scholarly journals An Underactuated Drift-Free Left Invariant Control System on a Specific Lie Group

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Camelia Pop Arieşanu
Keyword(s):  
Optimal Control ◽  
Control System ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Lie Group ◽  
Point Of View ◽  
Free Left ◽  
Invariant Control

The paper presents a geometrical overview on an optimal control problem on a special Lie group. The Hamilton-Poisson realization of the dynamics offers us the possibility to study the system from mechanical geometry point of view.

scholarly journals A Drift-Free Left Invariant Control System on the Lie Group SO(3)×R3×R3

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Camelia Pop
Keyword(s):  
Control System ◽  
Numerical Integration ◽  
Lie Algebra ◽  
Lie Group ◽  
Poisson Structure ◽  
Dual Space ◽  
Free System ◽  
Geometrical Properties ◽  
Free Left ◽  
Invariant Control

A controllable drift-free system on the Lie group G=SO(3)×R3×R3 is considered. The dynamics and geometrical properties of the corresponding reduced Hamilton’s equations on g∗,·,·- are studied, where ·,·- is the minus Lie-Poisson structure on the dual space g∗ of the Lie algebra g=so(3)×R3×R3 of G. The numerical integration of this system is also discussed.

scholarly journals Intrinsic Optimal Control for Mechanical Systems on Lie Group

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Chao Liu ◽  
Shengjing Tang ◽  
Jie Guo
Keyword(s):  
Optimal Control ◽  
Analytical Solution ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Lie Group ◽  
Control Method ◽  
Geodesic Distance ◽  
Infinite Horizon ◽  
Mechanical Systems ◽  
Programming Approach

The intrinsic infinite horizon optimal control problem of mechanical systems on Lie group is investigated. The geometric optimal control problem is built on the intrinsic coordinate-free model, which is provided with Levi-Civita connection. In order to obtain an analytical solution of the optimal problem in the geometric viewpoint, a simplified nominal system on Lie group with an extra feedback loop is presented. With geodesic distance and Riemann metric on Lie group integrated into the cost function, a dynamic programming approach is employed and an analytical solution of the optimal problem on Lie group is obtained via the Hamilton-Jacobi-Bellman equation. For a special case on SO(3), the intrinsic optimal control method is used for a quadrotor rotation control problem and simulation results are provided to show the control performance.

Solving the Problem of the Optimal Control System General Synthesis Based on Approximation of a Set of Extremals using the Symbol Regression Method

2020 ◽  
pp. 59-74
Author(s):  
S.V. Konstantinov ◽  
A.I. Diveev
Keyword(s):  
Optimal Control ◽  
Control System ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Function ◽  
Symbolic Regression ◽  
Regression Method ◽  
Synthesis Problem ◽  
Optimal Control System ◽  
Initial States

A new approach is considered to solving the problem of synthesizing an optimal control system based on the extremals' set approximation. At the first stage, the optimal control problem for various initial states out of a given domain is being numerically sold. Evolutionary algorithms are used to solve the optimal control problem numerically. At the second stage, the problem of approximating the found set of extremals by the method of symbolic regression is solved. Approach considered in the work makes it possible to eliminate the main drawback of the known approach to solving the control synthesis problem using the symbolic regression method, which consists in the fact that the genetic algorithm used in solving the synthesis problem does not provide information about proximity of the found solution to the optimal one. Here, control function is built on the basis of a set of extremals; therefore, any particular solution should be close to the optimal trajectory. Computational experiment is presented for solving the applied problem of synthesizing the four-wheel robot optimal control system in the presence of phase constraints. It is experimentally demonstrated that the synthesized control function makes it possible for any initial state from a given domain to obtain trajectories close to optimal in the quality functional. Initial states were considered during the experiment, both included in the approximating set of optimal trajectories and others from the same given domain. Approximation of the extremals set was carried out by the network operator method

AN OPTIMAL CONTROL PROBLEM ON THE LIE GROUP SO(4)

2008 ◽  
Vol 05 (03) ◽  
pp. 319-327 ◽  
Author(s):  
ANANIA ARON ◽  
IONEL MOŞ ◽  
ANIKO CSAKY ◽  
MIRCEA PUTA
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Lie Group ◽  
Controllable System ◽  
Geometrical Properties

An optimal control problem for a drift-free controllable system on the Lie group SO(4) is discussed and some of its dynamical and geometrical properties are pointed out.

scholarly journals Stability and Analytic Solutions of an Optimal Control Problem on the Schrödinger Lie Group

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 549-558
Author(s):  
Remus-Daniel Ene ◽  
Camelia Pop ◽  
Camelia Petrişor
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Nonlinear Stability ◽  
Lie Group ◽  
Asymptotic Method ◽  
Analytic Solutions ◽  
Optimal Homotopy Asymptotic Method

AbstractThe nonlinear stability and the existence of the periodic solutions for an optimal control problem on the Schrödinger Lie group are discussed. The analytic solutions via optimal homotopy asymptotic method of the dynamics and numerical simulations are presented, too.

scholarly journals The Delayed Doubly Stochastic Linear Quadratic Optimal Control Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Yan Chen ◽  
Jie Xu
Keyword(s):  
Optimal Control ◽  
Control System ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Riccati Equation ◽  
Control Variable ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Doubly Stochastic ◽  
Quadratic Optimal Control

In this paper, the delayed doubly stochastic linear quadratic optimal control problem is discussed. It deduces the expression of the optimal control for the general delayed doubly stochastic control system which contained time delay both in the state variable and in the control variable at the same time and proves its uniqueness by using the classical parallelogram rule. The paper is concerned with the generalized matrix value Riccati equation for a special delayed doubly stochastic linear quadratic control system and aims to give the expression of optimal control and value function by the solution of the Riccati equation.

scholarly journals Minirobots Moving at Different Partial Speeds

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1036
Author(s):  
Constantin Udrişte ◽  
Ionel Ţevy
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Initial Conditions ◽  
Control Policy ◽  
Point Of View ◽  
Planar Convex ◽  
Multi Temporal ◽  
Time Optimal ◽  
Hamilton Jacobi Bellman

In this paper, we present the mathematical point of view of our research group regarding the multi-robot systems evolving in a multi-temporal way. We solve the minimum multi-time volume problem as optimal control problem for a group of planar micro-robots moving in the same direction at different partial speeds. We are motivated to solve this problem because a similar minimum-time optimal control problem is now in vogue for micro-scale and nano-scale robotic systems. Applying the (weak and strong) multi-time maximum principle, we obtain necessary conditions for optimality and that are used to guess a candidate control policy. The complexity of finding this policy for arbitrary initial conditions is dominated by the computation of a planar convex hull. We pointed this idea by applying the technique of multi-time Hamilton-Jacobi-Bellman PDE. Our results can be extended to consider obstacle avoidance by explicit parameterization of all possible optimal control policies.

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