  Existing Nussbaum function based methods on the consensus of multi-agent
systems require (partial) identical unknown control directions of all agents
and cause dangerous dramatic control shocks. This paper develops a novel
saturated Nussbaum function to relax such limitations and proposes a Nussbaum
function based control scheme for the consensus problem of multi-agent systems
with arbitrary non-identical unknown control directions and safe control
progress. First, a novel type of the Nussbaum function with different
frequencies is proposed in the form of saturated time-elongation functions,
which provides a more smooth and safer transient performance of the control
progress. Furthermore, the novel Nussbaum function is employed to design
distributed adaptive control algorithms for linearly parameterized multi-agent
systems to achieve average consensus cooperatively without dramatic control
shocks. Then, under the undirected connected communication topology, all the
signals of the closed-loop systems are proved to be bounded and asymptotically
convergent. Finally, two comparative numerical simulation examples are carried
out to verify the effectiveness and the superiority of the proposed approach
with smaller control shock amplitudes than traditional Nussbaum methods.
