Finite-Time Stability of a Second-Order Bang–Bang Sliding Control Type

Abstract

This paper presents the double chain–integrator finite-time convergence in a closed loop with a second-order bang–bang sliding control. The direct Lyapunov method carried out the stability analysis and the reaching time estimation using a suitable non-smooth strong Lyapunov function. That is, the proposed energy function is strictly positive definite, with a strictly definite negative time derivative. Additionally, the proposed function estimates the reaching time in the presence of matching bounded perturbations. Numerical comparisons with well-known approaches were performed to assess the proposed strategy’s effectiveness.

1. Introduction

1.1. Motivation

Sliding modes (SMC) have been studied extensively for over 50 years and are widely used in practical applications due to their simplicity, robustness against parameter variations and disturbances, and finite-time convergence [1,2]. The fundamentals of this technique can be found in [1,3,4]. Sliding modes are nonlinear and usually have discontinuous right-hand sides. Their mathematical descriptions require the consideration of a special class of differential inclusions (see, [5,6]), and the stability analysis of such systems is a non-trivial task. For first-order sliding modes, Lyapunov stability is used to prove finite-time convergence and to obtain an upper bound for the reaching time. On the other hand, the stability analysis and the estimation of the reaching time for high-order sliding mode(s) (HOSM) have been studied using geometric proofs [7,8,9]. Nowadays, researchers are focused on finding suitable Lyapunov functions to prove the stability of HOSM [10,11]. In previous referenced works, a strict Lyapunov function was proposed for the super-twisting algorithm. The proposed Lyapunov functions ascertain a finite-time convergence, providing an estimate of the convergence time. At the same time, they ensure the robustness of the finite time or ultimate boundedness for a particular class of perturbations. The proposed functions are quadratic forms; the operations with them are as simple as for the linear time-invariant systems. In fact, finding such Lyapunov functions is an open problem for many classes of nonlinear sliding mode controllers. One common approach involves the solution of a set of partial differential equations [12]; unfortunately, the solution is complex to obtain. Another work dealing with this problem is [13], where a constructive method to find the non-smooth Lyapunov function was proposed based on the application of the characteristic method. Non-smooth Lyapunov functions are tools for developing the stability analysis of HOSM. For instance, in [10], the first strong Lyapunov function was presented for the super-twisting algorithm. This result is a breakthrough in proving finite-time stability for variable structure systems. Later, based on this work, several results have developed new non-smooth Lyapunov functions for discontinuous and adaptive systems. In [14], the extensions of strict Lyapunov functions for the arbitrary order Levant differentiator are presented. This function allows the design of the gains to obtain a suitable finite-time convergence. The authors of [15] developed Lyapunov functions incorporating in their structures the classical persistency of excitation conditions, showing global uniform asymptotic stability of the associated adaptive systems under sufficient and necessary conditions. These functions were applied to the parameter estimation problem. For the so-called integral sliding modes, the work in [16] proposes novel functions for the controls of mechanical systems. Notice that the flexibility of the Lyapunov approach allows the development of control algorithms to solve more particular problems. For saturated nonlinear systems, the work in [17] proposed a strong Lyapunov function to deal with the saturated super-twisting algorithm. Moreover, the implicit Lyapunov function methodology has also been applied to prove finite, and even more, fixed-time stability using the concept of homogeneity [18,19,20].

1.2. Contribution

In this work, the robustness property and finite-time convergence of a double integrator, controlled by a second-order sliding mode bang–bang type, is studied. This kind of sliding mode has been successfully used in the control community, both as an active disturbance rejection control and as a tracking differentiator [21,22,23], with some appealing properties, such as finite-time convergence and robustness to bounded perturbation terms. The stability analysis and the reaching time estimation are based on the Lyapunov method. To this end, we fit a subtle Lyapunov function, which allows estimating the finite-time convergence in the presence of uncertain matching-bounded perturbations. The main contribution of this work is the adaptation of a Lyapunov function to prove the finite-time convergence of the controller. The effectiveness of the proposed SMC was assessed via a numerical comparison with a well-known twisting algorithm for a particular underactuated manipulator.

1.3. Manuscript Organization

The remainder of this work is organized as follows. In Section 2, the problem statement and its corresponding motivation are introduced. Section 3 presents a strictly Lyapunov function for the stability analysis and the reaching time estimation. The numerical comparisons that allowed us to assess the proposed bang–bang type SMC’s effectiveness are presented in Section 4. The concluding remarks are given in Section 5. An Appendix is included where the corresponding proofs of some needed properties are developed.

1.4. Notation and Definitions

Consider the following differential equation, described by $\dot{x}=f\left(x\right)$, $x\in D\backslash N$, where $f\left(x\right)$ is defined and continuous on $D\backslash N$, and N is a set that has Lebesgue measure zero (where $f\left(x\right)$ can take multi-values at each point in N).

Definition 1.

The system $\dot{x}=f\left(x\right)$ is Finite-time stable if all trajectories start in a neighborhood of $x=0$ converge to $x=0$ in finite-time [24].

Definition 2.

A function $f:R^n\to R^n$ is called homogeneous with the degree m [24], with respect to the dilation ${\mathsf{\Lambda }}_{\mathbf{r}}:x$$\to diag\left({\lambda }^{r_i}\right)x$, where $\mathbf{r}={\left(r_1,\dots ,r_n\right)}^{\top }$; with $r_i>0$; for $i=\{1,\dots n\}$ (weighted homogeneity), if for any $\lambda >0$ the identity ${\lambda }^{-m}{\mathsf{\Lambda }}_{\mathbf{r}}^{-1}f\left({\mathsf{\Lambda }}_{\mathbf{r}}x\right)=f\left(x\right)$ holds (the function f can be a vector-set field or a vector field (see definition in [25])). During the development of this study, it is used

$$\mathrm{sign}\left[z\right]=\left\{\begin{array}{cc}1& ifz>0\\ [-1,1]& ifz=0\\ -1& ifz<0\end{array}\right.$$

2. Problem Statement

It is well known that for a double integral plant [26,27]

$$\begin{array}{c}\dot{x}=y\hfill \\ \dot{y}=u+f(x,y,t),\hfill \end{array}$$

(1)

with $\left|u\right|\le r$ and v is the desired value for x, the time-optimal solution is given by

$$u=-r\mathrm{sign}\left[x-v+\frac{y\left|y\right|}{2r}\right].$$

(2)

On the other hand, in the works [21,22,23], system (1) in the closed loop with (2) is referred to as a tracking differentiator. That is, for a given time-variant reference, v, the state variable, and y converge to $\dot{v}$ in a finite time when r is sufficiently large. Using this principle, it is interesting to study the stability properties of the following second-order uncertain system, defined by

$$\begin{array}{c}\dot{x}=y\hfill \\ \dot{y}=-r\mathrm{sign}\left[x+\frac{y\left|y\right|}{2r}\right]+f(x,y,t),\hfill \end{array}$$

(3)

where $x\in R$ and $y\in R$ are the states, $f(x,y,t)$ is an uncertain measurable but bounded function, i.e.,

$$\left|f(x,y,t)\right|\le Ct\ge 0,$$

with a known constant C $>0$. In general, the second-order sliding mode (3) is referred to as the minimum time system (MTS) [27]. Notice that the right-hand side of (3) is discontinuous. Therefore, their solutions are defined in the Filippov sense [5]. On the other hand, it is easy to see that system (3) has a sliding motion in the manifold $M=\{(x,y)\in R^2$: $x+y\left|y\right|/\left(2r\right)=0\}$. That is, all the trajectories of the system (3) converge towards the manifold M. Once the solution is inside of M, it converges asymptotically to the single equilibrium point $(x=0,y=0)$. It should be noted that the stability of the unperturbed system can be assured after using the following weak Lyapunov function

$$V_0(x,y)=r\left|x+\frac{y\left|y\right|}{2r}\right|+\frac{y^2}{2},$$

(4)

whose time derivative, along the trajectories of the system (3), leads to

$${\dot{V}}_0(x,y)=-r\left|y\right|.$$

Using function (4), only the stability in the Lyapunov sense can be assured. The global asymptotic stability is concluded by applying the extension of the invariance principle, made in ([6], Theorem 3.2), to the discontinuous system (3). Furthermore, it can be shown that the system is homogeneous with a degree $q=-1$ with respect to dilation $\mathbf{r}=\left[2,1\right]$ (see Appendix A.1), hence, according to the homogeneity principle, the global finite time stability of this system is additionally guaranteed by its global asymptotic stability ([6], Theorem 4.2). Reading [28] is also recommended.

Notice that, using the proposed weak Lyapunov function $V_0$, nothing can be said about the robustness properties or the estimation of the time of convergence of the solution of the system (3). This section is ended by introducing the main control problem of this work.

 Problem Statement:

The main control problem consists of finding a suitable strict Lyapunov function to assure the convergence to the origin, in a finite-time, of the system (3), despite a larger class of external perturbations. It is also interesting to compute an upper bound for the convergence time for both the perturbed and unperturbed cases.

3. Strictly Lyapunov Function for Stability Analysis

To simplify the stability test of this closed-loop system, it should be noted that it can be expressed as:

$$\begin{array}{c}\dot{s}=y-\left|y\right|\mathrm{sign}\left[s\right]+\frac{1}{r}f(.)\left|y\right|\hfill \\ \dot{y}=-r\mathrm{sign}\left[s\right]+f(.)\hfill \end{array}$$

(5)

where

$$s=x+\frac{y\left|y\right|}{2r},$$

and for simplicity $f(.)=f(x,y,t)$. The corresponding candidate Lyapunov function is proposed as:

$$V(s,y)={\gamma }_1\left(r\left|s\right|y^2+\frac{y^4}{4}+r^2{s}^2\right)+{\gamma }_{2}y\mathrm{sign}\left[s\right]{\left|s\right|}^{\frac{3}{2}}$$

(6)

where ${\gamma }_1$ and ${\gamma }_2$ are strictly positive constants.

Remark 1.

The traditional Lyapunov theorem cannot be applied because the proposed V is not locally Lipschitz [5,29]. On the other hand, as $V\left(\varphi (t,s_0,y_0)\right)$ is an absolutely continuous function of time along the state trajectories of the system (5), it is differentiable almost everywhere. Furthermore, function $V\left(\varphi (t,s_0,y_0)\right)$ is monotone, decreasing, and converges to zero, if the time derivative of $V(*)$ is negative definite almost everywhere. This fact assures the conditions needed to apply Zubov’s Theorem [30] (Theorem 20.2, p. 568). The basin idea of this study is the stability convergence analysis, which will be carried out based on these arguments.

To prove the finite-time stability property of the system (5), other useful properties related to the corresponding Lyapunov function are introduced.

P1

Function V is strictly positive definite for all

$$4{\gamma }_1^2{r}^3\ge {\gamma }_2^2$$

The proof of this property can be found in Appendix A.2.

P2

The time derivative of function V can be upper-bounded as follows:

$$V\le K{({\left|s\right|}^{\frac{1}{2}}+\left|y\right|)}^4,$$

(7)

where K can be estimated as:

$$K=\mu +\eta ,$$

where $\mu$ is any strictly positive constant, and $\eta$ is fixed, as

$$\eta =max\left\{\left({\gamma }_1{r}^2-\mu \right),\frac{{\gamma }_2-4\mu }{4},\frac{{\gamma }_{1}r-6\mu }{6},\left(\frac{{\gamma }_1}{4}-\mu \right)\right\},$$

(8)

where the set of positive constants $\{{\gamma }_1,{\gamma }_2,\mu ,r\}$ is provided, such that $\eta >0$. The proof of this property can be found in Appendix A.3.

P3

Function V is absolutely continuous, strictly positive definite, and differentiable for all $s\ne 0$. Given its obviousness, this property is not discussed here.

Now, we show that the proposed V is a strong Lyapunov function because its time derivative is strictly negative definite, assuring finite-time stability in the presence of bounded disturbances. Therefore, the time derivative of (6) is

$$\begin{array}{c}\dot{V}\left(t\right)={\gamma }_1\left(r\mathrm{sign}\left[s\right]\dot{s}{y}^2+2r\left|s\right|y\dot{y}+y^3\dot{y}+2r^{2}s\dot{s}\right)\\ +{\gamma }_2\left(\dot{y}{\mathrm{sign}\left[s\right]\left|s\right|}^{\frac{3}{2}}+\frac{3}{2}y\mathrm{sign}\left[s\right]{\left|s\right|}^{\frac{1}{2}}\dot{s}\right)\end{array}$$

Taking into account the trajectories of (5), one reaches the following equation

$$\begin{array}{c}\dot{V}\le -{\gamma }_2\left(r-C\right){\left|s\right|}^{\frac{3}{2}}+{\gamma }_2\left(\frac{3C}{2r}+3\right)y^2{\left|s\right|}^{\frac{1}{2}}\\ -2{\gamma }_{1}r\left(r-2C\right)\left|s\right|\left|y\right|-{\gamma }_1\left(r-2C\right){\left|y\right|}^3.\end{array}$$

(9)

Substituting the following equality

$$-\kappa {({\left|s\right|}^{\frac{1}{2}}+\left|y\right|)}^3+\kappa \left({\left|s\right|}^{\frac{3}{2}}+3\left|s\right|\left|y\right|+3{\left|s\right|}^{\frac{1}{2}}{y}^2+{\left|y\right|}^3\right)=0,$$

(10)

into inequality (9), for some $\kappa >0$, leads to

$$\begin{array}{cc}\hfill \dot{V}\le & -\kappa {({\left|s\right|}^{\frac{1}{2}}+\left|y\right|)}^3-\left({\gamma }_{2}r-{\gamma }_{2}C-\kappa \right){\left|s\right|}^{\frac{3}{2}}\hfill \\ \hfill & +\left(3\kappa +\frac{3{\gamma }_{2}C}{2r}+3{\gamma }_2\right)y^2{\left|s\right|}^{\frac{1}{2}}\hfill \\ \hfill & -\left(2{\gamma }_{1}r\left(r-2C\right)-3\kappa \right)\left|s\right|\left|y\right|\hfill \\ \hfill & -\left({\gamma }_1\left(r-2C\right)-\kappa \right){\left|y\right|}^3.\hfill \end{array}$$

(11)

To obtain an upper bound for $\dot{V}$ inequality (11) is rewritten as

$$\dot{V}\le -\left({\gamma }_{2}r-{\gamma }_{2}C-\kappa \right){\left|s\right|}^{\frac{3}{2}}-\left|y\right|z^{T}Nz-\kappa {({\left|s\right|}^{\frac{1}{2}}+\left|y\right|)}^3,$$

(12)

where $z={\left[{\left|s\right|}^{\frac{1}{2}},\left|y\right|\right]}^T$, and

$$N=\left[\begin{array}{cc}2{\gamma }_{1}r\left(r-2C\right)-3\kappa & -\left(\frac{3}{2}\kappa +\frac{3{\gamma }_{2}C}{4r}+\frac{3}{2}{\gamma }_2\right)\\ -\left(\frac{3}{2}\kappa +\frac{3{\gamma }_{2}C}{4r}+\frac{3}{2}{\gamma }_2\right)& {\gamma }_1\left(r-2C\right)-\kappa \end{array}\right].$$

(13)

Matrix N can be semi-definite positive, if ${\gamma }_1>1$ and $r>2C$ are sufficiently large, with $\kappa$ sufficiently small. On the other hand, to assure the negativeness of $\dot{V}$, the first term of (12) must satisfy that ${\gamma }_2(r-C)>\kappa$. In summarizing, if the set of constants $\left\{{\gamma }_1,{\gamma }_2,r,\kappa \right\}$ is selected, such that $N\ge 0$ and ${\gamma }_2(r-C)>\kappa$, then the following inequality is fulfilled:

$$\dot{V}\le -\kappa {({\left|s\right|}^{\frac{1}{2}}+\left|y\right|)}^3.$$

(14)

Evidently, from the last inequality, it can be concluded that s and y converge asymptotically to zero, even in the presence of bounded uncertainties. However, to conclude finite-time stability, the property P2 must be substituted into (14). This leads to the following inequality

$$\dot{V}\le -\frac{\kappa }{K^{\frac{3}{4}}}{V}^{\frac{3}{4}}=-\overline{k}{V}^{\frac{3}{4}}$$

(15)

Using simple integration over the last differential inequality, it can be concluded that an upper bound for the time convergence to zero, along the trajectories generated by (5), is given by

$$t_f\le \frac{4}{\overline{k}}{V}^{\frac{1}{4}}(s\left(0\right),y\left(0\right)).$$

(16)

This section ends by presenting the main proposition of this work.

Theorem 1.

Let the set of positive constants $\{{\gamma }_1,{\gamma }_2,\mu ,\kappa ,r\}$ such that

$$\begin{array}{ccc}r>2C& ;& {\gamma }_1^2{r}^3\ge {\gamma }_2^2/4,\end{array}$$

and

$$\begin{array}{ccc}N\ge 0& ;& \eta >0,\end{array}$$

where N and η are defined previously in (13) and (8), respectively. Then, all the trajectories of the perturbed system (5) converge to zero in a finite transient time with a maximal duration, given by (16).

Proof. 

Evidently, the theorem is a direct consequence of properties P1 and P3. □

Comment 1: The proposed Lyapunov function was previously introduced in [31]. This function allows us to tune the control gains to probe the finite-time stability of the closed-loop system in the presence of uniformly bounded uncertainties. That is, the proposed V qualifies as a strong Lyapunov function, as shown in the forthcoming developments.

Comment 2: The need to measure the two variables can be alleviated by applying an exact differentiator, such as the super-twisting. As we are working with mechanical systems, the complete forward property is fulfilled. We can work with a separate stability analysis, one for the differentiator and the second one developed in this paper for the controller. That is, variable y of Equation (3) can be estimated using a super-twisting observer.

4. Numerical Simulations

To assess the effectiveness of our control scheme, we designed two numerical experiments to accomplish system stabilization. The first simulation consists of a rest-to-rest control maneuver for an underactuated single-link flexible joint manipulator. To this end, we express this system in the Riachy form. To understand how well our controller performs, we compare it with the well-known twisting control algorithm. We consider the simple pendulum for the second simulation, where we accomplish system stabilization from the origin to some desired position. To this end, we compare the case where the whole state is available against the case where the state has to be estimated using a super-twisting algorithm.

4.1. Flexible Manipulator

Consider the underactuated system shown in Figure 1, formed by a flexible joint manipulator and described by the following set of equations

$$\begin{array}{c}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=-\frac{mgL}{I}sin{x}_1-\frac{k}{I}(x_1-x_3)\hfill \\ {\dot{x}}_3=x_4\hfill \\ {\dot{x}}_4=\frac{k}{J}(x_1-x_3)+\frac{1}{J}u+\frac{1}{J}w\hfill \end{array}$$

(17)

where $x_1$ is the angular link position, $x_2$ is the angular velocity, $x_3$ is the angular position of the motor shaft, $x_4$ is the corresponding angular velocity, and u is the torque input exerted by the motor. The joint’s flexibility is modeled as a linear torsion spring, with a stiffness coefficient given by k. Coefficients m, g, L, I, and J represent, respectively, the link mass, the gravity acceleration, the link length, the moment of inertia, and the motor moment of inertia. Function $w\left(t\right)$ lumped the unmodeled but bounded dynamic plus the external disturbances. That is, we assume that

$$\left|w\right|\le C.$$

The above system is underactuated because there is only a single input and two degrees of freedom. The control objective consists of designing a sliding mode controller to stabilize the angular link position to the rest position locally: $x_1={\overline{x}}_1$ and ${\overline{x}}_3=mgLsin{\overline{x}}_1+k{\overline{x}}_1$. The following constants are introduced to simplify the forthcoming developments

$$\begin{array}{ccc}{a}_1=\frac{mgL}{I};& a_2=\frac{k}{I}& S_1=sin{x}_1\\ b_1=\frac{k}{J}& b_2=\frac{1}{J}& C_1=cos{x}_1\end{array}.$$

(18)

To rewrite the system (17) into its regular form, previously presented by Riachy in [32], the following auxiliary variables are introduced

$$z_1=-a_1{S}_1-a_2{x}_1+a_2{x}_3+k_p(x_1-{\overline{x}}_1)+k_d{x}_2,$$

(19)

and

$$z_2=-a_1{C}_1{x}_2-a_2{x}_2+a_2{x}_4+k_p{x}_2-k_d\left(a_1{S}_1+a_2(x_1-x_3)\right).$$

(20)

It is easy to see that the following differential equation

$$\begin{array}{c}{\dot{z}}_1=z_2,\hfill \\ {\dot{z}}_2=\alpha \left(x\right)+b_{2}u+b_{2}w;\hfill \end{array}$$

(21)

holds, where $\alpha \left(x\right)$ is a deterministic function of the vector state x, defined as

$$\alpha \left(x\right)=-a_1{C}_1{\dot{x}}_2+a_1{x}_2{\mathrm{S}}_1{\dot{x}}_1.$$

Therefore, from the above equation, u can be proposed as

$$u=-\frac{1}{b_2}(v_i-\alpha \left(x\right));$$

where

$$v_i=r\mathrm{sign}\left[z_1+\frac{z_2\left|z_2\right|}{2r}\right],$$

(22)

and $r>0$ must be selected according to Theorem 1. It is easy to see that the regular system (21) satisfies the conditions pointed out in Theorem 1, found in [32]). That is, if $z_1$ and $z_2$ converge to zero, then the whole state of x also converges to zero.

To provide an intuitive idea of how good our control law is, we perform a numerical comparison between our controller, $v_i$, and the twisting algorithm $v_s$, defined as

$$v_s=-{\alpha }_1\mathrm{sign}\left[z_1\right]-{\alpha }_2\mathrm{sign}\left[z_2\right],$$

(23)

where ${\alpha }_1+{\alpha }_2\le r$, and $0<C<{\alpha }_2<{\alpha }_1-C$. To avoid numerical errors, the set of parameters defined in (18) is fixed as $a_i=1$ and $b_i=1$ for $i=\{1,2\}$. The external perturbation is set as $w=0.25sin(t/5)cos(t/3)+0.1$. Additionally, the function “sign” is numerically implemented as

$$\mathrm{sing}\left[x\right]=\frac{x}{\left|x\right|+ϵ};ϵ={10}^{-4}.$$

The experiment consisted of bringing the system from its initial rest position $x=0$, to the final rest position defined as

$$(x_1(\infty )=1\left[rad\right],x_3(\infty )=1.8415\left[rad\right]),$$

with the following set of control parameters $r=1.5$, ${\alpha }_1=1$ and ${\alpha }_2=0.5$.

The obtained results are shown in Figure 2. From this figure, it can be seen that the controller (22) has a better performance than the twisting algorithm (23). That is, the time convergence of $v_i$ is better, and the overshoots in the velocity states are noticeably small. Please note that it is not claimed that the second-order sliding mode of the bang–bang type is better than the twisting algorithm. In fact, a formal comparison is beyond the purposes of the study.

4.2. Simple Pendulum

Consider the simple pendulum represented by the following dynamics:

$$\begin{array}{c}{\dot{x}}_1=x_2\\ {\dot{x}}_2=-\frac{MgL}{2J}sin\left(x_1\right)-\frac{b}{J}{x}_2+\frac{1}{J}u\end{array}$$

where M is the pendulum’s mass, L is the longitude, g is the gravitational force, b is the friction coefficient, and J is the inertia. For simplicity, for simulation purposes $\frac{MgL}{2J}=1$, $\frac{b}{J}=1$ and $\frac{1}{J}=1$. The term $-\frac{MgL}{2J}sin\left(x_1\right)-\frac{b}{J}{x}_2$ is considered the unknown dynamics. The controller was simulated with the following parameters:

$$r=10,{\gamma }_1=0.5,{\gamma }_2=0.2.$$

With this selection, the condition given in Theorem 1 is fulfilled. Figure 3 depicts the obtained numerical results. The dotted line represents the bang–bang controller considering total access to the state vector. The disadvantage of having the necessity of the position and velocity can be alleviated by applying a robust differentiator. A classical one is a super-twisting algorithm. The continuous line represents the effect of the differentiator in the closed-loop dynamics. After a small time, both strategies stabilize the pendulum dynamics at $\frac{\pi }{2}$. Figure 4 compares the behavior of the Lyapunov functions in (4) and in (6). Notice that both arrive after 0.5 to the origin $V\left(t\right)=0$. However, the second graph emphasizes the advantages of using the strong Lyapunov function. The red line in both graphs represents Equation (4). The zone of convergence is above zero, which shows us that this function cannot guarantee finite-time convergence to zero in the presence of perturbations. The proposed Lyapunov function (represented by the continuous blue line) reaches the origin even though the presence of unknown perturbations. Finally, Figure 5 shows the estimation provided by the super-twisting algorithm.

Table 1. Controller behavior against different perturbation powers.

Perturbation $\mathit{\zeta }\left(\mathit{t}\right)$	${\displaystyle {\int }_0^{\mathit{t}}\mathit{e}\left(\mathit{\tau }\right)\mathit{d}\mathit{\tau }}$
$0.1sin\left(5t\right)cos\left(3t\right)$	2.087
$0.4sin\left(5t\right)cos\left(3t\right)$	1.939
$0.7sin\left(5t\right)cos\left(3t\right)$	1.851
$0.9sin\left(5t\right)cos\left(3t\right)$	1.996

shows the robustness properties of the controller. A perturbation with the structure $\zeta \left(t\right)=Asin\left({\omega }_{1}t\right)cos\left({\omega }_{2}t\right)$ has been simulated to see the effect in the numerical results. Notice that the accumulated error represented as the integral of the tracking error did not present many values. That means that the controller is robust against deterministic noises.

Additional simulations compare the classical twisting controller with the present approach. The twisting controller is defined as $u\left(t\right)=-k_1\mathrm{sign}\left(e_1\right)-k_2\mathrm{sign}\left(e_2\right)$. In this simulation, the gains were $k_1=10$ and $k_2=5$. $e_1=x_1-\pi /2$ and $e_2=x_2$. The velocity in the last term was obtained with the super-twisting algorithm to obtain a fair comparison. Figure 6 shows the convergence to the desired position. A better selection of the twisting gain controller can provide better convergence. The gains were selected by trial and error. Even when the convergence is better with the bang–bang controller, a better selection of the gains can improve the controller’s performance.

5. Conclusions

This paper introduces a strictly Lyapunov function to prove global finite-time convergence to the double integrator’s origin in a closed loop with a second-order sliding mode of the bang–bang type. The direct Lyapunov method was used to carry out the corresponding stability analysis and estimate the reaching time. To this end, a suitable non-smooth Lyapunov function was chosen. The Lyapunov function was proven to be a strong function for estimating the reaching time in the presence of a matching bounded perturbation. The control strategy performance was assessed through two numerical experiments. The first simulation considered an underactuated system, where a comparison between the proposed control scheme against the twisting algorithm was made; the second simulation consisted of stabilizing a simple pendulum from the origin to the desired rest. To this end, two scenarios were explored, one when the system position and velocity were available, while the other, the non-available velocity was computed using a differentiator of the super twisting type. The obtained results indicate that the second-order sliding mode of the bang–bang type outperforms the twisting algorithm and can stabilize the pendulum. As mentioned before, the second-order sliding mode of the bang–bang type is better than the twisting algorithm. In fact, a formal comparison is beyond the purposes of the study. Special mention has to be made about the robustness and the finite-time stability of this kind of controller. Further research should be oriented to develop an adaptive version of the bang–bang controller. In real applications, it would be desirable to impose constraints on the system states.