  Controllability is a fundamental property of control systems, serving as the
prerequisite for controller design. While controllability test is well
established in modelic (i.e., model-driven) control systems, extending it to
datatic (i.e., data-driven) control systems is still a challenging task due to
the absence of system models. In this study, we propose a general
controllability test method for nonlinear systems with datatic description,
where the system behaviors are merely described by data. In this situation, the
state transition information of a dynamic system is available only at a limited
number of data points, leaving the behaviors beyond these points unknown.
Different from traditional exact controllability, we introduce a new concept
called $\epsilon$-controllability, which extends the definition from
point-to-point form to point-to-region form. Accordingly, our focus shifts to
checking whether the system state can be steered to a closed state ball
centered on the target state, rather than exactly at that target state. On its
basis, we propose a tree search algorithm called maximum expansion of
controllable subset (MECS) to identify controllable states in the dataset.
Starting with a specific target state, our algorithm can iteratively propagate
controllability from a known state ball to a new one. This iterative process
gradually enlarges the $\epsilon$-controllable subset by incorporating new
controllable balls until all $\epsilon$-controllable states are searched.
Besides, a simplified version of MECS is proposed by solving a special shortest
path problem, called Floyd expansion with radius fixed (FERF). FERF maintains a
fixed radius of all controllable balls based on a mutual controllability
assumption of neighboring states. The effectiveness of our method is validated
in three datatic control systems whose dynamic behaviors are described by
sampled data.
