Study On Exact Order Reduction Approaches For Multidimensional Roesser State-space Model

One of the fundamental problems of multidimensional(n-D) systems is the establishment of a state-space model, e.g., the Roesser state-space model, for a given transfer function matrix of an n-D system which is known as the state-space realization of n-D systems and algebraically equivalent to the linear fractional representation(LFR) modeling problem of uncertain systems. However, statespace realization or LFR modeling often results in models with a high order,which increases the difficulty of the subsequent theoretical analysis or practical implementation, in particular when further interconnected in more complex systems. Therefore it is desirable to reduce the order of a given state space model with higher order. The main objective of this paper to study the problem of exact order reduction and two order reduction approaches will be proposed, e.g. the order reduction approach based on Jordan transformation, the order reduction approach based on eigenvalues.In the order reduction approach based on Jordan transformation, the paper proposes a novel elementary operation approach to order reduction for the Roesser state-space model of multidimensional(n-D) systems by introducing a new kind of transformation, i.e., the Jordan transformation, which guarantees the establishment of an objective matrix with more general structure than the existing one. Then two basic order reduction techniques are developed which can overcome the difficulty encountered by the existing methods and reveal, for the first time, the fact that the order reduction is still possible even when the column(or row) blocks in the related n-D polynomial matrix are full rank. Furthermore,based on the Jordan transformation, an equivalence relationship between two Roesser models after using the elementary operations among the different blocks will be clarified. Although these operations do not directly lower the total order of the model, the partial orders can be changed so that it may nevertheless yield a possibility for further order reduction. It turns out that this new approach includes our previous elementary operation order reduction approach just as a special case.In order to overcome the shortcomings of order reduction approach based on Jordan transformation, the paper presents the other order reduction approach based on eigenvalues. This approach is able to reduce the order of a given Roesser model directly without Jordan transformation. In addition, this approach also set up a system equivalence relation, by the equivalence relation we can reduce the order of a given Roesser model which cannot be reduced directly.