The topology of the space of linear systems

Denote by ¿n,m~ the set of all controllable linear systems (A,B) with m inputs and state space Rn. ¿n,m~ is an open subset of Rn×n × Rn×m and the general linear group GLn(R) acts on this class by the similarity transformations (A,B) ¿ (SAS-1, SB). The corresponding orbit space ¿n,m := ¿n,m~/GLn(R) is known to be a real analytic manifold of dimension nm, [1], [3]. We consider the problem to compute topological invariants like the Betti numbers of the orbit space ¿n,m. The obtained result is then applied to calculate some of the Betti numbers of the space Rat(n,m,p) of linear systems with m inputs, p outputs and McMillan degree n. These numbers have also been computed independently in the doctoral thesis of Delchamps [2]. This work will be part of the author's doctoral thesis at the University of Bremen.