Scaling and Balancing for High-Performance Computation of Optimal Controls

Ross, I. Michael; Gong, Qi; Karpenko, Mark; Proulx, Ronald J.

doi:10.2514/1.g003382

Cited by 32 publications

(23 citation statements)

References 37 publications

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“…It is also worth noting that DIDO does not require a "guess" to use the software. 30 In order words, the solution presented in Figures 1 through 4 is "unbiased" from a user's perspective. Because there was no clear difficulty in generating this…”

Section: A Pseudospectral Answer To the Challenge Problemmentioning

confidence: 99%

“…At NASA and elsewhere in the industry, it is not sufficient to generate a solution as presented in Figures 1 through 4 without subjecting it to a battery of mathematical and engineering tests. Furthermore, an argument that the residuals are small (e.g., 10 −8 ) is well-established to be irrelevant 15,17,30,31 because it is possible to produce a wrong answer -even an infeasible one -with very small "collocation errors." A standard industry practice in testing the accuracy of a control solution is to subject it to an established propagation technique and compute certain critical functions of the propagated state variables; see Refs.…”

Section: An Independent Verification Of Feasibilitymentioning

confidence: 99%

“…A standard industry practice in testing the accuracy of a control solution is to subject it to an established propagation technique and compute certain critical functions of the propagated state variables; see Refs. [27] and [30] for further details. In following this industrial rigor, we interpolate the control solution presented in Figure 2 to generate u(t) ∀ t ∈ [0, T ] and propagate the initial conditions through the ODE ẋ = f (x, u(t)) using ode45 in MATLAB.…”

Section: An Independent Verification Of Feasibilitymentioning

confidence: 99%

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A Pseudospectral Approach to High Index DAE Optimal Control Problems

Marsh,

Karpenko,

Gong

2018

Preprint

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Historically, solving optimal control problems with high index differential algebraic equations (DAEs) has been considered extremely hard. Computational experience with Runge-Kutta (RK) methods confirms the difficulties. High index DAE problems occur quite naturally in many practical engineering applications. Over the last two decades, a vast number of real-world problems have been solved routinely using pseudospectral (PS) optimal control techniques. In view of this, we solve a "provably hard," index-three problem using the PS method implemented in DIDO c , a state-of-the-art MATLAB optimal control toolbox. In contrast to RK-type solution techniques, no laborious index-reduction process was used to generate the PS solution. The PS solution is independently verified and validated using standard industry practices. It turns out that proper PS methods can indeed be used to "directly" solve high index DAE optimal control problems. In view of this, it is proposed that a new theory of difficulty for DAEs be put forth.1 Ph.D. candidate.

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Section: A Pseudospectral Answer To the Challenge Problemmentioning

confidence: 99%

Section: An Independent Verification Of Feasibilitymentioning

confidence: 99%

Section: An Independent Verification Of Feasibilitymentioning

confidence: 99%

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A Pseudospectral Approach to High Index DAE Optimal Control Problems

Marsh,

Karpenko,

Gong

2018

Preprint

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“…In practice, Algorithm 1 solves a dimensionless

P_{1}

for better numerical performance. Previous research has revealed scaling and balancing at the optimal control level contribute to increasing the efficiency of a solution algorithm 41 . Nondimensionalization is the scaling technique used in this article; details about this technique can be found in the reference 29 …”

Section: Hp‐pseudospectral Sequential Convex Programmingmentioning

confidence: 99%

Trajectory optimization with obstacles avoidance via strong duality equivalent and hp‐pseudospectral sequential convex programming

Xia

Wang

Gao

2021

Optim Control Appl Methods

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A trajectory optimization algorithm is developed in this article to compute a trajectory that avoids obstacles for an unmanned aerial vehicle and minimize the load factor. Quadratic and polyhedron obstacles are both considered in this article. To satisfy the safety distance requirements in engineering practice, a distance‐based obstacle avoidance constraint is formulated, which restricts the minimum distance that a vehicle can approach the obstacle. Strong duality equivalent converts the constraints into equivalent forms without introducing binary variables. Subsequently, the trajectory optimization problem is numerically solved via a hp‐pseudospectral sequential convex programming method, which iteratively solves a sequence of second‐order cone programming subproblems until convergence is attained. Furthermore, a modified trust‐region constraint and a customized nominal trajectory update rule are proposed to accelerate the convergence of the algorithm. A group of simulations are performed to demonstrate that the algorithm is computationally efficient and capable of generating a trajectory that satisfies safety distance requirements, preventing conservativeness in the trajectory, reducing the incidence of intersample constraints violation.

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“…Many of the advancements in the spectral algorithm are implemented in DIDO © [30,39], a stateof-the-art MATLAB optimal control toolbox for solving optimal control problems. As noted earlier, the condition number of the linear matrix equations involved in the spectral algorithm grows as N 2 k , k = 0, 1, .…”

Section: Solvementioning

confidence: 99%

Fast Mesh Refinement in Pseudospectral Optimal Control

Koeppen

Ross

Wilcox

et al. 2019

Journal of Guidance, Control, and Dynamics

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Mesh refinement in pseudospectral (PS) optimal control is embarrassingly easysimply increase the order N of the Lagrange interpolating polynomial and the mathematics of convergence automates the distribution of the grid points. Unfortunately, as N increases, the condition number of the resulting linear algebra increases as N 2 ; hence, spectral efficiency and accuracy are lost in practice. In this paper, we advance Birkhoff interpolation concepts over an arbitrary grid to generate well-conditioned PS optimal control discretizations. We show that the condition number increases only as √ N in general, but is independent of N for the special case of one of the boundary points being fixed. Hence, spectral accuracy and efficiency are maintained as N increases. The effectiveness of the resulting fast mesh refinement strategy is demonstrated by using polynomials of over a thousandth order to solve a low-thrust, long-duration orbit transfer problem.

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Scaling and Balancing for High-Performance Computation of Optimal Controls

Cited by 32 publications

References 37 publications

A Pseudospectral Approach to High Index DAE Optimal Control Problems

A Pseudospectral Approach to High Index DAE Optimal Control Problems

Trajectory optimization with obstacles avoidance via strong duality equivalent and hp‐pseudospectral sequential convex programming

Fast Mesh Refinement in Pseudospectral Optimal Control

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