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Research Express@NCKU Volume 25 Issue 5 - December 6, 2013 [ http://research.ncku.edu.tw/re/articles/e/20131206/2.html ]
Polynomial level-set method for attractor estimation
Ta-Chung Wang
1,*, Sanjay Lall
2, Matthew West
31 Institute of Civil Aviation, National Cheng Kung University 2 Department of Aeronautics and Astronautics, Stanford University
3 Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign tachung@mail.ncku.edu.tw
Journal of the Franklin Institute Volume 349, Issue 9, November 2012, Pages 2783–2798
W
e studied the problem of estimating the attractor, which is an invariant set of a dynamic system. Any initial system state starting in the domain of attractor of an attractor of an invariant set will eventually move inside the attractor. Therefore, system stability can be analyzed using attractors. Attractors could also be used for designing secure, private multiuser digital communication systems, provided that a clear knowledge of the attractor is available.Using invariant nature of attractors, estimation of an attractor can be carried out by flowing an initial set forward in time and observing how this initial set evolves. In this study, we present an advection algorithm for estimating the attractor. We use 0-sub-level set of polynomials to represent a set of system states. Semi-definite programs are solved to iteratively estimating the variation of the initial set. The method is proceeded as the following.
Given an autonomous system and a polynomial p(x), whose 0-sub-level set is an invariant set located in one attractor of the autonomous system, we would like to find a polynomial q(x), whose 0-sub-level set contains the points that were located in the 0-sub-level set of p(x) at previous time instant. Then the 0-sub-level set of q(x) are 0-sub-level set of p(x) advanced in time. Figure 1 shows the concept of the polynomial level-set method.
Figure 1.concept of the proposed level-set method
A polynomial approximation approach is used to achieve the aforementioned concept. Using the proposed approach, the shape of the famous Lorenz attractor is estimated using a polynomial of degree six. Figure 2 shows the result. The thick red and green curves are two simulated system trajectories, which show that the estimated set is indeed an invariant set.
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Figure 2.estimation of the Lorenz attractor
