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For control problems with control constraints, a local convergence rate is established for an hpmethod based on collocation at the Radau quadrature points in each mesh interval of the discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as either the number of collocation points or the number of mesh intervals increase, the discrete solution convergences to the continuous solution in the supnorm. The convergence is exponentially fast with respect to the degree of the polynomials on each mesh interval, while the error is bounded by a polynomial in the mesh spacing. An advantage of the hpscheme over global polynomials is that there is a convergence guarantee when the mesh is sufficiently small, while the convergence result for global polynomials requires that a norm of the linearized dynamics is sufficiently small. Numerical examples explore the convergence theory.more » « less

The problem of guidance and control of an airtosurface lowaltitude skidtoturn vehicle is considered. The objective is to steer the vehicle to a ground target from an initial state such that the vehicle remains at a constant low altitude for as long as possible and performs a bunt maneuver (negative sensedacceleration load) rapidly at the end of the trajectory in order to attain terminal target conditions. The vehicle is modeled as a point mass in motion over a flat Earth, and the vehicle is controlled using thrust magnitude, angle of attack, and sideslip angle. The guidance and control problem is posed as a decreasinghorizon optimal control problem and is resolved numerically at constant guidance cycles. The work described in this paper is the second part of a twopart sequence on trajectory optimization and guidance of a skidtoturn vehicle. In both cases, the objective is to minimize the time taken by the vehicle to complete a bunt maneuver subject to the following constraints: dynamic, boundary, state, path, and interiorpoint event constraints. In the second part of this twopart study, a numerical guidance law is employed to resolve the optimal control problem on a shrinking horizon. An assessment is made as to the time required to resolve the optimal control problem both in the absence and presence of a time delay, where the time delay is the amount of time required to solve the numerical approximation of the optimal control problem at the start of each guidance cycle. The results of this study identify that in the absence and presence of a time delay, the size of the mesh on subsequent guidance cycles decreases, the time required to solve the reducedhorizon optimal control problem is small compared to the guidance cycle duration time, and the terminal conditions are met with sufficient accuracy. Thus, the results show that the guidance law presented in this paper has the potential to solve optimal control problems in realtime.more » « less

The problem of airtosurface trajectory optimization for a lowaltitude skidtoturn vehicle is considered. The objective is for the vehicle to move level at a low altitude for as long as possible and perform a rapid bunt (negative sensedacceleration load) maneuver near the final time in order to attain terminal target conditions. The vehicle is modeled as a point mass in motion over a flat Earth, and the vehicle is controlled using thrust magnitude, angle of attack, and sideslip angle. The trajectory optimization problem is posed as a twophase optimal control problem using a weighted objective function. The work described in this paper is the first part of a twopart sequence on trajectory optimization and guidance of a skidtoturn vehicle. In both cases, the objective is to minimize the time taken by the vehicle to complete a bunt maneuver subject to the following constraints: dynamic, boundary, state, path, and interiorpoint event constraints. In the first part of this twopart study, the performance of thevehicle is assessed. In particular, the key features of the optimal reference trajectories and controls are provided. The results of this study identify that as greater weight is placed on minimizing the height of the bunt maneuver or as the maximum altitude constraint is raised, the time of the bunt maneuver decreases and the time of the problem solution increases. Also, the results of this study identify that as the allowable crossrange of the vehicle is reduced, the time and height of the bunt maneuver increases and the time of the problem solution decreasemore » « less

A new method is developed for solving optimal control problems whose solutions contain a nonsmooth optimal control. The method developed in this paper employs a modified form of the LegendreGaussRadau (LGR) orthogonal direct collocation method in which an additional variable and two additional constraints are included at the end of a mesh interval. The additional variable is the switch time where a discontinuity occurs. The two additional constraints are a collocation condition on each differential equation that is a function of control along with a control constraint at the endpoint of the mesh interval that defines the location of the nonsmoothness. These additional constraints modify the search space of the NLP in a manner such that an accurate approximation to the location of the nonsmoothness is obtained. An example with a nonsmooth solution is used throughout the paper to illustrate the improvement of the method over the standard LegendreGaussRadau collocation method.more » « less

A modified LegendreGaussRadau collocation method is developed for solving optimal control problems whose solutions contain a nonsmooth optimal control. The method includes an additional variable that defines the location of nonsmoothness. In addition, collocation constraints are added at the end of a mesh interval that defines the location of nonsmoothness in the solution on each differential equation that is a function of control along with a control constraint at the endpoint of this same mesh interval. The transformed adjoint system for the modified LegendreGaussRadau collocation method along with a relationship between the Lagrange multipliers of the nonlinear programming problem and a discrete approximation of the costate of the optimal control problem is then derived. Finally, it is shown via example that the new method provides an accurate approximation of the costate.more » « less