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- Glad, Torkel, 1947-, et al.
(författare)
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Mathematical Analysis of the Tippe Top
- 2005
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Ingår i: Regular and Chaotic Dynamics. - : Maik Nauka/Interperiodica. - 1560-3547. ; 10:4, s. 333-362
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Tidskriftsartikel (refereegranskat)abstract
- A rigorous, and possibly complete analysis of the phase space picture of the tippe top solutions for all initial conditions when the top does not jump and all relations between parameters α and γ, is for the first time presented here. It is based on the use the Jellett's integral of motion λ and the analysis of the energy function. Theorems about stability and attractivity of the asymptotic manifold are proved in detail. Lyapunov stability of (periodic) asymptotic solutions with respect to arbitrary perturbations is shown.
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- Sköldstam, Markus, 1975-
(författare)
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Analysis of the phase space, asymptotic behavior and stability for heavy symmetric top and tippe top
- 2004
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this thesis we analyze the phase of the heavy symmetric top and the tippe top. These tops are two examples of physical systems for which the usefulness of integrals of motion and invariant manifolds, in phase space picture analysis, can be illustratedIn the case of the heavy symmetric top, simplified proofs of stability of the vertical rotation have been perpetuated by successive textbooks during the last century. In these proofs correct perturbations of integrals of motion are missing. This may seem harmless since the deduced threshold value for stability is correct. However, perturbations of first integrals are essential in rigorous proofs of stability of motions for both tops.The tippe top is a toy that has the form of a truncated sphere equipped with a little peg. When spun fast on the spherical bottom its center of mass rises above its geometrical center and after a few seconds the top is spinning vertically on the peg. We study the tippe top through a sequence of embedded invariant manifolds to unveil the structure of the top's phase space. The last manifold, consisting of the asymptotic trajectories, is analyzed completely. We prove that trajectories in this manifold attract solutions in contact with the plane of support at all times and we give a complete description of their stability/instability properties for all admissible choices of model parameters and of the initial conditions.
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- Sköldstam, Markus, 1975, et al.
(författare)
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Modeling of Discrete Event Systems using Finite Automata With Variables
- 2007
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Ingår i: Proceedings of the 46th IEEE Conference on Decision and Control. - 0191-2216.
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Tidskriftsartikel (refereegranskat)abstract
- To get industrial acceptance of supervisory control theory, there is a need to bridge the gap between the signalbased industrial reality and the event-based supervisory control framework. This paper tries to do this by introducing a modeling formalism with automata extended with variables, guard expressions and action functions. The formalism is suitable for modeling plants and specifications in the supervisory control framework. An algorithm that transforms a set of extended automata into a set of ordinary automata with equivalent behavior, is presented. This allows the user to model complex behaviors with a compact representation, and at the same time use existing algorithms for analysis.
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| 7. |
- Sköldstam, Markus, 1975, et al.
(författare)
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Supervisory Control Applied to Automata Extended with Variables - Revised
- 2008
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- To get industrial acceptance of supervisory control theory, there is a need to bridge the gap between the signal-based industrial reality and the event-based supervisory control framework. This report tries to shorten this gap by introducing a modeling formalism with automata extended with variables, guard expressions and action functions. The formalism is suitable for modeling plants and specifications in the supervisory control framework. No restrictions are made on the sharing of variables between concurrent automata and don't care updating of shared variables is allowed. This leads to frame problems since unreachable states of subsystems can become reachable in the entire system. To define supervisory control problems in this general setting we introduce the concept of controllable languages with respect to the entire system which is a generalization of the classical definition of controllability. An algorithm that transforms supervisory control problems modeled by automata with shared variables into equivalent ordinary automata supervisory control problems, is presented. This allows the user to model complex behaviors with a compact representation, and at the same time use existing algorithms for synthesis and verification. The proposed approach has been implemented in the supervisory control tool, Supremica.
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