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Implicitization, Graph Ideals and Control Systems

We discuss how the external behavior of a polynomial automatic control system can be determined, i.e. how to find the differential equation relating the input and the output of the system given a state space description, focussing on algorithmic aspects. This problem is equivalent to what is known as implicitization in computational algebraic geometry and one way of doing this is to perform elimination in so called graph ideals. We compare different methods for implicitization with regard to computational complexity. Moreover, a bound for the degree of the input-output equation in terms of the degrees of the state equations is derived.