Convergence Time Calculation for Supertwisting Algorithm and Application for Nonaffine Nonlinear Systems

Zhang, Jianhua; Zhu, Quanmin; Li, Yang

doi:https://doi.org/10.1155/2019/6235190

Complexity

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Abstract Introduction Preliminaries Conclusion Appendix Data Availability Conflicts of Interest References Copyright Related Articles

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Advanced Control and Optimization for Complex Energy Systems

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Research Article | Open Access

Volume 2019 | Article ID 6235190 | https://doi.org/10.1155/2019/6235190

Convergence Time Calculation for Supertwisting Algorithm and Application for Nonaffine Nonlinear Systems

Jianhua Zhang,^1,2Quanmin Zhu,³and Yang Li^1,2

Guest Editor: Chun Wei

Received25 Jul 2019

Revised01 Sept 2019

Accepted07 Sept 2019

Published20 Oct 2019

Abstract

In this study, an accurate convergence time of the supertwisting algorithm (STA) is proposed to build up a framework for nonaffine nonlinear systems’ finite-time control. The convergence time of the STA is provided by calculating the solution of a differential equation instead of constructing Lyapunov function. Therefore, precise convergence time is presented instead of estimation of the upper bound of the algorithm’s reaching time. Regardless of affine or nonaffine nonlinear systems, supertwisting control (STC) provides a general solution based on virtual control law skill ensuring the output of the systems converges to the origin point at exact time. Benchmark tests are simulated to demonstrate the effectiveness and efficiency of the algorithm.

1. Introduction

Sliding mode control (SMC) has become one of the most efficient techniques to control uncertain complex systems and engineering [1–3]. Theoretically, such controllers are able to compensate and match disturbances by confining the systems’ trajectories in a properly chosen hypersurface (the so-called sliding manifold) [4–6] and, under the chosen surface, make the origin of the state space an asymptotically stable equilibrium point for the closed-loop systems [7–9]. Both the convergence to the sliding manifold and convergence to the origin are guaranteed in a finite time interval if the control action is large enough to counteract the effect of the uncertain terms and hypersurface is defined suitably [10–12].

STA is a well-known second-order sliding mode (SOSM) algorithm introduced in [13], and it is a possible solution for chattering reduction and widely used for control, observation, and robust exact differentiation. The reaching time estimation and the finite-time control method design are essentially complex problems with the sliding mode control systems’ study [14–16]. Finite-time convergence and robustness of the STA have been proved with some methods, such as geometrical methods [17], homogeneity properties of the algorithm [18], and Lyapunov methods [19]. The Lyapunov stability theorem and Lyapunov function provide a means of determining stability without explicit knowledge on system solutions [20–22]. Traditionally, quadratic Lyapunov functions are constructed to analyze and control design of nonlinear dynamic systems [23–25]. It should be noted that there also exist some other formats of Lyapunov functions, such as integral Lyapunov function, barrier Lyapunov function, and vector Lyapunov function. Such attempts have enhanced Lyapunov function applications in control system design. A strict Lyapunov function is provided to ascertain finite-time convergence, and it would provide an estimate of the convergence time, as well as the robustness of the finite-time or ultimate boundedness for the STA [19, 26]. By a detailed analysis of the Lyapunov function in finite time, robust convergence for the STA is proved, and it is not possible to provide necessary and sufficient conditions to estimate the convergence time from it.

However, the form of the estimate of the convergence time contained arbitrary positive matrixes which are related by the algebraic Lyapunov equation (ALE), making it difficult to operate with it for applications or further developments [24, 27, 28]. For nonlinear systems, numerical techniques (open solutions) have played a significant role in the controller design process [29–31]. As general solutions to the complicated nonlinear dynamic problem, the U model uses linear approaches to design nonlinear control [32, 33]. Specifically, it can be concluded that the applicable systems fall into classes of an increasing order of complexity: strict feedback, pure feedback, affine form, and nonaffine systems [34]. Nonaffine systems are difficult to control because of the complexity of the systems.

Motivated by the above observations, this paper presents accurate convergence time of the STA without and with perturbation and designs the nonaffine STA finite-time control. The main contributions are listed as follows:(1)Accurate convergence time is proposed for the STA based on the analytical solution of the differential equation. The main advantages are that the output of the systems converges to the origin point at exact time and the exact time is determined by the designer before the controller is implemented.(2)To overcome the main obstacle, the trajectory of the STA is analyzed by the analytical solution of the differential equation and LaSalle’s invariance principle instead of Lyapunov stability theory used commonly in the traditional sliding mode control.(3)Technically, for nonaffine nonlinear systems, sliding mode control based on the STA and backstepping achieves finite-time stabilization. The closed-loop control systems will achieve stability in finite time without violation of the constraint.

The rest of this study is organized as follows: In Section 2, the problem formulation and preliminaries are presented, which also contains some definitions and lemmas about the STA. They are presented to establish a basis for designing and analyzing the STA. In Section 3, accurate convergence time for finite-time convergence of the STA is developed by the parametric equation in different initial conditions. In Section 4, accurate convergence time is developed by the parametric equation in different initial conditions with perturbation. In Section 5, STA control is designed for nonaffine nonlinear systems based on the backstepping skill. The closed-loop systems’ trajectory is analyzed, and the output can be obtained effectively by the finite-time algorithm. In Section 6, three simulated case studies are conducted to initially demonstrate the efficiency and effectiveness of the procedure. In Section 7, the conclusions are given to summarize the study.

2. System Description and Preliminaries

The STA can be written aswhere are the scalar state variables: , and are gains to be designed.

If , the states of the system reach the origin in finite time ; therefore, the system undergoes finite-time convergence [19], using to indicate STA (1).

Lemma 1 (see [35]). For differential equation,The solution is in the parametric form:

More details about the solution of the parametric equation (3) are presented in Appendix A.

3. Finite-Time Convergence of STA

As mentioned, when , the accurate convergence time is considered. To show the trajectory of the STA, the analytic method is used to describe the system.

If the initial condition , then will reach the origin point in finite time at , where indicates the time of state when reaches the origin, and let ; if the initial condition , then will reach the origin point in finite time at , where indicates the time of state when reaches the origin, and let .

The main contribution are the following three facts: Fact 1. For the STS (1), when , and hold, the initial conditions are ; if holds, the system state arrives at the zero point in finite time. As shown in Figure 1, , and hold. Fact 2. This is the most important fact; for the STS (1), when , and hold, the initial conditions are ; if holds, the system states arrive at the zero point simultaneously. As shown in Figure 2, the initial conditions hold, in period ; then at time , hold. Fact 3. For the STS (1), when , and hold, the initial conditions are , and the system states arrive at the zero point simultaneously.

As shown in Figure 3, the initial conditions hold, and then system states change as in Figure 1 in period , where state converges to zero at time ; after that, as in Figure 2, states converge to zero in period at time , where hold.

To prove the facts above, in , let

Its derivative on the trajectories of (B.2) and (B.6) can be obtained as follows:where

The details about the parameters of equation (6) are presented in Appendix B.

Then, based on the parametric solution in Appendix A and general form of (5), define the solution in different situations by the parameter and define the trajectory in different parameters and different initial conditions.

Case I. For the supertwisting system, when , the trajectory iswhere the parameter is determined by initial conditions , , and as follows:Therefore, the initial parameter can be obtained asand then the parameter is obtained asThen, the whole trajectory can be obtained for the parametric equation, when , , and :The state will arrive at the origin point at where the following conditions are met:(a1)When , .(a2)When , if , .(a3)When , if , .Therefore, for the STS at time ,

Remark 1. If holds, in (12), and the system converges to the origin point at time ; if not, the system changes into another STA with initial conditions and because of the time invariance of the STA.

Case II. For the supertwisting system, when ,For the initial conditions, (13) becomesTherefore, the initial parameter can be obtained asand then the parameter can be obtained asThen, for the parametric equation, when and ,According to system (7) and convergence time (17), the states of the system reach the origin point at the same time:Therefore, the system reaches finite-time stability.

Remark 2. If the initial condition , based on the convergence time , both the states reach the origin point at the same time.
The supertwisting system is the time invariance system, and states reach finite-time stability when ; therefore, if the initial condition , based on the convergence time , by calculating both the states at , a new supertwisting system with the initial condition can be obtained. Combining process initial conditions and , the supertwisting system convergence time can be obtained.

Theorem 1. The supertwisting system (1) reaches finite-time stability, if gains satisfy and system’s initial condition ; the finite time satisfies , where is the time when state reaches the origin point in first time , and then time is elapsed, when both states reach the origin point at the same time; therefore, the STS reaches the origin point at time where the following conditions are met:(b1)When , .(b2)When , if , .(b3)When , if , .Here,

Proof of Theorem 1. Based on the supertwisting system and gains , if , the parametric solution can be obtained in the form of (7); based on the initial condition, parameters can be obtained by (9) and (10), and then can be obtained in following different situations: (b1)–(b3).
When , the states of the system change as in (12), and then the system will elapse the period , given by (17), which the parameter satisfies:whereThen, at , the states of the supertwisting system reach the origin point:This completes the proof of Theorem 1.

Case III. For the supertwisting system, when , the trajectory iswhere () and ().Therefore,Then, the whole trajectory can be obtained for the parametric equation, when , , and :The state will reach the origin point at , where the following conditions are met:(c1)When , .(c2)When , .(c3)When , if , .(c4)When , if , .Then, the states of the supertwisting system at time become

Case IV. For the supertwisting system, when , the trajectory isFor the initial conditions ,Therefore,for the parametric equationTherefore, the convergence time isTherefore, the system reaches finite-time stability.

Theorem 2. The supertwisting system (1), if gains satisfy and system initial conditions , reaches finite-time stability; the finite time satisfies , where is the time when state reaches the origin point in first time , and then time is elapsed, when both states reach the origin point at the same time.(d1)When , .(d2)When , .(d3)When , if , .(d4)When , if , .Here,

Proof of Theorem 2. Based on the supertwisting system and gains , if , the parametric solution can be obtained in the form of (23); based on the initial condition, parameters can be obtained by (25) and (26), and then can be obtained in following different situations: (d1)–(d4).
When , the states of the system change as in (29), and then the system will elapse period , given by (35), which the parameter satisfies:whereThen, , and the states of the supertwisting system reach the origin point at time :This completes the proof of Theorem 2.

Remark 3. From Theorem 1 and Theorem 2, the convergence time is and then exist in both and in the form of ; therefore, the gain parameter is not influenced by the convergence time.

4. Finite-Time Convergence of STA with Perturbation

The STA with perturbation can be written aswhere are the scalar state variables, with initial conditions ; are gains to be designed; and are the perturbation terms. It is well known that the STA is robustly stable to perturbations globally bounded, and the equilibrium point is ; therefore, suppose the perturbation terms of the system.

Assume that .

Letthen system (41) can be changed as

Next, letthen the system can be written as

The initial conditions are and parameters are , where , .

Under the conditions on STA without perturbation in Theorem 2, the robust finite time is presented for STA with perturbation. Assume , then . The finite time is presented by , where and can be obtained by Theorem 2 in different initial conditions and gains.

In Theorem 2, is dependent on and is then calculated in different situations to get the maximum. To compare the influence of perturbation, three STAs are built:

Without loss of generality, assume that , in time phasing .

For the perturbation of the STS at different gains and different initial conditions , assume states , , and as , , and ; therefore, and in .

The convergence time satisfies .

Assume that ; to compare the influence of perturbation, three STAs are built:

Choose a sliding mode surface as , , and , where ; therefore,because of

Therefore,

Because trajectories converge to origin in finite time, the trajectory reaches origin in finite time. Parameters are arbitrary positive constants; therefore, could be equivalent to . The convergence time satisfies

Theorem 3. For the supertwisting algorithm with perturbation (41), if and hold, the system converges to the origin point in finite time, and the maximum convergence time is , where and .

Proof of Theorem 3. For STA with perturbation, when the initial condition , in the first stage, the state would arrive at the origin point in finite time , and then when the system is in the second stage, both states arrive at the origin point in finite time:Then, the maximum finite time is presented as .

5. STC for Nonaffine Systems

To illustrate the STA, nonlinear system STC is proposed for the nonaffine system:

Using the coordinate transform,

The virtual control law is

Then,where

Choose the ideal control lawthen

Therefore, the system satisfies Theorem 1 and Theorem 2 by choosing the gain parameters .

To overcome the nonaffinity, the virtual control and backstepping skill are used. They are extremely important to ensure the successful design of the proposed STC.

6. Simulation

Several simulated examples were selected to conduct bench tests of the STA and STC. Example 1–Example 6 were testing simple STA, and the purpose of testing these examples was to investigate whether the convergence time is solved from the parametric equation. Examples 7–10 were designed for the test of STC for nonaffine nonlinear systems and demonstrated the superiority of STC and convergence time of the STA.

Example 1. Consider the STS given by (1), with , , , and in Case I; based on (9) and (10), we can get and , respectively, and then we can get ; therefore, the state reaches the zero point at time 1.0253.

Example 2. Consider the STS given by (1), with , , , and in Case II; based on (16) and (17), we can get and , respectively, and then we can get ; therefore, the states reach the zero point at time 9.0256.

Example 3. Consider the STS given by (1), with , , , and in Case III; based on (25) and (26), we can get and , respectively, and then we can get ; therefore, the state reaches the zero point at time 0.4641.

Example 4. Consider the STS given by (1), with , , , and in Case IV; based on (32) and (33), we can get and , respectively, and then we can get ; therefore, the states reach the origin point at time 3.4641.

Example 5. Consider the supertwisting system given bywith and initial conditions .
Based on Theorem 1, the system (60) reaches finite-time stability and calculates the time from parametric equation (7) step by step. Firstly, the initial parameter is calculated from (9) and from (10) because holds; therefore, and can be obtained from (11) as and in Figure 4. Secondly, is calculated from (17) in Figure 5. Finally, the reaching time in Figure 6.

Example 6. Consider the supertwisting system given bywith and initial conditions .
Firstly, the initial parameter is calculated from (25) and from (26) because , and hold; therefore, and can be obtained from (27) as and in Figure 7. Secondly, is calculated from (35) in Figure 8. Finally, the reaching time in Figure 9.

Example 7. Consider the STA with perturbation as given by (45), as :In Figure 10, states of three STAs are presented to indicate stability in finite time, and convergence time is , , and , respectively. The trajectory indicates states , , and with the same initial condition reach zero at different time because of different , , and . The trajectories are in between , respectively. State is the last state that reaches zero; therefore, the maximum convergence time . In addition, the maximum of the state is and absolutely necessary to estimate finite time in the next step.

Example 8. Consider the STA with perturbation as given by (45), as :In Figure 11, states of three STAs are presented to indicate stability in finite time, and convergence time is , , and , respectively. The trajectories are in between , respectively. State is the last state that reaches zero; therefore, the maximum convergence time . In addition, states of the sliding mode surface indicate the state is in between , providing evidence that is the maximum convergence time.

(a)

(b)

(c)

(d)

Example 9. Consider the supertwisting system with perturbation given bywhere are the scalar state variables, with initial conditions ; are gains; and and are perturbation terms. Therefore, the system (64) changes aswhere ; therefore, parameters are and , and initial conditions are ; based on Theorem 3, the convergence time can be obtained.
The whole convergence process can be split into three stages. Firstly, the state from initial condition reaches zero in finite time , and at this time, . Secondly, both states reach zero at the same time and elapse time in this stage. Finally, after last two stages, the states would stay at zero.
From Figure 12, it can be seen that, in the first stage, state from the initial condition reaches zero in finite time as in Figure 10 for Example 7, and the maximum convergence time is calculated, and in this stage, the maximum of state is . Therefore, in the next stage, the initial condition is , such as in Figure 11 for Example 8, and then the maximum convergence time is calculated. Finally, the total finite convergence time ; therefore, the system converges to zero after 12.3263.

(a)

(b)

Example 10. Consider a nonaffine nonlinear systemwhere are the output and input, respectively, and the reference output is taken as . The control objective is to design the STC for the system such that the output follows the given reference signal in finite time.
Choose states and with virtual control whereThe states satisfy the STA condition and initial conditions . Therefore, the states satisfy the condition in Figure 9 and output and reference output in Figure 10.
It can be observed that, for the nonaffine system, the closed-loop system has been controlled in finite time effectively in Figures 13–15, and time can be calculated and designed by control gain and initial condition.

7. Conclusion

A method to calculate the accurate convergence time of the STA has been provided, and an STC method for nonaffine nonlinear systems based on the STA is given. The trajectory of the STA is solved in the form of the parametric equation, and reaching time is calculated accurately. The result indicated the gain parameter is not influenced by the convergence time. For the application of the STA, STC is designed to solve the nonaffine systems’ control problem by backstepping skill.

Furthermore, after a certain prefixed gain and initial conditions, the reaching time of the STA is also useful to design STC. In the future, there still exist some problems to face with, such as in the STA and higher order twisting algorithm.

Appendix

A. Parameter Solution

Base on Lemma 1, the following solution can be obtained:where ; calculating the integral , we getand then calculating the integral , we get

Therefore, the solution of the parametric equation is discussed in different cases. Case I. ; based on equations (A.2) and (A.3), the solution can be obtained as To get finite time clearly using , equation (A.4) is changed as where is a new parameter variable. ase II. ; based on equations (A.2) and (A.3), the solution can be obtained as To get finite time clearly using , equation (A.6) is changed as where and because of and ; therefore, and . Case III. ; based on equations (A.2) and (A.3), the solution can be obtained as

B. Parameter Equation

To indicate (5) holds, three different situations are considered based on the sign of the initial condition .

Firstly, assume that , then , for . Let us discuss the trajectory when , , and , then based on the STA in

Substituting (B.1) into (1) leads to

Letthen its derivative on the trajectory of (B.2) can be obtained as follows:

Secondly, if the initial condition , then

Substituting (B.5) into (1) leads to

Letthen its derivative on the trajectory of (B.6) can be obtained as follows:

Finally, if because , , and when , then based on the STA in

Substituting (B.9) into (1) leads to

Letthen its derivative on the trajectory of (B.10) can be obtained as follows:

Based on (B.4), (B.8), and (B.12), the general form of (5) holds.

Data Availability

The data in this paper are simulated with MATLAB. No other data are present.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2019 Jianhua Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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