Domain of stability of synchronous generators by a cell mapping approach. (English) Zbl 0565.93053
Consider the dynamic system (vector field) \(V_ 1\) on \({\mathbb{R}}^ n\). Let p be the smooth diffeomorphism \({\mathbb{R}}^ n\) onto the open bounded rectangle K. Let, further, \(V_ 2\) be the vector field on K corresponding to \(V_ 1\) under p and let \(\psi: {\mathbb{R}}\times K\to K\) be its flow. Given the finite partition P of K into smaller pairwise disjoint rectangles (cells), the authors introduce the cell mapping \(C: P\to P\). If q is the centre of the cell \(c\in P\) then C(c) is defined to be the unique cell containing \(\psi\) (T,q). Here T is a fixed positive number. The authors propose to approximate a given dynamic system \(V_ 1\) by a dynamic system C on P. In particular, they study the domains of stability of asymptotically stable singular points of \(V_ 1\) in this way. Three two-dimensional examples are considered. It is shown (by these examples) that the proposed method gives better results than obtained by the second Lyapunov method and its modifications. The accuracy of this approximation is not studied theoretically. As the authors remark, the most time-consuming part of the computational program is the stage where the conversion of the given dynamic system to the cell mapping is accomplished.
Reviewer: L.Faibusovich
MSC:
| 93D99 | Stability of control systems |
| 93C10 | Nonlinear systems in control theory |
| 93C55 | Discrete-time control/observation systems |
| 37C10 | Dynamics induced by flows and semiflows |
| 37C75 | Stability theory for smooth dynamical systems |
| 58C25 | Differentiable maps on manifolds |
| 58K99 | Theory of singularities and catastrophe theory |
| 93B40 | Computational methods in systems theory (MSC2010) |
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