Dissipative Decomposition and Feedback Stabilization of Nonlinear Control Systems

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Hudon, Nicolas
Nonlinear Control , Nonlinear Systems
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.
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