Sinha S C, Dávid Alexandra
Nonlinear Systems Research Laboratory, Department of Mechanical Engineering, Auburn University, Auburn, AL 36839, USA.
Philos Trans A Math Phys Eng Sci. 2006 Sep 15;364(1846):2417-32. doi: 10.1098/rsta.2006.1832.
In this study, some techniques for the control of chaotic nonlinear systems with periodic coefficients are presented. First, chaos is eliminated from a given range of the system parameters by driving the system to a desired periodic orbit or to a fixed point using a full-state feedback. One has to deal with the same mathematical problem in the event when an autonomous system exhibiting chaos is desired to be driven to a periodic orbit. This is achieved by employing either a linear or a nonlinear control technique. In the linear method, a linear full-state feedback controller is designed by symbolic computation. The nonlinear technique is based on the idea of feedback linearization. A set of coordinate transformation is introduced, which leads to an equivalent linear system that can be controlled by known methods. Our second idea is to delay the onset of chaos beyond a given parameter range by a purely nonlinear control strategy that employs local bifurcation analysis of time-periodic systems. In this method, nonlinear properties of post-bifurcation dynamics, such as stability or rate of growth of a limit set, are modified by a nonlinear state feedback control. The control strategies are illustrated through examples. All methods are general in the sense that they can be applied to systems with no restrictions on the size of the periodic terms.
在本研究中，提出了一些用于控制具有周期系数的混沌非线性系统的技术。首先，通过使用全状态反馈将系统驱动到期望的周期轨道或固定点，在给定的系统参数范围内消除混沌。当希望将表现出混沌的自治系统驱动到周期轨道时，必须处理相同的数学问题。这可以通过采用线性或非线性控制技术来实现。在线性方法中，通过符号计算设计线性全状态反馈控制器。非线性技术基于反馈线性化的思想。引入一组坐标变换，从而得到一个可以用已知方法控制的等效线性系统。我们的第二个想法是通过采用时间周期系统的局部分岔分析的纯非线性控制策略，将混沌的起始延迟到给定参数范围之外。在这种方法中，通过非线性状态反馈控制修改分岔后动力学的非线性特性，如极限集的稳定性或增长速率。通过示例说明了控制策略。所有方法都具有通用性，因为它们可以应用于对周期项大小没有限制的系统。
