Michael A. Patterson

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.

Brief paper: A unified framework for the numerical solution of optimal control problems using pseudospectral methods

A unified framework is presented for the numerical solution of optimal control problems using collocation at Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points. It is shown that the LG and LGR differentiation matrices are rectangular and full rank whereas the LGL differentiation matrix is square and singular. Consequently, the LG and LGR schemes can be expressed equivalently in either differential or integral form, while the LGL differential and integral forms are not equivalent. Transformations are developed that relate the Lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem. The LG and LGR discrete costate systems are full rank while the LGL discrete costate system is rank-deficient. The LGL costate approximation is found to have an error that oscillates about the true solution and this error is shown by example to be due to the null space in the LGL discrete costate system. An example is considered to assess the accuracy and features of each collocation scheme.

GPOPS-II: A MATLAB Software for Solving Multiple-Phase Optimal Control Problems Using hp-Adaptive Gaussian Quadrature Collocation Methods and Sparse Nonlinear Programming

A general-purpose MATLAB software program called GPOPS--II is described for solving multiple-phase optimal control problems using variable-order Gaussian quadrature collocation methods. The software employs a Legendre-Gauss-Radau quadrature orthogonal collocation method where the continuous-time optimal control problem is transcribed to a large sparse nonlinear programming problem (NLP). An adaptive mesh refinement method is implemented that determines the number of mesh intervals and the degree of the approximating polynomial within each mesh interval to achieve a specified accuracy. The software can be interfaced with either quasi-Newton (first derivative) or Newton (second derivative) NLP solvers, and all derivatives required by the NLP solver are approximated using sparse finite-differencing of the optimal control problem functions. The key components of the software are described in detail and the utility of the software is demonstrated on five optimal control problems of varying complexity. The software described in this article provides researchers a useful platform upon which to solve a wide variety of complex constrained optimal control problems.

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.

Exploiting Sparsity in Direct Collocation Pseudospectral Methods for Solving Optimal Control Problems

Inadirectcollocationpseudospectralmethod,acontinuous-timeoptimalcontrolproblemistranscribedtoa finitedimensional nonlinear programming problem. Solving this nonlinear programming problem as efficiently as possible requires that sparsity at both the first- and second-derivative levels be exploited. In this paper, a computationally efficient method is developed for computing the first and second derivatives of the nonlinear programming problem functions arising from a pseudospectral discretization of a continuous-time optimal control problem. Specifically,in thispaper, expressions arederivedfor theobjective function gradient, constraint Jacobian, and Lagrangian Hessian arising from the previously developed Radau pseudospectral method. It is shown that the computation of these derivative functions can be reduced to computing the first and second derivatives of the functions in the continuous-time optimal control problem. As a result, the method derived in this paper reduces significantly the amount of computation required to obtain the first and second derivatives required by a nonlinear programming problem solver. The approach derived in this paper is demonstrated on an example where it is found that significant computational benefits are obtained when compared against direct differentiation of the nonlinear programming problem functions. The approach developed in this paper improves the computational efficiency of solving nonlinear programming problems arising from pseudospectral discretizations of continuous-time optimal control problems.

An efficient overloaded method for computing derivatives of mathematical functions in MATLAB

An object-oriented method is presented that computes without truncation the error derivatives of functions defined by MATLAB computer codes. The method implements forward-mode automatic differentiation via operator overloading in a manner that produces a new MATLAB code that computes the derivatives of the outputs of the original function with respect to the differentiation variables. Because the derivative code has the same input as the original function code, the method can be used recursively to generate derivatives of any order desired. In addition, the approach developed in this article has the feature that the derivatives are generated by simply evaluating the function on an instance of the class, thus making the method straightforward to use while simultaneously enabling differentiation of highly complex functions. A detailed description of the method is presented and the approach is illustrated and shown to be efficient on four examples.

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,

Optimal Trajectory and Control Generation for Landing of Multiple Aircraft in the Presence of Obstacles

Due to the large increase in the density of aircraft arrivals at airports in recent years, it has become important to optimize the scheduling of aircraft landings in order to reduce wait time, improve airport efficiency, and minimize fuel consumption while maintaining safety. As the density of aircraft arrivals increases, however, so too does the complexity of trajectory planning and generation. A key aspect of landing multiple aircraft on a single runway is conflict detection and resolution. In the context aircraft landing, a conflict is defined as the situation of loss of minimum safe separation between two aircraft Ref. 1. The conflict detection and resolution process consists of predicting, communicating to the pilot, and resolving the conflict. Typically, evaluating the likelihood of a conflict is based on the current position and velocity of an aircraft. The conflict is then resolved by determining a maneuver required by one or more aircraft to avoid the predicted conflict. The required information is then provided to the air traffic controller who communicates with the pilot to resolve the conflict. A great deal of research has been done on the problem of multiple-aircraft conflict detection and resolution and landing of multiple aircraft. Ref. 2 considers the problem of managing landing sequences for an arbitrary number of aircraft moving in the vicinity of a controlled aerodome. Ref. 3 considers how different airport landing sequencing algorithms affect both the arrival sequence of aircraft and air traffic control. Ref. 4 develops an approach for determining optimal trajectories to bring an unmanned aerial vehicle from a loitering state to a planted landing. Ref. 5 develops a queueing algorithm for routing aircraft on two airport runways. Ref. 6 considers a three-dimensional trajectory optimization algorithm is developed by to obtain a conflict-free flight path using a nonlinear point mass model with realistic operational constraints on individual aircraft. Ref. 7 considers the problem of optimal cooperative three-dimensional conflict resolution involving multiple aircraft, where the the initial and final locations of the aircraft are specified along with detailed point-mass aircraft dynamic models. The infinite-dimensional optimal control problem is then converted into a finite dimensional nonlinear program (NLP) using collocation on finite elements. Finally, Ref. 8

Utilizing the Algorithmic Differentiation Package ADiGator for Solving Optimal Control Problems Using Direct Collocation

ADiGator is a newly developed free MATLAB algorithmic differentiation package. In this paper, we study the use of the ADiGator algorithmic differentiation tool in order to supply the first and second derivatives of the NLP arising from direct collocation of optimal control problems. While the methods of this paper may be applied to multiple direct collocation schemes, we focus on an hp-adaptive Legendre-Gauss-Radau scheme which has been coded in the MATLAB optimal control software GPOPS-II. The methods required to indirectly compute the first and second derivatives of the NLP using ADiGator are presented and three test cases are given. In the test cases, the method of supplying derivatives via ADiGator is shown to be highly efficient when compared to the classical method of finite-differencing.