A novel second-order sliding mode control based on the Lyapunov method

Abstract

In this paper, a novel discontinuous second-order sliding mode control approach has been developed to handle sliding mode dynamics with a nonvanishing mismatched disturbance by using Lyapunov theory and a finite-time disturbance observer. Firstly, the finite-time disturbance observer is designed to estimate the nonvanishing mismatched disturbance. Secondly, a virtual controller has been constructed based on the estimated value such that the sliding variable can be stabilized to zero in a finite time. Then, the real discontinuous controller is designed to guarantee that the virtual controller can be well tracked in a finite time. Lyapunov analysis also verifies the finite-time stability of the closed-loop sliding mode control system. The developed discontinuous second-order sliding mode control method possesses two appealing features including strong robustness with respect to the matched and mismatched nonvanishing disturbances, and relaxation on the constant upper bound of uncertainties widely used in a conventional second-order sliding mode. Finally, an academic example is illustrated to verify the effectiveness of the proposed method.

Keywords

Disturbance observer second-order sliding mode finite-time stability

Introduction

There will always exist some discrepancies between an actual system and its mathematical model (Fu et al., 2015; Wang et al., 2018; Zhang et al., 2018). These discrepancies usually consist of exogenous disturbances, parameter perturbations and system uncertainties, which may bring some adverse effects on the control performance of closed-loop systems (Wang et al., 2017a; Xu et al., 2016; Yang and Ding, 2017). Aiming to diminish these negative effects, research on nonlinear systems with disturbances and uncertainties has been an active topic in past decades. Up to now, several control methods have been proposed, such as backstepping control (Xu et al., 2017; Yu and Zhong, 2011; Zhang and Shen, 2017), finite-time control (Fu et al., 2017; Shen et al., 2017), adaptive control (Sun et al., 2017), optimization (He and Gonzalez, 2017; Zhu et al., 2018), fuzzy control (Wang et al., 2017; Xie et al., 2014), nonlinear $H_{\infty}$ method (Wen et al., 2013), sliding mode control (SMC) (He et al., 2017; Utkin, 2013; Utkin et al., 1999; Yin et al., 2014), etc. Among these methods, SMC may be the most effective method for its robustness against system uncertainties and simplicity in implementation (Ding et al., 2017; Du et al., 2016; Yu and Long, 2015).

The design of a SM controller contains two major parts, constructing the SM surface and designing reaching law. The SM surface is constructed to ensure the closed-loop system can possess the desired dynamic behaviour, while the reaching law drives the system states to the SM surface and then to stay on it. It is worth noting that conventional SMC algorithms always employ a linear switching surface, which offers exponential convergence at best. Since the convergence performance is one of the important indexes for a control system, a finite-time control technique has also been introduced in SMC (Zhao et al., 2018). By constructing a nonlinear switching surface, the terminal sliding mode (TSM) was developed by Man and Yu (1997), where the states can converge to the origin along the sliding mode surface in a finite time.

However, there exist two potential restrictions in widespread applications of conventional first-order SMC (Ding et al., 2015). On the one hand, conventional first-order SMC requires the relative degree of the sliding variable to be one, which brings some obstacles for choosing the desired SM surface. On the other hand, the chattering problem always occurs in conventional first-order SMC. It may cause structure damage, especially in mechanical and aerospace industries (Utkin, 2013). One solution to reduce the chattering is to replace the sign function with a saturation function, while the price is that the robustness of the closed-loop system will be significantly diminished. It should be pointed out that the disturbance observer (DOB) based SMC can compensate disturbances and uncertainties, which can be regarded as an alternative approach to reduce chattering (Wang et al., 2016, 2017b). In addition, the second-order sliding mode (SOSM) is also an effective method to overcome the chattering.

The SOSM control methodology was developed by Emelyanov et al. (1986); Levant (2005a). The basic idea of SOSM can be summarized as follows. Regarding the derivative of a real controller (i.e. $\overset{\cdot}{u}$ ) as a new controller, one can design a discontinuous controller $\overset{\cdot}{u}$ such that the sliding variable will be driven to zero. This yields a continuous controller u since it is an integration of the discontinuous controller $\overset{\cdot}{u}$ . Consequently, the chattering will be significantly reduced. Meanwhile, the relative degree of a sliding variable in conventional SMC can be extended to be two. In recent decades, several SOSM control methods and their applications have been investigated, such as a twisting algorithm (Emelyanov et al., 1986), super-twisting algorithm (Nagesh and Edwards, 2014; Shtessel et al., 2013), suboptimal algorithm (Bartolini et al., 2003), quasi-continuous algorithm (Levant, 2005a), etc.

Nevertheless, there are also some problems in SOSM control methodology. Firstly, the dynamics of SOSM are always obtained by directly taking a two times derivative on the sliding variable. As a matter of fact, there always exists some system uncertainties and exogenous disturbances in the first-time derivative of a sliding variable (i.e. $\overset{\cdot}{s}$ ). The derivative directly on $\overset{\cdot}{s}$ will enlarge these uncertainties and disturbances. To this end, the switching gain of a SOSM controller should be selected to be very large in order to suppress these uncertainties and disturbances. However, the control gain cannot be tuned to be arbitrarily large due to the saturated limitation of its mechanical structure, which means that the control effort could not meet the requirement in general. Thus, it will bring much burden for controller design, even instability. Secondly, it can be noticed that a constant upper bound assumption is always imposed on the uncertainties in SOSM dynamics. However, the uncertainties in SM dynamics usually contain system states, which means that the constant upper bound condition for uncertainties is usually a local assumption.

Inspired by Ding and Li (2017); Ding et al. (2015), a novel SOSM control approach will be developed for a class of SM dynamics with nonvanishing matched and mismatched disturbances by using finite-time Lyapunov theory and Arie Levant’s differentiator (Levant, 2003; Shtessel et al, 2009). The detailed control design has three steps. Since the nonvanishing mismatched disturbance is always unknown, it will bring obstacles for control design. Arie Levant’s differentiator will be first used for estimating the nonvanishing mismatched disturbance in a SM dynamic system. And then, with the help of the estimated value, a virtual control law is constructed to compensate for the nonvanishing mismatched disturbance and stabilize the sliding variable in a finite time. Finally, the real controller is designed such that the virtual controller can be tracked in a finite time. Under the proposed discontinuous SOSM controller, the SM variable will be eventually stabilized to zero in a finite time. The contribution of this paper can be summarized as follows. Firstly, comparing with existing SOSM methods, the proposed discontinuous SOSM method can significantly reduce the uncertainties in the control channel such that the switching gain can be chosen to be a smaller one. Secondly, the given method can be used to deal with the nonvanishing mismatched disturbance existing in the SM dynamics. Thirdly, the uncertainties in SOSM dynamics considered in this paper are only required to be bounded by positive functions rather than positive constants.

The rest of the paper is organized as follows. The ‘Problem formulation’ section introduces the problem formulation briefly. The ‘Discontinuous SOSM controller design’ section shows the explicit steps of designing discontinuous SOSM controller. The ‘Example’ section gives an example system to verify the effectiveness of the proposed method. Finally, the ‘Conclusion’ section makes a concise conclusion.

Problem formulation

Consider the following nonlinear uncertain system described by

{\begin{matrix} \overset{\cdot}{x} = f (t, x) + g (t, x) u \\ s = s (t, x) \end{matrix}

(1)

where $x \in R^{n}$ and $u \in R$ are the system state and control input, $f (t, x)$ and $g (t, x)$ are unknown smooth functions and $s (t, x)$ is the sliding variable. To avoid finite-time escape, similar to the work by Levant and Michael (2009), it is also assumed that the trajectories of equation (1) are infinitely extendible in time for any Lebesgue-measurable control $u (x)$ bounded by a positive function.

For the system given by equation (1), we assume that the sliding variable s has a relative degree of two with respect to controller u. Then, taking directly a two times derivative of s, we can obtain

\overset{\cdot\cdot}{s} = a_{0} (t, x) + b_{0} (t, x) u

(2)

with smooth functions $a_{0} (t, x) = \overset{\cdot\cdot}{s} |_{u = 0}$ and $b_{0} (t, x) = \frac{\partial \overset{\cdot\cdot}{s}}{\partial u}$ being not exactly known, but satisfying the following assumption.

Assumption 1. There exist three positive constants $K_{m}$ , $K_{M}$ and C such that

| a_{0} (t, x) | \leq C, K_{m} \leq b_{0} (t, x) \leq K_{M}

As we know, the SOSM controller u is utilized to eliminate the effect of uncertainties $a_{0} (t, x)$ and $b_{0} (t, x)$ such that the sliding mode $s = \overset{\cdot}{s} = 0$ occurs in a finite time. Under Assumption 1, there have been proposed several well-known algorithms, including the twisting algorithm (Emelyanov et al., 1986), super-twisting algorithm (Levant, 1993), suboptimal algorithm (Bartolini et al., 2003), homogenous algorithm (Levant, 2005b; Tian et al., 2014a,b), etc.

It should be noted that $a_{0} (t, x)$ and $b_{0} (t, x)$ are usually linked with the state x, while Assumption 1 indicates that they should be bounded by positive constants. This implies that Assumption 1 may be a local condition in general. In this situation, a large initial value may violate Assumption 1 and the system states will diverge rapidly. To solve this problem, the local Assumption 1 has been extended to a global one by Ding et al. (2016) as follows.

Assumption 2. There exists positive constant $K_{m}$ and positive function $C (x)$ such that

| a_{0} (t, x) | \leq C (x), b_{0} (t, x) \geq K_{m}

It can be clearly seen from Assumption 2 that the uncertainty $a_{0} (t, x)$ is bounded by a positive function rather than a positive constant. Consequently, global control under Assumption 2 can be obtained.

Nevertheless, the two times derivative on s in SOSM dynamics will bring the certainties/uncertainties into the control channel. It implies that $a_{0} (t, x)$ and $b_{0} (t, x)$ may be very large. To this end, it will be a challenge to design a controller in the presence of the enlarged certainties/uncertainties. In order to deal with this problem, Ding and Li (2017) presented a novel method, which could preserve some terms in the dynamics of $\overset{\cdot}{s}$ . Thus, the uncertainties can be significantly reduced in the control channel. Specifically, by properly choosing a term $f_{1} (s_{1})$ and letting $s_{1} = s, s_{2} = \overset{\cdot}{s} - f_{1} (s_{1})$ , the system given by equation (2) can be rewritten as

{\begin{matrix} {\overset{\cdot}{s}}_{1} = s_{2} + f_{1} (s_{1}) \\ {\overset{\cdot}{s}}_{2} = a_{1} (t, x) + b_{1} (t, x) u \end{matrix}

(3)

where $s_{2}$ is measurable and $f_{1} (s_{1})$ is a known term. In addition, the following assumption holds

Assumption 3. There exists a positive constant $K_{m}$ and positive functions $C (x)$ , $ρ (s_{1})$ such that

| a_{1} (t, x) | \leq C (x), b_{1} (t, x) \geq K_{m}, | f_{1} (s_{1}) | \leq ρ (s_{1}) | s_{1} |^{\frac{1}{2}}

As a matter of fact, the uncertainties $a_{1} (t, x)$ and $b_{1} (t, x)$ in the system given by equation (3) are much smaller than $a_{0} (t, x)$ and $b_{0} (t, x)$ in the system given by equation (2), because the term $f_{1} (s_{1})$ has been kept in the dynamics of $\overset{\cdot}{s}$ .

The following is a simple example to illustrate the above descriptions.

Example 1. Consider an uncertain system

{\begin{matrix} {\overset{\cdot}{x}}_{1} = 2 x_{2} + x_{1}^{2} \\ {\overset{\cdot}{x}}_{2} = g (t, x) u + d_{1} (t, x) \\ s = s (t, x) \end{matrix}

(4)

where $x = (x_{1}, x_{2})^{T}$ are the system states, u is the control input and the smooth functions $d_{1} (t, x)$ and $g (t, x) \geq 1$ are system uncertainties including unmodelled dynamics and external disturbances. The smooth function $s (t, x)$ is the sliding variable, which has a relative degree of two with respect to the controller u. The goal is to design a controller u such that $x_{1}$ will track a constant signal $x_{c} .$

we can let $s = x_{1} - x_{c}$ . Then the dynamics of a sliding mode is given as follows

\overset{\cdot\cdot}{s} = a_{0} (t, x) + b_{0} (t, x) u

(5)

where $a_{0} (t, x) = 2 d_{1} (t, x) + 4 x_{1} x_{2} + 2 x_{1}^{3}$ and $b_{0} (t, x) = 2 g (t, x)$ .

As a comparison, we will use the method proposed by Ding and Li (2017) to obtain other SM dynamics. Let $s_{1} = x_{1} - x_{c}$ , $s_{2} = 2 x_{2} + x_{c}^{2}$ and $f_{1} (s_{1}) = (s_{1} + 2 x_{c}) s_{1}$ . It can be verified that $| f_{1} (s_{1}) | \leq ρ (s_{1}) | s_{1} |^{1 / 2}$ with $ρ (s_{1}) = | s_{1} + 2 x_{c} | | s_{1} |^{1 / 2}$ . Then the SM dynamics given by equation (5) can be rewritten as

{\begin{matrix} {\overset{\cdot}{s}}_{1} = s_{2} + f_{1} (s_{1}) \\ {\overset{\cdot}{s}}_{2} = a_{1} (t, x) + b_{1} (t, x) u \end{matrix}

(6)

with $a_{1} (t, x) = 2 d_{1} (t, x)$ , $b_{1} (t, x) = 2 g (t, x) .$

It can be clearly seen that the uncertainty $a_{1} (t, x)$ is far smaller than $a_{0} (t, x)$ . In this case, the controller design and implementation problem for the system given by equation (6) will be easier than that of the system given by equation (5).

However, it should be pointed out that the method proposed by Ding and Li (2017) cannot be applied in the presence of nonvanishing mismatched disturbances, because the disturbances will not vanish as the states approach zero. To tackle this problem, we further consider the following SOSM dynamics with a nonvanishing mismatched disturbance $c (t)$ described by

{\begin{matrix} {\overset{\cdot}{s}}_{1} = s_{2} + f_{1} (s_{1}) + c (t) \\ {\overset{\cdot}{s}}_{2} = a (t, x) + b (t, x) u \end{matrix}

(7)

where $s_{1} \in R$ and $s_{2} \in R$ are measurable sliding variables, the continuous function $f_{1} (s_{1})$ is known and the smooth functions $c (t), a (t, x)$ and $b (t, x)$ are unknown. Meanwhile, the following assumption holds.

Assumption 4. There exist positive constants $K_{m}, L$ and positive functions $C (x)$ and $ρ (s_{1})$ such that $| a (t, x) | \leq C (x)$ , $b (t, x) \geq K_{m}$ , $| \overset{\cdot}{c} (t) | \leq L$ and

| f_{1} (s_{1}) | \leq ρ (s_{1}) | s_{1} |^{\frac{1}{2}}

(8)

The goal of this paper is to design a suitable discontinuous SOSM controller for the system given by equation (7) such that $s_{1} = {\overset{\cdot}{s}}_{1} = 0$ can be kept in a finite time.

Finally, three lemmas that serve as the basis of the key tools for the main results are listed. To simplify the expression, we denote

{⌊ x ⌉}^{α} = sign (x) | x |^{α}

with any $α > 0$ .

Lemma 1. (Ding et al., 2018) If $p_{1} > 0$ and $0 < p_{2} \leq 1$ , then for $\forall x, y \in R$ , we have

| {⌊ x ⌉}^{p_{1} p_{2}} - {⌊ y ⌉}^{p_{1} p_{2}} | \leq 2^{1 - p_{2}} | {⌊ x ⌉}^{p_{1}} - {⌊ y ⌉}^{p_{1}} |^{p_{2}}

Lemma 2. (Qian and Lin, 2001) Let c and d be positive constants. Given any positive function $γ > 0$ , the following inequality holds

| x |^{c} | y |^{d} \leq \frac{c}{c + d} γ | x |^{c + d} + \frac{d}{c + d} γ^{- \frac{c}{d}} | y |^{c + d}, \forall x, y \in R

Lemma 3. (Hardy et al., 1952) Let p be a real number with $0 < p < 1$ . Then for $\forall x_{i} \in R, i = 1, \dots, n$ , we have

(| x_{1} | + \dots | x_{n} |)^{p} \leq | x_{1} |^{p} + \dots + | x_{n} |^{p}

Discontinuous SOSM controller design

In this section, the SOSM control scheme will be proposed for the sliding mode dynamics given by equation (7) by using a backstepping-like method. The sketch of the control design can be summarized as follows. First of all, design a disturbance observer to estimate the unknown disturbance $c (t)$ . Then, by utilizing the estimated value, construct a virtual controller $s_{2}^{*}$ such that the sliding variable $s_{1}$ can be stabilized to zero in a finite time and $c (t)$ can be compensated by the estimated value. Finally, design a discontinuous SOSM controller u to ensure that the sliding variable $s_{2}$ will track the virtual controller $s_{2}^{*}$ in a finite time. Figure 1 shows the block diagram of the controller design, where $x_{1}$ is the state variable and $x_{c}$ is the reference signal.

Figure 1.

Block diagram of the controller design.

Finite-time disturbance observer design

The finite-time disturbance observer can be designed by modifying Arie Levant’s differentiator (Levant, 2003; Shtessel et al, 2009) as follows

{\begin{matrix} {\overset{\cdot}{z}}_{0} = ν_{0} + s_{2} + f_{1} (s_{1}) \\ ν_{0} = - λ_{1} {⌊ z_{0} - s_{1} ⌋}^{1 / 2} + z_{1} \\ {\overset{\cdot}{z}}_{1} = - λ_{0} sign (z_{1} - ν_{0}) \end{matrix}

(9)

where $λ_{0}$ and $λ_{1}$ are coefficients. Then, we have the following lemma.

Lemma 4. If the disturbance observer is designed as equation (9), then there exists a time instant T such that $z_{1} \equiv c (t)$ can be achieved for $\forall t \geq T$ .

Proof. Denote $σ_{0} = z_{0} - s_{1}$ and $σ_{1} = z_{1} - c (t)$ . From the system given by equation (7), it can be easily verified that

\begin{matrix} {\overset{\cdot}{σ}}_{0} & = {\overset{\cdot}{z}}_{0} - {\overset{\cdot}{s}}_{1} \\ = - λ_{1} {⌊ z_{0} - s_{1} ⌋}^{1 / 2} + z_{1} - c (t) \\ = - λ_{1} {⌊ σ_{0} ⌋}^{1 / 2} + σ_{1} \end{matrix}

(10)

Similarly, we also have

\begin{matrix} {\overset{\cdot}{σ}}_{1} & = {\overset{\cdot}{z}}_{1} - \overset{\cdot}{c} (t) \\ = - λ_{0} sign (z_{1} - ν_{0}) - \overset{\cdot}{c} (t) \\ \in - λ_{0} sign (σ_{1} - {\overset{\cdot}{σ}}_{0}) + [- L, L] \end{matrix}

(11)

which is understood in the Filippov sense (Filippov, 1988). Combining equations (10) and (11), we have the following differential inclusion

{\begin{matrix} {\overset{\cdot}{σ}}_{0} = - λ_{1} {⌊ σ_{0} ⌋}^{1 / 2} + σ_{1} \\ {\overset{\cdot}{σ}}_{1} \in - λ_{0} sign (σ_{1} - {\overset{\cdot}{σ}}_{0}) + [- L, L] \end{matrix}

The rest of the proof actually does not differ much from that of Theorem 6.4 in the work by Shtessel et al. (2013) and thus is omitted here. This completes the proof.▪

Remark 1. According to Shtessel et al. (2013), the values of $λ_{0}$ and $λ_{1}$ can be chosen as $λ_{0} = 1.1 L$ and $λ_{1} = 1.5 L^{1 / 2}$ to guarantee convergence of the observer given by equation (9). Generally speaking, the faster the convergence rate the finite-time observer has, the larger the value $λ_{i}$ it requires. However, the large value of the parameters $λ_{i}$ could cause an excessive transient peaking. Consequently, the parameters $λ_{i}$ will be chosen in a compromising way.

Controller design

Note that $σ_{1} = z_{1} - c (t)$ . The system given by equation (7) can be rewritten as

{\begin{matrix} {\overset{\cdot}{s}}_{1} = s_{2} + f_{1} (s_{1}) + z_{1} - σ_{1} \\ {\overset{\cdot}{s}}_{2} = a (t, x) + b (t, x) u \end{matrix}

(12)

If we can design a SOSM controller to stabilize the system given by equation (12), then the system given by equation (7) can also be stabilized by the same controller.

From Lemma 4, it can be concluded that the observation error $σ_{1}$ will be equal to zero after a time instant T, which implies that when $t \geq T$ , the system given by equation (12) will reduce to

{\begin{matrix} {\overset{\cdot}{s}}_{1} = s_{2} + f_{1} (s_{1}) + z_{1} \\ {\overset{\cdot}{s}}_{2} = a (t, x) + b (t, x) u \end{matrix}

(13)

On this basis, we will propose the following main result of the paper.

Theorem 1. Under Assumption 4, there exist positive functions $β_{2} (x, s_{1}) \geq β_{1} (s_{1})$ and a positive constant $a \geq r_{1} = 2 r_{2} > 0$ such that the following discontinuous controller

u = - β_{2} (x, s_{1}) sign ({⌊ s_{2} + z_{1} ⌉}^{\frac{a}{r_{2}}} + β_{1}^{\frac{a}{r_{2}}} {⌊ s_{1} ⌉}^{\frac{a}{r_{1}}})

(14)

with $z_{1}$ being generated by equation (9), provides for the finite-time establishment of SOSM $s_{1} = {\overset{\cdot}{s}}_{1} = 0$ in the closed-loop system given by equations (7) and (14).

Proof. Note that under the observer given by equation (9), the system given by equation (7) will reduce to the system given by equation (13) when $t \geq T$ . Obviously, the control design can be divided into the following two parts. Firstly, the finite-time stability of the system given by equation (13) under the discontinuous controller given by equation (14) will be proved when $t \geq T$ . Secondly, it will be further proved that the system given by equation (7) will not escape to infinity during the time interval $[0, T]$ .

In the following, the discontinuous SOSM controller will be constructed by using the adding a power integrator technique (Qian and Lin, 2001). The proof will be given in three steps.

(1) Step 1. We choose a $C^{1}$ Lyapunov function as follows

V_{1} (s_{1}) = \frac{r_{1}}{2 ρ + r_{2}} | s_{1} |^{\frac{2 ρ + r_{2}}{r_{1}}}

with $ρ \geq a$ . With equation (8) and $r_{1} = 2 r_{2}$ in mind, taking derivative of $V_{1} (s_{1})$ along the system given by equation (13) produces

\begin{matrix} {\overset{\cdot}{V}}_{1} (s_{1}) & = {⌊ s_{1} ⌉}^{\frac{2 ρ - r_{2}}{r_{1}}} (s_{2} + f (s_{1}) + z_{1}) \\ \leq {⌊ s_{1} ⌉}^{\frac{2 ρ - r_{2}}{r_{1}}} (s_{2} - s_{2}^{*}) + {⌊ s_{1} ⌉}^{\frac{2 ρ - r_{2}}{r_{1}}} s_{2}^{*} + ρ (s_{1}) s_{1}^{\frac{2 ρ}{r_{1}}} + {⌊ s_{1} ⌉}^{\frac{2 ρ - r_{2}}{r_{1}}} z_{1} \end{matrix}

(15)

where $s_{2}^{*}$ is a virtual controller and can be designed as

s_{2}^{*} = - β_{1} (s_{1}) {⌊ ξ_{1} ⌉}^{r_{2} / a} - z_{1}

(16)

with $β_{1} (s_{1}) \geq ρ (s_{1}) + β_{0}, β_{0} > 0$ and $ξ_{1} = {⌊ s_{1} ⌉}^{\frac{a}{r_{1}}}$ . Submitting the virtual controller given by equation (16) into equation (15) obtains

\begin{matrix} {\overset{\cdot}{V}}_{1} (s_{1}) & \leq {⌊ s_{1} ⌉}^{\frac{2 ρ - r_{2}}{r_{1}}} (s_{2} - s_{2}^{*}) - β_{1} (s_{1}) ξ_{1}^{\frac{2 ρ}{a}} + ρ (s_{1}) s_{1}^{\frac{2 ρ}{r_{1}}} \\ \leq - β_{0} ξ_{1}^{\frac{2 ρ}{a}} + {⌊ ξ_{1} ⌉}^{\frac{2 ρ - r_{2}}{a}} (s_{2} - s_{2}^{*}) \end{matrix}

(17)

(2) Step 2. Constructing

W_{2} (s_{1}, {\bar{s}}_{2}) = \int_{{\bar{s}}_{2}^{*}}^{{\bar{s}}_{2}} {⌊ {⌊ κ ⌉}^{\frac{a}{r_{2}}} - {⌊ {\bar{s}}_{2}^{*} ⌉}^{\frac{a}{r_{2}}} ⌉}^{\frac{2 ρ}{a}} d κ

where ${\bar{s}}_{2} = s_{2} + z_{1}$ and ${\bar{s}}_{2}^{*} = s_{2}^{*} + z_{1}$ , then the Lyapunov function can be chosen as

V_{2} (s_{1}, {\bar{s}}_{2}) = V_{1} (s_{1}) + W_{2} (s_{1}, {\bar{s}}_{2})

(18)

Taking the derivative of $V_{2} (s_{1}, {\bar{s}}_{2})$ along the system given by equation (13) yields

\begin{matrix} {\overset{\cdot}{V}}_{2} (s_{1}, {\bar{s}}_{2}) = {\overset{\cdot}{V}}_{1} (s_{1}) + \frac{\partial W_{2} (s_{1}, {\bar{s}}_{2})}{\partial s_{1}} {\overset{\cdot}{s}}_{1} + \frac{\partial W_{2} (s_{1}, {\bar{s}}_{2})}{\partial {\bar{s}}_{2}} {\overset{\cdot}{\bar{}} s}_{2} \\ \leq - β_{0} ξ_{1}^{\frac{2 ρ}{a}} + {⌊ ξ_{1} ⌉}^{\frac{2 ρ - r_{2}}{a}} (s_{2} - s_{2}^{*}) + \frac{\partial W_{2} (s_{1}, {\bar{s}}_{2})}{\partial s_{1}} {\overset{\cdot}{s}}_{1} + {⌊ ξ_{2} ⌉}^{\frac{2 ρ}{a}} {\overset{\cdot}{\bar{s}}}_{2} \end{matrix}

(19)

where $ξ_{2} = {⌊ {\bar{s}}_{2} ⌉}^{\frac{a}{r_{2}}} - {⌊ {\bar{s}}_{2}^{*} ⌉}^{\frac{a}{r_{2}}}$ . Next, each term in the right hand side of equation (19) will be estimated.

Note that

| s_{2} - s_{2}^{*} | = | {\bar{s}}_{2} - {\bar{s}}_{2}^{*} | = | {⌊ {\bar{s}}_{2} ⌉}^{\frac{a}{r_{2}} \times \frac{r_{2}}{a}} - {⌊ {\bar{s}}_{2}^{*} ⌉}^{\frac{a}{r_{2}} \times \frac{r_{2}}{a}} |

From Lemma 1, one has

| s_{2} - s_{2}^{*} | \leq 2^{1 - \frac{r_{2}}{a}} | {⌊ {\bar{s}}_{2} ⌉}^{\frac{a}{r_{2}}} - {⌊ {\bar{s}}_{2}^{*} ⌉}^{\frac{a}{r_{2}}} |^{\frac{r_{2}}{a}} = 2^{1 - \frac{r_{2}}{a}} | ξ_{2} |^{\frac{r_{2}}{a}}

This implies that the following estimate holds

{⌊ ξ_{1} ⌉}^{\frac{2 ρ - r_{2}}{a}} (s_{2} - s_{2}^{*}) \leq 2^{1 - \frac{r_{2}}{a}} | ξ_{1} |^{\frac{2 ρ - r_{2}}{a}} {| ξ_{2} |}^{\frac{r_{2}}{a}}

(20)

Applying Lemma 2 to equation (20), it can be concluded that

{⌊ ξ_{1} ⌉}^{\frac{2 ρ - r_{2}}{a}} (s_{2} - s_{2}^{*}) \leq 2^{1 - \frac{r_{2}}{a}} (\frac{2 ρ - r_{2}}{2 ρ} γ | ξ_{1} |^{\frac{2 ρ}{a}} + \frac{r_{2}}{2 ρ} γ^{- \frac{2 ρ - r_{2}}{r_{2}}} | ξ_{2} |^{\frac{2 ρ}{a}})

with any $γ > 0$ . Let

2^{1 - \frac{r_{2}}{a}} \frac{2 ρ - r_{2}}{2 ρ} γ = \frac{1}{4} β_{0}

By a simple calculation, we have

{⌊ ξ_{1} ⌉}^{\frac{2 ρ - r_{2}}{a}} (s_{2} - s_{2}^{*}) \leq \frac{1}{4} β_{0} ξ_{1}^{\frac{2 ρ}{a}} + c_{1} (β_{0}) ξ_{2}^{\frac{2 ρ}{a}}

(21)

with

c_{1} (β_{0}) = 2^{1 - \frac{r_{2}}{a}} \frac{r_{2}}{2 ρ} {(\frac{(2 ρ - r_{2} {) 2}^{2 - r_{2} / a}}{ρ β_{0}})}^{\frac{2 ρ - r_{2}}{r_{2}}}

On the other hand, note that

\begin{array}{l} | \frac{\partial W_{2} (s_{1}, {\bar{s}}_{2})}{\partial s_{1}} {\dot{s}}_{1} | = \frac{2 ρ}{a} | \frac{\partial {⌊ {\bar{s}}_{2}^{*} ⌉}^{\frac{a}{r_{2}}}}{\partial s_{1}} {\dot{s}}_{1} \int_{{\bar{s}}_{2}^{*}}^{{\bar{s}}_{2}} {⌊ {⌊ κ ⌉}^{\frac{a}{r_{2}}} - {⌊ {\bar{s}}_{2}^{*} ⌉}^{\frac{a}{r_{2}}} ⌉}^{\frac{2 ρ}{a} - 1} d κ | \\ \leq \frac{2 ρ}{a} | {\bar{s}}_{2} - {\bar{s}}_{2}^{*} | | ξ_{2} |^{\frac{2 ρ}{a} - 1} | \frac{\partial {⌊ {\bar{s}}_{2}^{*} ⌉}^{\frac{a}{r_{2}}}}{\partial s_{1}} {\dot{s}}_{1} | \end{array}

By Lemma 1, we can get

| \frac{\partial W_{2} (s_{1}, {\bar{s}}_{2})}{\partial s_{1}} {\overset{\cdot}{s}}_{1} | \leq \frac{2 ρ}{a} 2^{1 - \frac{r_{2}}{a}} | ξ_{2} |^{\frac{r_{2}}{a} + \frac{2 ρ}{a} - 1} | \frac{\partial {⌊ {\bar{s}}_{2}^{*} ⌉}^{\frac{a}{r_{2}}}}{\partial s_{1}} {\overset{\cdot}{s}}_{1} |

(22)

Taking ${\bar{s}}_{2}^{*} = s_{2}^{*} + z_{1}$ and $s_{2}^{*} = - β_{1} (s_{1}) {⌊ ξ_{1} ⌉}^{r_{2} / a} - z_{1}$ into account, it can be calculated that

| \frac{\partial ({⌊ {\bar{s}}_{2}^{*} ⌉}^{\frac{a}{r_{2}}})}{\partial s_{1}} | \leq | \frac{\partial β_{1}^{\frac{a}{r_{2}}} (s_{1})}{\partial s_{1}} ξ_{1} | + \frac{a β_{1}^{\frac{a}{r_{2}}} (s_{1})}{r_{1}} | s_{1} |^{\frac{a}{r_{1}} - 1}

(23)

By using Lemma 3, we also have $| {\bar{s}}_{2} | \leq (| ξ_{2} |^{\frac{r_{2}}{a}} + β_{1} (s_{1}) | ξ_{1} |^{\frac{r_{2}}{a}}) .$ This, together with equation (23), Assumption 4 and the fact ${\overset{\cdot}{s}}_{1} = {\bar{s}}_{2} + f_{1} (s_{1})$ , implies

\begin{array}{l} | \frac{\partial ({⌊ {\bar{s}}_{2}^{*} ⌉}^{\frac{a}{r_{2}}})}{\partial s_{1}} {\dot{s}}_{1} | \leq (| \frac{\partial β_{1}^{\frac{a}{r_{2}}} (s_{1})}{\partial s_{1}} s_{1} | + \frac{a β_{1}^{\frac{a}{r_{2}}} (s_{1})}{r_{1}}) | ξ_{1} |^{1 - \frac{r_{1}}{a}} \\ (| ξ_{2} |^{\frac{r_{2}}{a}} + β_{1} (s_{1}) | ξ_{1} |^{\frac{r_{2}}{a}} + ρ (s_{1}) | ξ_{1} |^{\frac{r_{2}}{a}}) \end{array}

(24)

Applying Lemma 2 to equation (24), it can be concluded that there exist two positive functions $c_{2} (s_{1})$ and $c_{3} (s_{1})$ such that

| \frac{\partial ({⌊ {\bar{s}}_{2}^{*} ⌉}^{\frac{a}{r_{2}}})}{\partial s_{1}} {\overset{\cdot}{s}}_{1} | \leq c_{2} (s_{1}) | ξ_{1} |^{1 - \frac{r_{2}}{a}} + c_{3} (s_{1}) | ξ_{2} |^{1 - \frac{r_{2}}{a}}

(25)

From equations (22) and (25), one obtains

\begin{array}{l} | \frac{\partial W_{2} (s_{1}, {\bar{s}}_{2})}{\partial s_{1}} {\dot{s}}_{1} | \leq \frac{2 ρ}{a} 2^{1 - \frac{r_{2}}{a}} | ξ_{2} |^{\frac{r_{2}}{a} + \frac{2 ρ}{a} - 1} \\ (c_{2} (s_{1}) | ξ_{1} |^{1 - \frac{r_{2}}{a}} + c_{3} (s_{1}) | ξ_{2} |^{1 - \frac{r_{2}}{a}}) \end{array}

(26)

Applying Lemma 2 again to equation (26) yields

\begin{array}{l} | \frac{\partial W_{2} (s_{1}, {\bar{s}}_{2})}{\partial s_{1}} {\dot{s}}_{1} | \leq \frac{a - r_{2}}{a} 2^{1 - \frac{r_{2}}{a}} c_{2} (s_{1}) γ_{1} | ξ_{1} |^{\frac{2 ρ}{a}} \\ + \frac{2 ρ + r_{2} - a}{a} 2^{1 - \frac{r_{2}}{a}} γ_{1}^{- \frac{a - r_{2}}{2 ρ + r_{2} - a}} c_{2} (s_{1}) | ξ_{2} |^{\frac{2 ρ}{a}} + \frac{2 ρ}{a} 2^{1 - \frac{r_{2}}{a}} c_{3} (s_{1}) | ξ_{2} |^{\frac{2 ρ}{a}} \end{array}

with any $γ_{1} > 0$ . Similarly, letting

\frac{a - r_{2}}{a} 2^{1 - \frac{r_{2}}{a}} c_{2} (s_{1}) γ_{1} = \frac{1}{4} β_{0}

equation (26) can be estimated as

| \frac{\partial W_{2} (s_{1}, {\bar{s}}_{2})}{\partial s_{1}} {\overset{\cdot}{s}}_{1} | \leq \frac{1}{4} β_{0} ξ_{1}^{\frac{2 ρ}{a}} + c_{4} (s_{1}) ξ_{2}^{\frac{2 ρ}{a}}

(27)

with

c_{4} (s_{1}) = \frac{2 ρ + r_{2} - a}{a} 2^{1 - \frac{r_{2}}{a}} γ_{1}^{- \frac{a - r_{2}}{2 ρ + r_{2} - a}} c_{2} (s_{1}) + \frac{2 ρ}{a} 2^{1 - \frac{r_{2}}{a}} c_{3} (s_{1})

Finally, substituting equations (21) and (27) into equation (19) gives

\begin{array}{l} {\dot{V}}_{2} (s_{1}, {\bar{s}}_{2}) \leq - \frac{β_{0}}{2} ξ_{1}^{\frac{2 ρ}{a}} + c_{1} (β_{0}) ξ_{2}^{\frac{2 ρ}{a}} + c_{4} (s_{1}) ξ_{2}^{\frac{2 ρ}{a}} \\ + {⌊ ξ_{2} ⌉}^{\frac{2 ρ}{a}} (a (t, x) + b (t, x) u + {\dot{z}}_{1}) \end{array}

(28)

It can be easily verified from equation (9) that $| {\overset{\cdot}{z}}_{1} | \leq λ_{0}$ . Then the discontinuous controller can be constructed as

u = - β_{2} (x, s_{1}) sign (ξ_{2})

(29)

with

β_{2} (x, s_{1}) \geq \frac{c_{1} (β_{0}) + c_{4} (s_{1}) + \frac{β_{0}}{2} + C (x) + λ_{0}}{K_{m}}

Putting equation (29) into equation (28) obtains

{\overset{\cdot}{V}}_{2} (s_{1}, {\bar{s}}_{2}) \leq - \frac{β_{0}}{2} (ξ_{1}^{\frac{2 ρ}{a}} + ξ_{2}^{\frac{2 ρ}{a}})

By the fact that

\int_{{\bar{s}}_{2}^{*}}^{{\bar{s}}_{2}} {⌊ {⌊ κ ⌉}^{\frac{a}{r_{2}}} - {⌊ {\bar{s}}_{2}^{*} ⌉}^{\frac{a}{r_{2}}} ⌉}^{\frac{2 ρ}{a}} d κ \leq | {\bar{s}}_{2} - {\bar{s}}_{2}^{*} | | ξ_{2} |^{\frac{2 ρ}{a}} \leq 2^{1 - \frac{r_{2}}{a}} | ξ_{2} |^{\frac{2 ρ + r_{2}}{a}}

it can be verified that

V_{2} (s_{1}, {\bar{s}}_{2}) \leq 2 (| ξ_{1} |^{\frac{2 ρ + r_{2}}{a}} + | ξ_{2} |^{\frac{2 ρ + r_{2}}{a}})

Letting

c_{0} = \frac{β_{0}}{2 \times 2^{\frac{2 ρ}{2 ρ + r_{2}}}}

we can obtain

{\overset{\cdot}{V}}_{2} (s_{1}, {\bar{s}}_{2}) + c_{0} V_{2}^{\frac{2 ρ}{2 ρ + r_{2}}} (s_{1}, {\bar{s}}_{2}) \leq 0

Note that

\frac{2 ρ}{2 ρ + r_{2}} \in (0, 1)

It follows from the finite-time Lyapunov theory given by Bhat and Bernstein (2000) that the system (13) can be globally finite-time stabilized by the discontinuous controller (29). Note that the discontinuous controller (29) can be written as equation (14). It also implies that the reduced system (13) can be globally finite-time stabilized by the discontinuous controller (14).

(3) Step 3. Note that the system (7) can be rewritten as the system (12) by using the finite-time disturbance observer (9). Moreover, the disturbance $c (t)$ existing in the system (7) can be estimated precisely by the disturbance observer (9) after a finite time T. It implies that for $t \geq T$ , the system (12) will reduce to the system (13). Step 1 and Step 2 have proved that the system (13) can be globally finite-time stabilized by the proposed discontinuous controller. This also implies that the system (7) can be globally finite-time stabilized by the proposed discontinuous controller if the state (7) will not diverge to infinity before the time instant T. To this end, we should further prove that the finite-time escape phenomenon will not happen for the system (7).

Choose a Lyapunov function as

V_{3} (s) = \frac{1}{2} (s_{1}^{2} + s_{2}^{2})

Taking a derivative of $V_{3} (s)$ along the system given by equation (7) yields

\begin{matrix} {\overset{\cdot}{V}}_{3} (s) = s_{1} (s_{2} + f_{1} (s_{1}) + c (t)) + s_{2} (a (t, x) + b (t, x) u) \\ \leq s_{1} (s_{2} + f_{1} (s_{1}) + c (t)) + | s_{2} | (| a (t, x) | + b (t, x) | β_{2} (t, x) |) \end{matrix}

By the fact that the variables $x_{1}$ and $x_{2}$ are infinitely extendible in time, and $s_{1} = s (t, x)$ is a function of x and t. We can conclude that $a (t, x)$ , $b (t, x)$ , $s_{1}$ and $β_{2} (x, s_{1})$ are bounded during the time interval $[0, T]$ . On the other hand, by Assumption 4, we know $| \overset{\cdot}{c} (t) | \leq L$ , which implies $c (t)$ is bounded during $[0, T]$ . Therefore, there exists a constant $γ_{0}$ such that ${\overset{\cdot}{V}}_{3} (s) \leq γ_{0} V_{3} (s) + γ_{0},$ which implies

V_{3} (s (t)) \leq (V_{3} (s (0)) + 1) e^{γ_{0} t} - 1

(30)

It can be concluded from equation (30) that the system (12) will not escape to infinity during the time interval $[0, T]$ . This implies that the system (7) can be stabilized by the discontinuous controller given by equation (14) in a finite time and thus completes the proof.▪

Remark 2. It is clearly seen from Assumption 4 that the uncertainties in the SM dynamics are bounded by positive functions, while the uncertainties in conventional SOSM dynamics are always bounded by positive constants. This property leads to the global results in this paper instead of local results in most existing SOSM work.

Remark 3. Although the method proposed in this paper is inspired by Ding and Li (2017), there are several differences between this paper and the paper by Ding and Li (2017). First of all, the system given by Ding and Li (2017) can be considered as a special case of the system considered in this paper. As a matter of fact, by letting $c (t) = 0$ , the system (7) will reduce to the system considered by Ding and Li (2017). This also implies that the proposed result in this paper can be considered as an extension of the result in the work by Ding and Li (2017). Secondly, it can be noted that the observer used in this paper can precisely estimate the nonvanishing mismatched disturbance after a finite time instant T. However, it should be mentioned that the finite-time escape phenomenon may occur during the time interval $[0, T]$ . To this end, we have to prove that the states will not escape to infinity during the time interval $[0, T]$ in the presence of a nonvanishing mismatched disturbance. Finally, it can be clearly observed that the estimated value will be considered as a state variable in the virtual controller, which may violate the homogeneous condition of the controller design. Therefore, how to deal with the estimated value in the virtual controller is also an important issue, while this problem does not exist in the paper by Ding and Li (2017).

Example

Consider a nonlinear system with a nonvanishing mismatched disturbance as follows

{\begin{matrix} {\overset{\cdot}{x}}_{1} = 2 x_{2} + x_{1}^{2} + d_{2} (t) \\ {\overset{\cdot}{x}}_{2} = d_{1} (t, x) + g (t, x) u \end{matrix}

(31)

where $d_{1} (t, x)$ , $d_{2} (t)$ and $g (t, x)$ are uncertainties or disturbances. The task here is to design a discontinuous controller u such that the state $x_{1}$ will approach $1$ .

Choose the sliding variables as

{\begin{matrix} s_{1} = x_{1} - 1 \\ s_{2} = 2 x_{2} + 1 \end{matrix}

(32)

Based on equations (31) and equation (32), we can obtain the SOSM dynamics described by

{\begin{matrix} {\overset{\cdot}{s}}_{1} = s_{2} + f_{1} (s_{1}) + c (t) \\ {\overset{\cdot}{s}}_{2} = a (t, x) + b (t, x) u \end{matrix}

with $f (s_{1}) = (s_{1} + 1)^{2} - 1$ , $c (t) = d_{2} (t)$ , $a (t, x) = 2 g (t, x)$ and $b (t, x) = 2 d_{1} (t, x)$ . According to Theorem 1 the discontinuous controller can be designed as

u = - β_{2} (x, s_{1}) sign ({⌊ s_{2} + z_{1} ⌉}^{\frac{a}{r_{2}}} + β_{1}^{\frac{a}{r_{2}}} {⌊ s_{1} ⌉}^{\frac{a}{r_{1}}})

(33)

with $z_{1}$ being generated by

{\begin{matrix} {\overset{\cdot}{z}}_{0} = ν_{0} + s_{2} + f_{1} (s_{1}) \\ ν_{0} = - λ_{1} {⌊ z_{0} - s_{1} ⌉}^{1 / 2} + z_{1} \\ {\overset{\cdot}{z}}_{1} = - λ_{0} sign (z_{1} - ν_{0}) \end{matrix}

Meanwhile, the twisting algorithm is employed here to show a comparison. From the conventional SOSM theorem, by letting $s = x_{1} - 1$ , the standard SOSM dynamics is

\overset{\cdot\cdot}{s} = a_{0} (t, x) + b_{0} (t, x) u

with $a_{0} (t, x) = 2 d_{1} (t, x) + 4 x_{1} x_{2} + 2 x_{1}^{3} + 2 x_{1} d_{2} (t) + {\overset{\cdot}{d}}_{2} (t)$ and $b_{0} (t, x) = 2 g (t, x) .$ According to the reference (Shtessel et al., 2013), the twisting controller is chosen as

u = - p_{1} \cdot sign (z_{0}) - p_{2} \cdot sign (z_{1})

(34)

where $p_{1}, p_{2}$ are proper positive constants and $z_{0}, z_{1}$ are generated by the following differentiator (Shtessel et al., 2013)

{\begin{matrix} {\overset{\cdot}{z}}_{0} = - λ_{1} {⌊ z_{0} - s ⌉}^{1 / 2} + z_{1} \\ {\overset{\cdot}{z}}_{1} = - λ_{0} \cdot sign (z_{0} - s) \end{matrix}

with $λ_{0} = 1.1 L_{0}$ , $λ_{1} = 1.5 L_{0}^{1 / 2}$ .

Simulation with a bounded disturbance d₂(t)

It is assumed that $d_{1} (t, x) = 4 x_{1} \sin (\frac{1}{2} t)$ and $g (t, x) = 1 + \overset{2}{\sin} (x_{1})$ . The initial states of the system given by equation (31) are chosen as $x_{1} (0) = 1$ and $x_{2} (0) = - 2$ . The mismatched disturbance $d_{2} (t)$ is chosen as

d_{2} (t) = {\begin{matrix} 0, 0 \leq t < 4 \\ 2, 4 \leq t < 8 \\ 0, 8 \leq t < 12 \\ 1 + 8 \sin (t), t \geq 12 \end{matrix}

as shown in Figure 2. The parameters of the controller (33) are chosen as $β_{1} = 3$ , $β_{2} (x_{1}, s_{1}) = 6 + s_{1}^{2} + x_{1}^{2}$ , $a = 0.9$ , $r_{1} = 0.8$ , $r_{2} = 0.4$ , $λ_{0} = 1.1 L$ , $λ_{1} = 1.5 L^{1 / 2}$ , $L = 10$ . To obtain a satisfactory performance, the parameters of the twisting controller (34) are chosen as $p_{1} = 4.5$ , $p_{2} = 2.5$ and $L_{0} = 20$ . The simulation is carried out by the Euler method and the sampling time is $0.001$ s. The simulation results are given in Figures 3 to 6.

Figure 2.

Time history of the bounded disturbance $d_{2} (t)$ .

Figure 3.

Time history of a sliding variable under the controllers given by equations (33) (solid line) and (34) (dotted line) with initial states $x_{1} (0) = 1, x_{2} (0) = - 2$ .

Figure 4.

Time history of system states under the controllers given by equations (33) (solid line) and (34) (dotted line) with initial states $x_{1} (0) = 1, x_{2} (0) = - 2$ .

Figure 5.

Time history of the virtual controller given by equation (16) with initial states $x_{1} (0) = 1, x_{2} (0) = - 2$ .

Figure 6.

Time history of the controllers given by equations (33) and (34) with initial states $x_{1} (0) = 1, x_{2} (0) = - 2$ .

Figure 3 shows the response curves of the sliding variables and Figure 4 gives the response curves of the state variables. Figure 5 shows the response curve of the virtual controller given by equation (16), while Figure 6 shows the time history of the controllers. It is seen from Figure 3 that both controllers possess some robustness properties while the proposed controller given by equation (33) obtains better tracking performance than the twisting controller given by equation (34) in the presence of nonvanishing mismatched disturbance. Figure 3 also shows that the controller (33) provides a faster convergence rate and a smaller overshoot when compared with the twisting controller (34). These properties in Figure 3 can also be reflected in Figure 4.

On the other hand, the twisting algorithm is designed based on the assumption that the uncertainties should be bounded by some positive constants, which is usually a local condition. This implies that the twisting control method can only obtain a local result. In other words, if the initial states are very large, the system states may diverge to infinity. However, this issue will not happen in the proposed controller (33), since it is a global controller. Let the initial values be $x_{1} (0) = 3$ and $x_{2} (0) = - 8$ . And the parameters of both controllers are chosen to be the same as before. The simulation results are shown in Figures 7 and 8. It can be seen from Figures 7 and 8 that the states under the controller (34) start to diverge when their initial values are large, while the states under the controller (33) can still go back to their steady-state values even when disturbance is added to the system.

Figure 7.

Time history of system states under the controllers given by equations (33) (solid line) and (34) (dotted line) with initial states $x_{1} (0) = 3, x_{2} (0) = - 8$ .

Figure 8.

Time history of the controllers given by equations (33) and (34) with initial states $x_{1} (0) = 3, x_{2} (0) = - 8$ .

Simulation with an unbounded disturbance $d_{2} (t)$

It can be clearly observed that the above chosen disturbance $d_{2} (t)$ is bounded. However, the disturbance may also be unbounded. To this end, we further give the simulation under the following unbounded disturbance

d_{2} (t) = {\begin{matrix} 0, 0 \leq t < 4 \\ 2, 4 \leq t < 8 \\ 0, 8 \leq t < 12 \\ 0.5 t + 4 \sin (t), t \geq 12 \end{matrix}

as shown in Figure 9. With the unbounded disturbance, the initial states of the system (31) are still chosen as $x_{1} (0) = 1$ and $x_{2} (0) = - 2$ . The gain $β_{2} (x_{1}, s_{1})$ of the controller (33) is chosen as $β_{2} (x_{1}, s_{1}) = 4 + s_{1}^{2} + x_{1}^{2}$ and other parameters are the same as before. And the gains of the controller (34) are chosen as $p_{1} = 5.5$ and $p_{2} = 3.5$ . The simulation results under the controllers (33) and (34) are given in Figures 10 to 12. It can be clearly seen from Figures 10 and 11 that under the controller (33) the system states will go back to their steady values even when the system is affected by the unbounded disturbance, while the system states diverge to infinity when the unbounded disturbance is imposed on the system under the controller (34).

Figure 9.

Time history of the unbounded disturbance $d_{2} (t)$ .

Figure 10.

Time history of the sliding variable under the controllers given by equations (33) (solid line) and (34) (dotted line) in the presence of an unbounded disturbance $d_{2} (t)$ .

Figure 11.

Time history of system states under the controllers given by equations (33) (solid line) and (34) (dotted line) in the presence of an unbounded disturbance $d_{2} (t)$ .

Figure 12.

Time history of the controllers given by equations (33) and controller (34) in the presence of an unbounded disturbance $d_{2} (t)$ .

Conclusion

A control problem of sliding mode dynamics with nonvanishing mismatched disturbance has been addressed in this paper. Based on the Lyapunov-based technique, a backstepping-like discontinuous SOSM controller has been developed, which can be used to handle nonvanishing mismatched disturbance. Meanwhile, the finite-time Lyapunov analysis, instead of finite-time convergence, has been given for the closed-loop system. As a byproduct, the frequently-used constant-upper-bound hypotheses of conventional SOSM dynamics has been relaxed to the state-dependent hypotheses in this paper. The extension of the proposed SOSM control method to a higher-order case can be considered as our future research.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Science Foundation of China (grant numbers 61573170 and 31571571), the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Jiangsu Natural Science Foundation for Distinguished Young Scholars (grant number SBK2018010063), the Six Talent Peaks Project in Jiangsu Province(grant number XNYQC-006) and the Graduate Scientific Research innovation Program of Jiangsu Province (grant number KYCX17_1781).

References

Bhat

Bernstein

(2000) Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization 38: 751–766.

Bartolini

Pisano

Punta

et al . (2003) A survey of applications of second-order sliding mode control to mechanical systems. International Journal of Control 76: 875–892.

Ding

Levant

(2016) Simple homogeneous sliding-mode controller. Automatica 67: 22–32.

Ding

(2017) Second-order sliding mode controller design subject to a mismatched term. Automatica 77: 388–392.

Ding

Liu

Zheng

(2017) Sliding mode direct yaw-moment control design for in-wheel electric vehicles. IEEE Transactions on Industrial Electronics 64: 6752–6762.

Ding

Zheng

Sun

et al . (2018) Second-order sliding mode controller design and its implementation for buck converters. IEEE Transactions on Industrial Informatics 14: 1990–2000.

Ding

Wang

Zheng

(2015) Second-order sliding mode control for nonlinear uncertain systems bounded by positive functions. IEEE Transactions on Industrial Electronics 62: 5899–5909.

Chen

et al . (2016) Chattering-free discrete-time sliding mode control. Automatica 68: 87–91.

Emelyanov

Korovin

Levantovsky

(1986) Second order sliding modes in controlling uncertain systems. Soviet Journal of Computing and System Science 24: 63–68.

10.

Filippov

(1988) Differential Equations With Discontinuous Right-Hand Side. Netherlands: Kluwer Academic Publishers.

11.

Chai

(2015) Global finite-time stabilization of a class of switched nonlinear systems with the powers of positive odd rational numbers. Automatica 54: 360–373.

12.

Chai

(2017) Adaptive finite-time stabilization of a class of uncertain nonlinear systems via logic-based switchings. IEEE Transactions on Automatic Control 62: 5998–6003.

13.

Hardy

Littlewood

Polya

(1952) Inequalities. Cambridge: Cambridge University.

14.

Gonzalez

(2017) Numerical synthesis of pontryagin optimal control minimizers using sampling-based methods. IEEE conference on design and control, Melbourne, Australia, 12–15 December 2017, pp.733–738.

15.

Song

Liu

(2017) Robust finite-time bounded controller design of time-delay conic nonlinear systems using sliding mode control strategy. IEEE Transactions on Systems, Man, and Cybernetics: Systems 12 June 2017.

16.

Levant

(1993) Sliding order and sliding accuracy in sliding mode control. International Journal of Control 58: 1247–1263.

17.

Levant

(2003) Higher order sliding modes, differentiation and output-feedback control. International Journal of Control 76: 924–941.

18.

Levant

(2005a) Quasi-continuous high-order sliding mode controllers. IEEE Transactions on Automatic Control 50: 1812–1816.

19.

Levant

(2005b) Homogeneity approach to high order sliding-mode design. Automatica 41: 823–830.

20.

Levant

Michael

(2009) Adjustment of high-order sliding-mode controllers. International Journal of Robust Nonlinear Control 19: 1657–1672.

21.

Man

(1997) Terminal sliding mode control of MIMO linear systems. IEEE Transactions on Control Systems Technology 44: 1065–1070.

22.

Nagesh

Edwards

(2014) A multivariable super-twisting sliding mode approach. Automatica 50: 984–988.

23.

Qian

Lin

(2001) A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Transactions on Automatic Control 46: 1061–1079.

24.

Shen

Zhang

Xia

(2017) Continuous observer design for a class of multi-output nonlinear systems with multi-rate sampled and delayed output measurements. Automatica 75: 127–132.

25.

Shtessel

Dhkolnikov

Levant

(2009) Smooth second-order sliding modes: Missile guidance application. Automatica 45: 110–124.

26.

Shtessel

Edwards

Fridman

et al . (2013) Sliding Mode Control and Observation. Boston: Birkhäuser.

27.

Sun

Zhang

Wang

(2017) Adaptive disturbance attenuation for generalized high-order uncertain nonlinear systems. Automatica 80: 102–109

28.

Tian

Qian

(2014a) A generalized homogeneous solution for global stabilization of a class of non-smooth upper-triangular systems. International Journal of Control 87: 951–963.

29.

Tian

Zhang

Qian

et al . (2014b) Global stabilization of inherently nonlinear systems using continuously differentiable controllers. Nonlinear Dynamics 77: 739–752.

30.

Utkin

(2013) On convergence time and disturbance rejection of super-twisting control. IEEE Transactions on Automatic Control 58: 2013–2017.

31.

Utkin

Guldener

Shi

(1999) Sliding Modes Control in Electro-Mechanical Systems. London: Taylor and Francis.

32.

Wang

Gao

Karimi

et al . (2017a) Sliding mode control of fuzzy singularly perturbed systems with application to electric circuits. IEEE Transactions on Systems Man Cybernetics-Systems 24 August 2017. Epub ahead of print. DOI: 10.1109/TSMC.2017.2720968.

33.

Wang

Lam

(2016) Distributed active anti-disturbance output consensus algorithms for higher-order multi-agent systems with mismatched disturbances. Automatica 74: 30–37.

34.

Wang

et al . (2017b) Distributed active anti-disturbance consensus for leader-follower higher-order multi-agent systems with mismatched disturbances. IEEE Transactions on Automatic Control 62: 5795–5801.

35.

Wang

Pan

et al . (2018) Robust sliding mode control for robots driven by compliant actuators. IEEE Transactions on Control Systems Technology 13 February 2018.

36.

Wang

Shen

Karimi

et al . (2017) Dissipativity-based fuzzy integral sliding mode control of continuous-time T-S fuzzy systems. IEEE Transactions on Fuzzy System 26: 1164–1176.

37.

Wen

Zeng

Huang

(2013) Robust probabilistic sampling

H_{\infty}

output tracking control for a class of nonlinear networked systems with multiplicative noises. Journal of the Franklin Institute 350: 1093–1111.

38.

Xie

Yue

et al . (2014) Further studies on control synthesis of discrete-time T-S fuzzy systems via augmented multi-indexed matrix approach. IEEE Transactions on Cybernetics 44: 2784–2791.

39.

Peng

et al . (2017) Asynchronous dissipative state estimation for stochastic complex networks with quantized jumping coupling and uncertain measurements. IEEE Transactions on Neural Networks and Learning Systems 28: 268–277.

40.

Shi

et al . (2016) Finite-time distributed state estimation over sensor networks with Round-Robin protocol and fading channels. IEEE Transactions on Cybernetics 48: 336–345.

41.

Yang

Ding

(2017) Global output regulation for a class of lower triangular nonlinear systems: A feedback domination approach. Automatica 76: 65–69

42.

Yin

Chen

Zhong

(2014) Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems. Automatica 50: 3173–3181.

43.

Long

(2015) Finite-time consensus for second-order multi-agent systems with disturbances by integral sliding mode. Automatica 54: 158–165.

44.

Zhong

(2011) Semiglobal robust backstepping output tracking for strict-feedback form systems with nonlinear uncertainty. International Journal of Control 9: 366–375.

45.

Zhang

Yang

Ashwin

et al . (2018) Practically oriented finite-fime control design and implementation: Application to series elastic actuator. IEEE Transactions on Industrial Electronics 65: 4166–4176

46.

Zhang

Shen

(2017) Continuous sampled-data observer design for nonlinear systems with time delay larger or smaller than the sampling period. IEEE Transactions on Automatic Control 62: 5822–5829.

47.

Zhao

Yang

et al . (2018) Continuous output feedback TSM control for uncertain systems with a DC-AC inverter example. IEEE Transactions on Circuits and Systems II: Express Briefs 65: 71–75.

48.

Zhu

Xiang

Quan

et al . (2018) Multimode optimization design methodology for a flux-controllable stator permanent magnet memory motor considering driving cycles. IEEE Transactions on Industrial Electronics 65: 5353–5366.

A novel second-order sliding mode control based on the Lyapunov method

Abstract

Keywords

Introduction

Problem formulation

Discontinuous SOSM controller design

Finite-time disturbance observer design

Controller design

Example

Simulation with a bounded disturbance d2(t)

Simulation with an unbounded disturbance d 2 ( t )

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References

Simulation with a bounded disturbance d₂(t)

Simulation with an unbounded disturbance $d_{2} (t)$