Finite Dimensional Linear Stochastic System Identification

Abstract

The use of state space systems to represent dynamical systems of various types has a long history in mathematics and physics. Two fundamental examples are the well-known methods for analysing high order ordinary differential equations via the corresponding first order equations and the formulation of Hamiltonian mechanics—within which classical celestial mechanics forms a magnificent special case. Equally, in theoretical engineering, including the information and control sciences, state space methods have played and continue to play a fundamental rôle. From a mathematical viewpoint, the pure state evolution models of dynamical system theory undergo an enormous generalization in control theory to input-output systems, which then fall into equivalence classes according to their input-output behavior. Furthermore, from an engineering viewpoint, a vast array of problems are either initially posed in state space terms (aerospace engineering provides many examples) or are first presented in terms of input-output models which are then transformed into input-stateoutput form (process control being one source of examples). As a result, one of the basic subjects of study in mathematical system theory is the relationship between input-state-output systems and input–output systems. The generality of this relationship is conveyed by the Nerode Theorem which states that any non-anticipative (set-valued) input—(set valued) output system S possesses a state space realization Σ(S), that is to say, there exists an input-state-output system Σ(S) which generates the same input-output trajectories as S.