A Generalized Supertwisting Algorithm

The work proposes a generalized supertwisting algorithm (GSTA) and its constructive design strategy. In contrast with the conventional STA, the most remarkable characteristic of the proposed method is that the discontinuous term in the conventional STA is replaced with a fractional power term, which can fundamentally improve the performance of the conventional STA. It is shown that if the fractional power in the nonsmooth term becomes −1/2, the GSTA will reduce to the conventional STA. Under the GSTA, it will be rigorously verified by taking advantage of strict Lyapunov analysis that the sliding variables can finite-time converge to an arbitrarily small region in a neighborhood of the origin by tuning the gains and the fractional power. Finally, simulation studies are provided to demonstrate the superiority of the theoretically obtained results.


I. INTRODUCTION
A S IS well known, a variety of disturbances comprehensively emerges in the practical systems and gives rise to unfavorable influences on system performance and stability [1]. This indicates the disturbances have to be taken into account in modeling, controller design, and stability analysis [2]. During the last few decades, the research on nonlinear systems with disturbances has gained significant developments [3]- [5]. In this respect, abundant promising methods have been successively produced to handle uncertain nonlinear systems, such as the H ∞ control in [6], backstepping technique in [7], sliding-mode control (SMC) in [8], etc.
As one of the most efficient approaches to handle the uncertainties, SMC has raised ever-increasing interest in both academia and industry. This is on account of its strong robustness against disturbances, finite-time convergence, and computational simplicity [9]. It is noted that the execution of classical first-order SMC regretfully encounters the undesirable chattering problem. Frequently, the unexpected chattering will result in hazardous oscillation to dramatically degrade system performance [10]. Fortunately, the second-order sliding-mode (SOSM) algorithms (see, e.g., the twisting method [11] and the suboptimal algorithm [12]) can mitigate the adverse chattering effect with maintaining the robustness and accuracy of conventional sliding mode (SM) [13]- [17].
Among the existing SOSM algorithms, the supertwisting algorithm (STA) has played a special role and acquired widespread interest owing to its three merits. First, analogous to the traditional first-order SM (FOSM), the STA not only possesses the strong anti-disturbance ability but also can provide the finite-time convergence for SM dynamics [18]. For example, the STA was introduced in [19], where the proof of its convergence in terms of geometrical arguments for the first time was given. Furthermore, by means of the homogeneity property, the robustness of STA was analyzed in [20]. Since the proofs of STAs proposed in [19] and [20] rely on the geometric techniques or the homogeneity property, it is very problematic to estimate the convergence time. To address this, the Lyapunov methods were utilized to discuss the properties of STA for nonlinear systems with uncertainties [21]- [23]. By designing some explicit Lyapunov functions, the fruitful results on convergence conditions and reaching time estimation of STA can be found in the literature (e.g., see [24]- [26] and the references cited therein). Recently, a saturated STA was presented in [27] to regulate a first-order plant with uncertainties. Nevertheless, when the actuator saturation is inactive, the behavior of the controller proposed in [27] would be changed. To remedy the drawback, a conditioned STA was reported in [28] by virtue of the conditioning technique. Under stochastic perturbations, a stochastic STA was given in [29]. To achieve the minimization of the chattering effect, the adaptive STA was constructed in [30]. Moreover, an extension of STA to the multivariable scheme was developed in [31]. In [32], the multivariable STA was further investigated for a type of systems with uncertain input matrix and disturbances. Second, differing from the twisting algorithm [11], the prescribed convergence law [19], and the quasi-continuous algorithm [33], the STA merely needs the knowledge of the sliding variable, whilst the information concerning the derivative of the sliding variable is not required [34]. Last but not least, same as other SOSM algorithms, the STA is capable of efficiently weakening the unavoidable chattering produced by the high-frequency switching control action in conventional FOSM dynamic systems [21]- [26]. The above three This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ advantages of STA trigger its successful applications in a wide range of control engineering [35], [36].
It is worthy noticing that the aforesaid STAs have to include a discontinuous term under the integral, which is utilized to dominate the disturbance and ascertain the finite-time convergence. Nevertheless, as clearly elaborated in [37], since the discontinuous term under the integral is encompassed in the conventional STA, the chattering effect is not entirely eliminated but attenuated. This implies that the discontinuous term under the integral still generates some chattering, which will incur the increase of steady-state errors. Furthermore, in practical cases, the discontinuous term is severely influenced by the sampling time, which will result in great difficulty in implementation. As a consequence, how to reduce the adverse effect induced by the discontinuous term in the conventional STA is still a challenge to be tackled. In light of the foregoing remarkable shortcomings of the existing STAs, this work aims at the design of a novel generalized STA (GSTA). As compared with other related works, the contributions of the proposed GSTA are itemized as follows.
1) Analogous to the conventional STA developed by [19], the information of the sliding variable is solely demanded in the proposed GSTA.
2) The conventional STA itself involves the discontinuous term, which thereby gives rise to some unexpected steady-state errors. In order to address this obstacle, there have been several current research results, such as [38] and [39], where a discontinuous integral action is employed to generate a continuous control signal. Differing from the existing approaches, the proposed GSTA gets rid of the imperfection originated from the discontinuous term by using a nonsmooth term. 3) Distinguished from the Lyapunov functions utilized in [21] and [22], a new Lyapunov function is established to prove the stability of the presented GSTA. Meanwhile, with the aid of the constructed Lyapunov function, the convergence region of the sliding variable, which can be adjusted to be arbitrarily small, is explicitly given. The remaining parts of the work are outlined in the following manner. Section II states the problem formulation and preliminaries. The detailed steps on how to construct the GSTA are offered in Section III. Section IV shows an illustrative example. The study is summarized in Section V.
Notations: For any real numbers x and c, we use x c = |x| c · sign(x) in this article.

II. PROBLEM FORMULATION AND PRELIMINARIES
Consider a type of systems represented by where x ∈ R n is the state; u ∈ R is the controller; A(t, x) and B(t, x) are smooth functions; s is the sliding variable; and its relative degree is assumed to be one with regard to the control input u, that isṡ where d(t, x) =ṡ| u=0 is unknown and h(t, x) = (∂ṡ/∂u) = 0 is known. Meanwhile, the derivative of the disturbance d(t, x) also satisfies the following assumption. Assumption 1: There exists a positive constant K d > 0 such that |ḋ(t, x)| ≤ K d .
In order to steer the sliding variable s in system (2) to zero, the conventional STA control introduced by [19] is constructed as where λ 1 and λ 2 are the control gains to be selected.
It has been shown, for example, [21] and [40], that by properly choosing the gains λ 1 and λ 2 , the finite-time stability of system (4) can be assured. More precisely, an appropriate selection of λ 1 and λ 2 can render system (4) to exhibit a SOSM at s 1 =ṡ 1 = 0. It is worth pointing out that the differential equation (4), whose solutions are understood in the Filippov sense [41], is the so-called STA [42]. Nonetheless, it can be recognized that there exists a discontinuous term λ 2 sign(s 1 ) included in system (4). This will render the undesirable chattering phenomenon, affecting the performance of system (4). As a result, a natural question emerges: how to design a GSTA without the discontinuous term? In view of the above-mentioned issue, this article is devoted to the design of a GSTA.
Here, we list the definition of weighted homogeneity and four lemmas, which will be used for the proof of the main result.
Definition 1 [43]: For fixed coordinates (x 1 , . . . , x n ) ∈ R n and real numbers δ i > 0, i = 1, . . ., n, a function V : R n → R is said to be homogeneous of degree τ if there is a real number τ such that for ∀ε > 0 and ∀x ∈ R n \{0} with δ i being called as the weights of the coordinates.
Lemma 1 [44]: Provided that 0 < b ≤ 1, the following holds for ∀x, y ∈ R: Lemma 2 [44]: For any a ≥ 1, one has for ∀x, y ∈ R: Lemma 3 [45]: Let ϕ(x 1 , x 2 ) > 0 be a function of x 1 and x 2 . For any c 1 , c 2 > 0, there is Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. Lemma 4 [45]: If a function W(x) : R n → R and a positivedefinite function V(x) : R n → R have the same homogeneous degree with regard to the same dilation weight, then one can find a positive constantρ such that
Remark 1: It is worth pointing out that when α = (1/2), the proposed GSTA (6) is degenerated into the conventional STA (4). It can be seen that the Lyapunov function given in [21] for the conventional STA (4) is constructed as where ζ T = (ζ 1 , ζ 2 ) = ( s 1 (1/2) , s 2 ) and P is a symmetric and positive-definite matrix. With the aid of the Lyapunov function (7), the finite-time convergence for conventional STA (4) can be ensured. Nevertheless, it is obvious that the Lyapunov function (7) is inapplicable to the GSTA (6) in this article.
To show the effectiveness of the GSTA control (5), we only need to prove the stability of system (6).

B. Stability Analysis
In this section, we will first give the main result of this article formulated by a theorem. Afterward, a novel Lyapunov method will be designed to verify it.
Next, by (10), we rewrite system (9) as Stimulated by the work in [46], we select the Lyapunov function as where (13) .

Putting (24)-(26) into (19) obtainṡ
With the help of (18) and (27), (15) becomeṡ The constant gain c 1 can be designed as Putting (29) into (28) obtainṡ with H(e 1 , e 2 ) = |ξ 1 | (2−α+β/α) + |ξ 2 | (2−α+β/β) . From Assumption 1, one haṡ Next, it will be shown that the region Q will be finite-time reached, whose proof consists of two steps. A sketch of proof can be described as follows. In the first step, we will prove the fact thatV(e 1 , e 2 )| (13) < 0 once (e 1 , e 2 ) / ∈ 1 , where with k 1 > 0 and ε ∈ (0, 1) being an arbitrarily small positive constant. However, it cannot ensure the region 1 is an attractive region. This is because the states may escape from the region 1 , due to the fact thatV(e 1 , e 2 )| (13) < 0 will be no longer assured for ∀(e 1 , e 2 ) ∈ 1 . In the second step, we will find a larger region for a constant k 2 > 0 and prove that 2 is an attractive region for any initial state (e 1 (0), e 2 (0)) ∈ 2 . This implies that there exists a finite time t * 1 such that (e 1 (t), e 2 (t)) ∈ 2 for ∀t ≥ t * 1 . Then, the convergence region of the sliding variables s 1 and s 2 can be obtained directly from the region 2 . In the following, we will give the detailed proof.
Step 2: It will be proved thatV(e 1 , e 2 )| (13) < 0 once (e 1 , e 2 ) / ∈ 2 . Actually, from the previous step, we only need to prove that 1 is a subset of 2 . The proof is given as follows. From Definition 1, it can be easily verified that both V (2−α+β/2) (e 1 , e 2 ) and H(e 1 , e 2 ) are homogeneous of degree 2 − α + β with respect to the dilation weight (1, α). Thereby, from Lemma 4, there exists a constant k 2 > 0 such that For any (e 1 , e 2 ) ∈ 1 , one has from (34) that That is to say (e 1 , e 2 ) ∈ 2 , that is, 1 ⊂ 2 . Accordingly, once (e 1 , e 2 ) / ∈ 2 , thenV(e 1 , e 2 )| (13) < 0. Since 2 is a level set of the Lyapunov function, then there exists a finite time t * 1 such that (e 1 , e 2 ) ∈ 2 , that is, V(e 1 , e 2 ) ≤ m for ∀t ≥ t * 1 . When t ≥ t * 1 , it is concluded that With c 1 = λ 1 and e 2 = (1/λ 1 )s 2 in mind, we can obtain from (36) that Next, let us estimate s 1 . Since for ∀t ≥ t * 1 , it is deduced from the inequality that when t ≥ t * (38) which, together with the fact e 1 = s 1 and (37), yields Therefore, it can be concluded from (37) and (39) that the region Q will be reached within a finite time. That ends the proof of Theorem 1. Remark 2: It should be pointed out that by adjusting control gains λ 1 and/or λ 2 to be large enough, the attractive region Q can be rendered to be as small as expected. Nonetheless, the selections of λ 1 and λ 2 are subjected to the considerations of system stability and control saturation constraint, and hence they cannot be chosen sufficiently large. In this case, an additional parameter α can be tuned to improve the performance of the closed-loop system without increasing λ 1 and λ 2 to be sufficiently large. For instance, supposing , we can select α to approximate to (1/2) such that the fractional power (1/β) in (8) is larger enough than 1.
Remark 3: It can be recognized that ifḋ(t, x) = 0, system (6) will reduce to (e 1 , e 2 )| (13) . (41) Note that (2 − α + β/2) ∈ (0, 1). It follows from the finitetime Lyapunov theory in [47] that system (13) [i.e., system (6)] is finite-time stable. Meanwhile, it can be found that the region Q will naturally vanish. Remark 4: The defects of the proposed scheme mainly include the following three points. First, the upper bound of the derivative for the disturbance d(t, x) in system (2) must be known a priori. Second, the parameter α in the presented algorithm must satisfy the condition (1/2) < α < 1. Third, the nonlinear control gain h(t, x) in system (2) must be known a priori as well. The aforementioned inherent disadvantages may limit the application scope of the presented GSTA. These issues will be dealt with in our future research.
Note that in [48], an SMC design method was presented for a class of uncertain linear impulsive systems via switching gains. In [49], an impulsive SOSM control in the reduced information environment was proposed. Also, an impulsive adaptive STA was reported in [50]. The STA and GSTA can be extended to control the uncertain nonlinear impulsive systems.
Remark 5: Notice that the selection range of parameter α is ([1/2], 1). Interestingly, when the parameter α tends to (1/2), the proposed GSTA will become the conventional STA. Because of the effect of sampling time on the digital implementation of the controller, the anti-disturbance performance of the conventional STA may not be better than that of the proposed GSTA in practical applications. Moreover, in practice, it is difficult to implement the discontinuous switching function −λ 2 · sign(s 1 ) in the conventional STA. Nevertheless, the nonsmooth term −λ 2 s 1 β in the proposed GSTA could be easily carried out by DSP. Hence, the extra parameter α seems not to increase the complexity for implementation.
IV. SIMULATION STUDY Consider a system described bẏ where x ∈ R is the system state, u ∈ R denotes the control input, and φ(t, x) refers to the perturbation. The purpose here is to construct a controller u such that x will converge to zero. In order to achieve the goal, we define the output variable s = x, whose time derivative along system (42) iṡ In accordance with Theorem 1, the presented GSTA control can be designed as where α ∈ ([1/2], 1), β = 2α − 1, and λ 1 and λ 2 are two proper positive constants.
In the following, we will consider two scenarios to analyze the performance of controller (44). One is for system (42) in the absence of disturbances, and the other is for system (42) conducted under disturbances.

A. System (42) Without Disturbances
In this scenario, we will, respectively, discuss how λ 1 , λ 2 and α influence the performance of controller (44) without disturbances. The initial state is chosen as x(0) = 0.7. Under the circumstance, the following three different cases are investigated.
Case 1 (For Different Values of Parameter λ 1 ): The parameter λ 1 is selected as λ 1 = 2.3, 2.9, 6.2, respectively. Let λ 2 = 2 and α = 0.75. The derived simulation results are shown by Fig. 2. Fig. 2 displays the time history of x and u for system (42) in the absence of disturbances with λ 1 = 2.3, 2.9, 6.2. Apparently, one can observe from Fig. 2 that the states can converge to zero within a finite time. Moreover, it can also be seen that when λ 2 and α are fixed, the larger λ 1 insinuates the shorter settling time of states and the smaller overshoot.
The response curves of x and u for system (42) without disturbances with λ 2 = 1.5, 3.2, 5.1 are exhibited in Fig. 3. Clearly, one can see from this figure that when λ 1 and α are fixed, the bigger λ 2 indicates the slower convergence speed of states and the larger overshoot.

Case 3 (for different values of parameter α):
The parameter α is given as 0.7, 0.75, 0.8, respectively. We select λ 1 = 6.2 and λ 2 = 1.5. The attained simulation results are exhibited through Fig. 4.  Fig. 4 shows the time history of x and u for system (42) in the absence of disturbances with α = 0.7, 0.75, 0.8. Obviously, it can be recognized from Fig. 4 that as λ 1 and λ 2 are fixed, the smaller α means the faster convergence speed of state x.
In summary, the aforementioned simulations results in this scenario demonstrate the conclusions given in Remark 3.
It can be clearly seen from Fig. 5 that the state x can converge to a region. Moreover, we can also observe that when the same types of disturbances with different magnitudes are taken into account, bigger magnitude of the disturbance brings larger steady-state errors.
To summarize, on the basis of the above simulation results, the validity of Theorem 1 is substantiated.
On the other hand, for comparison investigations, we also construct a conventional STA control and a smooth controller, which are, respectively, given as with λ 1 and λ 2 being two proper positive constants.
In what follows, the superiority of the proposed control approach will be testified.
From Fig. 6, it can be seen that under the different categories of disturbances, the steady-state errors under the conventional STA control (45) are the smallest among the three controllers. This implies that the anti-disturbance ability of the conventional STA control (45) is the best among the three controllers. Nevertheless, one can observe from Fig. 7 that the oscillation of the conventional STA control (45) is the heaviest.
It should be emphasized that in comparison with the conventional STA control (45) and the smooth controller (46), in terms of the anti-disturbance performance, the proposed GSTA control (44) is superior to the smooth controller (46), while from the viewpoint of control oscillation, the proposed controller (44) is better than the conventional STA control (45). In summary, on the basis of the aforementioned simulation  results, when the anti-disturbance performance and the control oscillation are concurrently taken into consideration, the proposed GSTA control (44) exhibits a performance better than controllers (45) and (46). Remark 6: The parameter α in the proposed controller (44) can be flexibly selected between (1/2) and 1. In accordance with the analysis of the above simulation results, one can see that the parameter α brings great effects on control performances. It can also be observed that if the parameter α equals (1/2), the proposed controller (44) will reduce to the conventional STA control (45). In addition, if the parameter α is selected as α = 1, the proposed controller (44) will become the smooth controller (46). This implies that the presented controller (44) is between the conventional STA control (45) and the smooth controller (46). Based on the above simulation results, it can be seen that the proposed controller (44) can not only make the closed-loop system have good convergence performance and strong anti-disturbance ability but also avoid the adverse effects caused by discontinuous control.

V. CONCLUSION
In the article, a novel design framework for GSTA has been proposed. Rigorous mathematical analysis by means of the Lyapunov theory demonstrates that the developed GSTA control can assure that the states of the closed-loop system will converge to a region as small as desired around the origin within a finite time. The relationship between the convergence domain of the states and the control parameters has been established. This will be advantageous since by tuning the control parameters, a satisfactory disturbance rejection performance of the considered system can be obtained. In future work, we will extend the derived results for practical applications and control of uncertain nonlinear impulsive systems.