Observability and external description of linear time-varying singular control systems

We define total and smooth observability for linear time varying singular control systems, {dollar}E(t)xspprime{dollar} + {dollar}F(t)x{dollar} = {dollar}B(t)u{dollar} {dollar}y{dollar} = {dollar}C(t)x{dollar}. The dynamics {dollar}Exspprime{dollar} + {dollar}Fx{dollar} = {dollar}Bu{dollar} are a solvable singular differential equation. We characterize these observability properties by rank and column independence conditions on derivative arrays generated from the original system coefficients. Our approach has computational advantages even in the classical nonsingular case. Smooth observability is stronger than total observability, but weaker than uniform observability when {dollar}E{dollar} is nonsingular. For singular systems with analytic coefficients, total observability is equivalent to smooth observability. We use observability results to define an external description which characterizes the input-output behavior of a system. For nonobservable singular systems of any index, we develop an observable-unobservable decomposition from the output-nulling space of the unforced system. Two components describe the space of observable states complementary to the unobservable output-nulling space. Both output and input information determine one component, and the complete determination of the other components by the input is a consequence of singular system structure. We define the three system components by projection operators associated with a natural completion of the system's singular differential equation. Ordinary differential equations (ODE's) describe the dynamics of the three state components in {dollar}Rsp{lcub}n{rcub}{dollar} for a given input and appropriate initial conditions. The system's derivative array information permits pointwise computation of the output-nulling space, natural completion, and associated projectors for a large class of systems. Thus, we generate a transformation to a local canonical form with respect to observability.; Finally, we identify an "index one" restriction implied by the constant rank assumptions of recent nonsingular state realization theory based on geometric control concepts. This enables us to illustrate how solvable singular systems theory may provide a more computationally feasible path to state realization for higher index problems.