scholarly journals Fundamentals of Synthesized Optimal Control

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 21
Author(s):  
Askhat Diveev ◽  
Elizaveta Shmalko ◽  
Vladimir Serebrenny ◽  
Peter Zentay
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Control Method ◽  
Direct Method ◽  
Equilibrium Points ◽  
Control Object ◽  
Optimal Control Method ◽  
Optimal Value ◽  
New Formulation ◽  
The Mathematical Model

This paper presents a new formulation of the optimal control problem with uncertainty, in which an additive bounded function is considered as uncertainty. The purpose of the control is to ensure the achievement of terminal conditions with the optimal value of the quality functional, while the uncertainty has a limited impact on the change in the value of the functional. The article introduces the concept of feasibility of the mathematical model of the object, which is associated with the contraction property of mappings if we consider the model of the object as a one-parameter mapping. It is shown that this property is sufficient for the development of stable practical systems. To find a solution to the stated problem, which would ensure the feasibility of the system, the synthesized optimal control method is proposed. This article formulates the theoretical foundations of the synthesized optimal control. The method consists in making the control object stable relative to some point in the state space and to control the object by changing the position of the equilibrium points. The article provides evidence that this approach is insensitive to the uncertainties of the mathematical model of the object. An example of the application of the method for optimal control of a group of robots is given. A comparison of the synthesized optimal control method with the direct method on the model without disturbances and with them is presented.

Symmetric Control Systems

2021 ◽  
pp. 37-51
Author(s):  
A.I. Diveev ◽  
E.A. Sofronova
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Systems ◽  
Control Object ◽  
Terminal State ◽  
Initial State ◽  
Phase Constraints ◽  
The Mathematical Model

The paper focuses on the properties of symmetric control systems, whose distinctive feature is that the solution of the optimal control problem for an object, the mathematical model of which belongs to the class of symmetric control systems, leads to the solution of two problems. The first optimal control problem is the initial one; the result of its solution is a function that ensures the optimal movement of the object from the initial state to the terminal one. In the second problem, the terminal state is the initial state, and the initial state is the terminal state. The complexity of the problem being solved is due to the increase in dimension when the models of all objects of the group are included in the mathematical model of the object, as well as the emerging dynamic phase constraints. The presence of phase constraints in some cases leads to the target functional having several local extrema. A theorem is proved that under certain conditions the functional is not unimodal when controlling a group of objects belonging to the class of symmetric systems. A numerical example of solving the optimal control problem with phase constraints by the Adam gradient method and the evolutionary particle swarm method is given. In the example, a group of two symmetrical objects is used as a control object

scholarly journals Discrete Optimal Control Method Based on the Optimal Strategy of Fishing

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Lichun Zhang ◽  
Qingdao Huang
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Control Method ◽  
Logistic Function ◽  
Optimal Size ◽  
Discrete Optimal Control ◽  
Optimal Control Method ◽  
Optimal Value ◽  
Principle Of Maximum

Consideration was given to the discrete optimal control method for the optimal fishing strategy. Our method is new and efficient for discrete optimal control problem, which is different from the other optimal methods such as the traditional variational method, the Pontryagin principle of maximum, and the dynamic programming. The basic construction of the model is the traditional logistic function relating to the growth of fry. The discrete optimal control method for optimal fishing strategy was used to construct the optimal rate of each fishing strategy; the main focus of our work is on the rigorous mathematical analysis of the optimal control problem. The analysis allows one to obtain the optimal initial investment amount of the fry and the optimal size of the total catch. Furthermore, when the initial investment amount of the fry is below or above the optimal value and the intrinsic growth rate of fishRis too small, we derive that fishing operations should not be started in the last few years to make the overall fishing amount optimal. At last, several typical examples are given to illustrate the obtained results.

A Class of Optimizations and Energy-Saving Application

2011 ◽  
Vol 135-136 ◽  
pp. 10-14
Author(s):  
Fu Lai Yao ◽  
He Xu Sun
Keyword(s):  
Optimal Control ◽  
Energy Saving ◽  
Control Method ◽  
Minimum Energy ◽  
Energy Optimization ◽  
Total Load ◽  
Optimal Control Method ◽  
Optimal Value

This paper presents a class of optimization functions, and gives the optimal value. The optimal conclusion is applied to energy optimization for general devices. When the total load is fixed and the devices are used with same model, the optimal control method is given: adjusting each device to the same load, the minimum energy is required.

CALCULATING THE BOUNDARIES OF THE AIRCRAFT EXIT ZONE TO A SET POINT

2021 ◽  
pp. 3-11
Author(s):  
O. N. Korsun ◽  
A. V. Stulovsky ◽  
S. V. Nikolaev
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Direct Method ◽  
Vertical Plane ◽  
Control Object ◽  
Aircraft Design ◽  
Continuous Functions ◽  
Cubic Splines ◽  
Parameters Estimation ◽  
Parameter Estimates

The article considers the method of calculating the boundaries of the exit zone of the aircraft to a given point based on the optimal control method. To find the optimal control, it is proposed to use a direct method based on parameterization of the desired control signals using third-order Hermitian splines. The choice of Hermitian cubic splines was motivated by the fact that these splines and their first order derivatives are smooth and continuous functions, on the one hand, and, on the other, do not require the additional solution of algebraic equations to meet the specific conditions in spline nodes which is obligatory for classic cubic splines. Spline parameters estimation is achieved through solution of the unconditional multiparametric optimization problem. The target functional includes the squares of mismatches between the desired output signals and the object model output signals. In this paper the parameter estimates are obtained using the widely known numerical optimization algorithm – the particle swarm method. The paper considers the aircraft motion in the vertical plane, for which a mathematical model of the control object is formed and the target functional is formulated. The proposed solution is advisable to apply when calculating the optimal trajectories and flight profiles of aircraft when planning their functioning for the designed purpose. The developed method allows solving a number of tasks in the process of modern aircraft design and flight tests. The application of the proposed method, the required structure of the mathematical model of the object and the features of the formation of the minimized functional are shown in a specific example.

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 Research of Trajectory Optimization Approaches in Synthesized Optimal Control

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 336
Author(s):  
Askhat Diveev ◽  
Elizaveta Shmalko
Keyword(s):  
Optimal Control ◽  
Equilibrium Point ◽  
Trajectory Optimization ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Control Object ◽  
Symbolic Regression ◽  
Optimal Location ◽  
Stable Equilibrium Point ◽  
Stabilization System

This article presents a study devoted to the emerging method of synthesized optimal control. This is a new type of control based on changing the position of a stable equilibrium point. The object stabilization system forces the object to move towards the equilibrium point, and by changing its position over time, it is possible to bring the object to the desired terminal state with the optimal value of the quality criterion. The implementation of such control requires the construction of two control contours. The first contour ensures the stability of the control object relative to some point in the state space. Methods of symbolic regression are applied for numerical synthesis of a stabilization system. The second contour provides optimal control of the stable equilibrium point position. The present paper provides a study of various approaches to find the optimal location of equilibrium points. A new problem statement with the search of function for optimal location of the equilibrium points in the second stage of the synthesized optimal control approach is formulated. Symbolic regression methods of solving the stated problem are discussed. In the presented numerical example, a piece-wise linear function is applied to approximate the location of equilibrium points.

Sign in / Sign up

Export Citation Format

Share Document