Terminal time regulator‐based exact‐time sliding mode control for uncertain nonlinear systems

Abstract

This article proposes a sliding mode control scheme with exact convergence time for uncertain nonlinear systems. Terminal time regulator is defined for designing the global exact‐time time‐varying sliding mode. The proposed method has the following advantages: first, the reaching phase is absolutely eliminated from the initial time such that the sliding mode invariance is identically guaranteed on an entire dynamics; second, on the sliding mode, the system state converges to the origin exactly at the predefined settling time instant, rather than at an uncertain time instant with boundedness as obtained by existing methods; third, the control signal of the proposed controller is bounded even when the time approaches to the predefined settling time; and fourth, low‐pass filtering is utilized to suppress the signal chattering in the discontinuous control while maintaining the system transient behaviors by selecting a proper time constant. Lyapunov analysis verifies that for any initial condition the closed‐loop system can terminally converge to zero exactly at the predefined time instant. The proposed control is finally generalized for Euler–Lagrange (EL) systems.

Adaptive Sliding Mode Control for Unmanned Surface Vehicles with Predefined-Time Tracking Performances

“…where ϕ(t) ∈ R 3 , ϕ j (t) is the j-th element of ϕ(t), b j ∈ (0, 1), T p is the user-defined constant and ϕ 0j and ϕ ∞j are the initial and terminal values of ϕ j (t), respectively. Differentiating (21) with respect to time yields…”

“…For the convenience of analysis, only the scalar form of the tracking error is considered in the following. With the control objective in mind, the tracking error is required to remain within the preselected behavioral boundaries as −α j ϕ j (t) < e 1j (t) < β j ϕ j (t), (24) where ϕ j (t) is given in (21) and α j and β j are the parameters to be designed for j = 1, 2, 3. To guarantee that the tracking error e 1j can always remain within the behavioral boundary (24), it is necessary to map the tracking error to an unconstrained equivalent ver-sion.…”

“…In order to facilitate readers to better understand the design process of the controller ( 33)-( 35), the control block diagram of the USV is shown in Figure 1. Firstly, a behavioral boundary (24) of the tracking error is given based on the performance function (21). Second, an error transformation function ( 25) is used to map the tracking error into an unconstrained version of the transformation error θ.…”

“…Based on this idea, a series of predefined-time sliding mode manifolds were developed [20]. The other is to establish a predefined-time sliding mode manifold based on the terminal time regulator [21]. Then, a predefined-time sliding mode controller can be designed by constructing auxiliary functions.…”

“…Inspired by [24][25][26], the predefined-time prescribed performance control is used to simultaneously achieve the tracking performance and convergence time constraints of USVs in this paper. Furthermore, by integrating the barrier function given in [14] and the auxiliary function proposed in [21] into the control design, the predefined-time predefined-bounded control of USVs is achieved for the first time.…”