arXiv subjects

Eduardo M. G. Vila

Publications and source records attributed to Eduardo M. G. Vila.

4 recordsLinked to original sources

arXiv2026

Tight Bounds on Polynomials and Its Application to Dynamic Optimization Problems

This paper presents a pseudo-spectral method for Dynamic Optimization Problems (DOPs) that allows for tight polynomial bounds to be achieved via flexible sub-intervals. The proposed method not only rigorously enforces inequality constraints, but also allows for a lower cost in comparison with non-flexible discretizations. Two examples are provided to demonstrate the feasibility of the proposed method to solve optimal control problems. Solutions to the example problems exhibited up to a tenfold reduction in relative cost.

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arXiv2024

Mesh Refinement with Early Termination for Dynamic Feasibility Problems

We propose a novel early-terminating mesh refinement strategy using an integrated residual method to solve dynamic feasibility problems. As a generalization of direct collocation, the integrated residual method is used to approximate an infinite-dimensional problem into a sequence of finite-dimensional optimization subproblems. Each subproblem in the sequence is a finer approximation of the previous. It is shown that these subproblems need not be solved to a high precision; instead, an early termination procedure can determine when mesh refinement should be performed. The new refinement strategy, applied to an inverted pendulum swing-up problem, outperforms a conventional refinement method by up to a factor of three in function evaluations.

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arXiv2022

Solving optimal control problems with non-smooth solutions using an integrated residual method and flexible mesh

Solutions to optimal control problems can be discontinuous, even if all the functionals defining the problem are smooth. This can cause difficulties when numerically computing solutions to these problems. While conventional numerical methods assume state and input trajectories are continuous and differentiable or smooth, our method is able to capture discontinuities in the solution by introducing time-mesh nodes as decision variables. This allows one to obtain a higher accuracy solution for the same number of mesh nodes compared to a fixed time-mesh approach. Furthermore, we propose to first solve a sequence of suitably-defined least-squares problems to ensure that the error in the dynamic equation is below a given tolerance. The cost functional is then minimized subject to an inequality constraint on the dynamic equation residual. We demonstrate our implementation on an optimal control problem that has a chattering solution. Solving such a problem is difficult, since the solution involves infinitely many switches of decreasing duration. This simulation shows how the flexible mesh is able to capture discontinuities present in the solution and achieve superlinear convergence as the number of mesh intervals is increased.

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arXiv2022

Fast and accurate method for computing non-smooth solutions to constrained control problems

Introducing flexibility in the time-discretisation mesh can improve convergence and computational time when solving differential equations numerically, particularly when the solutions are discontinuous, as commonly found in control problems with constraints. State-of-the-art methods use fixed mesh schemes, which cannot achieve superlinear convergence in the presence of non-smooth solutions. In this paper, we propose using a flexible mesh in an integrated residual method. The locations of the mesh nodes are introduced as decision variables, and constraints are added to set upper and lower bounds on the size of the mesh intervals. We compare our approach to a uniform fixed mesh on a real-world satellite reorientation example. This example demonstrates that the flexible mesh enables the solver to automatically locate the discontinuities in the solution, has superlinear convergence and faster solve times, while achieving the same accuracy as a fixed mesh.

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