Control of stochastic chaos using sliding mode method

Abstract

a b s t r a c tStabilizing unstable periodic orbits of a deterministic chaotic system which is perturbed by a stochastic process is studied in this paper. The stochastic chaos is modeled by exciting a deterministic chaotic system with a white noise obtained from derivative of a Wiener process which eventually generates an Ito differential equation. It is also assumed that the chaotic system being studied has some model uncertainties which are not random. The sliding mode controller with some modifications is used for stochastic chaos suppression. It is shown that the system states converge to the desired orbit in such a way that the error covariance converges to an arbitrarily small bound around zero. As some case studies, the stabilization of 1-cycle and 2-cycle orbits of chaotic Duffing and Φ 6 Van der Pol systems is investigated by applying the proposed method to their corresponding stochastically perturbed systems. Simulation results show the effectiveness of the method and the accuracy of the statements proved in the paper.

Chaos suppression in a fractional order financial system using intelligent regrouping PSO based fractional fuzzy control policy in the presence of fractional Gaussian noise

“…Some authors have looked at noise induced chaos [41] or stochastic chaos [42]. Others like Lalley [43] have looked at ways in which the noise can be filtered to obtain only the chaotic dynamics.…”

“…(6) and (7), γ is the order of the differintegration and (2N + 1) is the order of the filter. The present study considers a fifth-order Oustaloup's rational approximation [42] for the FO elements within the frequency range ω ∈ {10 −2 , 10 2 } rad/time-units.…”

Passivity-Sliding Mode Control of Uncertain Chaotic Systems with Stochastic Disturbances

“…It has many attractive features such as fast response, good transient response, and insensitivity to variations. Some related results have been presented [16,17]. In [16], control of stochastic chaos via sliding mode control is investigated.…”

“…Some related results have been presented [16,17]. In [16], control of stochastic chaos via sliding mode control is investigated. In [17], chaos synchronization of nonlinear gyros with stochastic excitation is considered by using sliding mode control.…”

Two Dimensional Sliding Mode Control

“…At a glance in the literature, it is understood that there are many works in the field of SMC for 1-D continuous and discrete time systems. (see Utkin, 1977;Asada & Slotine, 1986;Hung et al, 1993;DeCarlo et al, 1988;Wu and Gao, 2008;Furuta, 1990;Gao et al, 1995;Wu & Juang, 2008;Lai et al, 2006;Young et al, 1999;Furuta & Pan, 2000;Proca et al, 2003;Choa et al, 2007;Li & Wikander, 2004;Hsiao et al, 2008;Salarieh & Alasty, 2008) Furthermore SMC has been contributed to various control methods (see for example Hsiao et al, 2008;Salarieh & Alasty, 2008) and several experimental works (Proca et al, 2003). Recently, a SMC design for a 2-D system in RM model has been presented (Wu & Gao, 2008) in which the idea of a 1-D quasi-sliding mode (Gao et al, 1995) has been extended for the 2-D system.…”