  This paper study the parameter selection of predefined-time sliding mode and
try to design a general nonsingular predefined-time terminal sliding mode. 1).
On parameter selection: Some existing predefined-time sliding modes are
designed to focus on the reaching time and ignore the characterization of the
equilibrium point i.e. x_e= 0. In actual engineer, the system can only converge
to a small neighborhood near the equilibrium point because of the existence of
uncertainty i.e. x_e \to 0. The actual equilibrium point should be deduced by
taking the limit but not directly solve the equilibrium point. Hence, the
actual selection of exponential term is suggested to be not 1 to make system
convergence to the equilibrium point within predefined time. 2). On
singularity-avoidance: Based on the exponential feature of predefined-time
stability systems, a mathematic concept of switching sliding mode
\cite{su1994adaptive} is applied in the design of nonsingular predefined-time
terminal sliding mode. However, this class of sliding mode switching method
will cause a singular problem, which can be described as control input u \to
\infty if the system state x \to 0 and any-order time derivative of x is not 0.
Hence, a non-singular selection condition for each recursive sliding mode
surface is explored in this paper to realize the global finiteness of control
input. In addition, an unnecessary chattering of control input will also be
caused by the sliding mode switching frequency, and a novel switching condition
will be applied in the controller to reduce this chattering. Simulations will
be carried out to validate the effect of the novel nonsingular predefined-time
sliding mode control method to some extent.
