Empirical observability Gramian rank condition for weak observability of nonlinear systems with control

We provide a rank condition for weak observability by defining a modified empirical observability Gramian that explicitly incorporates system control. Based on this rank condition, we derive an upper bound on the local unobservability index of the system that guarantees weak observability. We also show a connection between the Fisher information matrix and the control explicit empirical observability Gramian. The utility of the control explicit Gramian is demonstrated in examples.