Observability of nonlinear differential algebraic systems

Local observability of nonlinear differential-algebraic equations (DAEs) from the linearization along a trajectory

“…We indicated by an example the importance of these results in overcoming the inadequacy of time-invariant linearizations for determining observability of nonlinear DAEs. Our sufficient conditions for smooth observability are verifiable and are strong enough to guarantee that the full system observability Jacobian satisfies the smooth observability condition in [13]. The results of this note can provide a basis for the future development of algorithms for determining observability of nonlinear DAE systems.…”

“…For example, the analysis in [13] shows that the hypotheses in Theorem 2 will hold for Hessenberg DAE systems supplied with an effective output function, that is, an output for which O j; k in (11) is 1-full with respect to x on I .…”

“…Thus, these results can be the basis for algorithms for determining observability of nonlinear DAEs. These new results also draw a nice connection between the results of [5] and [13]. Specifically, we consider the following question: If the observability Jacobian of the linearization along a trajectory satisfies the observability condition of [5], is the nonlinear system locally observable in a neighborhood of the trajectory; in particular, does the Jacobian of the nonlinear system satisfy the generalized observability condition of [13]?…”

“…Theorem 1 [13]: Given equations (5), suppose A1)-A4) hold in a set W (which is open in R (j+2)n+1 and) containing a given extended trajectory p(t) 1…”

“…This condition generalized classical matrix rank criteria for observability of nonsingular systems. The condition was extended in [13] to give an observability test for systems described by a general class of nonlinear DAEs.…”