Global analysis and synthesis of oscillations: a dissipativity approach
Stan G.B.
March 2005Abstract
The main theme of this research concerns the global (as opposed to local) analysis of stable limit cycle oscillations in dynamical systems. The global analysis of oscillations in systems and networks of interconnected systems is a longstanding problem. Dynamical systems that exhibit robust nonlinear oscillations are called oscillators. Oscillators are ubiquitous in physical, biological, biochemical, and electromechanical systems. Detailed models of oscillators abound in the literature, most frequently in the form of a set of nonlinear differential equations whose solutions robustly converge to a limit cycle oscillation. Local stability analysis of oscillators is possible by means of Floquet theory but global stability analysis is usually restricted to low-dimensional (second order) models. For these low-dimensional models, global analysis is performed by using specific planar tools (phase plane methods, Poincaré-Bendixon theorem, etc.) which do not generalize easily to high-dimensional models. As a consequence, global limit cycle analysis of models of dimension higher than two is quite hard since there currently exists no general analysis method. This lack of general analysis methods typically forces complex models of oscillators to be studied through numerical simulation methods. Although numerical simulations of these models may give an insight into their behaviour, a more in-depth understanding is generally impeded by the complexity of the models and the challenge of their rigorous global stability analysis. Moreover, even in the case of low-dimensional models, the planar methods used for their (global) analysis do not generalize to the analysis of their interconnection into a network. These considerations show the need for developing dimension independent methods that allow the global analysis of oscillators, either isolated or in interconnection.
In particular, this thesis shows that dissipativity theory introduced by Willems can be extended to allow (global) stability analysis of limit cycles in Lure-type models of oscillators and networks of oscillators. These Lure-type models of oscillators have been named passive oscillators.
From the global analysis point of view, the main results of this research deal with the implications of this extended dissipativity theory for
(1) the global stability analysis of limit cycle oscillations in isolated passive oscillators; (2) the global stability analysis of limit cycle oscillations in networks of passive oscillators; (3) the global analysis of synchrone limit cycle oscillations in networks of identical passive oscillators.
From the synthesis point of view, we show that the structure of passive oscillators suggests a method for the design of a nonlinear parametric proportional-integral controller aimed at the generation of global limit cycle oscillations in stabilizable nonlinear systems.
BibTex Entry
- @PhdThesis{,
- author = {Stan G.B.},
- school = {University of Liege},
- title = {Global analysis and synthesis of oscillations: a dissipativity approach},
- year = {2005},
- month = {March}
- }
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