Stability, exponential

The above remark is of the utmost importance for evaluating the potential of the proposed observer in a real setup. In fact, exponential stability would ensure robustness of the state estimation against bounded and/or vanishing model uncertainties and disturbances [35], due to inaccurate and/or incomplete knowledge of reaction kinetics and to usual simplifying assumptions adopted for the model derivation (e.g., perfect mixing). 

Remarks 5.1 and 5.2 on the exponential stability of the estimation error dynamics can be extended to the overall controller-observer scheme as well. Hence, robustness with respect to effects due to modeling uncertainties and/or disturbances is guaranteed. Moreover, the following remarks can be stated. 

It can be concluded that the exponential stability property confers to the adaptive model-based scheme a satisfactory degree of robustness. Therefore, even in the presence of large model uncertainties, its performance is comparable with or better than that of model-free approaches. 

Tikhonov s theorem (Theorem 2.1) indicates a further requirement that must be fulfilled by the controllers in the fast time scale in order for the time-scale decomposition developed above to remain valid, these controllers must ensure the exponential stability of the fast dynamics. From a practical point of view, this is an intuitive requirement one cannot expect stability and control performance at the process level if the operation of the process units is not stable. 

Abstract These lectures are devoted to the main results of classical perturbation theory. We start by recalling the methods of Hamiltonian dynamics, the problem of small divisors, the series of Lindstedt and the method of normal form. Then we discuss the theorem of Kolmogorov with an application to the Sun-Jupiter-Saturn problem in Celestial Mechanics. Finally we discuss the problem of long-time stability, by discussing the concept of exponential stability as introduced by Moser and Littlewood and fully exploited by Nekhoroshev. The phenomenon of superexponential stability is also recalled. 

The rest of this section is devoted to the discussion of the theory of complete stability, exponential stability and superexponential stability. 

In its simplest formulation, exponential stability is the result of making quantitative the heuristic estimates above. Essentially, the problem is to evaluate the r-dependence of the constant Br in (25). In this form the result has been first stated by Moser (1955) and Littlewood (1959a and 1959b). 

Giorgilli, A. and Zehnder, E. (1992). Exponential stability for time dependent potentials. ZAMP, 43(5) 827-855. 

Giorgilli, A. (2003). Notes on exponential stability of Hamiltonian systems. Pubbli-cazioni della Classe di Scienze, Scuola Normale Superiore, Pisa, Centro di Ricerca Matematica Ennio De Giorgi . 

In quasi-integrable systems we do not find only KAM tori, but also resonant motions, and among resonant motions we find the chaotic ones. If e is small and h satisfies a suitable geometric condition (convexity of h is sufficient) the Nekhoroshev theorem proves the exponential stability of the actions for all initial conditions, including the resonant ones. More precisely, there exist positive constants eo, a,b,Io,to such that if < so, for any (I(0),ip(0)) e B x T it is I(t) — /(0) < Io a for any time t satisfying the exponential estimate ... 

Canudas de Wit, C., Sordalen, 0. J. (1991). Exponential stabilization of mobile robots with nonholomic constraints. In Proceedings of the IEEE Conference on Decision and Control. Brighton, UK IEEE Press. 

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