87,638 research outputs found
Computational alternatives to obtain time optimal jet engine control
Two computational methods to determine an open loop time optimal control sequence for a simple single spool turbojet engine are described by a set of nonlinear differential equations. Both methods are modifications of widely accepted algorithms which can solve fixed time unconstrained optimal control problems with a free right end. Constrained problems to be considered have fixed right ends and free time. Dynamic programming is defined on a standard problem and it yields a successive approximation solution to the time optimal problem of interest. A feedback control law is obtained and it is then used to determine the corresponding open loop control sequence. The Fletcher-Reeves conjugate gradient method has been selected for adaptation to solve a nonlinear optimal control problem with state variable and control constraints
Orthogonal Polynomial Approximation in Higher Dimensions: Applications in Astrodynamics
We propose novel methods to utilize orthogonal polynomial approximation in higher dimension spaces, which enable us to modify classical differential equation solvers to perform high precision, long-term orbit propagation. These methods have immediate application to efficient propagation of catalogs of Resident Space Objects (RSOs) and improved accounting for the uncertainty in the ephemeris of these objects. More fundamentally, the methodology promises to be of broad utility in solving initial and two point boundary value problems from a wide class of mathematical representations of problems arising in engineering, optimal control, physical sciences and applied mathematics. We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10^−9ms^−2, globally) are required near the Earths surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is feasible, with both speed and storage efficiency op- timized using radial adaptation. The second class of problems addressed includes orbit propagation and solution of associated boundary value problems. The successive Chebyshev-Picard path approximation method is shown well-suited to solving these problems with over an order of magnitude speedup relative to known methods. Furthermore, the approach is parallel-structured so that it is suited for parallel implementation and further speedups. Used in conjunction with orthogonal Finite Element Model (FEM) gravity approximations, the Chebyshev-Picard path approximation enables truly revolutionary speedups in orbit propagation without accuracy loss
A recursively feasible and convergent Sequential Convex Programming procedure to solve non-convex problems with linear equality constraints
A computationally efficient method to solve non-convex programming problems
with linear equality constraints is presented. The proposed method is based on
a recursively feasible and descending sequential convex programming procedure
proven to converge to a locally optimal solution. Assuming that the first
convex problem in the sequence is feasible, these properties are obtained by
convexifying the non-convex cost and inequality constraints with inner-convex
approximations. Additionally, a computationally efficient method is introduced
to obtain inner-convex approximations based on Taylor series expansions. These
Taylor-based inner-convex approximations provide the overall algorithm with a
quadratic rate of convergence. The proposed method is capable of solving
problems of practical interest in real-time. This is illustrated with a
numerical simulation of an aerial vehicle trajectory optimization problem on
commercial-of-the-shelf embedded computers
Successive Convexification of Non-Convex Optimal Control Problems and Its Convergence Properties
This paper presents an algorithm to solve non-convex optimal control
problems, where non-convexity can arise from nonlinear dynamics, and non-convex
state and control constraints. This paper assumes that the state and control
constraints are already convex or convexified, the proposed algorithm
convexifies the nonlinear dynamics, via a linearization, in a successive
manner. Thus at each succession, a convex optimal control subproblem is solved.
Since the dynamics are linearized and other constraints are convex, after a
discretization, the subproblem can be expressed as a finite dimensional convex
programming subproblem. Since convex optimization problems can be solved very
efficiently, especially with custom solvers, this subproblem can be solved in
time-critical applications, such as real-time path planning for autonomous
vehicles. Several safe-guarding techniques are incorporated into the algorithm,
namely virtual control and trust regions, which add another layer of
algorithmic robustness. A convergence analysis is presented in continuous- time
setting. By doing so, our convergence results will be independent from any
numerical schemes used for discretization. Numerical simulations are performed
for an illustrative trajectory optimization example.Comment: Updates: corrected wordings for LICQ. This is the full version. A
brief version of this paper is published in 2016 IEEE 55th Conference on
Decision and Control (CDC). http://ieeexplore.ieee.org/document/7798816
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