Representation of a Class of Polynomial MIMO systems via Positive Realizations

In several technological frameworks only positive state space realizations of signal processing algorithms (filters or control laws) can be implemented. On the other hand, the imposition of an a priori positivity constraint on the processing algorithm is a too strong design limitation. For this reason, many authors studied the problem of state-space realization of generic stationary linear filters through an Internally Positive Realization (IPR), such as a combination of positive filters. The IPR problem for discrete-time single-input/single-output (SISO) linear systems has been widely investigated, and important results are available in the literature. Recently, theoretical contributions to the IPR problem for multi-input/multi-output (MIMO) linear systems case have also appeared. The IPR of nonlinear systems has been never investigated. In this paper the IPR problem of polynomial MIMO systems and filters is formulated and a straightforward method, based on the Kronecker algebra, for the construction of IPR's is proposed. The local stability properties of the resulting positive realization are also investigated. The importance of this work lies in the fact that the behavior of any nonlinear system can be accurately approximated through polynomial systems.