Xu, Daolin; (1996) Flexible control of chaotic systems. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Chaos is a typical phenomenon in nonlinear dynamical systems. Until recently, the extreme sensitivity of chaotic dynamics to initial conditions has been regarded as a harmful property to be avoided in engineering design. For many years the concept that chaotic systems are neither predictable nor controllable was widely accepted. However, this opinion altered when in 1990 Ott, Grebogi and Yorke demonstrated that a chaotic system could be directed onto embedded unstable periodic orbits. Their work suggested that chaotic properties can be usefully utilised to achieve the special advantages that small controls can lead to large effects and flexible switching between many different orbits can be realised by means of control without changing the global configuration of the system. A great potential in applications of controlling chaos has emerged in electronics, lasers, chemical reactions, communications and biological systems, strongly indicating that the theory of controlling chaos can evolve into a very useful technology to solve engineering problems. The aim of this thesis is to develop improved methods of controlling chaos. To achieve this aim various different aspects have been considered, namely: to firstly overcome some deficiencies of the well-known OGY method; to enable the stabilisation of highly unstable periodic orbits (saddles or repellors); allow control in noisy environments; produce stabilisation with high accuracy; and achieve control in a global scale for systems of arbitrary dimension. Basically four new methods have been developed. Firstly a one-step optimal control method is introduced which is applicable to local scale dynamics. The method does not require knowledge of eigenvalues, eigenvectors and stable manifolds associated with the desired unstable periodic orbit. As a consequence the method is much easier to apply to Hamiltonian systems, compared to the OGY method, is also able to stabilise repellors. Secondly a multiple section control approach is developed in conjunction with the one step optimal control scheme reducing the time for the growth of errors. The method of control on multiple sections may greatly enhance the effectiveness for stabilising highly unstable orbits with large eigenvalues in very noisy environments. Thirdly, a control scheme is introduced based on the contraction mapping theorem in the form of variable feedback control. This method achieves control on a global scale, regardless of whether or not a system state is close to the desired state. The convergence is guaranteed by the contraction mapping theorem. This variable feedback scheme overcomes drawbacks in other parametric control schemes where accessible control parameters cannot be found. Finally, a self- locating control method is developed which can automatically find the location of a desired orbit without knowing any explicit details of the system dynamics. The control process can self-correct an approximate orbit (which is initially given) using an output sequence. This method is particularly useful when stabilisation with high accuracy is required. The robustness of each control scheme is also discussed including the effects of noise and the stability of the control algorithms. A number of numerical applications have been carried out which illustrate control in high dimensional systems, control in relatively noisy environments and stabilisation of unstable periodic orbits with large eigenvalues. The feasibility of these new approaches in both maps and flows has been confirmed.

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