  This paper considers a leader-following formation control problem for
heterogeneous, second-order, uncertain, input-affine, nonlinear multi-agent
systems modeled by a directed graph. A tunable, three-layer neural network (NN)
is proposed with an input layer, two hidden layers, and an output layer to
approximate an unknown nonlinearity. Unlike commonly used trial and error
efforts to select the number of neurons in a conventional NN, in this case an
\textit{a priori} knowledge allows one to set up the number of neurons in each
layer. The NN weights tuning laws are derived using the Lyapunov theory. The
leader-following and formation control problems are addressed by a robust
integral of the sign of the error (RISE) feedback and a NN-based control. The
RISE feedback term compensates for unknown leader dynamics and the unknown,
bounded disturbance in the agent error dynamics. The NN-based term compensates
for the unknown nonlinearity in the dynamics of multi-agent systems, and
semi-global asymptotic tracking results are rigorously proven using the
Lyapunov stability theory. The results of the paper are compared with two
previous results to evaluate the efficiency and performance of the proposed
method.
