Christopher L. Darby

Corrigendum: Algorithm 902: GPOPS, a MATLAB software for solving multiple-phase optimal control problems using the gauss pseudospectral method

An algorithm is described to solve multiple-phase optimal control problems using a recently developed numerical method called the Gauss pseudospectral method. The algorithm is well suited for use in modern vectorized programming languages such as FORTRAN 95 and MATLAB. The algorithm discretizes the cost functional and the differential-algebraic equations in each phase of the optimal control problem. The phases are then connected using linkage conditions on the state and time. A large-scale nonlinear programming problem (NLP) arises from the discretization and the significant features of the NLP are described in detail. A particular reusable MATLAB implementation of the algorithm, called GPOPS, is applied to three classical optimal control problems to demonstrate its utility. The algorithm described in this article will provide researchers and engineers a useful software tool and a reference when it is desired to implement the Gauss pseudospectral method in other programming languages.

Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method

A method is presented for direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using global collocation at Legendre-Gauss-Radau (LGR) points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem to the costates of the optimal control problem. More precisely, it is shown that the dual multipliers for the discrete scheme correspond to a pseudospectral approximation of the adjoint equation using polynomials one degree smaller than that used for the state equation. The relationship between the coefficients of the pseudospectral scheme for the state equation and for the adjoint equation is established. Also, it is shown that the inverse of the pseudospectral LGR differentiation matrix is precisely the matrix associated with an implicit LGR integration scheme. Hence, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Numerical results show that the use of LGR collocation as described in this paper leads to the ability to determine accurate primal and dual solutions for both finite and infinite-horizon optimal control problems.

Direct Trajectory Optimization Using a Variable Low-Order Adaptive Pseudospectral Method

A variable-order adaptive pseudospectral method is presented for solving optimal control problems. The method developed in this paper adjusts both themesh spacing and the degree of the polynomial on eachmesh interval until a specified error tolerance is satisfied. In regions of relatively high curvature, convergence is achieved by refining the mesh, while in regions of relatively low curvature, convergence is achieved by increasing the degree of the polynomial. An efficient iterativemethod is then described for accurately solving a general nonlinear optimal control problem. Using four examples, the adaptive pseudospectral method described in this paper is shown to be more efficient than either a global pseudospectral method or a fixed-order method.

Minimum-Fuel Low-Earth Orbit Aeroassisted Orbital Transfer of Small Spacecraft

The problem of small spacecraft minimum-fuel heat-rate-constrained aeroassisted orbital transfer between two lowEarth orbits with inclination change is considered. Assuming impulsive thrust, the trajectory design is described in detail and the aeroassisted orbital transfer is posed as a nonlinear optimal control problem. The optimal control problem is solved using an hp-adaptive pseudospectral method, and the key features of the optimal trajectories are identified. It was found that the minimum impulse solutions are obtained when the vehicle enters the atmosphere exactly twice. Furthermore, even for highly heat-rate-constrained cases, the final mass fraction of the vehicle was fairly large. Finally, the structural loads on the vehiclewere quite reasonable, even in the caseswhere the heating rate was unconstrained.

Direct Trajectory Optimization and Costate Estimation of General Optimal Control Problems Using a Radau Pseudospectral Method

A method is presented for direct trajectory optimization and costate estimation using global collocation at Legendre-Gauss-Radau (LGR) points. The method is formulated first by casting the dynamics in integral form and computing the integralfrom the initialpoint to the interior LGR points and the terminal point. The resulting integration matrix is nonsingular and thus can be inverted so as to express the dynamics in inverse integral form. Then, by appropriate choice of the approximation for the state, a pseudospectral (i.e., differential) form that is equivalent to the inverse integral form is derived. As a result, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Moreover, the formulation derived in this paper enables solving general finite-horizon problems using global collocation at the LGR points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem (NLP) to the costates of the optimal control problem. Finally,

Costate Estimation using Multiple-Interval Pseudospectral Methods

A method is presented for costate estimation in nonlinear optimal control problems using multiple-interval collocation at Legendre-Gauss (LG) or Legendre-GaussRadau (LGR) points. Transformations from the Lagrange multipliers of the nonlinear programming problem to the costate of the continuous-time optimal control problem are given. When the optimal costate is continuous, the transformed adjoint systems of the nonlinear programming problems are discrete representations of the continuous-time first-order optimality conditions. If, however, the optimal costate is discontinuous, then the transformed adjoint systems are not discrete representations of the continuous-time first-order optimality conditions. In the case where the costate is discontinuous, the accuracy of the costate approximation depends on the locations of the mesh points. In particular, the accuracy of the costate approximation is found to be significantly higher when mesh points are located at discontinuities in the costate. Two numerical examples are studied and demonstrate the effectiveness of using the multiple-interval collocation approach for estimating costate in continuous-time nonlinear optimal control problems.

A State Approximation-Based Mesh Refinement Algorithm for Solving Optimal Control Problems Using Pseudospectral Methods

A state approximation-mesh refinement algorithm is presented for determining solutions to optimal control problems using pseudospectral methods. In the method presented in this paper the polynomial approximation of the state is used to assess the difference between consecutive selections of the number of segments and the number of collocation point within each segment to determine better combinations of segments and collocation points on the subsequent grid. This process of computing the difference between polynomial approximations is continued until the difference lies below a user-specified threshold. Because the method developed in this paper combines dividing the problem into segments and determining the best number of collocation points within each segment, the approach conceived here is termed a hybrid global/local collocation method. The user-specified parameters of the method are provided and the approach is demonstrated successfully on four optimal control problems of varying complexity. It is found that the hybrid approach developed in this paper leads to a greater accuracy with lower overall computational time as compared to using a purely global approach.

An Initial Examination of Using Pseudospectral Methods for Timescale and Differential Geometric Analysis of Nonlinear Optimal Control Problems

An initial examination of the development of a framework for analyzing the time-scale and differential geometric structure of nonlinear optimal control problems is considered. The framework is synthesizedby combining a recently developeddirect collocationmethod called the Gauss pseudospectral method (GPM) with concepts from differentialgeometry. In particular, the GPM is known to provide optimal state and costate information, thus enabling the computation of accurate Hamiltonian phase space trajectories. Using the optimal Hamiltonian phase space trajectories from the GPM, it is possible to analyze the timescale and differential geometric structure by computing the finite-time Lyapunov exponents and Lyapunov vectors. The Lyapunov exponents provide information about both the stable/unstableand slow/fast behavior of the Hamiltonian system along the optimal trajectory. Furthermore, the directions in the phase space along which these different behaviors act are isolated by decomposing the tangent space of the Hamiltonian system using the finitetime Lyapunov vectors. The approach is demonstrated successfully on two examples. The main contribution of this paper is to demonstrate the effectiveness of combining the Gauss pseudospectral method with results from differential geometry to assess the structure of optimally controlled systems, but without having to solve the Hamiltonian boundary-value problem that arises from the calculus of variations.