Representations and realizations of singular systems: the tortoise and the hare revisited

The notion of system description or representation is reconciled with system realization or implementation. The latter is defined as a causal construct for obtaining the solution. The realization starts from a particular description, where dynamical and algebraic equations are separated, and the dynamical states correspond to integrator outputs. Clearly such a description is nonunique. The optimal realization problem is then to find the particular realization in the orbit of a particular group H, which minimizes the sensitivity measure introduced by W.S. Gray and E.I. Verriest (Proc. of 28th IEEE Conf. on Decision and Control, Tampa, FL, Dec. 1989). This group H is a subgroup, leaving the (separation) structure invariant, of the group characterizing the orbits under restricted system equivalence. A practical implementation is one for which asymptotically the solution is obtained with a causal regular realization. The asymptotics are obtained via the use of nonstandard analysis methods. The behavior of the regularizing parameter is determined by the structure at infinity of the original singular system. As examples, a differentiator and a purely algebraic set of equations are discussed