Global prescribed-time regulation of the chain of integrators subject to external disturbances

This paper studies the global prescribed-time regulation problem for the chain of integrators subject to matched disturbances. New integral controllers consisting of an integral dynamic are proposed to make all states of the systems bounded and prescribed-time convergent to zero. First, by introducing an integral dynamic and a novel state transformation, the chain of integrators with disturbances are converted into a class of generalized feedback nonlinear systems with mismatched disturbances. Second, by virtue of the dynamic gain technique, the problem of designing prescribed-time regulating controllers for the generalized feedback nonlinear systems is transformed into that of designing a dynamic parameter for a class of new systems. At last, following Lyapunov stability theorem and the characteristic of the dynamic parameter, the boundness of the new systems yields the prescribed-time regulation of the generalized feedback nonlinear systems, which indicates the global prescribed-time regulation of the considered systems. The effectiveness of integral controllers is illustrated from simulation results.