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This dissertation considers the problem of approximate dissipative potentials construction and their use in smooth feedback stabilization of nonlinear control systems. For mechanical systems, dissipative potentials, usually a generalized Hamiltonian function, can be derived from physical intuition. When a dissipative Hamiltonian is not available, one can rely on dissipative Hamiltonian realization techniques, as proposed recently by Cheng and coworkers. Extensive results are available in the literature for (robust) stabilization based on the obtained potential. For systems of interest in chemical engineering, especially systems with mass action kinetics, energy is often ill-defined. Moreover, realization techniques are difficult to apply, due to the nonlinearities associated with the reaction terms. Approximate dissipative realization techniques have been considered by many researchers for analysis and feedback design of controllers in the context of chemical processes. The objective of this thesis is to study the construction of local dissipative potentials and their application to solve stabilization problems. The present work employs the geometric stabilization approach proposed by Jurdjevic and Quinn, refined by Faubourg and Pomet, and by Malisoff and Mazenc, for the design of stabilizing feedback laws. This thesis seeks to extend and apply the Jurdjevic--Quinn stabilization method to nonlinear stabilization problems, assuming no a priori knowledge of a Lyapunov function. A homotopy-based local decomposition method is first employed to study the dissipative Hamiltonian realization problem, leading to the construction of locally defined dissipative potentials. If the obtained potential satisfies locally the weak Jurdjevic--Quinn conditions, it is then shown how to construct feedback controllers using that potential, and under what conditions a Lyapunov function can be constructed locally for time-independent control affine systems. The proposed technique is then used for the construction of state feedback regulators and for the stabilization of periodic orbits based on a construction proposed by Bacciotti and Mazzi. In the last chapter of the thesis, stabilization of time-dependent control affine systems is considered, and the main result is used for the stabilization of periodic solutions using asymptotic feedback tracking. Low-dimensional examples are used throughout the thesis to illustrate the proposed techniques and results. / Thesis (Ph.D, Chemical Engineering) -- Queen's University, 2010-06-17 10:13:42.201

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Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2007. / Includes bibliographical references (leaves 153-177). Also available in electronic version.

Dissertation submitted in compliance with the requirements for Masters Degree in Technology in Electrical Engineering (Light Current) at Technikon Natal, 1998. / Designing compensators in the frequency domain is a complicated problem even for linear systems that have dead time. The situation is far more difficult if the system is also nonlinear. This study introduces a new method for the design of compensators for time-invariant, nonlinear systems that have dead time. The method is based on an optimization approach and utilizes large signal linearization methodology. / M

It is found here that the state of a network of coupled ordinary differential equations is partially localizable through a pair of contractive ideal subspaces, chosen from dual complete lattices related to the synchrony and synchronization of cells within the network. The first lattice is comprised of polydiagonal subspaces, corresponding to synchronous activity patterns that arise from functional equivalences of cell receptive fields. This lattice is dual to a transdiagonal subspace lattice ordering subspaces transverse to these network-compatible synchronies. Combinatorial consideration of contracting polydiagonal and transdiagonal subspace pairs yields a rich array of dynamical possibilities for structured networks. After proving that contraction commutes with the lattice ordering, it is shown that subpopulations of cells are left at fixed potentials when pairs of contracting subspaces span the cells' local coordinates - a phenomenon named glyph formation here. Treatment of mappings between paired states then leads to a theory of network-compatible sequence generation. The theory's utility is illustrated with examples ranging from the construction of a minimal circuit for encoding a simple phoneme to a model of the primary visual cortex including high-dimensional environmental inputs, laminar speficicity, spiking discontinuities, and time delays. In this model, glyph formation and dissolution provide one account for an unexplained anomaly in electroencephalographic recordings under periodic flicker, where stimulus frequencies differing by as little as 1 Hz generate responses varying by an order of magnitude in alpha-band spectral power. Further links between coupled-cell systems and neural dynamics are drawn through a review of synchronization in the brain and its relationship to aggregate observables, focusing again on electroencephalography. Given previous theoretical work relating the geometry of visual hallucinations to symmetries in visual cortex, periodic perturbation of the visual system along a putative symmetry axis is hypothesized to lead to a greater concentration of harmonic spectral energy than asymmetric perturbations; preliminary experimental evidence affirms this hypothesis. To conclude, connections drawn between dynamics, sensation, and behavior are distilled to seven hypotheses, and the potential medical uses of the theory are illustrated with a lattice depiction of ketamine xylazine anaesthesia and a reinterpretation of hemifield neglect.

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The drill-strings used in drilling operate under extreme condi-tions, therefore, an accurate understanding of their dynamics is necessary and has attracted much interest. Although a bottom hole assembly (BHA) is to a great ex- tent responsible for the dynamics of the system, the in uence of the drill-pipes has been increasingly neglected by current models. Their dynamics and geometrical behaviour should be better analysed for a deeper understanding of underlying phe- nomena. For example, under stick-slip oscillations, the torque on the drill-string may cause torsional buckling of the drill-pipes, incurring in helical con guration, in which the apparent length is reduced, a ecting the forces at the bit{rock interface. With such behaviour and interactions in mind, this work focuses on elaborating a comprehensive mathematical model to investigate the dynamics of drill-strings, with attention to the drill-pipes section. Firstly, lower dimensional models are used to analyse the stick-slip limit cycle and its limits of existence. Then, a model developed for MEMS is used as a base for a comprehensive model using the formu- lation of Cosserat rods. Relevant boundary conditions are applied and a numerical simulation procedure is established. Simulations are performed for a range of sce- narios under stick-slip occurrence, and the behaviour of the drill-pipes is analysed. Focus is then given to axial vibrations with bit-bounce and the in uence on stick- slip, later to lateral vibrations with whirling motion of the drill-pipes, and nally to helical con gurations, taken by the drill-string under combined torsional, axial and lateral loads, showing the consequent shortening of the drill-string.