  In this paper, we restudy how to modify the model-free adaptive control
(MFAC) to reject the disturbance both in single-input single-output (SISO)
systems and multiple-input multiple-output (MIMO) systems, with the aim to pave
the way for the development of this interesting controller in the future. First
of all, in order to accurately describe the nonlinear system model at each
time, we compensate the equivalent dynamic linearization model (EDLM) with
disturbance and prove it through the definition of differentiability and the
Taylor series. Then based on modified EDLM, we redesign MFAC compensated with
disturbance and firstly reanalyze the discrete-time nonlinear system through
the closed-loop system equation at each time. This is all possible because some
nonlinear system functions can be accurately described by the EDLM compensated
with disturbance according to Taylor series. At last, several examples are
given to verify the theorem.
