Global structure of families of multivariable linear systems with an application to identification

The identification problem for linear stochastic systems may be stated roughly as follows: given observations on two stochastic processes which are the input and output of some unknown linear system, determine some estimate of the parameters of the system. A set of candidate linear systems which contains the “true” system is introduced, and probabilistic assumptions on the two stochastic processes turn the identification problem into the deterministic problem of minimizing some objective function over this candidate model set. If this set is a manifold, the existence of globally convergent identification algorithms hinges on the critical point behavior of the objective functions which it carries. By way of Morse Theory, the critical point behavior of objective functions on a manifold has implications with regard to the topology of the manifold. This paper analyzes the topology and critical point behavior of objective functions on a specific manifold of linear systems which appears frequently as the candidate model set in identification problems. This manifold is the set Σ of allm-input,p-output linear systems of fixed McMillan degree with real or complex coefficients. Over this manifold sits the principal bundle (tilde Sigma) of minimal realizations of systems in Σ It is shown that there exist three natural analytic metrics on the associated vector bundle. It is also shown that, in the real case, the first Stiefel-Whitney class of the bundle (tilde Sigma) has min(m, p)-1 nonvanishing powers; the same conclusion is drawn about the first Chern class of (tilde Sigma) in the complex case. These results, which follow from Morse Theory and some elementary homotopy and homology theory, imply that the category of the bundle (tilde Sigma) is at least min(m, p), and hence that the Lusternik-Schnirelmann category of Σ is at least min(m, p). It follows that canonical forms (i.e. sections of (tilde Sigma) ) may exist only when min(m, p) = 1 and that any objective function on Σ with compact sublevel sets has at least min(m, p) critical points. In particular, there exist on Σ no globally convergent gradient algorithms when min(m, p) > 1.