Realization algorithms and approximation methods of bilinear systems

High-dimensional mathematical models of bilinear control systems are often not amenable due to the difficulty in implementation. In this paper, we address the problem of order-reduction for both discrete and continuous time bilinear systems. Two model-reduction algorithms are presented; one is based on the singular value decomposition of the generalized Hankel matrix (the Hankel Approach) and the other is based on the eigenvalue / eigenvector decomposition of the product of reachability and observability Gramians (the Gramian Approach). Equivalence between these two algorithms is established. The main result of this paper is a systematic approach for obtaining reduced-order bilinear models. Furthermore, the bilinear reachabiiity and observability Gramians are shown to be obtainable from the solutions of generalized Lyapunov equations. Computer simulations of a neutron-kinetic system are presented to illustrate the effectiveness of the proposed model-reduction algorithms.