Control of chaos in nonlinear systems with time-periodic coefficients

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Sep 2006

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.

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Control of chaos in nonlinear systems with time-periodic coefficients

S.C Sinha () Alexandra Dvid Receive free email alerts when new articles cite this article - sign up in the box at the top right-hand corner of the article or click here - Email alerting service Control of chaos in nonlinear systems with time-periodic coefficients BY S. C. SINHA* AND ALEXANDRA DA VID 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. 1. Introduction The phenomenon of chaos in nonlinear systems has been investigated extensively in the last two decades. Recently, there has been a significant interest in controlling chaotic systems. Among the control strategies used to suppress chaos, perhaps the most well known is the OttGrebogiYorke method (Ott et al. 1990). This method relies on the facts that chaotic systems are very sensitive to initial conditions and that there is typically an infinite number of unstable periodic orbits embedded in the chaotic attractor. One of these unstable orbits is stabilized by controlling perturbations. The method uses the Poincare map of the system. Some of its limitations are that the parameter changes can only be discrete and only certain embedded orbits can be stabilized. Another control One contribution of 15 to a Theme Issue Exploiting chaotic properties of dynamical systems for their control. method has been suggested by Pyragas (1992). In his approach, the stabilization is achieved either by periodic external perturbation or by feedback control with time delay. A detailed review of chaos control can be found in the work by Fradkov & Evans (2002). Most of the work, however, has dealt with autonomous systems. In the physics literature, a non-autonomous system is often converted into an autonomous, discrete system using a Poincare map in which the fixed points of the Poincare map correspond to periodic orbits of the system. In this case, since it is not easy to obtain an analytical expression for the Poincare map, one often needs to obtain an approximate expression based on numerical methods or experimental data. Local stabilization of periodic orbits in chaotic systems by feedback control has been dealt with in previous works by several authors (Cheng & Dong 1993a,b; Chen 1996). However, in all these papers, the non-autonomous nature of the problems was ignored and the RouthHurwitz criterion was applied as if the systems were autonomous. Since the problem of stabilization of a periodic orbit leads to a non-autonomous system with time-periodic coefficients, Floquet theory must be used to guarantee stability of the system. Sinha et al. (2000) have proposed a general approach in the design of active controllers for nonlinear systems exhibiting chaos. In this method, it is shown that a system exhibiting chaos can be driven to a desired periodic motion by combining a feed-forward and a feedback controller. The feedback controller is designed using a method proposed by Sinha & Joseph (1994), where a time-invariant auxiliary system is constructed and stabilized with pole placement method. However, this method uses a least-square approximation approach and in certain parameter ranges, the stability of the periodic orbits may not be guaranteed. In this paper, three approaches are suggested for local chaos control in nonlinear systems with time-periodic coefficients. The first one is based on the symbolic computation of the Floquet transition matrix (FTM) associated with the linear part of the system. By using the technique introduced by Sinha & Butcher (1997), the state transition matrix (STM) of the resulting system is calculated symbolically. The symbolic expression of the STM contains the unknown control gains and these can be assigned by placing the Floquet multiplier (...truncated)


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S.C Sinha, Alexandra Dávid. Control of chaos in nonlinear systems with time-periodic coefficients, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2006, pp. 2417-2432, 364/1846, DOI: 10.1098/rsta.2006.1832