Abstract
This paper proposes a robust supertwisting algorithm (STA) design for nonlinear systems where both matched and unmatched uncertainties are considered. The main contributions reside primarily to conceive a novel structure of STA, in order to ensure the desired performance of the uncertain nonlinear system. The modified algorithm is formed of double closed-loop feedback, in which two linear terms are added to the classical STA. In addition, an integral sliding mode switching surface is proposed to construct the attractiveness and reachability of sliding mode. Sufficient conditions are derived to guarantee the exact differentiation stability in finite time based on Lyapunov function theory. Finally, a comparative study for a variable-length pendulum system illustrates the robustness and the effectiveness of the proposed approach compared to other STA schemes.
1. Introduction
Sliding Mode Control (SMC) strategy is considered an effective methodology for control uncertain systems. This strategy gives a major objective in control system design to attain stability in the presence of uncertainties [1–5]. The design of the SMC systems mainly consists of two steps: the choice of the sliding mode switching surface and the design of the sliding mode controller. Moreover, SMC has believed significant amount of interest due to several advantages, such that fast convergence, high robustness, and invariance to certain internal system parameter variations and its implementation are easy [5–9]. On the other hand, the worst disadvantage of the SMC methodology is the chattering phenomenon. Thus, to reduce this problem of the chattering effect, numerous techniques are proposed in literature [10–13]; one of them is the supertwisting algorithm (STA) method. The STA has become the prototype of Second-Order Sliding Mode Control (SOSMC) algorithm, which has the capability of system robust stabilization, finite time convergence to the sliding surface, and chattering reduction even in the presence of uncertainties [14–16]. Also, it is able to enforce that the system states converge to the sliding variable [17, 18]. Nevertheless, the more disadvantage in the supertwisting algorithm is difficulty of designing the gains of the signum function, which leads a very slow convergence and slowly setting time response [14]. In this context, several works have been presented recently proving the stability of the STA using Lyapunov theory and presenting an easy synthesis method of these gains [18]. In addition, great effort has been devoted to enhance the convergence and the robustness of the traditional STA. We can quote some methods: paper [19] proposes the addition of a new term in the classical STA, which leads to improved convergence. In [18, 20], the regulation mechanism has been modified by adding a linear term of the sliding variable to the traditional STA. To this end, the aforementioned methods can be only improving the speed convergence of the sliding variable in zero, but they lead a large overshoot of the system response.
In this paper we focus on developing a new modified structure of STA with bounded uncertainty in order to limit the overshoot and shorten the settling time of the system response. This new structure has double closed-loop feedback terms. The first one consists of an outer loop negative feedback to accelerate the sliding variable to close to zero and the second feedback is a correction term about an auxiliary variable to reduce the overshoot. Compared with the existing results, the main contributions of this paper are highlighted as follows:(i)Apply the proposed method for an uncertain nonlinear system, considering two types of uncertainties such as matched and unmatched.(ii)An integral sliding surface is designed to construct the reachability of the sliding mode.(iii)A variable-length pendulum system is included to illustrate the applicability of the proposed STA and a comparative study is established with other STA schemes.
The rest of this paper is organized as follows: Section 2 describes the mathematical system description and the problem formulation. The proposed approach is detailed in Section 3. In Section 4, the proving of reaching condition using the Lyapunov function is given. Simulation results are presented in Section 5, and conclusion remarks are in Section 6.
2. Mathematical System Description and Problem Formulation
Consider a second-order uncertain nonlinear system described by the state equation:where , are the state variables of the system, is the state vector, is the input signal control, and denotes the output vector. and represent the nonlinear dynamic function and the nonlinear control function, respectively. and are the corresponding unknown uncertainties of nonlinear vector which can be regarded as satisfying the following assumptions.
Assumption 1. The matched uncertainty is assumed to be bounded by the unknown scalar such that
Assumption 2. There exists an unknown nonnegative nonlinear function such that the unmatched uncertainty is bounded as
The main objective of this paper is to design a control input signal for the nonlinear uncertain system (1), which satisfies assumptions (2) and (3), such that the sliding variable converges to zero in finite time.
Let us consider the classical STA with matched uncertainty used for the design of second-order sliding mode controller [18, 19]:where and are, respectively, the sliding variable and the auxiliary variable, and are some positive constants, and the uncertainty can be expressed as follows: where is the amplitude of uncertainty. Its value should satisfy the following inequalities:where and are positive constants.
In order to accelerate the convergence of the sliding variable to zero, the gain of signum function must have large values. On the other hand, the values of and should be as small as possible to reduce the chattering phenomenon [19]. To avoid the conflict, many researches [18, 20, 21] propose the following STA system:where and are positive constants.
The modified structure (7) is used to obtain a faster convergence of . However, if the absolute value of increased, then increased also; consequently this produces a long setting time and overshoot of sliding variable. To solve this problem, we propose a novel modified STA in the next section.
3. Proposed Approach
The objective of this paper is to ameliorate the structure of supertwisting algorithm in order to improve the convergence of sliding variable. However, the proposed method includes a new structure to limit the absolute value of defined in (7), in which a negative feedback term about to is added. The new modified algorithm with double closed-loop feedback can be formulated as follows:where are positive constants. The parameters of the novel modified STA can be selected according to the matched uncertainty (5). and can be fitted to reduce the chattering phenomenon and and can be adjusted to guarantee the convergence of the sliding variable . Using relation (8) and the initial conditions , we can calculate the regulation mechanism as follows:From (8) and (9), we can note that the overshoot of the sliding variable can be reduced by and the linear correction term makes the system faster. Indeed, the performance advantages of the new structure of ST are achieved.
3.1. Stability Analysis of the Novel Structure of STA
In this section, the task is to determinate sufficient conditions to ensure the robustness of the modified algorithm with matched uncertainty (5). Relation (8) can be driven as a mathematical model:where .
To facilitate further development, we take .
The derivative of the vector using (10) is given bywhere Using the fact that , , , and are positive scalars, then the matrix is Hurwitz.
Choose and , where and . The stability of the novel modified STA can be designed according to the following theorem.
Theorem 3. Consider the new modified STA (8) with the matched uncertainty (5). If the parameters , , , and are positive constants, then the sliding mode of system (1) will be built in finite time; that is, the sliding variable will converge to the sliding surface in finite time.
Proof. Propose the following candidate Lyapunov function with respect to the vector :where is a symmetric positive definite matrix asIt is easy to show that the Lyapunov function (13) can be bounded from both sides [17] bywhere and are, respectively, the minimum and maximum eigenvalues of and is the Euclidean norm of .
Using relation (15), it results thatAccording to (17), it turns out thatThe time derivative of can be calculated as follows:whereTo guarantee is negative definite, (19) can be handled as follows:withwhere and are symmetrical matrices.
The time derivative of is bounded as follows:where are, respectively, the minimum eigenvalue of and.
Using (16)–(18), the inequality (23) can be expressed as follows:where and are positive constants.
Hence, the time derivative of is bounded as follows:Equation (25) shows that the derivative of the Lyapunov function is negative definite . We can conclude that the new modified STA (8) with matched uncertainty (5) converges to zero in finite time.
3.2. Comparative Study
A comparative analysis has been made to compare the convergence performances between the different structures of supertwisting algorithm: the classical STA, the modified STA (MSTA) (see references [18–21]), and the novel modified STA. In order to obtain smaller amplitude chattering, the parameters and should be adjusted as small as possible, and the parameters and are selected to guarantee the convergence of the sliding variable. These parameters are chosen as , , , and . Figure 1 shows the dynamic of the sliding variable and the auxiliary variable with the following initial conditions: and .
It can be observed in Figure 1 and Table 1 that a faster speed of the sliding variable approaching zero is obtained in modified STA and the novel modified STA, as they add the same linear correction term compared to classical STA. Moreover, by the effect of the linear feedback term added in the novel modified structure, the sliding variable obtains small overshoot (1.1%). Thus, the new structure of STA proposed achieves the best convergence performances in terms of settling time and overshoot.
4. Sliding Mode Reachability
To reduce the steady error, an integral term of tracking error is introduced, which makes up the integral sliding surface [22, 23] as follows:where is the tracking error, denotes the reference trajectory, and and are positive constants.
The time derivative of (26) yields thatUsing (1), the above equation can be rewritten asSubstituting (8) into (28), the control law is given by
Theorem 4. Consider a dynamic uncertain second-order system (1) subject both matched and unmatched uncertainties. If the sliding mode surface is selected as (26) and the control input is designed as (29); then, the states variables converge to the trajectories signal and the sliding variable will reach the sliding surface in finite time.
Proof. Consider the Lyapunov function as follows:To ensure that the sliding mode is reached in finite time, the derivative of with respect to time must be negative definite.
Take the time derivative of Lyapunov function as follows:Substituting (28) into (31), we can obtainSubstituting equation (29) into (32), it results thatUsing (2) and (9), (33) can be calculated asand then can be rewritten aswhere . It is clear from (35) that if , then is negative definite.
Therefore, we conclude that the sliding mode can be reached in finite time and the control law as defined in (29) would guarantee that when .
5. Simulation Results and Discussions
In this section, the effectiveness of our developed algorithm will be illustrated using a pendulum system example. We compare our results with those obtained by other existing methods such as classical and modified STA (MSTA). The pendulum system is driven by an engine installed on the top side, which is called control torque (see Figure 2) [24]. Thus, the task is tracking some function in real time by the oscillation angle of the road.
The dynamic equation of the pendulum system is expressed byThe parameters of this system are presented in Table 2.
We consider and ; the state space of the pendulum system can be described byThe unmatched and matched uncertainties are chosen by the following equations:According to Assumptions 1 and 2, it can be verified that and . The bounded uncertainty of the supertwisting algorithm can be selected as .
In the simulation, the initial values for the state system (37) are selected as and and the following parameters are used as , , , , , and .
Figures 3 and 4 show, respectively, the system state variables and of the pendulum system with both matched and unmatched uncertainties using different approaches. It is easy to see that the angular coordinate and the angular velocity of the system for the three methods converge to desired trajectories without chattering phenomenon despite the presence of uncertainties. For comparison, we can note that the performance results of our proposed method are better than what is reported in [18] and [21].
However, the results of Figures 5 and 6 show that the proposed structure of STA outperform the MSTA. Indeed, Figure 5 illustrates the evolution responses of the sliding variable for the different structures of supertwisting algorithm. We can note the novel modified STA given a faster convergence of sliding variable and smallest tracking error (Figure 6) compared to other results obtained by especially using the classical and modified STA [18–21]. A similar analysis can be seen in Table 3. The performance indexes of different methods listed in this table confirm that the novel modified STA has a faster settling time and shorting overshoot. This indicates that the proposed approach is more resistant to uncertainties.
The time evolution of the control signals shown in Figure 7 clearly demonstrates that the case of using the novel structure is producing small vibrations. In addition, the superiority of the new modified STA is shown in Table 4. Consequently, the proposed control method has good performance qualities for the nonlinear model of pendulum system in the presence of matched and unmatched uncertainties.
6. Conclusion
In this paper, a new structure of supertwisting algorithm is proposed and applied to an uncertain nonlinear system subject to matched and unmatched uncertainties. The main idea consists of adding two closed-loop feedback terms to the traditional supertwisting with bounded uncertainty in order to ameliorate the performances of the system response. Theoretical analysis is achieved to guarantee the stability using the Lyapunov function. Therefore, a comparative study demonstrates that the proposed approach can improve the convergence of the sliding variable and makes the system faster despite the presence of uncertainties. The proposed approach shows favorable results compared with the methods reported in the literature.
Conflicts of Interest
There are no any conflicts of interest related to this paper.