospectral methods. This result indicates that the method can take advantage of the properties of both direct and indirect formulations, and allows for the costates to be estimated directly from the Lagrange multipliers of the NLP. The method has been shown empirically to have very fast convergence (exponential) in the states, controls, and costates, for problems with analytic solutions. This convergence rate of the proposed method is significantly faster than traditional finite difference methods, and has been demonstrated with many example problems. The initial costate estimate from the proposed method can be used to define an optimal feedback law for real time optimal control of nonlinear problems. The application and effectiveness of this approach has been demonstrated with the simulated trajectory optimization of a launch vehicle.by David Benson.Ph.D

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

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

Practical application of pseudospectral optimization to robot path planning

To obtain minimum time or minimum energy trajectories for robots it is necessary to employ planning methods which adequately consider the platform’s dynamic properties. A variety of sampling, graph-based or local receding-horizon optimisation methods have previously been proposed. These typically use simplified kino-dynamic models to avoid the significant computational burden of solving this problem in a high dimensional state-space. In this paper we investigate solutions from the class of pseudospectral optimisation methods which have grown in favour amongst the optimal control community in recent years. These methods have high computational efficiency and rapid convergence properties. We present a practical application of such an approach to the robot path planning problem to provide a trajectory considering the robot’s dynamic properties. We extend the existing literature by augmenting the path constraints with sensed obstacles rather than predefined analytical functions to enable real world application

Concurrent Learning Adaptive Model Predictive Control with Pseudospectral Implementation

This paper presents a control architecture in which a direct adaptive control technique is used within the model predictive control framework, using the concurrent learning based approach, to compensate for model uncertainties. At each time step, the control sequences and the parameter estimates are both used as the optimization arguments, thereby undermining the need for switching between the learning phase and the control phase, as is the case with hybrid-direct-indirect control architectures. The state derivatives are approximated using pseudospectral methods, which are vastly used for numerical optimal control problems. Theoretical results and numerical simulation examples are used to establish the effectiveness of the architecture.Comment: 21 pages, 13 figure