Nearest stable system
(2013)
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- Abstract
- (en) Stability is a universal concept which we experience in our everyday lives. It plays a central role in the study of dynamical systems and is still the subject of intensive research by the control community. The question of finding the nearest stable system appears in system identification where one sometimes faces unstable models created from the perturbed data of a stable system. In many cases, solving the problem in the spectral space does not make sense due to the sensitivity of the poles of the system to perturbations. We thus investigate algorithms for finding a nearest stable system in the space of coefficients. Very few existing methods are available to solve the problem. This is due to the nonsmooth, nonconvex nature of the set of stable systems. Moreover, the methods that can solve the problem are not always able to actually guarantee that the solution will be stable. Our work contains the original description of two methods for matrices and two methods for polynomials which substantially expand the number of optimization methods available to the user for the resolution of the nearest stable system. These methods cover all the possible variations of the problem : for polynomials or for matrices, in continuous-time or in discrete-time, for real or complex data. We prove the convergence of the methods to a stationary point of the formulation. The various approaches proposed in the thesis will provide new tools and inspiration for the resolution of closely related problems in a field in constant evolution and full of challenges.
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Citations
Orban de Xivry, F.-X. (2013). Nearest stable system. https://hdl.handle.net/2078.5/203655