  This paper introduces a novel recursive distributed estimation algorithm
aimed at synthesizing input and state interval observers for nonlinear
bounded-error discrete-time multi-agent systems. The considered systems have
sensors and actuators that are susceptible to unknown or adversarial inputs. To
solve this problem, we first identify conditions that allow agents to obtain
nonlinear bounded-error equations characterizing the input. Then, we propose a
distributed interval-valued observer that is guaranteed to contain the
disturbance and system states. To do this, we first detail a gain design
procedure that uses global problem data to minimize an upper bound on the
$\ell_1$ norm of the observer error. We then propose a gain design approach
that does not require global information, using only values that are local to
each agent. The second method improves on the computational tractability of the
first, at the expense of some added conservatism. Further, we discuss some
possible ways of extending the results to a broader class of systems. We
conclude by demonstrating our observer on two examples. The first is a unicycle
system, for which we apply the first gain design method. The second is a
145-bus power system, which showcases the benefits of the second method, due to
the first approach being intractable for systems with high dimensional state
spaces.
