Pole Assignment for Symmetric Quadratic Dynamical Systems: An Algorithmic Method

In this article an algorithmic method is proposed for the solution of the pole assignment problem which is associated with a symmetric quadratic dynamical system, when it is completely controllable. The problem is shown to be equivalent to two subproblems, one linear and the other multi-linear. Solutions of the linear problem must be decomposable vectors, i.e. they must lie in an appropriate Grassmann variety. The proposed method computes a reduced set of quadratic Plucker relations, with only three terms each, which describe completely the specific Grassmann variety. Using these relations one can solve the multi-linear problem and consequently calculate the feedback matrices which give a solution to the pole assignment problem. An illustrative example of the proposed algorithmic procedure is given. The main advantage of our approach is that the complete set of feedback solutions is obtained, over which further optimisation can be carried out, if desired. This is important for problems with structural constraints (e.g. decentralization) or norm-constraints on the feedback gain-matrix.