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Set arithmetic and the enclosing problem in dynamics

100%
Mrozek M., Zgliczyński P.
Annales Polonici Mathematici
|
2000
|
tom 74
|
nr 1
237-259
EN
We study the enclosing problem for discrete and continuous dynamical systems in the context of computer assisted proofs. We review and compare the existing methods and emphasize the importance of developing a suitable set arithmetic for efficient algorithms solving the enclosing problem.
2
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On Computer-Assisted Proving The Existence Of Periodic And Bounded Orbits

100%
Srzednicki R.
Annales Mathematicae Silesianae
|
2015
|
tom 29
|
nr 1
7-17
EN
We announce a new result on determining the Conley index of the Poincaré map for a time-periodic non-autonomous ordinary differential equation. The index is computed using some singular cycles related to an index pair of a small-step discretization of the equation. We indicate how the result can be applied to computer-assisted proofs of the existence of bounded and periodic solutions. We provide also some comments on computer-assisted proving in dynamics.
3
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Verification techniques for sensitivity analysis and design of controllers for nonlinear dynamic systems with uncertainties

88%
Rauh A., Minisini J., Hofer E.
International Journal of Applied Mathematics and Computer Science
|
2009
|
tom 19
|
nr 3
425-439
EN
Control strategies for nonlinear dynamical systems often make use of special system properties, which are, for example, differential flatness or exact input-output as well as input-to-state linearizability. However, approaches using these properties are unavoidably limited to specific classes of mathematical models. To generalize design procedures and to account for parameter uncertainties as well as modeling errors, an interval arithmetic approach for verified simulation of continuoustime dynamical system models is extended. These extensions are the synthesis, sensitivity analysis, and optimization of open-loop and closed-loop controllers. In addition to the calculation of guaranteed enclosures of the sets of all reachable states, interval arithmetic routines have been developed which verify the controllability and observability of the states of uncertain dynamic systems. Furthermore, they assure asymptotic stability of controlled systems for all possible operating conditions. Based on these results, techniques for trajectory planning can be developed which determine reference signals for linear and nonlinear controllers. For that purpose, limitations of the control variables are taken into account as further constraints. Due to the use of interval techniques, issues of the functionality, robustness, and safety of dynamic systems can be treated in a unified design approach. The presented algorithms are demonstrated for a nonlinear uncertain model of biological wastewater treatment plants.
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