It has been noted that there are numerous similarities between the behaviour of chaotic and stochastic systems. The theoretical links between chaotic and stochastic systems are investigated based on the evolution of the density of dynamics and an equivalency relationship based on the invariant measure of an ergodic system. It is shown that for simple chaotic systems an equivalent stochastic model can be analytically derived when the initial position in state space is only known to a limited precision. Based on this a new methodology for the modelling of complex nonlinear time series displaying chaotic behaviour with stochastic models is proposed. This consists of using a stochastic model to learn the evolution of the density of the dynamics of the chaotic system by estimating initial and transitional density functions directly from a time series. A number of models utilising this methodology are proposed, based on Markov chains and hidden Markov models. These are implemented and their performance and characteristics compared using computer simulation with several standard techniques.

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A new theory is proposed for the full-information finite and infinite horizontime robust H∞ control that is equivalently effective for the regulation and/or tracking problems of the general class of time-varying nonlinear systems under the presence of exogenous disturbance inputs. The theory employs the sequence of linear-quadratic and time-varying approximations, that were recently introduced in the optimal control framework, to transform the nonlinear H∞ control problem into a sequence of linearquadratic robust H∞ control problems by using well-known results from the existing Riccati-based theory of the maturing classical linear robust control. The proposed method, as in the optimal control case, requires solving an approximating sequence of Riccati equations (ASRE), to find linear time-varying feedback controllers for such disturbed nonlinear systems while employing classical methods. Under very mild conditions of local Lipschitz continuity, these iterative sequences of solutions are known to converge to the unique viscosity solution of the Hamilton-lacobi-Bellman partial differential equation of the original nonlinear optimal control problem in the weak form (Cimen, 2003); and should hold for the robust control problems herein. The theory is analytically illustrated by directly applying it to some sophisticated nonlinear dynamical models of practical real-world applications. Under a r -iteration sense, such a theory gives the control engineer and designer more transparent control requirements to be incorporated a priori to fine-tune between robustness and optimality needs. It is believed, however, that the automatic state-regulation robust ASRE feedback control systems and techniques provided in this thesis yield very effective control actions in theory, in view of its computational simplicity and its validation by means of classical numerical techniques, and can straightforwardly be implemented in practice as the feedback controller is constrained to be linear with respect to its inputs.