  This paper considers a leader-following problem for a group of heterogeneous
linear time invariant (LTI) followers that are interacting over a directed
acyclic graph. Only a subset of the followers has access to the state of the
leader in specific sampling times. The dynamics of the leader that generates
its sampled states is unknown to the followers. For interaction topologies in
which the leader is a global sink in the graph, we propose a distributed
algorithm that allows the followers to arrive at the sampled state of the
leader by the time the next sample arrives. Our algorithm is a practical
solution for a leader-following problem when there is no information available
about the state of the leader except its instantaneous value at the sampling
times. Our algorithm also allows the followers to track the sampled state of
the leader with a locally chosen offset that can be time-varying. When the
followers are mobile agents whose state or part of their state is their
position vector, the offset mechanism can be used to enable the followers to
form a transnational invariant formation about the sampled state of the leader.
We prove that the control input of the followers to take them from one sampled
state to the next one is minimum energy. We also show in case of the
homogeneous followers, after the first sampling epoch the states and inputs of
all the followers are synchronized with each other. Numerical examples
demonstrate our results.
