Ensemble controllability of time-invariant linear systems

In this paper, we study the control of an ensemble of structurally similar time-invariant linear systems. In particular, we derive explicit necessary and sufficient controllability conditions for such systems in terms of the rank of the system matrices. We present examples to demonstrate these rank conditions, and construct optimal controls for steering a linear ensemble system between states of interest by using an optimization-free computational method based on the singular value decomposition. This work extends our previous results in ensemble control of time-varying linear systems, where the established controllability conditions are implicit and are defined by the singular system of the linear operator that characterizes the system dynamics.