I. Michael Ross

COSTATE ESTIMATION BY A LEGENDRE PSEUDOSPECTRAL METHOD

We present a Legendre pseudospectral method for directly estimating the costates of the Bolza problem encountered in optimal control theory. The method is based on calculating the state and control variables at the Legendre‐Gauss‐Lobatto (LGL) points. An Nth degree Lagrange polynomial approximation of these variables allows a conversion of the optimal control problem into a standard nonlinear programming (NLP) problem with the state and control values at the LGL points as optimization parameters. By applying theKarush ‐Kuhn‐Tucker (KKT) theorem to the NLP problem, we show that the KKT multipliers satisfy a discrete analog of the costate dynamics including the transversality conditions. Indeed, we prove that the costates at the LGL points are equal to the KKT multipliers divided by the LGL weights. Hence, the direct solution by this method also automatically yields the costates by way of the Lagrange multipliers that can be extracted from an NLP solver. One important advantage of this technique is that it allows a very simple way to check the optimality of the direct solution. Numerical examples are included to demonstrate the method.

Direct Trajectory Optimization by a Chebyshev Pseudospectral Method

A Chebyshev pseudospectral method is presented in this paper for directly solving a generic optimal control problem with state and control constraints. This method employs N t h degree Lagrange polynomial approxiniations for the state and control variables with the values of these variables at the Chebyshev-GaussLobatto (CGL) points as the expansion coefficients. This process yields a nonlinear programming problem (NLP) with the state and control values at the CGL points as unknown NLP parameters. Numerical examples demonstrate this method yields more accurate results than those obtained from the traditional collocation methods.

Pseudospectral Knotting Methods for Solving Optimal Control Problems

A class of computational methods for solving a wide variety of optimal control problems is presented; these problems include nonsmooth, nonlinear, switched optimal control problems, as well as standard multiphase problems. Methods are based on pseudospectral approximations of the differential constraints that are assumed to be given in the form of controlled differential inclusions including the usual vector field and differential-algebraic forms. Discontinuities and switches in states, controls, cost functional, dynamic constraints, and various other mappings associated with the generalized Bolza problem are allowed by the concept of pseudospectral (PS) knots. Information across switches and corners is passed in the form of discrete event conditions localized at the PS knots. The optimal control problem is approximated to a structured sparse mathematical programming problem. The discretized problem is solved using off-the-shelf solvers that include sequential quadratic programming and interior point methods. Two examples that demonstrate the concept of hard and soft knots are presented.

Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control

Abstract In recent years, many practical nonlinear optimal control problems have been solved by pseudospectral (PS) methods. In particular, the Legendre PS method offers a Covector Mapping Theorem that blurs the distinction between traditional direct and indirect methods for optimal control. In an effort to better understand the PS approach for solving control problems, we present consistency results for nonlinear optimal control problems with mixed state and control constraints. A set of sufficient conditions is proved under which a solution of the discretized optimal control problem converges to the continuous solution. Convergence of the primal variables does not necessarily imply the convergence of the duals. This leads to a clarification of the Covector Mapping Theorem in its relationship to the convergence properties of PS methods and its connections to constraint qualifications. Conditions for the convergence of the duals are described and illustrated. An application of the ideas to the optimal attitude control of NPSAT1, a highly nonlinear spacecraft, shows that the method performs well for real-world problems.

Pseudospectral Methods for Infinite-Horizon Optimal Control Problems

A central computational issue in solving infinite-horizon nonlinear optimal control problems is the treatment of the horizon. In this paper, we directly address this issue by a domain transformation technique that maps the infinite horizon to a finite horizon. The transformed finite horizon serves as the computational domain for an application of pseudospectral methods. Although any pseudospectral method may be used, we focus on the Legendre pseudospectral method. It is shown that the proper class of Legendre pseudospectral methods to solve infinite-horizon problems are the Radau-based methods with weighted interpolants. This is in sharp contrast to the unweighted pseudospectral techniques for optimal control. The Legendre-Gauss-Radau pseudospectral method is thus developed to solve nonlinear constrained optimal control problems. An application of the covector mapping principle for the Legendre-Gauss-Radau pseudospectral method generates a covector mapping theorem that provides an efficient approach for the verification and validation of the extremality of the computed solution. Several example problems are solved to illustrate the ideas.

A review of pseudospectral optimal control: From theory to flight

Abstract The home space for optimal control is a Sobolev space. The home space for pseudospectral theory is also a Sobolev space. It thus seems natural to combine pseudospectral theory with optimal control theory and construct “pseudospectral optimal control theory”, a term coined by Ross. In this paper, we review key theoretical results in pseudospectral optimal control that have proven to be critical for a successful flight. Implementation details of flight demonstrations onboard NASA spacecraft are discussed along with emerging trends and techniques in both theory and practice. The 2011 launch of pseudospectral optimal control in embedded platforms is changing the way in which we see solutions to challenging control problems in aerospace and autonomous systems.

Spectral Algorithm for Pseudospectral Methods in Optimal Control

Recent convergence results with pseudospectral methods are exploited to design a robust, multigrid, spectral algorithm for computing optimal controls. The design of the algorithm is based on using the pseudospectral differentiation matrix to locate switches, kinks, corners, and other discontinuities that are typical when solving practical optimal control problems. The concept of pseudospectral knots and Gaussian quadrature rules are used to generate a natural spectral mesh that is dense near the points of interest. Several stopping criteria are developed based on new error-estimation formulas and Jacksons theorem. The sequence is terminated when all of the convergence criteria are satisfied. Numerical examples demonstrate the key concepts proposed in the design of the spectral algorithm. Although a vast number of theoretical and algorithmic issues still remain open, this paper advances pseudospectral methods along several new directions and outlines the current theoretical pitfalls in computation and control.

Issues in the real-time computation of optimal control

Under appropriate conditions, the dynamics of a control system governed by ordinary differential equations can be formulated in several ways: differential inclusion, control parametrization, flatness parametrization, higher-order inclusions and so on. A plethora of techniques have been proposed for each of these formulations but they are typically not portable across equivalent mathematical formulations. Further complications arise as a result of configuration and control constraints such as those imposed by obstacle avoidance or control saturation. In this paper, we present a unified framework for handling the computation of optimal controls where the description of the governing equations or that of the path constraint is not a limitation. In fact, our method exploits the advantages offered by coordinate transformations and harnesses any inherent smoothness present in the optimal system trajectories. We demonstrate how our computational framework can easily and efficiently handle different cost formulations, control sets and path constraints. We illustrate our ideas by formulating a robotics problem in eight different ways, including a differentially flat formulation subject to control saturation. This example establishes the loss of convexity in the flat formulation as well as its ramifications for computation and optimality. In addition, a numerical comparison of our unified approach to a recent technique tailored for control-affine systems reveals that we get about 30% improvement in the performance index.

Hybrid Optimal Control Framework for Mission Planning

With the progressive sophistication of future missions, it has become increasingly apparent that a new framework is necessary for efficient planning, analysis, and optimization of various concepts of operations (CONOPS). In recognizing that CONOPS involve categorical variables, we propose a hybrid optimal control framework that mathematically formalizes such problems. Hybrid optimal control theory extends ordinary optimal control theory by including categorical variables in the problem formulation. The proposed formalism frees mission planners to focus on high-level decision making by automating and optimizing the details of the inner loops. The eventual goal of this formalism is to develop efficient tools and techniques to support the objective of increasing autonomy for future systems. In using the pseudospectral knotting method to solve hybrid optimal control problems, we generate a mixed-variable programming (MVP) problem. A simple, feasible integer programming subproblem is identified that reduces the combinatorial complexity of solving the MVP. In addition to developing the framework using various examples from aerospace engineering, we provide details for a two-agent benchmark problem associated with a multiagent launch system. The entire process is illustrated with a numerical example.

Low-Thrust, High-Accuracy Trajectory Optimization

J OURNAL OF G UIDANCE , C ONTROL , AND D YNAMICS Vol. 30, No. 4, July–August 2007 Low-Thrust, High-Accuracy Trajectory Optimization I. Michael Ross, ∗ Qi Gong, † and Pooya Sekhavat ‡ Naval Postgraduate School, Monterey, California 93943 DOI: 10.2514/1.23181 Multirevolution, very low-thrust trajectory optimization problems have long been considered difficult problems due to their large time scales and high-frequency responses. By relating this difficulty to the well-known problem of aliasing in information theory, an antialiasing trajectory optimization method is developed. The method is based on Bellman’s principle of optimality and is extremely simple to implement. Appropriate technical conditions are derived for generating candidate optimal solutions to a high accuracy. The proposed method is capable of detecting suboptimality by way of three simple tests. These tests are used for verifying the optimality of a candidate solution without the need for computing costates or other covectors that are necessary in the Pontryagin framework. The tests are universal in the sense that they can be used in conjunction with any numerical method whether or not antialiasing is sought. Several low-thrust example problems are solved to illustrate the proposed ideas. It is shown that the antialiased solutions are, in fact, closed-loop solutions; hence, optimal feedback controls are obtained without recourse to the complexities of the Hamilton–Jacobi theory. Because the proposed method is easy to implement, it can be coded on an onboard computer for practical space guidance. the field to exchange ideas over several workshops. These workshops, held over 2003–2006, further clarified the scope of the problems, and ongoing efforts to address them are described in [12]. From a practical point of view, the goal is to quickly obtain verifiably optimal or near-optimal solutions to finite- and low-thrust problems so that alternative mission concepts can be analyzed I. Introduction C ONTINUOUS-THRUST trajectory optimization problems have served as one of the motivating problems for optimal control theory since its inception [1–4]. The classic problem posed by Moyer and Pinkham [2] is widely discussed in textbooks [1,3,4] and research articles [5–7]. When the continuity of thrust is removed from such problems, the results can be quite dramatic as illustrated in Fig. 1. This trajectory was obtained using recent advances in optimal control techniques and is extensively discussed in [8]. In canonical units, the problem illustrated in Fig. 1 corresponds to doubling the semimajor axis (a 0 1, a f 2), doubling the eccentricity (e 0 0:1, e f 0:2), and rotating the line of apsides by 1 rad. Note that the extremal thrust steering program for minimizing fuel is not tangential over a significant portion of the trajectory. Furthermore, the last burn is a singular control as demonstrated in Fig. 2 by the vanishing of the switching function. Although such finite-thrust problems can be solved quite readily nowadays, it has long been recognized [9–11] that as the thrust authority is reduced, new problems emerge. These well-known challenges chiefly arise as a result of a long flight time measured in terms of the number of orbital revolutions. Consequently, such problems are distinguished from finite-thrust problems as low-thrust problems although the boundary between finite thrust and low thrust is not altogether sharp. Although ad hoc techniques may circumvent some of the low- thrust challenges, it is not quite clear if the solutions generated from such methods are verifiably optimal. As detailed in [8], the engineering feasibility of a space mission is not dictated by trajectory generation, but by optimality. This is because fuel in space is extraordinarily expensive as the cost of a propellant is driven by the routine of space operations, or the lack of it, and not the chemical composition of the fuel. In an effort to circumvent ad hoc techniques to efficiently solve emerging problems in finite- and low-thrust trajectory optimization, NASA brought together leading experts in al Or iti bit In it Fin rb al O Transfer Trajectory Fig. 1 A benchmark minimum-fuel finite-thrust orbit transfer problem. Thrust Acceleration, u s = 0 Switching Function, s Received 13 February 2006; accepted for publication 21 August 2006. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Copies of this paper may be made for personal or internal use, on condition that the copier pay the