Piecewise fast multi-power reaching law: Basis for sliding mode control algorithm

Abstract

A piecewise fast multi-power reaching law (PFMPRL) is proposed aiming at the problems of chattering and slow convergence in the reaching phase of sliding mode control (SMC). In this paper, the fast power reaching law and the double power reaching law are combined, and a nonlinear function is introduced to design the exponential term in PFMPRL. The proposed method ensures the characteristic of fast convergence of the system at all the phases of tendency. The characteristic of fixed-time convergence has also been satisfied. The study proves that the system state can converge to steady-state error bounds within a finite time in the presence of system uncertainty and bounded external disturbance. Compared with the existed methods, the proposed method has shorter convergence time and smaller steady-state error bound. To suppress the influence of model uncertainty and disturbance in system control, a non-linear disturbance observer (NDO) is introduced, and combined with the reaching law-based non-singular terminal sliding mode control (NTSMC), is applied to the cart inverted pendulum system. Simulation results and numerical analysis verify the effectiveness and superiority of this approach.

Introduction

SMC algorithms is a powerful technique and a variable structure method which has the advantages of strong robustness and fast response speed, and is widely used in the stability control of uncertain systems.^1–5 The existence of the switching item of sliding mode control gives it some anti-interference capabilities to model uncertainty and external disturbance, but it also causes serious chattering phenomenon. Chattering phenomenon not only affects the system control accuracy, but also results in a waste of energy in practical application, and can even cause instability in the system by exciting the high frequency dynamics of the unmodeled part. Generally, sliding mode control can be split into two regions—the reaching phase and the sliding phase. As long as the system state reaches the sliding surface, it is invariant to parameter uncertainty and external interference^6,7; however, it will still be affected by disturbances during the reaching phase. Therefore, how to eliminating chattering while also increasing the reaching rate has been an important research topic in the field of sliding mode control.^8–12

To solve the chattering problem in sliding mode control systems, many effective researches have been carried out by scholars both in China and abroad. Boundary layer method,^13–15 also known as quasi-sliding mode method, uses saturation function instead of sign function to keep the system state near the field of sliding mode surface, forming a “quasi-sliding mode.” Although it effectively weakens chattering, it also affects the control accuracy. High-order sliding mode control^16–20 places the switching term of the sliding mode surface function in the first or high-order derivative of the control input, thus greatly reducing chattering. However, it is difficult to obtain the first or high-order derivative of the sliding mode variable, and it is difficult to apply it in the first-order system. Terminal sliding mode control^21–23 can not only effectively eliminate chattering, but also ensure the convergence of system state in a finite time. However, its convergence speed is slower compared with the traditional exponential reaching law of sliding mode control. The SMC method,^24–27 based on the reaching law, is similar to the terminal sliding mode control in its mathematical structure, ²⁸ except that the latter is mainly aimed at the dynamic quality of the sliding mode. Compared with the above control methods, the method with the reaching law improves the dynamic performance of the system in the reaching stage of the sliding process while weakening chattering. The design of the approach law has an important influence on the convergence speed, control accuracy and stability of the system.

Gao ²⁹ proposed isokinetic reaching law, exponential reaching law (ERL), power reaching law (PRL) and general reaching law for the chattering phenomenon in sliding mode system control, which laid the foundation for the study of approach law. In ref., ³⁰ combining the advantages of ERL and PRL, a fast power reaching law is proposed (FPRL), which is distant from the sliding mode surface and can enter into the sliding mode smoothly, thus weakening chattering. According to the double power reaching law (DPRL) proposed in ref., ³¹ Li and Cai ³² found a specific class of DPRL and established the convergence formula of the DPRL, which further accelerated the convergence speed when it was far from the equilibrium point, but the dynamic property near the equilibrium point was not as good as the fast power approach law. Liao et al. ³³ combined the fast power and double power reaching laws through the fal function, and proposed a double power combination function reaching law (DPCFRL), and which has the former’s characteristics when it is close to the equilibrium point and the latter’s characteristics when it is far from the equilibrium point, but the convergence speed of the reaching law function is quite different in the two stages. In ref., ³⁴ according to the structure of fast power reaching law, a new variable exponential power reaching law (VPRL) is proposed, which realizes the adaptive adjustment of the approaching speed through the design of the power term and improves the system’s approaching speed. However, it cannot maximize the approaching speed because it is limited by the form of the fast power reaching law when it is far from the equilibrium point. In order to consider the approaching speed when both moving away from and near the equilibrium point, the convergent law in the form of combinatorial variable exponential is a more effective choice.

In this paper, in order to improve the convergence speed in each stage of the reaching phase, a piecewise fast multi-power reaching law is proposed. The main contributions of this paper are summarized below:

1. By designing a nonlinear function, we divide the approaching process of the system into two stages. When it is remote from the equilibrium point, it is characterized as the property of double powers reaching law, and when it is near the equilibrium point, it is characterized as the property of fast powers reaching law; the coefficients of the exponential term will be adjusted according to the different phases. It greatly improves the approach rate of the system throughout the process approach.

2. The fixed-time convergence characteristics and steady-state error bounds of the proposed reaching law are analyzed. The theoretical analysis proves its superiority and effectiveness.

3. In consideration of the presence of disturbances on the system, it will affect the control performance, a disturbance observer based on tracking differentiator is proposed to estimate the uncertainty of system parameters and external disturbance, and its effectiveness is analyzed theoretically.

4. Simulation results and numerical analysis show that the convergence speed and steady-state error bounds of the proposed approach law are better than other approaches, and the proposed PFMPRL has better dynamic quality than the existing reaching law method when applied to the stability control of the inverted pendulum system by combining the non-singular terminal synovial control.

In the rest parts of this paper, in section “Piecewise fast multi-power reaching law,” the piecewise fast multi-power reaching law (PFMPRL) is proposed. In section “Dynamic performance analysis,” the performance of the reaching law is analyzed, including the characteristic of fixed-time convergence and the steady-state error bounds. In section “Simulation,” the proposed reaching law is compared with the existing reaching law by simulation and numerical analysis, and it is applied to the inverted pendulum system. Section “Conclusion” concludes the paper.

Piecewise fast multi-power reaching law

Related work

In refs.^31,33 fast power approximation law and double power approximation law were proposed respectively:

$$\stackrel{·}{s}=-k_{1}s-k_2{\left|s\right|}^{\gamma }\mathrm{sgn}\left(s\right)$$

(1)

$$\stackrel{·}{s}=-k_1{\left|s\right|}^{{\gamma }_1}\mathrm{sgn}\left(s\right)-k_2{\left|s\right|}^{\gamma }\mathrm{sgn}\left(s\right)$$

(2)

Where, $k_1>0$ , $k_2>0$ , $0<\gamma <1$ , ${\gamma }_1>1$ , $s$ is the sliding mode surface.

If external disturbances are not considered, the above two reaching laws converge the sliding mode surface $s$ and its first derivative $\stackrel{·}{s}$ to 0 within a finite time. The reaching process of the system can be seen as two phases: away from the sliding surface phase, and close to the sliding surface phase. In this paper, we suppose 1 as the borderland of $\left|s\right|$ in two phases. When $\left|s\right|>1$ , the previous term of both equations plays a leading role; And, when $\left|s\right|\to 0$ , the latter term of the two equations plays a leading role.

Take the Lyapunov function $V=s^2$ , and when the initial state of the sliding mode surface function $s_0>1$ , substitute equations (1) and (2) into the Lyapunov function respectively and take the derivative:

$$\stackrel{·}{V}=-2k_{1}V-2k_2{V}^{\frac{\gamma +1}{2}}$$

(3)

$$\stackrel{·}{V}=-2k_1{V}^{{\gamma }_1+1/2}-2k_2{V}^{\frac{\gamma +1}{2}}$$

(4)

The following two cases are discussed:

1) $s\left(t\right)>1,$

At this point, the sliding mode state $s\left(t\right)$ is remote from the sliding mode surface. From equations (3) and (4), it can be seen that the previous term on the right side of the equation of the two equations plays a leading role, while the second term on the right side is the same and has little influence on the approaching speed. Therefore, the difference in the approaching speed between the two equations is mainly reflected in the first term:

$$\begin{array}{c}\multicolumn{1}{c}{t_{11}=\frac{1}{4k_1}\ln V_0}\\ \multicolumn{1}{c}{=\frac{1}{k_1(1-{\gamma }_1)}\ln (1-z)}\end{array}$$

(5)

$$\begin{array}{c}\multicolumn{1}{c}{t_{21}=-\frac{1}{k_1(1-{\gamma }_1)}(1-{V_0}^{1-{\gamma }_1/4})}\\ \multicolumn{1}{c}{=-\frac{z}{k_1(1-{\gamma }_1)}}\end{array}$$

(6)

Where, $z=1-{V_0}^{\raisebox{1ex}{$1-{\gamma }_1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$ , due to $V_0=s_0^2>1$ , ${\gamma }_1>1$ , therefore, $0<z<1$ , $\frac{\ln (1-z)}{-z}>1$ , which shows that $t_{11}>t_{21}$ ; when $s\left(t\right)>1$ , the convergence time of the DPRL is shorter.

2) $s\left(t\right)<1,$

At this point, the sliding mode state $s\left(t\right)$ is next to the sliding mode surface, and the initial state is $s_0=1$ . In refs.^35,36 in combination with equations (3) and (4), the convergence time of fast power and double power reaching laws respectively is:

$$\begin{array}{c}\multicolumn{1}{c}{t_{12}=-\frac{1}{k_1(1-\gamma )}\ln (1+\frac{k_1}{k_2}{\left|s_0\right|}^{1-\gamma })}\\ \multicolumn{1}{c}{=-\frac{1}{k_1(1-\gamma )}\ln (1+\frac{k_1}{k_2})}\end{array}$$

(7)

$$\begin{array}{c}{t}_{22}=-\frac{{\left|s_0\right|}^{1-{\gamma }_1}{k_1}^{\raisebox{1ex}{$1-{\gamma }_1$}\!\left/ \!\raisebox{-1ex}{${\gamma }_1$}\right.}}{\left(1-{\gamma }_1\right)}\\ \times F\left(1,\frac{{\gamma }_1-1}{{\gamma }_1-\gamma };\frac{2{\gamma }_1-\gamma -1}{{\gamma }_1-\gamma };-\frac{K_2}{K_1}{\left|s_0\right|}^{\gamma -{\gamma }_1}\right)\\ =-\frac{{k_1}^{\raisebox{1ex}{$1-{\gamma }_1$}\!\left/ \!\raisebox{-1ex}{${\gamma }_1$}\right.}}{\left(1-{\gamma }_1\right)}\times F\left(1,\frac{{\gamma }_1-1}{{\gamma }_1-\gamma };\frac{2{\gamma }_1-\gamma -1}{{\gamma }_1-\gamma };-\frac{K_2}{K_1}\right)\end{array}$$

(8)

Where, $F(·)$ is gauss hypergeometric function. Further calculations can be obtained $\raisebox{1ex}{$t_{12}$}\!\left/ \!\raisebox{-1ex}{$t_{22}$}\right.<1$ , therefore, when $s\left(t\right)\to 0$ , the FPRL has a shorter convergence time.

From the above, when the system state is remote from the sliding mode surface, the DPRL has a faster reaching rate, and when the system state is next to the sliding mode surface, the FPRL has a faster reaching rate. Further analysis of the convergence time equations (5)–(8) of the two reaching laws shows that the values of $\gamma$ and ${\gamma }_1$ have a great influence on the approach rate. At different stages, the power terms have different effects.

In ref. ³⁴ the variable exponential power reaching law is shown as follows:

$$\stackrel{·}{s}=-k_{1}s-k_2{\left|s\right|}^{\lambda }\mathrm{sgn}\left(s\right)$$

(9)

Where, $\lambda =\{\begin{array}{c}\alpha ,\left|s\right|<1\hfill \\ \multicolumn{1}{c}{\left|s\right|,\left|s\right|\ge 1}\end{array}$ , $k_1>0$ , $k_2>0$ , $0<\alpha <1$ . Due to the addition of variable exponential term, the value of $\lambda$ can be changed adaptively, so that the system can reach a faster approach rate in different stages.

Design of piecewise fast multi-power reaching law

According to the analysis of the reaching law method proposed above, combining the advantages of fast power and double power reaching laws in different stages, and the idea of variable exponential reaching law for adaptive adjustment of exponential term, a piecewise fast multi-power reaching law is proposed:

$$\stackrel{·}{s}=-k_1{r}_{1}s-k_2{\left|s\right|}^{r_2}\mathrm{sgn}\left(s\right)-k_3{\left|s\right|}^{r_3}\mathrm{sgn}\left(s\right)$$

(10)

Where, $k_1>0$ , $k_2>0$ , $k_3>0$ . Consider a nonlinear function which is designed to divide the reaching law into two stages and make the exponential parameters adjust according to different stages:

$$r_1=\{\begin{array}{c}{\alpha }_1+{\beta }_1\tanh \left(\kappa (\mathrm{\Delta }-\left|s\right|)\right),\left|s\right|\ne \mathrm{\Delta }\hfill \\ \multicolumn{1}{c}{{\alpha }_1+{\beta }_1,\left|s\right|=\mathrm{\Delta }}\end{array}$$

(11)

$$r_2=\{\begin{array}{c}{\alpha }_2+{\beta }_2\tanh \left(\kappa (\left|s\right|-\mathrm{\Delta })\right),\left|s\right|\ne \mathrm{\Delta }\hfill \\ \multicolumn{1}{c}{{\alpha }_2-{\beta }_2,\left|s\right|=\mathrm{\Delta }}\end{array}$$

(12)

$$r_3=\{\begin{array}{c}{\alpha }_3+{\beta }_3\tanh \left(\kappa (\left|s\right|-\mathrm{\Delta })\right),\left|s\right|\ne \mathrm{\Delta }\hfill \\ \multicolumn{1}{c}{{\alpha }_3-{\beta }_3,\left|s\right|=\mathrm{\Delta }}\end{array}$$

(13)

Where ${\alpha }_1={\beta }_1=\frac{1}{2}$ , ${\alpha }_2\ge 1$ , $0<{\beta }_2<1$ , ${\alpha }_3\ge 1$ , $\frac{1}{2}<{\beta }_2<1$ , $\mathrm{\Delta }=1$ . The reaching law with fixed power term parameters must consider the reaching rate when the state is remote from or next to the sliding mode surface at the same time, the designed nonlinear function takes advantage of the characteristics of the hyperbolic tangent function $\tanh (\kappa ·x)=\frac{e^{\kappa ·x}-e^{-\kappa ·x}}{e^{\kappa ·x}+e^{-\kappa ·x}}$ , which is smooth and continuous, the value range is [−1,1], and it is approximately a linear function at the equilibrium point ³⁷ ; the parameters of the power term of the reaching law can be adjusted according to the sliding mode function, so as to take into account the approaching speed of the states in different stages. If $\kappa \to \infty$ , the value of $r_1$ , $r_2$ , and $r_3$ can be divided into two situations:

$$\begin{array}{cc}\left(1\right)\hfill & \left|s\right|>\mathrm{\Delta },\hfill \\ \hfill & \{\begin{array}{c}{r}_1={\alpha }_1-{\beta }_1\hfill \\ \multicolumn{1}{c}{r_2={\alpha }_2+{\beta }_2}\\ \multicolumn{1}{c}{r_3={\alpha }_3+{\beta }_3}\end{array}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\left(2\right)\hfill & \left|s\right|\le \mathrm{\Delta },\hfill \\ \hfill & \{\begin{array}{c}{r}_1={\alpha }_1+{\beta }_1\hfill \\ \multicolumn{1}{c}{r_2={\alpha }_2-{\beta }_2}\\ \multicolumn{1}{c}{r_3={\alpha }_3-{\beta }_3}\end{array}\hfill \end{array}$$

(15)

When $\left|s\right|>\mathrm{\Delta }$ , according to equation (14), equation (10) can be equivalent to the form of double power reaching law (2); when $\left|s\right|\le \mathrm{\Delta }$ , according to equation (15), equation (10) can be similarly equivalent to the form of fast power reaching law (1), as shown in equations (16) and (17), respectively. The exponential parameter selection is also more flexible. According to the convergence time formulas of equations (7) and (8), appropriate parameters can be selected to make the convergence time shorter.

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{s}=-k_2{\left|s\right|}^{r_2}\mathrm{sgn}\left(s\right)-k_3{\left|s\right|}^{r_3}\mathrm{sgn}\left(s\right)}\\ \multicolumn{1}{c}{=-k_2{\left|s\right|}^{{\alpha }_2+{\beta }_2}\mathrm{sgn}\left(s\right)-k_3{\left|s\right|}^{{\alpha }_3+{\beta }_3}\mathrm{sgn}\left(s\right)}\end{array}$$

(16)

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{s}=-k_1{r}_{1}s-k_2{\left|s\right|}^{r_2}\mathrm{sgn}\left(s\right)-k_3{\left|s\right|}^{r_3}\mathrm{sgn}\left(s\right)}\\ \multicolumn{1}{c}{=-k_{1}s-k_2{\left|s\right|}^{{\alpha }_2-{\beta }_2}\mathrm{sgn}\left(s\right)-k_3{\left|s\right|}^{{\alpha }_3-{\beta }_3}\mathrm{sgn}\left(s\right)}\end{array}$$

(17)

The proposed new reaching law method combines the advantages of the FPRL and DPRL; whether it is remote from or next to the sliding mode surface, it has the characteristics of rapid approach and no chattering. Through the design of the nonlinear function, the power term parameters are adjusted according to different stages, which further improves the approach rate, and there is a maximum convergence time independent of the initial state. When there is a bounded disturbance, the state of the system can also converge within the bound of steady-state error in finite time.

Dynamic performance analysis

The characteristic of fixed-time convergence

In order to facilitate the analysis, the definition and lemma of fixed-time convergence are first introduced before the characteristic analysis of the reaching law ³⁸ :

Definition 1

The origin of a system is called as fixed-time attractive if it is uniformly finite-time attractive with an attraction domain $\nu$ and the settling time function $T(t_0,x_0)$ is bounded on $\mathbb{R}\times \nu$ , that is there exists a number $T_{max}\in {\mathbb{R}}_{+}$ such that $T(t_0,x_0)<T_{max}$ if $t_0\in \mathbb{R}$ and $x_0\in \nu$ .

Lemma 1

Let a continuous function $V\left(x\right)$ : ${\mathbb{R}}^n\to \mathbb{R}$ be proper on an open connected set $\Omega$ : 0 ∈ $\mathrm{int}\left(\Omega \right)$ . If for some numbers $\mu \in (0,1)$ , $\upsilon \in {\mathbb{R}}_{+}$ , ${\mathrm{\Upsilon }}_{\mu }\in {\mathbb{R}}_{+}$ , ${\mathrm{\Upsilon }}_{\upsilon }\in {\mathbb{R}}_{+}$ the following inequality can be obtained,

$$\stackrel{·}{V}=\{\begin{array}{c}-{\mathrm{\Upsilon }}_{\mu }{V}^{1-\mu },V\le 1\hfill \\ \multicolumn{1}{c}{-{\mathrm{\Upsilon }}_{\upsilon }{V}^{1+\upsilon },V\ge 1}\end{array}$$

(18)

then the origin of the system can be called global fixed-time convergence equilibrium point, furthermore, the maximum settling time is estimated by:

$$T\left(x_0\right)\le T_{max}\le \frac{1}{\mu {\mathrm{\Upsilon }}_{\mu }}+\frac{1}{\upsilon {\mathrm{\Upsilon }}_{\upsilon }}$$

(19)

Based on the above lemma 1, ³⁸ the following theorem is given to demonstrate the characteristic of fixed time convergence of the proposed algorithm.

Theorem 1

For system (10), system states $s$ and $\stackrel{·}{s}$ converge to 0 within a fixed time $T_{max}$ , that is, after a finite time $T\left(s_0\right)$ , $s=\stackrel{·}{s}=0$ , and the convergence time $T\left(s_0\right)$ has an upper bound $T_{max}$ independent of the initial value of the state:

$$T_{max}=\frac{1}{2k_2({\beta }_2-{\alpha }_2+1)}+\frac{1}{2k_3({\alpha }_3+{\beta }_3-1)}$$

(20)

Proof

Define the Lyapunov function as follows:

$$V=s^2$$

(21)

It is clear that $V\left(0\right)=0$ , when $s>\mathrm{\Delta }$ , system (10) is in the form of equation (16). According to Mei and Wang, ³¹ the system (16) can reaches the steady-state in a finite time, that is, $s=0$ is the finite-time equilibrium point. When $s\le \mathrm{\Delta }$ , system (10) is in the form of equation (17), and according to Liao et al. and Zhang et al.^33,39 it can be deduced that the system (17) can reaches the steady-state in a finite time. Thus, for system (10), $s=0$ is the equilibrium point for global finite time convergence, and the first condition of lemma 1 is satisfied.

Find the time derivative of Lyapunov function $V$ and substitute equation (10) into it:

$$\begin{array}{c}V=2s\stackrel{.}{s}\\ =\{\begin{array}{l}-2k_1{s}^2-2k_2{\left|s\right|}^{r_2+1}\\ -2k_3{\left|s\right|}^{r_3+1},V\le {\mathrm{\Delta }}^2\\ -2k_2{\left|s\right|}^{r_2+1}\\ -2k_3{\left|s\right|}^{r_3+1},V>{\mathrm{\Delta }}^2\end{array}\\ =\{\begin{array}{l}-2k_{1}V-2k_2{V}^{\raisebox{1ex}{$r_2+1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ -2k_3{V}^{\raisebox{1ex}{$r_3+1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},V\le {\mathrm{\Delta }}^2\\ -2k_2{V}^{\raisebox{1ex}{$r_2+1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\\ -2k_3{V}^{\raisebox{1ex}{$r_3+1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},V>{\mathrm{\Delta }}^2\end{array}\\ \multicolumn{1}{c}{=\{\begin{array}{c}-2k_{1}V-2k_2{V}^{{\alpha }_2-{\beta }_2+1/2}\hfill \\ \multicolumn{1}{c}{-2k_3{V}^{{\alpha }_3-{\beta }_3+1/2},V\le {\mathrm{\Delta }}^2}\\ \multicolumn{1}{c}{-2k_2{V}^{{\alpha }_2-{\beta }_2+1/2}}\\ \multicolumn{1}{c}{-2k_3{V}^{{\alpha }_3+{\beta }_3+1/2},V>{\mathrm{\Delta }}^2}\end{array}}\\ \multicolumn{1}{c}{\le \{\begin{array}{c}-4k_2{V}^{1-({\beta }_2-{\alpha }_2+1/2)},V\le {\mathrm{\Delta }}^2\hfill \\ \multicolumn{1}{c}{-2k_3{V}^{1-({\alpha }_3-{\beta }_3+1/2)},V>{\mathrm{\Delta }}^2}\end{array}}\end{array}$$

(22)

Where, ${\alpha }_2-{\beta }_2\ge {\alpha }_3-{\beta }_3$ , $k_2\le k_3$ , the fixed time of the state $s$ can be obtained according to the corresponding parameter relationships in equations (18) and (19), and ${\mathrm{\Upsilon }}_{\mu }=4k_2$ , ${\mathrm{\Upsilon }}_{\upsilon }=2k_3$ , $\mu =\frac{{\beta }_2-{\alpha }_2+1}{2}$ , $\upsilon =\frac{{\alpha }_3+{\beta }_3-1}{2}$ . According to lemma 1, $s=0$ is the equilibrium point of global fixed time convergence of system (10), and the convergence time $T\left(s_0\right)$ satisfies:

$$\begin{array}{c}\multicolumn{1}{c}{T\left(s_0\right)\le T_{max}}\\ \multicolumn{1}{c}{=\frac{1}{\mu {\mathrm{\Upsilon }}_{\mu }}+\frac{1}{\upsilon {\mathrm{\Upsilon }}_{\upsilon }}}\\ \multicolumn{1}{c}{=\frac{1}{2k_2({\beta }_2-{\alpha }_2+1)}+\frac{1}{k_3({\alpha }_3+{\beta }_3-1)}}\end{array}$$

(23)

In conclusion, theorem 1 is proved.

Analysis of steady-state error bound

The above analysis shows that PFMPRL can converge to the equilibrium point $s=0$ within a global fixed time, and gives the upper bound of the convergence of which independent initial state values. If the system has parameter uncertainty and external disturbance, the states $s$ and $\stackrel{·}{s}$ can only converge to a domain of equilibrium in finite time.

Lemma 2

$x\in D\subset R^n$ , $\stackrel{·}{x}=f\left(x\right)$ , $f:R^n\to R^n$ is a continuous function defined in the domain $D$ of equilibrium. Suppose a continuous function $V$ : $D\to R$ holds under the following conditions ⁴⁰ :

1. $V$ is a continuously differentiable positive definite function;

2. When $x\ne 0$ , $\stackrel{·}{V}$ is negative definite;

3. There are real numbers $\sigma >0$ , $\phi >0$ and a certain neighborhood $N\subset D$ , such that $\stackrel{·}{V}+\sigma V^{\phi }\le 0$ .

Then the function $\stackrel{·}{x}=f\left(x\right)$ converges into the domain in a finite time about the equilibrium point.

According to Lemma 2, when considering the parameter uncertainty and external disturbance of the system (10), the following theorem is produced:

Theorem 2

Consider systems with uncertainties and external disturbances:

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{s}=-k_1{r}_{1}s-k_2{\left|s\right|}^{r_2}\mathrm{sgn}\left(s\right)}\\ \multicolumn{1}{c}{-k_3{\left|s\right|}^{r_3}\mathrm{sgn}\left(s\right)+d\left(t\right)}\end{array}$$

(24)

Where $d\left(t\right)$ is a bounded disturbance, and there is a positive constant $\delta >0$ , such that $d\left(t\right)$ satisfies $\left|d\left(t\right)\right|\le \delta$ , then the state $s$ of the system (24) and its first derivative $\stackrel{·}{s}$ can converge to the neighborhood in finite time as follows:

$$\left|s\right|\le \min \left\{\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_1{r}_1$}\right.,{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_2$}\right.\right)}^{\frac{1}{r_2}},{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_3$}\right.\right)}^{\frac{1}{r3}}\right\}$$

(25)

$$\begin{array}{c}\left|\stackrel{.}{s}\right|\le \min \left\{\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$r_1$}\right.,k_1{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_2$}\right.\right)}^{\frac{1}{r_2}},k_1{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_3$}\right.\right)}^{\frac{1}{r3}}\right\}\\ +\min \left\{\delta ,k_2{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_1{r}_1$}\right.\right)}^{{\mathrm{r}}_2},k_2{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_3$}\right.\right)}^{\frac{{\mathrm{r}}_2}{r_3}}\right\}\\ +\min \left\{\delta ,k_3{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_1{r}_1$}\right.\right)}^{{\mathrm{r}}_3},k_3{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_2$}\right.\right)}^{\frac{{\mathrm{r}}_2}{r_3}}\right\}+\delta \end{array}$$

(26)

Proof

Define the following Lyapunov function:

$$V=\frac{1}{2}{s}^2$$

(27)

Find the time derivative for (27), and substitute equation (24) into it:

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}=s\stackrel{·}{s}}\\ \multicolumn{1}{c}{=-k_1{r}_1{s}^2-k_2{\left|s\right|}^{r_2+1}-k_3{\left|s\right|}^{r_3+1}+d\left(t\right)s}\\ \multicolumn{1}{c}{\le -k_1{r}_1{s}^2-k_2{\left|s\right|}^{r_2+1}-k_3{\left|s\right|}^{r_3+1}+d\left(t\right)\left|s\right|}\\ \multicolumn{1}{c}{\le -k_1{r}_1{s}^2-k_2{\left|s\right|}^{r_2+1}-k_3{\left|s\right|}^{r_3+1}+\delta \left|s\right|}\end{array}$$

(28)

Where, as can be seen from equation (11), the term of $-k_1{r}_1{s}^2$ exists if and only if $\left|s\right|\le \mathrm{\Delta }$ , equation (28) can be changed to:

$$\{\begin{array}{c}\stackrel{·}{V}+k_1{r}_1{s}^2\le -k_2{\left|s\right|}^{r_2+1}-k_3{\left|s\right|}^{r_3+1}+\delta \left|s\right|\hfill \\ \multicolumn{1}{c}{\stackrel{·}{V}+k_2{\left|s\right|}^{r_2+1}\le -k_1{r}_1{s}^2-k_3{\left|s\right|}^{r_3+1}+\delta \left|s\right|}\\ \multicolumn{1}{c}{\stackrel{·}{V}+k_3{\left|s\right|}^{r_3+1}\le -k_2{\left|s\right|}^{r_2+1}-k_1{r}_1{s}^2+\delta \left|s\right|}\end{array}$$

(29)

For equation (29), if satisfies one of $k_2{\left|s\right|}^{r_2}\ge \delta$ , $k_3{\left|s\right|}^{r_3}\ge \delta$ , $k_1{r}_1\left|s\right|\ge \delta$ , there are:

$$\stackrel{·}{V}+\sigma V^{\phi }\le 0$$

(30)

Where, corresponding to equation (29), $\sigma =k_1{r}_1$ , $\phi =1$ or $\sigma =k_2$ , $\phi =\frac{r_2+1}{2}$ or $\sigma =k_3$ , $\phi =\frac{r_3+1}{2}$ , according to lemma 2, the state $s$ can converge to the following neighborhood:

$$\left|s\right|\le {\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_2$}\right.\right)}^{\frac{1}{r_2}}$$

(31)

$$\left|s\right|\le {\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_3$}\right.\right)}^{\frac{1}{r3}}$$

(32)

$$\left|s\right|\le \raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_1{r}_1$}\right.$$

(33)

Because only one of these conditions has to be satisfied, state $s$ converges to the domain (25) in finite time, and equation (25) is substituted into equation (24) to obtain:

$$\begin{array}{c}\left|\stackrel{.}{s}\right|\le k_1{r}_{1}s+k_2{\left|s\right|}^{r_2}\mathrm{sgn}\left(s\right)+k_3{\left|s\right|}^{r_3}\mathrm{sgn}\left(s\right)+\left|d\left(t\right)\right|\\ \le k_1{r}_{1}s+k_2{\left|s\right|}^{r_2}+k_3{\left|s\right|}^{r_3}+\delta \\ \le k_1\min \left\{\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_1{r}_1$}\right.,{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_2$}\right.\right)}^{\frac{1}{r_2}},{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_3$}\right.\right)}^{\frac{1}{r_3}}\right\}\\ +k_2\min {\left\{\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_1{r}_1$}\right.,{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_2$}\right.\right)}^{\frac{1}{r_2}},{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_3$}\right.\right)}^{\frac{1}{r3}}\right\}}^{r_2}\\ +k_3\min {\left\{\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_1{r}_1$}\right.,{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_2$}\right.\right)}^{\frac{1}{r_2}},{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_3$}\right.\right)}^{\frac{1}{r3}}\right\}}^{r_3}+\delta \\ =\min \left\{\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$r_1$}\right.,k_1{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_2$}\right.\right)}^{\frac{1}{r_2}},k_1{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_3$}\right.\right)}^{\frac{1}{r3}}\right\}\\ +\min \left\{\delta ,k_2{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_1{r}_1$}\right.\right)}^{{\mathrm{r}}_2},k_2{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_3$}\right.\right)}^{\frac{{\mathrm{r}}_2}{r_3}}\right\}\\ +\min \left\{\delta ,k_3{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_1{r}_1$}\right.\right)}^{{\mathrm{r}}_3},k_3{\left(\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$k_2$}\right.\right)}^{\frac{{\mathrm{r}}_2}{r_3}}\right\}+\delta \end{array}$$

(34)

In conclusion, theorem 2 is proved.

Design of disturbance observer

As can be seen from section “Analysis of steady-state error bound,” when there is a bounded interference in the system, the sliding mode variable $s$ no longer converges to 0 in a finite time, but converges to a neighborhood near the equilibrium point. In practical applications, in order to suppress the impact of disturbance on the system and improve the control accuracy, a nonlinear disturbance observer (NDO) can be used to estimate the interference. According to the tracking differentiator proposed by Dong and Zhang and Bu et al.,^41,42 a NDO based on the hyperbolic tangent tracking differentiator (TANHTD-NDO) is designed.

Without loss of generality, we consider the second order nonlinear control system of the form:

$$\stackrel{·}{x}=f(x)+g\left(x\right)·u\left(t\right)+n\left(t\right)$$

(35)

Where, $x\in \mathbb{R}$ is the system state, and $u\in \mathbb{R}$ is the control input. $n\left(t\right)\in \mathbb{R}$ is the disturbance term which including uncertainty and external disturbance.

Lemma 3

For the following system ⁴² :

$$\{\begin{array}{c}\stackrel{·}{\hat{x}}=f(t,x)+g(t,x)u\left(t\right)+\hat{n},\hfill \\ \multicolumn{1}{c}{\stackrel{·}{\hat{n}}=R^{2}F(\hat{x}-x,\hat{n}/R),}\end{array}$$

(36)

where, $\hat{x}$ is the estimate of system state $x$ , and $\hat{n}$ is the estimate of the lumped disturbance $n$ , $F(·)$ is a nonlinear function. If $R>0$ , $T>0$ , the solution to equation (36) satisfies $\underset{R\to \infty }{lim}{\int }_0^T|\hat{x}-x|\mathrm{dt}=0$ , namely $\hat{x}$ converges to $x$ ; furthermore, according to equations (35) and (36), $\hat{n}$ converges to $n$ .

Lemma 1 provides a general design method for a disturbance observer combined with a tracking differentiator. Therefore, it is possible to further applied to the design of disturbance observer by constructing a suitable tracking differentiator.

The hyperbolic tangent tracking differentiator is simple in form, with few parameters to be set, and, because its function is smooth and continuous, it can effectively suppress the output chattering. The tracking differentiator can guarantee the stability and convergence of the proposed disturbance observer, and has excellent tracking performance. The outstanding advantage of the proposed disturbance observer is that it can estimate almost all types of disturbances and does not need the prior information of disturbances. The design of a TANHTD-NDO is shown below:

$$\{\begin{array}{c}\stackrel{·}{\hat{x}}=f(t,x)+g(t,x)·u+\hat{n}\hfill \\ \multicolumn{1}{c}{\stackrel{·}{\hat{n}}=R^2[-a_1\tanh \left(w_1(\hat{x}-x)\right)}\\ \multicolumn{1}{c}{-a_2\tanh (w_2\hat{n}/R)]}\end{array}$$

(37)

where $\hat{x}$ and $\hat{n}$ are the estimates of $x$ and $n$ , $R>0$ , $a_1>0$ , $a_2>0$ , $w_1>0$ , $w_2>0$ is the design parameter.

The defined variable $\xi =n-\hat{n}$ represents the error between the estimated disturbance of the nonlinear disturbance observer and the actual disturbance. Suppose the observation error $\xi$ satisfies the following relation: There are positive constant $\zeta >0$ , such that $\left|\xi \right|\le \zeta$ . At this time, system (24) can be written as:

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{s}\le -k_1{r}_{1}s-k_2{\left|s\right|}^{r_2}\mathrm{sgn}\left(s\right)}\\ \multicolumn{1}{c}{-k_3{\left|s\right|}^{r_3}\mathrm{sgn}\left(s\right)+\xi }\end{array}$$

(38)

According to theorem 2, the states $s$ and $\stackrel{·}{s}$ of system (24) converge to the neighborhood in finite time as follows:

$$\left|s\right|\le \min \left\{\raisebox{1ex}{$\zeta $}\!\left/ \!\raisebox{-1ex}{$k_1{r}_1$}\right.,{\left(\raisebox{1ex}{$\zeta $}\!\left/ \!\raisebox{-1ex}{$k_2$}\right.\right)}^{\frac{1}{r_2}},{\left(\raisebox{1ex}{$\zeta $}\!\left/ \!\raisebox{-1ex}{$k_3$}\right.\right)}^{\frac{1}{r_3}}\right\}$$

(39)

$$\begin{array}{c}\left|\stackrel{.}{s}\right|\le \min \left\{\raisebox{1ex}{$\zeta $}\!\left/ \!\raisebox{-1ex}{$r_1$}\right.,k_1{\left(\raisebox{1ex}{$\zeta $}\!\left/ \!\raisebox{-1ex}{$k_2$}\right.\right)}^{\frac{1}{r_2}},k_1{\left(\raisebox{1ex}{$\zeta $}\!\left/ \!\raisebox{-1ex}{$k_3$}\right.\right)}^{\frac{1}{r3}}\right\}\\ +\min \left\{\zeta ,k_2{\left(\raisebox{1ex}{$\zeta $}\!\left/ \!\raisebox{-1ex}{$k_1{r}_1$}\right.\right)}^{{\mathrm{r}}_2},k_2{\left(\raisebox{1ex}{$\zeta $}\!\left/ \!\raisebox{-1ex}{$k_3$}\right.\right)}^{\frac{{\mathrm{r}}_2}{r_3}}\right\}\\ +\min \left\{\zeta ,k_3{\left(\raisebox{1ex}{$\zeta $}\!\left/ \!\raisebox{-1ex}{$k_1{r}_1$}\right.\right)}^{{\mathrm{r}}_3},k_3{\left(\raisebox{1ex}{$\zeta $}\!\left/ \!\raisebox{-1ex}{$k_2$}\right.\right)}^{\frac{{\mathrm{r}}_2}{r_3}}\right\}+\zeta \end{array}$$

(40)

Where the nonlinear disturbance observer based on the tracking differentiator has good tracking performance to the disturbance and can accurately estimate the disturbance, disturbance observation error $\xi \to 0$ , the upper bound $\zeta \to 0$ . From this we can see steady-state error bound $\left|s\right|\to 0$ , $\left|\stackrel{·}{s}\right|\to 0$ . In conclusion, when there are bounded disturbances, the nonlinear disturbance observer can greatly improve the control accuracy of the approach law and improve the dynamic characteristics of the system.

Simulation

Platform introduction

• 1. Consider a SISO system:

$$\stackrel{·}{s}=u+d\left(t\right)$$

(41)

Where $u$ is the control input to the system, $d\left(t\right)\le \delta$ is the time-varying bounded disturbance, $s$ is the sliding mode variable. The PFMPRL proposed in this paper was compared and simulated with fast power reaching law (FPRL), ³⁰ double power reaching law (DPRL), ³¹ double power combination function reaching law (DPCFRL) ³³ and variable power reaching law (VPRL). ³⁴ All of the above reaching law can be represented as follow:

• a. FPRL

$$u_1=-k_{1}s-k_2{\left|s\right|}^r\mathrm{sgn}\left(s\right)$$

(42)

• b. DPRL

$$u_2=k_1{\left|s\right|}^{\rho }\mathrm{sgn}\left(s\right)-k_2{\left|s\right|}^{\gamma }\mathrm{sgn}\left(s\right)$$

(43)

• c. DPCFRL

$$\begin{array}{c}{u}_3=k_1\mathrm{fal}-k_2{\left|s\right|}^{\gamma }\mathrm{sgn}\left(s\right)\hfill \\ \mathrm{fal}=\{\begin{array}{c}{\left|s\right|}^{\rho }\mathrm{sgn}\left(s\right),\left|s\right|>\mathrm{\Delta }\hfill \\ \multicolumn{1}{c}{\frac{s}{{\mathrm{\Delta }}^{1-\rho }},\left|s\right|\le \mathrm{\Delta }}\end{array}\hfill \end{array}$$

(44)

• d. VPRL

$$\begin{array}{c}{u}_4=k_{1}s-k_2{\left|s\right|}^{\lambda }\mathrm{sgn}\left(s\right)\hfill \\ \lambda =\{\begin{array}{c}\gamma ,\left|s\right|<1\hfill \\ \multicolumn{1}{c}{\left|s\right|,\left|s\right|\ge 1}\end{array}\hfill \end{array}$$

(45)

• e. PFMPRL

$$u_5=-k_1{r}_{1}s-k_2{\left|s\right|}^{r_2}\mathrm{sgn}\left(s\right)-k_3{\left|s\right|}^{r_3}\mathrm{sgn}\left(s\right)$$

(46)

Among them, in order to more objectively compare and simulate each reaching law without loss of generality, the parameters in the literature were slightly modified. The parameter settings of each reaching law are shown in Table 1.

• 2. Consider a cart inverted pendulum system, where the speed and smoothness of its dynamic response are the main performance indicators. In order to verify that the proposed method also has fast convergence performance in practical application, and can ensure its superiority, the model of the inverted pendulum system could be described as ⁴³ :

$$\begin{array}{c}\{\begin{array}{c}{\stackrel{·}{x}}_1(t)=x_2(t)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2(t)=f(x)+b(x)u_{\mathrm{di}}(t)+d}\end{array}\hfill \\ y=x_1(t)\hfill \end{array}$$

(47)

Table 1.

The parameters of the reaching laws.

Parameters	$u_1$	$u_2$	$u_3$	$u_4$	$u_5$
$k_1$	5	5	5	5	5
$k_2$	1	1	1	1	1
$k_3$	–	–	–	–	5
$\gamma$	1/2	1/2	1/2	1/2	–
$\rho$	–	3/2	3/2	–	–
$\mathrm{\Delta }$	–	–	1	–	1
${\alpha }_1$ / ${\beta }_1$	–	–	–	–	1/2/1/2
${\alpha }_2$ / ${\beta }_2$	–	–	–	–	1/1/2
${\alpha }_3$ / ${\beta }_3$	–	–	–	–	1/1/4

Where $x(t)={[x_1(t),x_2(t)]}^{\mathrm{T}}$ is the state variable, represents the angle and angular velocity of the inverted pendulum respectively; $y=x_1(t)$ is the output of the system; $f(x),b(x)$ is a nonlinear function; $u_d(t)$ is the control input; $d$ is the disturbance including system parameter uncertainty and external interference, $f_1(x),b_1(x)$ as follows ⁴³ :

$$\begin{array}{c}f(x)=\frac{m_{t}g\sin \left(x_1\right)-m_p\mathrm{Lsin}\left(x_1\right)\cos \left(x_1\right)x_2^2}{L((4/3)m_t-m_p{\cos }^2\left(x_1\right))}\hfill \\ b(x)=\frac{\cos \left(x_1\right)}{L((4/3)m_t-m_p{\cos }^2\left(x_1\right))}\hfill \end{array}$$

(48)

For the simulation purpose, the parameters of the cart-pole system are considered as $L=0.5\mathrm{m}$ , $m_c=1\mathrm{kg}$ , $m_p=0.1\mathrm{kg}$ , $g=9.8\mathrm{m}/{\mathrm{s}}^2$ .

The sliding surface is designed as $s=x_1+\beta x_2^{\frac{p}{q}}$ ; ⁴⁴ in the formula, $\beta >0$ is the parameter of the non-singular terminal sliding surface to be designed, $p$ and $q$ are positive odd number, and satisfies $1<\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.<2$ . The control objective is to make the state of system (41) converge to 0 in a finite time. By combining the above reaching laws with the NTSMC method, the control input of the system can be expressed by the following equation:

$$u_{\mathrm{di}}(t)=-\frac{1}{b\left(x\right)}[\frac{q}{\beta p}{x}_2^{2-p/q}-f\left(x\right)+u_i]$$

(49)

Where $u_i(i=1,2,3,4,5)$ is the reaching law presented in section “Platform introduction.”

Simulation of SISO system

Simulation comparison of convergence characteristics

For the system (41), when disturbance $d\left(t\right)=0$ , the initial values of sliding mode state are set as $s_0=1$ , $s_0=5$ , $s_0=50$ , control input $u$ uses the control laws (42)–(46) separately. The comparative simulation of the five control laws is shown in Figures 1 to 3.

Figure 1.

Comparison of convergence conditions of $s$ and $\left|\stackrel{·}{s}\right|$ when $s_0=1$ .

Figure 2.

Comparison of convergence conditions of $s$ and $\left|\stackrel{·}{s}\right|$ when $s_0=5$ .

Figure 3.

Comparison of convergence conditions of $s$ and $\left|\stackrel{·}{s}\right|$ when $s_0=50$ .

Analysis of Figures 1 to 3 and the results can be seen, all the reaching laws can make the sliding mode variable $s$ and $\left|\stackrel{·}{s}\right|$ reach the equilibrium point without chattering in a finite time, but the convergence speed of the proposed method is obviously faster than that of the other reaching laws. Tables 2 to 4 show the convergence time of each approach law at $s_0=1$ , $s_0=5$ and $s_0=50$ . As can be seen from Table 2, when $s_0=1$ , FPRL, DPCFRL and VPRL have the same form of reaching law, so their convergence time is the same, while the convergence speed of the DPRL is the slowest. When $s_0=5$ , the convergence rate of the DPRL is still the slowest, while VPRL is slightly faster than DPCFRL; when $s_0=100$ , FPRL has the slowest convergence rate. The proposed PFMRPL has a significantly faster convergence speed regardless of $s_0=1$ , $s_0=5$ or $s_0=100$ .

Table 2.

Comparison of convergence time of each approach law at $s_0=1$ .

Reaching law	Convergence time/s
FPRL	0.89
DPRL	1.25
DPCFRL	0.89
VPRL	0.89
PFMPRL	0.45

Table 3.

Comparison of convergence time of each approach law at $s_0=5$ .

Reaching law	Convergence time/s
FPRL	1.28
DPRL	1.36
DPCFRL	1.16
VPRL	1.00
PFMPRL	0.67

Table 4.

Comparison of convergence time of each approach law at $s_0=50$ .

Reaching law	Convergence time/s
FPRL	1.56
DPRL	1.24
DPCFRL	1.2
VPRL	1.08
PFMPRL	0.68

In order to show the influence of parameter a on the approach rate, since the exponential parameter of the proposed approach law method is designed by the nonlinear function with reference to parameter a, the values of a are 0.1, 0.5, and 0.8, respectively. In the case of $s_0=5$ , the comparison simulation is shown in the Figure 4, it can be seen that as the value increases, the convergence speed increases, but too large a value may increase chattering.

Figure 4.

Comparison of convergence conditions of $s$ and $\left|\stackrel{·}{s}\right|$ with different values of ${\beta }_2$ .

By comparing Figures 1 to 3 and Tables 1 to 3, it can be seen that FPRL is sensitive to the initial value of the state, so it cannot be seen that there is an upper limit of convergence time; the proposed reaching law and the others are less affected by initial state values, according to theorem 1, there is an upper limit of maximum time convergence, and within the range described in equation (23). In conclusion, the proposed approach law has better dynamic characteristics regardless of whether the initial value of the sliding mode state is close to or far from the equilibrium point.

Simulation comparison of steady-state error

For the system (41), parameter uncertainty and external disturbance are considered as:

$$d\left(t\right)=3\sin \left(t\right)+2\cos \left(2t\right)$$

(50)

At this point, set the initial value of sliding mode state as $s_0=6$ , the upper bound of the disturbance $\delta =5$ . Comparing and simulating each reaching law represented by equations (42)–(46), the resulting steady-state error bound is shown in Figures 5 to 7. Comparison of convergence conditions of $s$ with disturbance.

Figure 5.

Comparison of convergence conditions of $s$ with disturbance.

Figure 6.

Convergence condition of $s$ of the proposed method with disturbance.

Figure 7.

Convergence condition of $\left|\stackrel{·}{s}\right|$ of the proposed method with disturbance.

Figures 6 and 7 show the convergence of sliding mode variable $s$ and its first derivative; it is easy to see that, in the presence of disturbance $d\left(t\right)$ , the proposed approach law can converge to the steady-state error boundary in finite time and remain within the range expressed in equation (34). It can be seen from Figure 5 and Table 5 that all of the reaching law methods can converge to the steady-state error bounds, but the range of the proposed reaching law is significantly smaller than that of other methods.

Table 5.

Comparison of steady-state error bounds for each reaching law.

Reaching law	Steady-state error bounds
FPRL	0.79
DPRL	0.87
DPCFRL	0.78
VPRL	0.78
PFMPRL	0.38

Simulation of inverted pendulum system

For the system (47), when disturbance $d\left(t\right)=0$ , set the initial state of the system $x_0={[-{30}^{°},0]}^{\mathrm{T}}$ , the control input $u_{\mathrm{di}}(t)$ is shown in Equation (49), and the controller parameter is set to $p=21$ , $q=19$ , $\beta =0.7$ . The swing angle, angular velocity, control input, sliding mode surface curve and three-dimensional curve of the system phase trajectory are demonstrated in Figures 8 to 11.

Figure 8.

Pendulum angle and angular velocity response curves of each reaching law method.

Figure 9.

Control input response curve of each reaching law method.

Figure 10.

Sliding mode s response curve of each reaching law method.

Figure 11.

Three-dimensional curve of system phase trajectory.

It is easy to see from Figures 8 and 11 that the proposed PFMPRL combined with non-singular terminal sliding mode convergence has the fastest speed, which can make the system in a stable condition within a fixed time with less chattering. As can be seen from the curve of phase trajectory, the control process of this method is smooth and without jitter, which is consistent with the response curve of system pendulum angle and angular velocity, and has strong robustness. Figures 10 and 11 represent the control input curve and the sliding mode surface curve. It is observed that the control input and the sliding mode surface of the proposed method converge in a shorter time, remain stable with almost no chattering and have better dynamic characteristics.

For the system (47), when the disturbance $d\left(t\right)=0.7\sin \left(2t\right)+0.3\cos \left(2\pi t\right)$ , according to theorem 2, the state of the system can converge to the neighborhood near the equilibrium point in a finite time under the action of the controller. Excessive disturbance may even make the system unstable. In practical application, if there is a high requirement for control accuracy, the convergence of system state in a neighborhood often fails to meet the actual needs. Therefore, the disturbance observer is considered to be used to estimate the disturbance and compensate it to the controller, in order to suppress the impact of the disturbance on the system control and weaken the chattering. Figures 12 to 15 show the dynamic characteristics of the proposed reaching laws with or without nonlinear disturbance observer. Figures 16 and 18 show the dynamic quality of each reaching law under the action of nonlinear disturbance observer.

Figure 12.

Comparison of system state response curves with or without disturbance observer.

Figure 13.

Comparison of control input response curves with or without disturbance observer.

Figure 14.

Comparison of sliding mode s response curves with or without disturbance observer: (a) (without NDO), and (b) (with NDO).

Figure 15.

Three-dimensional curve of system phase trajectory.

Figure 16.

Pendulum angle and angular velocity response curves of each reaching law method with NDO.

Figure 17.

Sliding mode s response curve of each reaching law method with NDO.

Figure 18.

HC method for the sign function.

It can be seen from Figures 12 to 15 that, without the NDO, the system state $x_1$ and $x_2$ and the sliding mode surface $s$ converge to a field described in equation (39) in a finite time; the phase trajectories are not smooth and there is chattering, which cannot meet the control accuracy requirements in practical applications. After adding the disturbance observer based on tracking differentiator, the simulation results show that the nonlinear disturbance observer accurately estimates the disturbance, and the influence of the disturbance on the system is reduced. The system state $x_1$ and $x_2$ and the sliding mode surface variable $s$ can converge to the equilibrium point in a finite time, and the phase trajectory curve is smooth without chattering. It can be seen from Figures 16 and 17 that, after adding the NDO, the system has an obvious effect on suppressing chattering, but the control effect of the proposed reaching law is better and the chattering is smaller.

In sliding mode control, the presence of a sign function $\mathrm{sgn}\left(s\right)$ may cause chattering, and this is a common phenomenon in sliding mode control. This issue can be solved by the boundary control and the HC (hysteresis control) methods.^45,46 To weaken system chattering, the sign function $\mathrm{sgn}\left(s\right)$ is designed by HC method as shown in Figure 18.

The sign function designed by the HC method is applied to the control of the inverted pendulum system of the cart, and the proposed reaching law method is adopted. In the case of $d\left(t\right)=0$ and $d\left(t\right)=0.7\sin \left(2t\right)+0.3\cos \left(2\pi t\right)$ , compare and simulate with the control method using $\mathrm{sgn}\left(s\right)$ , as shown in Figures 19 and 20.

Figure 19.

Comparison of HC method and $\mathrm{sgn}\left(s\right)$ without interference.

Figure 20.

Comparison of HC method and $\mathrm{sgn}\left(s\right)$ with interference.

It can be seen from Figures 19 and 20, in the absence of interference, the HC method can effectively reduce system chattering and steady-state error is smaller, but weakens the convergence speed. In the presence of interference, under the action of NDO, using the HC method will cause excessive system steady-state errors, which will seriously affect the system control accuracy.

Conclusion

This paper proposes a design scheme of piecewise fast multi-power reaching law, a nonlinear function is designed to divide the reaching law into two stages and make the exponential parameters adjust according to different stages. The proposed method combines the advantages of FPRL, DPRL, and VPRL. It has a fast approach rate whether it is remote form or close to the sliding mode. The theoretical analysis shows that the new reaching law has the characteristic of fixed-time convergence and has a maximum convergence time independent of the initial value of the sliding mode state. When there is parameter uncertainty and external interference, the sliding mode variable $s$ can converge to a domain near the equilibrium point in a finite time. Considering the requirement of system control accuracy in practical application, a TANHTD-NDO is designed to estimate the model uncertainty and bounded disturbance, and the influence of the disturbance on the system is suppressed, so that the system can reach the steady-state in a finite time. Simulation analysis shows that, compared with the existing methods of FPRL, DPRL, DPCFRL, and VPRL, the proposed reaching law has faster convergence speed, stronger anti-interference ability, smaller steady-state error bounds, and better robustness and dynamic characteristics. Finally, the addition of a nonlinear interference observer has a positive interference suppression effect and improves the dynamic characteristics of the system in the presence of disturbances.

Supplemental Material

About_repetition_rate – Supplemental material for Piecewise fast multi-power reaching law: Basis for sliding mode control algorithm

Supplemental material, About_repetition_rate for Piecewise fast multi-power reaching law: Basis for sliding mode control algorithm by Guang-Yu Yang and Si-Yi Chen in Measurement and Control

Supplemental Material

Piecewise_Fast_Multi-Power_Reaching_Law_Basis_for_Sliding_Mode_Control_AlgorithmMarked_version – Supplemental material for Piecewise fast multi-power reaching law: Basis for sliding mode control algorithm

Supplemental material, Piecewise_Fast_Multi-Power_Reaching_Law_Basis_for_Sliding_Mode_Control_AlgorithmMarked_version for Piecewise fast multi-power reaching law: Basis for sliding mode control algorithm by Guang-Yu Yang and Si-Yi Chen in Measurement and Control