Efficient and More Accurate Representation of Solution Trajectories in Numerical Optimal Control

Abstract

We show via examples that, when solving optimal control problems, representing the optimal state and input trajectory directly using interpolation schemes may not be the best choice. Due to the lack of considerations for solution trajectories in-between collocation points, large errors may occur, posing risks if this solution is to be applied. A novel solution representation method is proposed, capable of yielding a solution of much higher accuracy for the same discretization mesh. This is achieved by minimizing the integral of the residual error for the overall trajectory, instead of forcing the errors to be zero only at collocation points. In this way, the requirement for mesh resolution can be significantly reduced, leaving the problem dimensions relatively small. This particular formulation also avoids some of the drawbacks found in the earlier work of integrated residual minimization, leading to more efficient computations.

Solving Dynamic Optimization Problems to a Specified Accuracy: An Alternating Approach using Integrated Residuals

“…Based on the same concept of minimizing the integrated residual, our earlier work [12] presented a solution representation method that is able to obtain solutions of much higher accuracy than collocation methods, while maintaining non-increasing objective values. This approach effectively treats computing an approximate solution of the DOP as a multi-objective optimization problem.…”

“…The method retains the same decision variables as in ( 4), namely Z := (χ, υ, p, t 0 , t f ), and uses the interpolation polynomial formula x(•), ẋ(•) and ũ(•) for the computation and integration of various elements of the discretized DOP. The interpolation formulation is provided in more detail in our previous work on solution representation methods [12].…”

“…2) Solution representation method: In our early work [12], we proposed an optimization formulation for representing continuous DOP trajectories with higher accuracy from discretized direct collocation NLP solutions. The DAIR residual minimization problem would be a more suitable candidate for the purpose of solution representation, with the following benefits:…”

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

“…The idea of using integrated residuals as part of the transcription process overcomes some of the limitations of collocation [9], [10]. Compared to classical time-marching schemes (shooting methods) or point-wise residual minimisation (collocation), integrated residual methods have the benefit of producing a solution with a more uniform error over the whole time domain.…”

Minimum-Fuel Ascent of Hypersonic Vehicle considering Control Constraint Using the Improved Pigeon-Inspired Optimization Algorithm

“…Due to the limitation of control performance, the optimal trajectory that only considers trajectory performance may not be realized, so it is necessary to add control constraints to trajectory optimization. At present, the numerical method is widely studied and applied in trajectory optimization [12][13][14], and it can be divided into indirect method and direct method. The indirect method uses optimal-control theory to obtain an analytical solution to the trajectory optimization problem.…”