改进的保群算法及其在混沌系统中的应用

混沌系统的跟踪控制是近年来非线性控制领域研究的热点之一. 本文提出了一种基于快速下降控制方法的保群算法, 此方法使受控混沌系统能够快速稳定到相空间的一个不动点; 另外提出一种基于滑模控制方法的保群算法, 此方法使受控混沌系统能够快速跟踪一个确定的运动轨迹. 应用这两种新方法分别对两个经典的混沌系统(Lorenz系统和Duffing系统)进行了相应的数值实验, 实验结果表明这两种方法都具用较高的精度和稳定性.

Modified group preserving methods and applications in chaotic systems

The tracking control of chaotic system has been one of the research focus areas of nonlinear control in recent years, in which the vital problem is to enable chaotic system to stabilize to an equilibrium point or to track a deterministic trajectory quickly. The conventional chaos control methods make the control power unnecessarily large and generate the phenomenon of chattering easily, resulting in the instabilities of the systems.The problems above can be transformed into the solutions of differential algebraic equations effectively. Considering that the group preserving scheme not only approximates the original system, but also preserve as much as possible the geometric structure and invariants of the original system, this paper takes advantage of the group preserving method to study the control method in chaotic system from two different perspectives.A new group preserving scheme based on the fast descending control method is presented, which enables chaotic system to stabilize to an equilibrium point quickly. Firstly, we introduce a novel approach to replace the optimal control problem of nonlinear system by directly specifying a time-decaying Lagrangian function, which helps us to transform the optimal control problem into a system of differential algebraic equations. Then we derive a modified group preserving scheme for the system.Similarly, we propose a new group preserving scheme based on the sliding mode control method for chaotic system to track a deterministic trajectory quickly. Owing to numerical discretization errors, signal noises and structural uncertainties in dynamical systems, the conventional sliding mode control method cannot guarantee to maintain the trajectories on the sliding surface, unless the numerical integration method is designed to do so. On the other hand, the conventional sliding mode control method easily induces high frequency chattering of the control force. Therefore, we modify the conventional sliding mode control method and use the modified group preserving scheme to find the control force.The above two methods are the combination of traditional control method and the Lie-group method. An invariant manifold is properly designed, and the original system is transformed into the differential algebraic system, in which the modified group preserving scheme can be used to find the control force. The resulting controlled system is stable.Finally, the proposed methods are applied to the classic Lorenz system and Duffing system correspondingly. Numerical experimental results show that the new approaches are very accurate and stable. Since the two controlled methods are fast in convergence and chattering-free, each of them has a good application prospect in the tracking control of chaotic systems.