This paper provides a solution to generalize the integrator and the
integral control action. It is achieved by defining two function sets to
generalize the integrator and the integral control action, respectively,
resorting to a stabilizing controller and adopting Lyapunov method to analyze
the stability of the closed-loop system. By originating a powerful Lyapunov
function, a universal theorem to ensure regionally as well as semi-globally
asymptotic stability is established by some bounded information. Consequently,
the justification of two propositions on the generalization of integrator and
integral control action is verified. Moreover, the conditions used to define
the function sets can be viewed as a class of sufficient conditions to design
the integrator and the integral control action, respectively.
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