Predefined‐time‐synchronized backstepping control of strict‐feedback nonlinear systems

Abstract

Fixed-time-synchronized control using norm-normalized sign functions has been recently studied, where all state variables converge to the origin at the same time and the bound of the settling time is independent of the initial states. However, the existing fixed-time-synchronized control results are only applicable to second-order systems with matched nonlinearities. Thus, this article pays attention to an extension problem of time-synchronized control to high-order systems with unmatched nonlinearities. We develop a predefined-time-synchronized backstepping tracker design for multi-input multi-output strict-feedback nonlinear systems, where all output tracking errors converge to the origin at the same time, regardless of unmatched nonlinearities, and the bound of the synchronized settling time is independent of the initial errors and selected a priori. Sufficient conditions on continuously differentiable and nonsingular virtual and actual controller designs are presented for the recursive design. It is shown that the predefined-time-synchronized stability is ensured recursively and the singularity problem of the proposed tracker can be avoided. Finally, simulation comparison results with the existing fixed-time backstepping control are provided to illustrate the effectiveness of the theoretical approach.

Unified Sufficient Conditions for Predefined-Time Stability of Non-Linear Systems and Its Standard Controller Design

“…M(q) q + C(q, q) q + G(q) + F u = u, (16) where q ∈ R n , q ∈ R n ; q ∈ R n denote the generalized position, velocity, and acceleration vectors, respectively; M(q) ∈ R n×n is a positive definite moment of inertia matrix; C(q, q) ∈ R n×n is the centripetal Coriolis matrix; G(q) ∈ R n is the gravity vector; F u ∈ R n represents the uncertainty vector; and u ∈ R n denotes the control input. For the convenience of controller design, let x 1 = q and x 2 = q.…”

“…For the convenience of controller design, let x 1 = q and x 2 = q. Therefore, the Lagrangian system (16) with n = 1 can be rewritten as follows:…”

“…Lyapunov theory-an effective instrument for analyzing the stability of control systemsis often combined with sliding mode control, backstepping control, and adding power integrator techniques to design controllers to ensure asymptotic [9], finite-time [10][11][12], fixed-time [13][14][15], and predefined-time [16,17] stability of the controlled system. In particular, the sliding mode control technique is usually used to design a controller to guarantee the controlled system's finite-time stability [17][18][19][20][21][22].…”

Design of Backstepping Control Based on a Softsign Linear–Nonlinear Tracking Differentiator for an Electro-Optical Tracking System

“…The backstepping control method used in this paper is a commonly used control method [11,12], the basic idea of which is decomposing a complex system into subsystems that do not exceed the order of the system. As shown in Equation (1) above, the system can be divided into n subsystems, and then a Lyapunov function V n and an intermediate virtual control quantity x (i+1)d can be designed, respectively, for each subsystem x i+1 .…”

“…Furthermore, an electro-optical tracking system is limited by the sensor frequency and the performance of the driving mechanism, which leads to the problems of a low tracking accuracy and slow error convergence. To address this problem, backstepping control techniques [11][12][13] have been developed rapidly in recent years. Backstepping control is not only characterized by an ease of design and implementation but also by the ability to measure the state of the system in real time and adjust the inputs according to the difference between the target output and the actual output, thus achieving highly accurate tracking.…”