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Feedback from observations introduced into midcourse guidance correction program, showing optimal random feedback solution obtained for deterministic optimal feedback control problems
The Magnetospheric Multiscale Mission (MMS) is a NASA mission intended to make fundamental advancements in our understanding of the Earth s magnetosphere. There are three processes that MMS is intended to study including magnetic reconnection, charged particle acceleration, and turbulence. There are four phases of the MMS mission and each phase is designed to study a particular region of the Earth's magnetosphere. The mission is composed of a formation of four spacecraft that are nominally in a regular tetrahedron formation. In this work, we present optimal orbit designs for Phase I and II. This entails designing reference orbits such that the spacecraft dwell-time in the region of interest is a maximum. This is non-trivial because the Earth's magnetosphere is dynamic and its shape and position are not constant in inertial space. Optimal orbit design for MMS also entails designing the formation so that the relative motion of the four spacecraft yields the greatest science return. We develop performance metrics that are directly related to the science return, and use Sequential Quadratic Programming (SQP) to determine optimal relative motion solutions. While designing for optimal science return, we also consider practical constraints such as maximum eclipse time and minimum inter-spacecraft separation distances. Data are presented that illustrates how long we can ensure that the formation remains in the relevant region of the Earth's magnetosphere. We also draw general conclusions about where in the orbit acceptable tetrahedron configurations can be provided and for how long.
A multi-objective optimal control technique is modified to accommodate changing cost function weights and is used to control a flexible wing aircraft model. Variation of the weights is used to adjust the relative importance of each objective according to either a prescribed function of time or of the state. Several techniques for obtaining a practical approximation to the optimal control solution are presented, and stability of a specific weight structure with the optimal controller is demonstrated. Functionality of the multi-objective control design with weight variation is demonstrated in simulation of a flexible wing transport aircraft and is shown to improve performance over the fixed weight version both at a constant flight condition and across changing flight conditions.
We analyze the problem of suppressing the unwanted component of a time-harmonic acoustic field (noise) on a predetermined region of interest. The suppression is rendered by active means, i.e., by introducing the additional acoustic sources called controls that generate the appropriate anti-sound. Previously, we have obtained general solutions for active controls in both continuous and discrete formulations of the problem. We have also obtained optimal solutions that minimize the overall absolute acoustic source strength of active control sources. These optimal solutions happen to be particular layers of monopoles on the perimeter of the protected region. Mathematically, minimization of acoustic source strength is equivalent to minimization in the sense of L(sub 1). By contrast. in the current paper we formulate and study optimization problems that involve quadratic functions of merit. Specifically, we minimize the L(sub 2) norm of the control sources, and we consider both the unconstrained and constrained minimization. The unconstrained L(sub 2) minimization is certainly the easiest problem to address numerically. On the other hand, the constrained approach allows one to analyze sophisticated geometries. In a special case, we call compare our finite-difference optimal solutions to the continuous optimal solutions obtained previously using a semi-analytic technique. We also show that the optima obtained in the sense of L(sub 2) differ drastically from those obtained in the sense of L(sub 1).
Genetic algorithms, which simulate evolution in natural systems, have been used to find solutions to optimization problems that seem intractable to standard approaches. In this study, the feasibility of using a GA to find an optimum, fixed profile nozzle for a pulse detonation engine (PDE) is demonstrated. The objective was to maximize impulse during the detonation wave passage and blow-down phases of operation. Impulse of each profile variant was obtained by using the CFD code Mozart/2.0 to simulate the transient flow. After 7 generations, the method has identified a nozzle profile that certainly is a candidate for optimum solution. The constraints on the generality of this possible solution remain to be clarified.
Control processes and optimization problems solutions by stochastic differential equations, discussing dynamic models and programming, linear filtering and optimal feedback
The transonic wing design process has been vastly improved at Lockheed-Georgia. The revised design procedure enhances useability and reliability by combining numerical optimization and inverse design into a single wing design code with transonic analysis provided by a modified version of FLO22. A more versatile set of geometric decision variables has been integrated into the optimization portion for geometry perturbations. An automatic restart feature permits the interchangeability of solutions between optimization and inverse design as the design progresses. In combination, these improvements enable practical utilization of a VAX 11/780 computer and significantly reduce the elapsed time required to complete a transonic wing design.
In this work, we present composite Bernstein polynomials as a direct collocation method for approximating optimal control problems. An analysis of the convergence properties of composite Bernstein polynomials is provided, and beneficial properties of composite Bernstein polynomials for the solution of optimal control problems are discussed. The efficacy of the proposed approximation method is demonstrated through a bang-bang example. Lastly, we apply this method to a motion planning problem, offering a practical solution that emphasizes the ability of this method to solve complex optimal control problems.
This paper is concerned with a trust region approximation management framework (AMF) for solving the nonlinear programming problem in general and multidisciplinary optimization problems in particular The intent of the AMF methodology is to facilitate the solution of optimization problems with high-fidelity models. While such models are designed to approximate the physical phenomena they describe to a high degree of accuracy, their use in a repetitive procedure, for example, iterations of an optimization or a search algorithm, make such use prohibitively expensive. An improvement in design with lower-fidelity, cheaper models, however, does not guarantee a corresponding improvement for the higher-fidelity problem. The AMF methodology proposed here is based on a class of multilevel methods for constrained optimization and is designed to manage the use of variable-fidelity approximations or models in a systematic way that assures convergence to critical points of the original high-fidelity problem.
Special purpose rapid computer for providing an on-line solution to the sampled-data time-optimal- control problem in which inputs are subjected to amplitude constraints
The trajectory optimization technique described in this paper provides several distinct advantages over previous formulations. First, fully numerically integrated trajectory modeling is used. That is, no approximations to the trajectory are made and the inclusion of any level of complicated force models desired is allowed. Second, only trajectory propagation is used so there is no requirement for optimization. This is accomplished by the novel method of splitting the trajectory into independent legs, which are then subjected to constrained optimization. Third, each of the trajectory legs may be specified by any convenient set of parameters particularly useful for that leg. Any of these parameters may then be subject to constraints. Fourth, the nonlinear optimization problem is solved by solving a sequence of linear problems which converges to the optimal nonlinear solution. Fifth, the robustness of this formulation requires little or no user interaction with the optimization once a feasible problem has been posed.
The Programming System for Structural Synthesis software package, which couples structural analysis and optimization, has been distributed over a network of work stations for use in Space Shuttle Solid Rocket Booster joint redesign optimization. Finite difference computing techniques were applied to the optimization gradients in parallel execution, allowing several work stations to simultaneously contribute to the problem's solution. An optimal joint shape was obtained which achieves minimum weight while keeping the gap between joints well closed and limiting structural stresses. The optimization cycle was reduced from two hours to one-half hour.
The first part of the paper presents some closed-form solutions to the optimal two-impulse transfer between fixed position and velocity vectors on Keplerian orbits when some constraints are imposed on the magnitude of the initial and final impulses. Additionally, a numerically-stable gradient-free algorithm with guaranteed convergence is presented for the minimum delta-v two-impulse transfer. In the second part of the paper, cooperative bargaining theory is used to solve some two-impulse transfer problems when the initial and final impulses are carried by different vehicles or when the goal is to minimize the delta-v and the time-of-flight at the same time.
In many singular perturbation solutions of optimal control problems, the most difficult numerical task is to solve the boundary-layer equations. However, these equations have a special structure that may often be used to expedite their solution. This paper begins by noting the general nature of the boundary-layer equations for optimal control problems. These results are then applied to the aircraft minimum time-to-climb problem. A specific numerical example is considered to illustrate the characteristics of the solution of the boundary-layer equations for this problem.
The problem of designing optimal inputs in the identification of multi-input multi-output linear systems with unknown time-varying parameters is considered using a Bayesian approach. A sensitivity index gives a measure of performance for the closed-loop system inputs. The computation of the optimal closed-loop mappings is shown to be a nontrivial exercise in stochastic control with no analytic solution, but optimal open-loop and affine laws yield much more tractable problems. For time-invariant systems, the sensitivity index considered is shown to be equivalent to the trace of the (strictly positive definite) information matrix associated with the system. Numerical examples are given. A Kalman filter is used to estimate the parameters. A necessary condition for the Kalman filter not to diverge when applying linear feedback is also given.
This article shows how rational analysis can be used to minimize learning cost for a general class of statistical learning problems. We discuss the factors that influence learning cost and show that the problem of efficient learning can be cast as a resource optimization problem. Solutions found in this way can be significantly more efficient than the best solutions that do not account for these factors. We introduce a heuristic learning algorithm that approximately solves this optimization problem and document its performance improvements on synthetic and real-world problems.
A discrete steepest ascent method which allows controls which are not piecewise constant (for example, it allows all continuous piecewise linear controls) was derived for the solution of optimal programming problems. This method is based on the continuous steepest ascent method of Bryson and Denham and new concepts introduced by Kelley and Denham in their development of compatible adjoints for taking into account the effects of numerical integration. The method is a generalization of the algorithm suggested by Canon, Cullum, and Polak with the details of the gradient computation given. The discrete method was compared with the continuous method for an aerodynamics problem for which an analytic solution is given by Pontryagin's maximum principle, and numerical results are presented. The discrete method converges more rapidly than the continuous method at first, but then for some undetermined reason, loses its exponential convergence rate. A comparsion was also made for the algorithm of Canon, Cullum, and Polak using piecewise constant controls. This algorithm is very competitive with the continuous algorithm.
Optimal control theory is used to examine a specific class of spacecraft trajectory problems where high- and low-thrust propulsion systems are utilized. These problems assume a spacecraft is in an established orbit about a planet. It is desired to execute an intercept of a pre-determined position in space in a specified amount of time using an optimal high-thrust program. The spacecraft then returns to the original orbit station using the low-thrust propulsion system in an optimal fashion. A minimum fuel solution is sought using the linearized equations of motion, known as the CW equations, which simplify the necessary computations. Solutions are obtained for problems with a fixed final time. However, for the time-open case, the optimal solution is for the final time to be infinite. With a weighted function of the final time in the performance index, a limited range of optimal single impulse solutions for the time -open case can also be found.