  In this paper, we consider the problem of safe control synthesis of general
controlled nonlinear systems in the presence of bounded additive disturbances.
Towards this aim, we first construct a governing augmented state space model
consisting of the equations of motion of the original system, the integral
control law and the nonlinear disturbance observer. Next, we propose the
concept of Disturbance Observer based Integral Control Barrier Functions
(DO-ICBFs) which we utilize to synthesize safe control inputs. The
characterization of the safe controller is obtained after modifying the
governing integral control law with an additive auxiliary control input which
is computed via the solution of a quadratic problem. In contrast to prior
methods in the relevant literature which can be unnecessarily cautious due to
their reliance on the worst case disturbance estimates, our DO-ICBF based
controller uses the available control effort frugally by leveraging the
disturbance estimates computed by the disturbance observer. By construction,
the proposed DO-ICBF based controller can ensure state and input constraint
satisfaction at all times. Further, we propose Higher Order DO-ICBFs that
extend our proposed method to nonlinear systems with higher relative degree
with respect to the auxiliary control input. Finally, numerical simulations are
provided to validate our proposed approach.
