John T. Betts

Practical methods for optimal control using nonlinear programming

Preface 1. Introduction to nonlinear programming 2. Large, sparse nonlinear programming 3. Optimal control preliminaries 4. The optimal control problem 5. Optimal control examples Appendix A. Software Bibliography, Index.

Application of sparse nonlinear programming to trajectory optimization

The most effective numerical techniques for the solution of trajectory optimization and optimal control problems combine a nonlinear iteration procedure with some type of parametric approximation to the trajectory dynamics. Early methods attempted to parameterize the dynamics using a small number of variables because the iterative search procedures could not successfully solve larger problems. With the development of more robust nonlinear programming algorithms, it is now feasible and desirable to consider formulations of the trajectory optimization problem incorporating a large number of variables and constraints. The purpose of this paper is to address the manner in which a trajectory is parameterized and the design of the nonlinear programming algorithm to effectively deal with this formulation.

A sparse nonlinear optimization algorithm

One of the most effective numerical techniques for solving nonlinear programming problems is the sequential quadratic programming approach. Many large nonlinear programming problems arise naturally in data fitting and when discretization techniques are applied to systems described by ordinary or partial differential equations. Problems of this type are characterized by matrices which are large and sparse. This paper describes a nonlinear programming algorithm which exploits the matrix sparsity produced by these applications. Numerical experience is reported for a collection of trajectory optimization problems with nonlinear equality and inequality constraints.

Mesh refinement in direct transcription methods for optimal control

SUMMARY The direct transcription method for solving optimal control problems involves the use of a discrete approximation to the original problem. This paper describes a technique for changing the discretization in order to improve the accuracy of the approximation. An integer programming technique is used to minimize the maximum error during the refinement iterations. The eƒciency of the method is illustrated for an application with path inequality constraints. ( 1998 John Wiley & Sons, Ltd.

Very low-thrust trajectory optimization using a direct SQP method

Abstract The direct transcription or collocation method has demonstrated notable success in the solution of trajectory optimization and optimal control problems. This approach combines a sparse nonlinear programming algorithm with a discretization of the trajectory dynamics. A challenging class of optimization problems occurs when the spacecraft trajectories are characterized by thrust levels that are very low relative to the vehicle weight. Low-thrust trajectories are demanding because realistic forces due to oblateness, aerodynamic drag, and third-body perturbions often dominate the thrust. Furthermore because the thrust is so low, significant changes to the orbits require very long duration trajectories. When a collocation method is applied to a problem of this type, the resulting nonlinear program is very large because the trajectories are long, and very nonlinear because of the perturbing forces. This paper describes the application of the transcription method to the solution of very low-thrust orbit transfers. The vehicle dynamics are defined using a modified set of equinoctial coordinates, and the trajectory modeling is described using these dynamics. A solution is presented for a representative transfer using a spacecraft with a thrust acceleration of approximately 1.25×10 −7 km/s 2 . This transfer requires over 578 revolutions, and leads to a sparse optimization problem with 416 123 variables and 249 674 constraints. Issues related to the numerical conditioning and problem formulation are discussed.

Application of Direct Transcription to Commercial Aircraft Trajectory Optimization

One of the most effective numerical techniques for the solution of trajectory optimization and optimal control problems is the direct transcription method. This approach combines a nonlinear programming algorithm with a discretization of the trajectory dynamics. When the resulting mathematical programming problem is solved using a sparse sequential quadratic programming algorithm, the technique produces solutions very rapidly and has demonstrated considerable robustness when applied to atmospheric and orbital trajectories. This paper describes the application of the direct transcription technique to the optimal design of a commercial aircraft trajectory, subject to realistic constraints on the aircraft flight path. A primary result of the paper is to demonstrate that the transcription formulation leads to a very natural treatment of realistic Federal Aviation Administration (FAA) imposed path constraints within a high fidelity simulation. A second important result is to demonstrate that modeling tabular data using smooth approximations significantly improves the speed of convergence.

Trajectory optimization on a parallel processor

The design of a trajectory for an aerospace vehicle involves choosing a set of variables to optimally shape the path of the vehicle. Typically the trajectory is simulated by numerically solving the differential equations describing the dynamics of the vehicle. The optimal trajectory is usually determined by using a nonlinear programming (parameter optimization) algorithm to select the variables. Problems that require choosing control functions are usually reduced to choosing a finite set of parameters. The computational expense of a trajectory optimization is dominated by two factors: the cost of simulating a trajectory and the cost of computing gradient information for the optimization algorithm. This paper presents a technique for using a parallel processor to reduce the cost of these calculations. The trajectory is broken into phases, which can be simulated in parallel, thereby reducing the cost of an individual trajectory. This multiple shooting technique has been suggested by a number of authors. The nonlinear optimization problem that results from this formulation produces a Jacobian matrix that is sparse. The Jacobian is computed using sparse finite differencing, which is also performed in parallel, thereby reducing the cost of obtaining gradient information for the optimization algorithm. This paper describes the application of sparse finite differencing to a multiple shooting formulation of the two-point boundary-value problem, in a manner suitable for implementation on a parallel processor. Computational experience with the algorithm as implemented on the BBN GPIOOO (Butterfly) parallel processing computer is described.

Convergence of Nonconvergent IRK Discretizations of Optimal Control Problems with State Inequality Constraints

It has been observed that optimization codes are sometimes able to solve inequality state constrained optimal control problems with discretizations which do not converge when used as integrators on the constrained dynamics. Understanding this phenomenon could lead to a more robust design for direct transcription codes as well as better test problems. This paper examines how this phenomenon can occur. The difference between solving index 3 differential algebraic equations (DAEs) using the trapezoid method in the context of direct transcription for optimal control problems and a straightforward implicit Runge--Kutta (IRK) formulation of the same trapezoidal discretization is analyzed. It is shown through numerical experience and theory that the two can differ as much as O(1/h3) in the control. The optimization can use a small sacrifice in the accuracy of the states in the early stages of the trapezoidal method to produce better accuracy in the control, whereas more precise solutions converge to an incorrect solution. Convergence independent of the index of the constraints is then proven for one class of problems. The theoretical results are used to explain computational observations.

Solving the nonlinear least square problem: Application of a general method

An algorithm for solving the general nonlinear least-square problem is developed. An estimate for the Hessian matrix is constructed as the sum of two matrices. The first matrix is the usual first-order estimate used by the Gauss method, while the second matrix is generated recursively using a rank-one formula. Test results indicate that the method is superior to the standard Gauss method and compares favorably with other methods, especially for problems with nonzero residuals at the solution.

Compensating for order variation in mesh refinement for direct transcription methods

Abstract The numerical theory for Implicit Runge–Kutta methods shows that there can be order reduction when these methods are applied to either stiff or differential algebraic equations. This paper discusses how this theory can be utilized in direct transcription trajectory optimization by modifying a currently used mesh refinement strategy.