Sum‐of‐Squares‐Based Finite‐Time Adaptive Sliding Mode Control of Uncertain Polynomial Systems With Input Nonlinearities

Abstract

This paper proposes a novel adaptive sliding mode control (ASMC) for a class of polynomial systems comprising uncertain terms and input nonlinearities. In this approach, a new polynomial sliding surface is proposed and designed based on the sum-of-squares (SOS) decomposition. In the proposed method, an adaptive control law is derived such that the finite-time reachability of the state trajectories in the presence of input nonlinearity and uncertainties is guaranteed. To do this, it is assumed that the uncertain terms are bounded and the input nonlinearities belong to sectors with positive slope parameters. However, the bound of the uncertain terms is unknown and adaptation law is proposed to effectively estimate the uncertainty bounds. Furthermore, based on a novel polynomial Lyapunov function, the finite-time convergence of the sliding surface to a pre-chosen small neighborhood of the origin is guaranteed. To eliminate the time derivatives of the polynomial terms in the stability analysis conditions, the SOS variables of the Lyapunov matrix are optimally selected. In order to show the merits and the robust performance of the proposed controller, chaotic Chen system is provided. Numerical simulation results demonstrate chattering reduction in the proposed approach and the high accuracy in estimating the unknown parameters.

Application of Sum of Squares Method in Nonlinear H∞ Control for Satellite Attitude Maneuvers

“…there will exist the multiplication of P and F, which does not constitute a convex problem. Here, we can use a common approach in the LMI method to take the inverse matrix of P matrix in the solution [18]. is approach involves the following steps.…”

“…In solving the controller, the optimization problem is decomposed into a series of feasible semidefinite programming problems using a relaxation method, and then the optimal L 2 gain is obtained. In literature [18], a novel SOSbased adaptive sliding mode control for polynomial systems consisting of uncertainties and input nonlinearities is proposed. Vafamand et al [19] propose an approach that uniquely considers the stability of input saturated polynomial systems together with the nonlinear control law.…”

Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive Control

“…wherê,̂,̂,̂,̂, and̂are estimations of the unknown constant parameters , , , , , and , respectively; 1 , 2 , and 3 are positive constant gains selectable by the designer. By substituting (48) into (47), the closed-loop synchronization error dynamics is obtained aṡ1…”

“…In technical literature, various strategies have been proposed for control of chaotic systems [41][42][43][44][45][46][47][48]. Three approaches have commonly been used when both the structure and the parameters of the system are known: linear feedback control [49][50][51][52][53], nonlinear control [54][55][56][57][58], and active control [59][60][61][62][63][64][65][66][67][68].…”

Super‐Twisting Observer‐Based Integral Sliding Mode Control for Tracking the Rapid Acceleration of a Piston in a Hybrid Electro‐Hydraulic and Pneumatic System

“…Numerous studies for improving control quality such as sliding mode control [19], adaptive nonlinear control [12,13,20], and intelligent control [21,22,31,55] have been proposed. Sliding mode control (SMC) [36,37,[60][61][62][63][64][65][66][67][68] has reliable robustness like other robust controllers [38][39][40][41][42] with ability of reduce the impacts of disturbance, so that it can be applied for a wide range of complex systems [43]. Moreover, it can be stably combined with complicated adaptive algorithms [44][45][46][47][48][49], and be easily integrated with other control techniques [50][51][52] in both mathematical theory and practical experiment.…”