| For dynamic systems, it is imperative to know the behavior of the system. Several crucial characteristics such as stability, controllability and observability are usually of interest. For linear systems, these are well studied properties. However, to analyze those properties of a nonlinear system is a much harder problem. Meshing methods are usual numerical tools for analyzing the behavior of a general system. However, the required number of meshing points grows exponentially with the dimension of the system.; In this dissertation, a polynomial level-set method for advecting a semi-algebraic set is presented. This method uses the sub-level set of a polynomial function to represent sets. The problem of flowing these sets under the advection map of a dynamic system is converted to a semidefinite program, which is used to compute the coefficients of the polynomials. Only the coefficients of the polynomials need to be stored. Hence, the proposed method reduces the time-complexity of the problem since the number of coefficients required grows more slowly than the number of mesh points. The potential applications for this method include the estimation of reachable sets, the analysis of system behavior with unknown system parameters, the estimation of the domain-of-attraction, and the estimation of the attractor. |
