State Synchronization for a Class of n-Dimensional Nonlinear Systems with Sector Input Nonlinearity via Adaptive Two-Stage Sliding Mode Control

Abstract

<jats:p>This study addresses an adaptive two-stage sliding mode control (SMC) scheme for the state synchronization between two identical systems, which belong to a kind of <jats:italic>n</jats:italic>-dimensional chaotic system, by considering the appearance of lumped system uncertainties and external disturbances. The controlled system is assumed to be attached to sector nonlinearity for the control input. The proposed adaptive control scheme involves time-variable state-feedback gains, which are updated in accordance with the appropriate adapted rules without foreknown the certain information of nonlinear system dynamics, bounds of lumped system uncertainties and external disturbances, and sector input nonlinearity. The derivation of the control scheme is found based on the introduced sequence of two sliding functions. The stage 1 sliding function is defined by the states of the error dynamical system, where the asymptotical stability is inherent. Then, the stage 2 sliding function is formed by the stage 1 function, where the finite-time stabilization is guaranteed. The proposed adaptive control scheme can cope with the effect of sector nonlinearity for the control to meet the control goal. The sufficient conditions of the stability of the error dynamical system are proven mathematically by means of the Lyapunov theorem. Besides, the capacity of the present scheme is carried out by the numerical studies.</jats:p>

Estimation of Synchronization Errors between Master and Slave Chaotic Systems with Matched/Mismatched Disturbances and Input Uncertainty

“…Theorem 1. Consider the uncertain error dynamics (3) with bounded input uncertainty, the system trajectory controlled by the SMC controller (17) can asymptotically converge to the sliding manifold S(t) = 0.…”

“…From the theoretical point of view, the discontinuous sign function in control input (17) may cause chattering. To reduce the chattering resulted from the discontinuous sign function, we can introduce the continuous saturation function described by…”

“…Therefore, designing a robust control scheme to asymptotically suppress or eliminate the influence of unknown disturbances is a very important issue when discussing synchronization control for chaotic systems [16], and some research works have been proposed by studying the effects of input uncertainties [17][18][19]. By considering the uncertainties for control inputs, the state synchronization controller for two identical n-dimensional chaotic systems was proposed in [17], and the robust adaptive finite-time controller was reported for synchronization in [18]. For second-order chaotic systems, the synchronization controller was proposed by the adaptive sliding mode control [19].…”

“…However, the uncertainty in the control input is not considered. In [17][18][19], researchers took into account the effects of input uncertainties for synchronization controller design. However, only special types of chaotic systems and matched disturbances were considered.…”

Function Projective Synchronization of Two Complex Networks with Unknown Sector Nonlinear Input and Multiple Time-Varying Delay Couplings

“…Wang and Liu [38] researched the sliding mode control of the master-slave chaotic systems with sector nonlinear input. Yang et al [39] addressed an adaptive two-stage sliding mode control to realize the synchronization for a class of n -dimensional nonlinear systems with sector nonlinearity input. Although the researches on sector nonlinear input have achieved certain results, most existing studies mainly focus on a single system rather than complex networks.…”