Establishing local strong accessibility of large-scale nonlinear systems by replacing the Lie algebraic rank condition

Within a recent development of algorithms to establish local structural identifiability, local observability and local strong accessibility of nonlinear systems, it turned out that sensitivities, governed by linear time-varying dynamics, are fundamental. As to local strong accessibility of nonlinear systems, the algorithm essentially checks controllability of linearizations along trajectories of the nonlinear system. In the literature concerning local controllability of nonlinear systems on the other hand, examples are regularly presented illustrating that controllability of linearizations is a stronger property than local strong accessibility. This paper clarifies these apparently contradicting results by using an important theoretical result from the literature and several illustrative examples. These reveal that the Lie algebraic rank condition (LARC), that is currently used to check local strong accessibility, may indeed be replaced by a rank condition based on sensitivities (SERC), that essentially checks controllability of linearizations along trajectories, provided that these trajectories are taken to be non-singular. This replacement is important for two reasons. One is that the computation of LARC requires a finite, but a-priori unknown number of steps, which may be very large. The other is especially important for large-scale nonlinear systems, for which the large number of symbolic differentiations involved in LARC results in excessive computation times or even renders the calculation infeasible. Both phenomena are also illustrated with examples.