Predefined-time smooth stability analysis of nonlinear chaotic systems with applications in the PMSM system and Hindmarsh-Rose neuron model

Abstract

This article investigates the predefined-time stabilization of nonlinear chaotic systems with applications in the permanent magnet synchronous motor (PMSM) system and Hindmarsh-Rose neuron model. Distinguished from the traditional predefined-time control methods, this investigation develops the smooth control protocols, in which the discontinuous absolute value and signum functions are not used anymore, so that the unfavorable chattering phenomenon can be avoided effectively. By the Lyapunov stability analysis, the sufficient condition is derived to achieve the predefined-time stable for nonlinear chaotic systems, in which the upper-bound time estimation (TE) of arriving at the stable state is explicit in contrast to the traditional finite-/fixed-time convergence. Specifically, the analytical results are successfully applied into stabilizing the PMSM system and Hindmarsh-Rose neuron model within the predefined-time. Finally, the numerical simulations for stabilizing the chaotic PMSM system and Hindmarsh-Rose neuron model verify the effectiveness and advantages of theoretical analysis.

Predefined-time stabilization of Lorenz system with applications for stabilizing and synchronizing chaotic finance systems

“…Therefore, according to lemma 2, the PtS of Lorenz system equation (1) is realized within any given moment T c by adopting the designed control scheme equation (4). , Remark 1.…”

“…Besides, the signals of controllers u i , i = 1, 2, 3, in equation (4) are depicted in figure 3(a), which will tend to 0 no more than T c . Next, we test the finite-time stabilization of Lorenz system deduced in corollary 1 and the fixed-time stabilization for the case that k 2 = 1 in equation (4). Other values of parameter and initial state are not changed.…”

“…The nonlinear Lorenz system was observed and officially proposed in 1963 [1], which is one of the classical chaotic models and has been studied widely in recent decades [2]. Meanwhile, other chaotic systems have been discovered one after another, like the permanent magnet synchronous motor (PMSM) [3,4], food web system [5], chaotic finance system (CFS) [6], and so on [7]. Due to the extreme sensitivity of chaotic systems to initial states, their orbits are difficult to be predicted accurately.…”

“…In order to resolve the issue, the predefined-time smooth control scheme has been developed for a class of nonlinear systems [51]. Besides that, for getting better convergence performance and explicit UbCT, the strategy of PtS has been further applied into robotic manipulators [52], rigid spacecrafts [53], space continuum robot [54], PMSM [4,55], memristive chaotic circuit [56], and others. Apart from the deterministic systems, the problem of stochastic PtS has been explored by considering the unknown disturbances [57], and the theoretical results have been also applied into unmanned aerial vehicles (UAVs) [58].…”