The Adjoint Gradient Method for Time-Optimal Control of a Moon Landing: Ascent, Descent, and Abort

2020 ◽  
Author(s):  
Philipp Eichmeir ◽  
Karin Nachbagauer ◽  
Wolfgang Steiner
Keyword(s):  
Optimal Solution ◽  
Initial Guess ◽  
Optimal Controls ◽  
Novel Approach ◽  
End Conditions ◽  
Time Optimal ◽  
Gradient Based ◽  
Adjoint Gradient ◽  
Moon Landing

Abstract This article illustrates a novel approach for the determination of time-optimal controls for dynamic systems under observance of end conditions. Such problems arise in robotics, e.g. if the control of a robot has to be designed such that the time for a rest-to-rest maneuver becomes a minimum. So far, such problems have been considered as two-point boundary value problems, which are hard to solve and require an initial guess close to the optimal solution. The aim of this contribution is the development of an iterative, gradient based solution strategy for solving such problems. As an example, a Moon-landing as in the Apollo program, will be considered. In detail, we discuss the ascent, descent and abort maneuvers of the Apollo Lunar Excursion Module (LEM) to and from the Moon’s surface in minimum time. The goal is to find the control of the thrust nozzle of the LEM to minimize the final time.

Time-Optimal Control of Dynamic Systems Regarding Final Constraints

2020 ◽  
Author(s):  
Philipp Eichmeir ◽  
Karin Nachbagauer ◽  
Thomas Lauß ◽  
Karim Sherif ◽  
Wolfgang Steiner
Keyword(s):  
Dynamic Systems ◽  
Boundary Value ◽  
Optimal Solution ◽  
Initial Guess ◽  
Optimal Controls ◽  
Cost Functional ◽  
Novel Approach ◽  
End Conditions ◽  
Time Optimal ◽  
The Cost

Abstract Within the framework of this article, we pursue a novel approach for the determination of time-optimal controls for dynamic systems under observance of end conditions. Such problems arise in robotics, e.g. if the control of a robot has to be designed such that the time for a rest-to-rest maneuver becomes a minimum. So far, such problems have been generally considered as two-point boundary value problems, which are hard to solve and require an initial guess close to the optimal solution. The aim of this work is the development of an iterative, gradient based solution strategy which can be applied even to complex multibody systems. The so-called adjoint method is a promising way to compute the direction of the steepest descent, i.e. the variation of a control signal causing the largest local decrease of the cost functional. The proposed approach will be more robust than solving the underlying boundary value problem, as the cost functional will be minimized iteratively while approaching the final conditions. Moreover, so-called influence differential equations are formulated to relate the changes of the controls and of the final conditions. In order to meet the end conditions, we introduce a descent direction that, on the one hand, approaches the optimum of the constrained cost functional and, on the other hand, reduces the error in the prescribed final conditions.

An Adaptive Hierarchical Multiscale Algorithm for Estimation of Optimal Well Controls

SPE Journal ◽  
2014 ◽  
Vol 19 (05) ◽  
pp. 909-930 ◽  
Author(s):  
D.F.. F. Oliveira ◽  
A.. Reynolds
Keyword(s):  
Correlation Length ◽  
Ad Hoc ◽  
Net Present Value ◽  
A Priori ◽  
Optimal Solution ◽  
Optimal Controls ◽  
Simple Method ◽  
Gradient Based ◽  
Multiscale Algorithm ◽  
Adjoint Gradient

Summary In determining the optimal well controls by maximizing net present value (NPV) for the remaining life of a reservoir, one typically defines the length of the control steps a priori. Moreover, these control steps are often the same for all wells. We provide a scale-splitting/merging method for adaptively selecting the number and the lengths of control steps as the overall optimization proceeds. We start with a reasonably small number of control steps and find the associated optimal controls by maximizing NPV. Both the adjoint-gradient-based steepest-ascent method and ensemble-based optimization (EnOpt) are considered as optimization algorithms. Because the correlation length used to generate the ad hoc covariance matrix indigenous to EnOpt affects the results, we implement a simple method to reduce the correlation length as the optimization proceeds. This enables EnOpt to generate a good approximation for well-control problems in which the optimal solution is bang-bang. The adaptive approach is applied to two example problems, and the results are compared with those obtained with a predetermined number of control steps.

scholarly journals An iterative method for time optimal control of dynamic systems

2011 ◽  
Vol 21 (1) ◽  
pp. 5-23 ◽  
Author(s):  
Navvab Kashiri ◽  
Mohammad Ghasemi ◽  
Morteza Dardel
Keyword(s):  
Optimal Control ◽  
Iterative Method ◽  
Dynamic Systems ◽  
Minimum Principle ◽  
Minimum Energy ◽  
Optimal Solution ◽  
Initial Guess ◽  
Open Loop ◽  
Time Optimal Control ◽  
Time Optimal

An iterative method for time optimal control of dynamic systemsAn iterative method for time optimal control of a general type of dynamic systems is proposed, subject to limited control inputs. This method uses the indirect solution of open-loop optimal control problem. The necessary conditions for optimality are derived from Pontryagin's minimum principle and the obtained equations lead to a nonlinear two point boundary value problem (TPBVP). Since there are many difficulties in finding the switching points and in solving the resulted TPBVP, a simple iterative method based on solving the minimum energy solution is proposed. The method does not need finding the switching point so that the resulted TPBVP can be solved by usual algorithms such as shooting and collocation. Also, since the solution of TPBVPs is sensitive to initial guess, a short procedure for making the proper initial guess is introduced. To this end, the accuracy and efficiency of the proposed method is demonstrated using time optimal solution of some systems: harmonic oscillator, robotic arm, double spring-mass problem with coulomb friction and F-8 aircraft.

scholarly journals On the Optimal Control of Stationary Fluid–Structure Interaction Systems

Fluids ◽  
2020 ◽  
Vol 5 (3) ◽  
pp. 144
Author(s):  
Leonardo Chirco ◽  
Sandro Manservisi
Keyword(s):  
Optimal Control ◽  
Fluid Structure Interaction ◽  
Optimal Solution ◽  
Adjoint System ◽  
Optimality System ◽  
Necessary Condition ◽  
Optimal Controls ◽  
Fluid Structure ◽  
Structure Interaction ◽  
Gradient Based

Fluid–structure interaction (FSI) systems consist of a fluid which flows and deforms one or more solid surrounding structures. In this paper, we study inverse FSI problems, where the goal is to find the optimal value of some control parameters, such that the FSI solution is close to a desired one. Optimal control problems are formulated with Lagrange multipliers and adjoint variables formalism. In order to recover the symmetry of the stationary state-adjoint system an auxiliary displacement field is introduced and used to extend the velocity field from the fluid into the structure domain. As a consequence, the adjoint interface forces are balanced automatically. We present three different FSI optimal controls: inverse parameter estimation, boundary control and distributed control. The optimality system is derived from the first order necessary condition by taking the Fréchet derivatives of the augmented Lagrangian with respect to all the variables involved. The optimal solution is obtained through a gradient-based algorithm applied to the optimality system. In order to support the proposed approach and compare these three optimal control approaches numerical tests are performed.

A novel method for the determination of linting propensity of paper

TAPPI Journal ◽  
2012 ◽  
Vol 11 (10) ◽  
pp. 9-17
Author(s):  
ALESSANDRA GERLI ◽  
LEENDERT C. EIGENBROOD
Keyword(s):  
Ash Content ◽  
The Novel ◽  
Offset Printing ◽  
Excellent Correlation ◽  
End Conditions ◽  
Bottom Side ◽  
Novel Method ◽  
Total Fraction

A novel method was developed for the determination of linting propensity of paper based on printing with an IGT printability tester and image analysis of the printed strips. On average, the total fraction of the surface removed as lint during printing is 0.01%-0.1%. This value is lower than those reported in most laboratory printing tests, and more representative of commercial offset printing applications. Newsprint paper produced on a roll/blade former machine was evaluated for linting propensity using the novel method and also printed on a commercial coldset offset press. Laboratory and commercial printing results matched well, showing that linting was higher for the bottom side of paper than for the top side, and that linting could be reduced on both sides by application of a dry-strength additive. In a second case study, varying wet-end conditions were used on a hybrid former machine to produce four paper reels, with the goal of matching the low linting propensity of the paper produced on a machine with gap former configuration. We found that the retention program, by improving fiber fines retention, substantially reduced the linting propensity of the paper produced on the hybrid former machine. The papers were also printed on a commercial coldset offset press. An excellent correlation was found between the total lint area removed from the bottom side of the paper samples during laboratory printing and lint collected on halftone areas of the first upper printing unit after 45000 copies. Finally, the method was applied to determine the linting propensity of highly filled supercalendered paper produced on a hybrid former machine. In this case, the linting propensity of the bottom side of paper correlated with its ash content.

Stable and Supported Semantics in Continuous Vector Spaces

2020 ◽  
Author(s):  
Yaniv Aspis ◽  
Krysia Broda ◽  
Alessandra Russo ◽  
Jorge Lobo
Keyword(s):  
Logic Program ◽  
Matrix Multiplication ◽  
Vector Spaces ◽  
Continuous Space ◽  
Continuous Vector ◽  
Novel Approach ◽  
Gradient Based ◽  
Normal Logic ◽  
Parameter Values ◽  
Normal Logic Program

We introduce a novel approach for the computation of stable and supported models of normal logic programs in continuous vector spaces by a gradient-based search method. Specifically, the application of the immediate consequence operator of a program reduct can be computed in a vector space. To do this, Herbrand interpretations of a propositional program are embedded as 0-1 vectors in $\mathbb{R}^N$ and program reducts are represented as matrices in $\mathbb{R}^{N \times N}$. Using these representations we prove that the underlying semantics of a normal logic program is captured through matrix multiplication and a differentiable operation. As supported and stable models of a normal logic program can now be seen as fixed points in a continuous space, non-monotonic deduction can be performed using an optimisation process such as Newton's method. We report the results of several experiments using synthetically generated programs that demonstrate the feasibility of the approach and highlight how different parameter values can affect the behaviour of the system.

The Use of TMAH to Etch Silicon and Expose Metal Bridging Failures

2000 ◽  
Author(s):  
Mark Morris ◽  
James Mohr ◽  
Esteban Ortiz ◽  
Steven Englebretson
Keyword(s):  
Failure Mechanism ◽  
Schottky Diodes ◽  
Tetramethylammonium Hydroxide ◽  
Novel Approach ◽  
Failure Location ◽  
Metal Etching ◽  
Plastic Mold

Abstract Determination of metal bridging failures on plastic encapsulated devices is difficult due to the metal etching effects that occur while removing many of the plastic mold compounds. Typically, the acids used to remove the encapsulation are corrosive to the metals that are found within the device. Thus, decapsulation can result in removal of the failure mechanism. Mechanical techniques are often not successful due to damage that results in destruction of the die and failure mechanism. This paper discusses a novel approach to these types of failures using a silicon etch and a backside evaluation. The desirable characteristics of the technique would be to remove the silicon and leave typical device metals unaffected. It would also be preferable that the device passivation and oxides not be etched so that the failure location is not disturbed. The use of Tetramethylammonium Hydroxide (TMAH), was found to fit these prerequisites. The technique was tested on clip attached Schottky diodes that exhibited resistive shorting. The use of the TMAH technique was successful at exposing thin solder bridges that extruded over the edge of the die resulting in failure.

The Determination of Maximum Range for an Unpowered Atmospheric Vehicle

1964 ◽  
Vol 68 (638) ◽  
pp. 111-116 ◽  
Author(s):  
D. J. Bell
Keyword(s):  
Calculus Of Variations ◽  
Lagrange Multipliers ◽  
Digital Computer ◽  
Necessary Conditions ◽  
Lagrange Equations ◽  
Initial Portion ◽  
End Conditions ◽  
Maximum Range ◽  
Isothermal Atmosphere

SummaryThe problem of maximising the range of a given unpowered, air-launched vehicle is formed as one of Mayer type in the calculus of variations. Eulers’ necessary conditions for the existence of an extremal are stated together with the natural end conditions. The problem reduces to finding the incidence programme which will give the greatest range.The vehicle is assumed to be an air-to-ground, winged unpowered vehicle flying in an isothermal atmosphere above a flat earth. It is also assumed to be a point mass acted upon by the forces of lift, drag and weight. The acceleration due to gravity is assumed constant.The fundamental constraints of the problem and the Euler-Lagrange equations are programmed for an automatic digital computer. By considering the Lagrange multipliers involved in the problem a method of search is devised based on finding flight paths with maximum range for specified final velocities. It is shown that this method leads to trajectories which are sufficiently close to the “best” trajectory for most practical purposes.It is concluded that such a method is practical and is particularly useful in obtaining the optimum incidence programme during the initial portion of the flight path.

scholarly journals A Novel Approach for Solving Nonsmooth Optimization Problems with Application to Nonsmooth Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Hamid Reza Erfanian ◽  
M. H. Noori Skandari ◽  
A. V. Kamyad
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Piecewise Linear Function ◽  
Nonsmooth Equations ◽  
Smooth Functions ◽  
Generalized Derivative ◽  
Novel Approach

We present a new approach for solving nonsmooth optimization problems and a system of nonsmooth equations which is based on generalized derivative. For this purpose, we introduce the first order of generalized Taylor expansion of nonsmooth functions and replace it with smooth functions. In other words, nonsmooth function is approximated by a piecewise linear function based on generalized derivative. In the next step, we solve smooth linear optimization problem whose optimal solution is an approximate solution of main problem. Then, we apply the results for solving system of nonsmooth equations. Finally, for efficiency of our approach some numerical examples have been presented.

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