Abstract
The paper considers the problem of variable structure control for nonlinear systems with uncertainty and time delays under persistent disturbance by using the optimal sliding mode surface approach. Through functional transformation, the original time-delay system is transformed into a delay-free one. The approximating sequence method is applied to solve the nonlinear optimal sliding mode surface problem which is reduced to a linear two-point boundary value problem of approximating sequences. The optimal sliding mode surface is obtained from the convergent solutions by solving a Riccati equation, a Sylvester equation, and the state and adjoint vector differential equations of approximating sequences. Then, the variable structure disturbance rejection control is presented by adopting an exponential trending law, where the state and control memory terms are designed to compensate the state and control delays, a feedforward control term is designed to reject the disturbance, and an adjoint compensator is designed to compensate the effects generated by the nonlinearity and the uncertainty. Furthermore, an observer is constructed to make the feedforward term physically realizable, and thus the dynamical observer-based dynamical variable structure disturbance rejection control law is produced. Finally, simulations are demonstrated to verify the effectiveness of the presented controller and the simplicity of the proposed approach.
Highlights
Various approaches have been proposed to solve disturbance rejection problems, such as H∞ control [1], adaptive control [2], internal model control [3], variable structure control (VSC) [4], and optimal control [5,6,7]
We will reduce the controller design problem of original system (1) to that of a delay-free one referring to the method proposed in [20] which applied this method to a linear time-delay system
A1 = e−AσA1, B1 = e−AτB1, in which A is referred to as the characteristic matrix equation and the method of its solution can be found in the researches of Fiagbedzi and Pearson [20, 21] and Zheng, Cheng, and Gao [22], and so forth, system (1) is assumed spectral controllable, which implies that system (5a) and (5b) is completely controllable [20]
Summary
Various approaches have been proposed to solve disturbance rejection problems, such as H∞ control [1], adaptive control [2], internal model control [3], variable structure control (VSC) [4], and optimal control [5,6,7]. The contributions of this study are as follows: firstly, the functional transformation method is applied, which transforms the time-delay system to a delay-free one and reduces the original problem from an infinite-dimension space to a finite-dimension one; subsequently, the optimal sliding mode surface (OSMS) is designed by using approximating sequence method, which simplifies the nonlinear OSMS design problem to a linear two-point boundary value (TPBV) one; the corresponding compensators are designed in the variable structure disturbance rejection control (VSDC) so that the nonlinearity, uncertainty, and the time-delay effects are entirely compensated consequentially; the disturbance effect is reduced by the feedforward compensation term; to realize its physical implementation, a reduced-order observer is constructed to reconstruct the disturbance state vectors, and the observer-based dynamical VSDC is produced; the designed VSDC is employed to a quarter-car suspension model which possesses the nonlinear, uncertain, and timedelay properties, and by comparing the system responses with the open-loop system (OLS), the effectiveness and the simplicity of the designed control are demonstrated.
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