  A solution is provided in this note for the adaptive consensus problem of
nonlinear multi-agent systems with unknown and non-identical control directions
assuming a strongly connected underlying graph topology. This is achieved with
the introduction of a novel variable transformation called PI consensus error
transformation. %that allows for decoupled consensus control design of
multi-agent systems. The new variables include the position error of each agent
from some arbitrary fixed point along with an integral term of the weighted
total displacement of the agent's position from all neighbor positions. It is
proven that if these new variables are bounded and regulated to zero, then
asymptotic consensus among all agents is ensured. The important feature of this
transformation is that it provides input decoupling in the dynamics of the new
error variables making the consensus control design a simple and direct task.
Using classical Nussbaum gain based techniques, distributed controllers are
designed to regulate the PI consensus error variables to zero and ultimately
solve the agreement problem. The proposed approach also allows for a specific
calculation of the final consensus point based on the controller parameter
selection and the associated graph topology. Simulation results verify our
theoretical derivations.
