Control scheme for LTI systems with Lipschitz non‐linearity and unknown time‐varying input delay

Abstract

Abstract:In this paper, we propose a control structure for a class of systems with Lipschitz nonlinearity and unknown time-varying input delay. This scheme considers the worst case scenario in control design with Truncated Prediction Feedback (TPF) approach, and takes into account the information of the lower bound of delay in the stability analysis. A finite-dimensional controller is constructed, requiring neither the nonlinear function nor the exact delay function. The truncated prediction deviation is minimized by employing the delay range, and then bounded by integral construction and related techniques. Within the framework of LyapunovKrasovskii functionals, sufficient delay-range-dependent conditions are derived for the closed-loop system to guarantee the global stability. Two numerical examples are given to validate the proposed control design. IntroductionIn Note that all the results above are built based on the assumption that the exact delay information is known, which may be irrealizable in many circumstances. The predictor-based feedback control methods suffer from a difficulty in practical implementation when the delay function is unknown. A Lyapunov-based adaptive control design is presented in [25] to deal with unknown constant delay for a class of linear systems. However, few results are available on stabilization for nonlinear systems with unknown time-varying input delay. The TPF control also encounters the same barrier since the prediction is based on the exact knowledge of the delay function [19]-[24].Inspired by the above observation and based on our previous results [26]-[30], in this paper, we propose a control scheme for a class of systems with unknown time-varying input delay and Lipschitz nonlinearity. The key contributions include: (i) in contrast to [13,16,[21][22][23], we propose a new control structure by merely using the delay upper bound, which greatly reduces the computation burden and improves the practical implementation; (ii) we extend the results on the truncated predictor feedback for linear systems to a class of Lipschitz nonlinear systems with unknown time-varying input delay and the derived conditions are less conservative with respect to the existing results [24]; (iii) the information of the lower bound of delay is taken into account in the stability analysis, which benefits the derived conditions. The remainder of this paper is organized as follows. Section 2 presents the problem formulation and a few preliminary results for the stability analysis. Section 3 presents the main results on the controller design and stability analysis. Simulation results are given in Section 4. Section 5 concludes the paper. Problem statement and preliminariesWe consider the systeṁwhere x(t) ∈ R n is the state, u(t) ∈ R p is the input, A ∈ R n×n and B ∈ R n×p are constant matrices with (A, B) being controllable, ζ(t) is a continuously differentiable function that incorporates the actuator delay, and ϕ : R n → R n , ϕ(0) = 0, is a Lipschitz nonlinear function with a Lipschitz constant γ. Fo...

Predictive-based sliding mode control for mitigating torsional vibration of drill string in the presence of input delay and external disturbance

“…Regarding nonlinearity and parametric uncertainty nature of the drill string system, robust control approaches for nonlinear processes can be beneficial in the presence of the input delay. The TPF for a class of linear time-invariant systems with Lipschitz nonlinearity and unknown timevarying input delay is proposed in Wang et al (2017). Also, the asymptotic stability of a nonlinear system in the presence of external disturbances and input delay using a Smith predictor-based sliding mode controller is presented in Moosapour et al (2019).…”