Prescribed-Time Second-Order Sliding Mode Controller Design Subject to Mismatched Term

This brief investigates the second-order sliding mode (SOSM) control for uncertain nonlinear systems subject to unknown mismatched terms. A new prescribed-time SOSM controller is developed by applying a time-varying coordinate transformation. The proposed controller can allow to prescribe the convergence time a priori and irrespective of initial conditions even in the presence of mismatched terms. Moreover, all the upper bounds of matched and mismatched uncertainties are not required to be exactly known. Finally, a simulation example with different initial conditions is given to show the effectiveness of the proposed controller.


I. INTRODUCTION
S ECOND-ORDER sliding mode (SOSM) control has been proved to be an effective method to overcome the essential shortcomings of the conventional first-order sliding mode control such as the chattering problem and the restriction problem of relative degree [1], [2]. Over the past three decades, many well-known SOSM algorithms have been developed such as the twisting controller [3], the super-twisting controller [4], the suboptimal controller [5], the quasi-continuous controller [6], the relay polynomial controller and the quasi-continuous polynomial controller [7]. In the case that the derivative of the sliding variable is unmeasurable, robust exact differentiator can be utilized such that output-feedback controllers can be implemented [8].
It is noted that most of the existing SOSM control algorithms can only achieve finite-time convergence of the sliding surface. For some practical applications [9], one often needs to achieve the control objective in a short and finite time with some small uncertainties. Therefore, these controllers that allow the user to prescribe the convergence time Manuscript  a priori and irrespective of initial conditions offer a clear advantage over those that do not. By selecting the parameters, many existing controllers or observers with finite or fixed-time stability can allow for prescription of the convergence time a priori [10], [11], [12], [13], [14], to just name a few. However, the convergence time of finite-time stability is dependent on initial conditions, and only an upper bound of the convergence time can be obtained for fixedtime stability. Moreover, the gain adjustments of all these works to prescribe the convergence time are seldom simple. In [15], an entirely new methodology for solving finite-time regulation with a prescribed regulation time independent of initial conditions is developed. Based on this methodology, a new prescribed-time controller is designed for norm nonlinear systems in the presence of matched uncertainties. Since then, many results about prescribed-time control or estimation have been developed [16], [17], [18], [19]. However, few results about prescribed-time SOSM control have been reported. Therefore, to develop a new SOSM algorithm with prescribed convergence time is of important significance. It can also be observed from the above literature that the conventional SOSM control needs to take a direct two times derivative of the sliding variable and thus can only deal with matched uncertainties. As has been indicated in [20] that the direct derivative method may also lead to the disadvantages such as large input uncertainties, noise enlargement, etc. All the aforementioned problems can be avoided by redesigning the first derivative of the sliding variable as a new state variable plus a mismatched term. Then, a new SOSM dynamic system subject to mismatched term can be obtained. Aiming at the new SOSM dynamic system, some new SOSM algorithms have been developed in the recent papers [20], [21], [22], [23], [24]. However, all these results can only achieve finite or fixed-time convergence of the sliding surface, and the upper bound of the mismatched term is usually required to be known in advance. To the best of our knowledge, the prescribed-time SOSM controller design subject to mismatched term with unknown bound has not been reported.
Motivated by the above analysis, the prescribed-time SOSM controller design problem will be solved in this brief. The contributions are mainly twofold. First of all, a novel prescribedtime SOSM controller, that prescribes the convergence time a priori and irrespective of initial conditions, is developed for uncertain nonlinear systems subject to mismatched terms for the first time. Secondly, it provides a SOSM algorithm handling the nonlinear systems with uncertainties bounded by unknown positive functions rather than frequently-used known positive functions or even known constant upper bounds.

A. Prescribed-Time Scaling Function and Definition
The basis for our controller design is the use of a monotonically increasing function μ 1 which is borrowed from [15]. The function has the properties that μ 1 (0, T) = 1 and μ 1 (T, T) = +∞. Define on t ∈ [0, T) with positive integers r, m. Note that in addition to the properties μ(0, T) = 1 and μ(T, T) = +∞, the function μ(t, T) can grow more quickly than μ 1 (t, T) through the positive integers r (the relative degree of the system) and m ≥ 1 (a design parameter).

Definition 1 [15] (FT-ISS+C):
The systemẋ = f (x, t, d) (of arbitrary dimension of x and d) is said to be fixed-time input-to-state stable and convergent to zero (FT-ISS+C) in time T if there exist class KL functions β and β f , and a class K function γ , such that for all t ∈ [0, T),

B. Problem Statement
Consider the following uncertain nonlinear system: where x ∈ R n , u ∈ R are the state and control input, respectively; s is a measurable output (i.e., the sliding variable); a(t, x) and b(t, x) are unknown nonlinear functions.
Generally, the relative degree of the output s with respect to u is assumed to be r = 2 (see, e.g., [8]). In this case, the following input-output relation with matched uncertainty can be obtainedṡ where x) = 0 are not exactly known but satisfy the following assumption: Note that the conventional SOSM controllers designed for system (3) usually take a direct derivative ofṡ = ∂s ∂t + ∂s xẋ no matter what it contains, which may bring some disadvantages such as large input uncertainties, enlargement of noise inṡ, etc. (see, e.g., [20], [23]). To overcome these disadvantages, we divide the termṡ = ∂s ∂t + ∂s ∂xẋ into two parts: the properly chosen term c(t, s) depending on s as the mismatched term and the rest known term s 2 as the new variable. Then, a new SOSM dynamic with mismatched term can be obtaineḋ where s 1 = s and ∂s 1 ∂t Assumption 2: For the functions g 2 (t, x), h 2 (t, x) and c(t, s 1 ), the following inequalities hold: where g * 2 > 0 andh 2 (t, x) are known constant lower bound and time-varying nonnegative function, respectively; The goal of this brief is to design a new SOSM controller for system (4) such that s 1 , s 2 can be driven to zero in any prescribed-time T free of system initial conditions.

A. Design Process
To design the controller, we use the time-varying scaling function (2) with r = 2, i.e., and define ν(t, T) with the properties that v(0, t) = 1 and v(T, T) = 0. It can be easily calculated thaṫ Then, we use μ in the transformation [s 1 , with which the original system (4) can be converted intȯ In the following, the design process of the controller u will be divided into two steps.
Proof: Solving the differential inequality (15) gives which implies that It can be clearly observed from (35) that w 1 -system is ISS (though not FT-ISS) w.r.t. the inputs (σ, d 1 ).
It can be calculated from (9) and (23) that Define . Then, it follows from (48) that Selectq ≥ |Q(ν)| and by using (47) and (49), one has Hence, the proof is completed. Remark 1: The proposed prescribed-time SOSM controller is only valid for t ∈ [0, T), which may limit its application. An alternative way to make it valid for t ∈ [0, ∞) is to switch the proposed controller to some existing SOSM controllers that can maintain the sliding variables at origin when t ≥ T. It should be noted that when s 1 → 0 as t → T, the mismatched term will also tend to zero and thus the SOSM dynamic system (4) with mismatched term will become to the conventional SOSM dynamic system (3) with matched uncertainty. If the bound of the matched uncertainty is known in advance, many existing SOSM controllers can be applied.

V. CONCLUSION
In this brief, we have proposed a new class of prescribed-time SOSM controller for uncertain nonlinear systems subject to mismatched terms. The proposed controllers allow to prescribe the convergence time a priori and irrespective of initial conditions even when the bounds of the mismatched and matched terms are not exactly known. How to extend this algorithm to high order cases will be considered in the future.