Стабилизация неустойчивых точек равновесия и циклов в нелинейных динамических системах / Stabilization of Unstable Equilibrium Points and Cycles in Non-linear Dynamical Systems тема диссертации и автореферата по ВАК РФ 05.13.18, кандидат наук Шалби Лина Ахмед Сайед Хамис

• Шалби Лина Ахмед Сайед Хамис
• кандидат наук
• 2020

Оглавление диссертации кандидат наук Шалби Лина Ахмед Сайед Хамис

Contents

Introduction

1 Dynamical Systems and Their Stability

1.1 Discrete-Time Dynamical Systems

1.2 Continuous-Time Dynamical Systems

1.3 Examples on Unstable Dynamical Systems

1.3.1 The Logistic Map

1.3.2 Henon Map

1.3.3 Lorenz Model

1.3.4 Rossler 4D Model

1.3.5 Restricted Three Body Problem

2 Stabilization of Unstable Systems by Predictive Feedback Control

2.1 PFC Method and Other Methods for

Discrete-Time Chaotic Systems

2.2 Extended PFC Method to Stabilize Chaotic Continuous-Time Systems

3 Stabilization by Optimal Control

3.1 Control with ^-Minimization Method

3.2 Application: Spacecraft Fuel Consumption

3.2.1 Fuel Optimal Control Strategy

3.2.2 Trajectory Control at Lagrangian Points

Conclusion

Bibliography

Введение диссертации (часть автореферата) на тему «Стабилизация неустойчивых точек равновесия и циклов в нелинейных динамических системах / Stabilization of Unstable Equilibrium Points and Cycles in Non-linear Dynamical Systems»

Introduction

Dynamical systems are widely used in many scientific fields. The main interest arises when a dynamical system has strange or unpredicted behaviour. In the last decades, many researchers concerned with how to control unstable behaviour of dynamical systems. The aim is to bring an orbit or trajectory close enough to a desired location by using a very small perturbation. This target is reachable by applying some control, so the unstable equilibrium points and cycles of periodic or chaotic behaviour are either theoretically stabilized or be convergent for the trajectory by using optimal control methods. Many researches focus on stabilizing special unstable system describing their study case. Few others researches, we are concerning with them, are focusing on stabilizing chaos generally. In 1990, E. Ott [43] was the first researcher who concerned with stabilizing chaos. This method has been introduced to control discrete-time data, so continuous-time systems should firstly be discretized by Poincare map. It was followed by many other publications improving his method or introducing new

methods (see [4,63]). For instance one of the promising methods is Predictive Feedback Control (PFC) proposed in [45] for discrete-time systems. However the systematic comparison of various stabilization algorithms (both for discrete-time and continuous-time systems) is an open task.

Optimal control theory is one of the fields of applied mathematics that deals with setting a suitable control law for a dynamical system with an objective function required to be minimized or maximized. Traditionally optimal control is focused on problems with smooth objective. However in stabilization methods nonsmooth objectives often arise. In [57], Tabak and Kuo introduced the optimal control method with l1 function which could be used to determine the minimum fuel consumption for a spacecraft optimal trajectory planning. We continue this line of research for holding a spacecraft in a neighborhood of an unstable Lagrangian point (libration point) in Earth-Moon system. There are numerous approaches to solving this problem, see e.g. the recent survey [55]. However most of them are oriented on design of particular trajectories moving around the point (e.g. Halo and Lissajous orbits). We propose different techniques based on optimal control with li objective and alternating controlled/uncontrolled steps.

Motivations

In this dissertation, stabilization of unstable equilibrium points is achieved with different methods. The first group of methods is to imply small control actions to make the equilibrium point or the orbit stable. This could be helpful either if the goal is to change the behavior of the system (to avoid chaotic trajectories) or to find new equilibrium points or orbits (which cannot be discovered by direct calculations). Predictive feedback control (PFC) is the typical tool for such purposes. The second group of methods relies on optimal control techniques. The important application is space research, where it is needed to keep a spacecraft or a satellite inside a very small neighborhood of the equilibrium point.

Aim of The Work

The general aim of the work is to study non-linear systems, find unstable points or orbits and stabilize them by using very small control actions. In more details, the aim of this research is summarized as following:

1. Exploit predictive feedback control method for discrete-time systems and compare its efficiency with popular stabilization methods for chaotic discrete-time dynamical systems.

2. Extend predictive feedback control method to stabilize unstable equilibrium points of continuous-time systems.

3. Hold trajectory inside small neighborhood of equilibrium point via optimal control, and apply this approach for a spacecraft close to Lagrangian points of Earth-Moon system.

Scientific Results

In this research, the results are summarized in the following points:

1. Predictive feedback control (PFC) method is studied for numerous examples of chaotic discrete-time systems, and it is easy implemented by using our published on-line codes in Matlab1 to stabilize an unstable fixed point to be stable of period-1. Comparison between different chaos control methods is performed, and the high efficiency of PFC method is illustrated.

2. A new control method to stabilize continuous-time system is proposed extending predictive feedback control method and its stabilization is proved theoretically.

3. The stability of chaotic attractors of Lorenz 3D system and Rossler 4D system is achieved by using the extended PFC method. The

1 https://www.mathworks.com/matlabcentral/fileexchange/73911-predictive-feedback-control-method-for-discrete-time-systems

designed commands for PFC method by using Matlab program are published on-line, for Lorenz system2 and Rossler 4D system3.

4. Novel approach for holding trajectory in a vicinity of the unstable equilibrium point is proposed. It is based on optimal control methods with l1 objective function and alternating controlled/uncontrolled steps.

5. Optimal control li-minimization is used to control the trajectory of spacecraft at collinear Lagrangian points L1, L2 and L3. This technique characterized by impulsive control which minimizes the fuel consumed for its instant nonzero values.

Scientific Novelty

1. A comparison is done to the recent control methods OGY, TDAS, ETDAS, PFC, and three control methods based on SOMA and DE. The methods are applied to control 100 orbits of Logistic and Henon maps generated randomly. They are compared according to the percent of successfully controlled period-1 orbit, maximum absolute autocorrelation, minimum and maximum re-

https://www.mathworks.com/matlabcentral/fileexchange/73912-epfc-method-applied-to-lorenz-3d-continuous-time-system

https://www.mathworks.com/matlabcentral/fileexchange/73913-epfc-method-applied-to-rossler-4d-continuous-time-system

2

3

quired number of iteration to reach accuracy less than 10 5, and maximum absolute value of additive control term.

2. New predictive feedback control method is introduced to stabilize chaotic continuous-time systems. The method stability is verified. Unstable attractors of Lorenz system and Rossler hyperchaos system are stabilized by the extended PFC method.

3. Novel approach is proposed to keep trajectory inside a small neighborhood of the unstable equilibrium point. This approach is based on ^-minimization optimal method and switching between controlled and uncontrolled parts of trajectory.

Theoretical and Practical Importance

The dissertation is theoretical. The detected unstable equilibrium points are stabilized by control methods which can be used for any dynamical system described by map or autonomous differential equations to stabilize its attractors of cyclic or chaotic behaviour. On the other hand the proposed optimal control strategy by using ^-technique may be modified for implementations in real life space companies and any other which have problems deal with fuel consumption of vehicle dynamics, such as satellite and statecraft, required to be directed to reach a specified position.

Methodology and Research Methods

In the dissertation, different control methods are used to stabilize and control unstable fixed points of discrete-time and continuous-time non-linear systems. The methods OGY, TDAS, ETDAS, and PFC are used to stabilize the discrete-time non-linear systems and to compare their efficiency by autocorrelation and maximum absolute value of the inserted control. And new PFC method is used by adding small control to stabilize continuous-time systems.

Optimal control method with ^-minimization is used to control the trajectory of a spacecraft with minimum required fuel. The technique determines the minimum control impulses given to the acceleration of spacecraft to direct it towards Lagrangian points L1, L2 and L3.

Propositions for The Defense

1. The unstable attractors of different dynamical systems are determined and their behaviour is illustrated.

2. Systematic comparison between discrete-time stabilization methods is performed, comparing their efficiency to control chaotic orbits at the fixed points.

3. A new control method to stabilize continuous-time systems is proposed using predictive feedback control (PFC) and its validation is proved theoretically.

4. The stability of chaotic attractors of 3D Lorenz system and 4D Roossler system is achieved by using PFC method.

5. Novel approach is proposed to hold trajectory inside a close neighborhood of unstable equilibrium point, depending on optimal control methods with l1 objective function and alternating between control and uncontrol.

6. Optimal control ^-minimization is used to control the trajectory of spacecraft close to unstable collinear Lagrangian points L1, L2 and L3 of Earth-Moon system.

7. The fuel consumption of a spacecraft to be kept as close as possible to Lagrangian point is minimized by using ^-technique with switching control on and off leaving the spacecraft to move freely without control forming a cycle around the point.

Presentations and Seminars of The Results

The results of the dissertation were given in a conference: XIII All-Russian Conference on Automatic Control (VSPU-2019), ICS

RAS, Moscow, June 17-20, 2019;

One seminar was presented at V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences Moscow Seminar on Automatic Control, Moscow, October 01, 2019.

List of Publications

The main results of the dissertation are represented in 3 papers, published in three Scopus indexed journals with one included in Web-of-Science:

1. Polyak B. T. and Shalby L. A., "Stabilization of Spacecraft at the Collinear Lagrangian Points with Minimum Fuel Consumption", Automation and Remote Control, V.80 No. 12, pp.2217-2228, 2019.

2. Shalby L. A., "Comparison between chaos-control methods efficiency for discrete systems", WSEAS Transactions on Systems and Control, V. 14, 2019, Art. 36, pp. 284-290.

3. Shalby L. A., "Predictive feedback control method for stabilization of continuous time systems". Advances in Systems Science and Applications. V.17, No.4, pp. 1-13, 2017.

Заключение диссертации по теме «Математическое моделирование, численные методы и комплексы программ», Шалби Лина Ахмед Сайед Хамис

Conclusion

In this dissertation, the unstable fixed points and equilibrium points of popular dynamical systems , Logistic 1D and Henon 2D maps, Lorenz 3D, Rossler 4D and Restricted three body problem in 2D systems, are determined and their behaviour is studied. Then the stabilization and control of unstable equilibrium points have been achieved with different methods.

A systematic comparison is done to the recent chaos control methods, OGY, TDAS, ETDAS, PFC, and three control laws set for Henon map, to stabilize discrete-time systems. The methods are discussed briefly and then applied to control 100 orbits of Logistic and Henon maps generated randomly. They are compared according to the percent of successfully controlled period-1 orbit, maximum absolute autocorrelation, minimum and maximum required number of iteration to reach accuracy less than 10—5, and maximum absolute value of additive control term. TDAS and PFC methods showed the most capability to control chaos of discrete-time systems with high efficiency, but PFC

method has less value of maximum absolute control term and higher autocorrelation.

A new control method to stabilize continuous-time systems is proposed by using predictive feedback control (PFC) method. The system is discretized numerically using Euler method, and then the PFC method is applied. The PFC method is used by adding small control term to the numerical solution of the system, this term is the multiplication of a controlling matrix by the difference between two consecutive predicted iterates. We have also discussed the stability analysis of the continuous-time system compared to its numerical discrete form, and the choice of the control matrix is set depending on the eigenvalues of the system. We have proved that the method is stable. The method has been applied to the most popular systems, Lorenz 3D system and Rossler hyperchaos 4D system. Their trajectories have been controlled to each equilibrium point after a few iterations. The extended PFC method is easy implemented and highly efficient method, as it is generalized to stabilize chaotic systems either discrete-time or continuous-time system.

Optimal control method ^-minimization is used to control the trajectory of a spacecraft at collinear Lagrangian points. The minimum control impulses are given to the acceleration of spacecraft to direct it towards Lagrangian points Li, L2 and L3. To apply ^-minimization technique, the spacecraft equations of motion within Earth-Moon sys-

tem, described by restricted three-body problem, are linearized after setting Lagrangian point at the origin, and then discretized numerically. The control values of l1-minimization are substituted to the numerical solution of continuous-time system. The control is achieved by solving the system numerically concerning the instant control values, and the proposed strategy is successfully applied several times to keep the spacecraft close to L1, L2 and L3, over time with minimum consumed fuel. The minimum fuel consumption at L3 is the smallest value because it is the farthest point and has the least gravity effect.

This novel approach for holding trajectory in a neighborhood of unstable equilibrium point is based on the optimal control method, l1-minimization objective function and alternating controlled/uncontrolled steps. And the fuel consumption of a spacecraft to be kept as close as possible to Lagrangian point is minimized by switching control on and off leaving the spacecraft to move freely without control forming cycles around the point.

Список литературы диссертационного исследования кандидат наук Шалби Лина Ахмед Сайед Хамис, 2020 год

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