scholarly journals Optimal control of propellant consumption during vertical lifting of rocket in homogeneous atmosphere using regularized solution of integral equation of the first kind

2020 ◽  
Vol 12 (S) ◽  
pp. 221-231
Author(s):  
Aleksey G. VIKULOV
Keyword(s):  
Optimal Control ◽  
Integral Equation ◽  
Optimal Trajectory ◽  
Regularization Parameter ◽  
Motion Equations ◽  
Suborbital Flights ◽  
Propellant Consumption ◽  
Minimal Consumption ◽  
Regularized Solution ◽  
Differential And Integral Equations

The article is devoted to the study of the optimal control of propellant consumption during vertical lifting of rocket in homogeneous atmosphere using regularized solution of integral equation of the first kind. The problem of lifting of a rocket into desired height along optimal trajectory in the view of minimal consumption of propellant leads to solving the set of differential and integral equations. Problem of optimal control of propellant consumption during lifting of rocket in homogeneous atmosphere is solved using regularized solution of integral equation of the first kind which is solution of corresponding Euler equation on discrete time net. Influence of the regularization parameter and some additional parameters on precision of discreted problem is investigated. Considered algorithm is summed up easily to the case of non-homogeneous atmosphere by introducing dependence of the ballistic coefficient on altitude of flight and to problem of putting spacecraft into determined orbit and suborbital flights by setting desired altitude and velocity and modifying of motion equations.

scholarly journals Optimal control of propellant consumption during insertion of rocket into a circle orbit of the Earth

2020 ◽  
Vol 18 (4) ◽  
pp. 705-712
Author(s):  
Aleksey Vikulov
Keyword(s):  
Optimal Control ◽  
Integral Equations ◽  
Euler Equations ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Rocket Engine ◽  
Motion Equations ◽  
The Earth ◽  
Suborbital Flights ◽  
Propellant Consumption

The problem of launching a rocket into the Earth's orbit has already been solved using the regularization method in previous studies. But the regularization method remains relevant for application to solving integral equations of the first kind, which determine the components of speed and acceleration. The problem of optimal control of propellant consumption during the insertion of a rocket into a circle orbit of the Earth is solved using regularized solutions of integral equations of the first kind which are solutions of corresponding Euler equations on discrete-time net. The influence of the regularization parameter and some additional parameters on precision of discredited problem is investigated. Calculations are carried out for existing chemical rocket engine and promising plasmic one. Considered algorithm is summed up easily to problem of suborbital flights by setting desired coordinate system and modifying motion equations. Conclusions were drawn about the required speed for the lowest fuel consumption, as well as about the problem for a single-stage rocket. Thus, the development of a plasma rocket engine with an exhaust velocity is more than ten times higher than that of a chemical one.

The human-like trajectory planning for autonomous vehicles based on optimal control in a test track environment

2021 ◽  
pp. 095440702110090
Author(s):  
Xing Xu ◽  
Minglei Li ◽  
Feng Wang ◽  
Ju Xie ◽  
Xiaohan Wu ◽  
...  
Keyword(s):  
Optimal Control ◽  
Autonomous Vehicles ◽  
Optimal Trajectory ◽  
Control Method ◽  
Autonomous Vehicle ◽  
Trajectory Data ◽  
Test Track ◽  
Optimal Control Method ◽  
The Future ◽  
On The Road

A human-like trajectory could give a safe and comfortable feeling for the occupants in an autonomous vehicle especially in corners. The research of this paper focuses on planning a human-like trajectory along a section road on a test track using optimal control method that could reflect natural driving behaviour considering the sense of natural and comfortable for the passengers, which could improve the acceptability of driverless vehicles in the future. A mass point vehicle dynamic model is modelled in the curvilinear coordinate system, then an optimal trajectory is generated by using an optimal control method. The optimal control problem is formulated and then solved by using the Matlab tool GPOPS-II. Trials are carried out on a test track, and the tested data are collected and processed, then the trajectory data in different corners are obtained. Different TLCs calculations are derived and applied to different track sections. After that, the human driver’s trajectories and the optimal line are compared to see the correlation using TLC methods. The results show that the optimal trajectory shows a similar trend with human’s trajectories to some extent when driving through a corner although it is not so perfectly aligned with the tested trajectories, which could conform with people’s driving intuition and improve the occupants’ comfort when driving in a corner. This could improve the acceptability of AVs in the automotive market in the future. The driver tends to move to the outside of the lane gradually after passing the apex when driving in corners on the road with hard-lines on both sides.

scholarly journals Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ulrich Langer ◽  
Olaf Steinbach ◽  
Huidong Yang
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Optimal Control Problems ◽  
Regularization Parameter ◽  
Target Function ◽  
Mesh Size ◽  
Turbulence Theory ◽  
Diffusion Reaction ◽  
Control Problems ◽  
Element Approximation

Abstract We consider elliptic distributed optimal control problems with energy regularization. Here the standard L 2 {L_{2}} -norm regularization is replaced by the H - 1 {H^{-1}} -norm leading to more focused controls. In this case, the optimality system can be reduced to a single singularly perturbed diffusion-reaction equation known as differential filter in turbulence theory. We investigate the error between the finite element approximation u ϱ ⁢ h {u_{\varrho h}} to the state u and the desired state u ¯ {\overline{u}} in terms of the mesh-size h and the regularization parameter ϱ. The choice ϱ = h 2 {\varrho=h^{2}} ensures optimal convergence the rate of which only depends on the regularity of the target function u ¯ {\overline{u}} . The resulting symmetric and positive definite system of finite element equations is solved by the conjugate gradient (CG) method preconditioned by algebraic multigrid (AMG) or balancing domain decomposition by constraints (BDDC). We numerically study robustness and efficiency of the AMG preconditioner with respect to h, ϱ, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver.

A Multiplier Rule for a Functional Subject to Certain Integrodifferential Constraints

1969 ◽  
Vol 91 (2) ◽  
pp. 185-189 ◽  
Author(s):  
M. Wittler ◽  
C. N. Shen
Keyword(s):  
Optimal Control ◽  
Integral Equation ◽  
Calculus Of Variations ◽  
Necessary Conditions ◽  
Inequality Constraint ◽  
Fundamental Lemma ◽  
First Order ◽  
Multiplier Rule ◽  
Nuclear Rocket ◽  
Equation Constraint

A problem in the optimal control of a nuclear rocket requires the minimization of a functional subject to an integral equation constraint and an integrodifferential inequality constraint. A theorem giving first-order necessary conditions is derived for this problem in the form of a multiplier rule. The existence of multipliers and the arbitrariness of certain variations is shown. The fundamental lemma of the calculus of variations is applied. A simple example demonstrates the applicability of the theorem.

scholarly journals High order iterative methods for matrix inversion and regularized solution of fredholm integral equation of first kind with noisy data

2018 ◽  
Vol 22 ◽  
pp. 01002
Author(s):  
Suzan Cival Buranay ◽  
Ovgu Cidar Iyikal
Keyword(s):  
Integral Equation ◽  
Iterative Methods ◽  
Fredholm Integral Equation ◽  
Recurrence Formula ◽  
Noisy Data ◽  
Matrix Inversion ◽  
High Order ◽  
The Given ◽  
High Order Iterative Methods ◽  
Regularized Solution

The motivation of the present work is to propose high order iterative methods with a recurrence formula for approximate matrix inversion and provide regularized solution of Fredholm integral equation of first kind with noisy data by an algorithm using the proposed methods. From the given family of methods of orders p = 7,11,15,19 are applied to solve problems of Fredholm integral equation of first kind. From the literature, iterative methods of same orders are used to solve the considered problems and numerical comparisons are shown through tables and figures.

Solving optimal control problems using the Picard’s iteration method

2020 ◽  
Vol 54 (5) ◽  
pp. 1419-1435
Author(s):  
Abderrahmane Akkouche ◽  
Mohamed Aidene
Keyword(s):  
Optimal Control ◽  
Iteration Method ◽  
Optimal Trajectory ◽  
Optimal Control Problems ◽  
Approximate Analytical Solution ◽  
Minimum Principle ◽  
Necessary Optimality Conditions ◽  
Control Law ◽  
Control Problems ◽  
Objective Functional

In this paper, the Picard’s iteration method is proposed to obtain an approximate analytical solution for linear and nonlinear optimal control problems with quadratic objective functional. It consists in deriving the necessary optimality conditions using the minimum principle of Pontryagin, which result in a two-point-boundary-value-problem (TPBVP). By applying the Picard’s iteration method to the resulting TPBVP, the optimal control law and the optimal trajectory are obtained in the form of a truncated series. The efficiency of the proposed technique for handling optimal control problems is illustrated by four numerical examples, and comparison with other methods is made.

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