A Fast Reaching Law in Sliding Mode Control with Application to an Inverted Pendulum Robot

Abstract

Sliding mode control (SMC) is an effective and robust technique for managing uncertain nonlinear systems. The conventional SMC approach integrates a constant-rate reaching law with the boundary layer method to regulate the system. However, it does not address scenarios in which the initial state variables are significantly distant from the boundary layer. To expedite the process of reaching the sliding surface, this study introduces a fast reaching law in SMC, ensuring a fixed control time for reaching the sliding mode surface. The proposed fast reaching law is applied to an inverted pendulum robot, demonstrating its effectiveness in this typical system. In addition, we propose a qualitative evaluation method to compare various existing reaching law methods. The simulation results indicate that the proposed reaching law outperforms current approaches, substantiating its effectiveness.

1. Introduction

Many methods are available for controlling nonlinear systems, such as feedback linearization [1], adaptive control [2], sliding mode control (SMC) [3,4,5], and backstepping control [6,7]. Among these, SMC, which has been studied extensively owing to its robustness and simplicity, which involves designing a sliding mode surface and an effective control law to drive the system states toward it [8]. Once the system reaches the sliding mode surface, it moves towards the equilibrium point and remains unaffected by external disturbances [1]. Due to the robustness, fast convergence, and ability to handle nonlinearities, sliding mode control is widely applied in fields such as multi-area power systems [9], energy storage [10], and hybrid-source power systems [11].

In many nonlinear systems, reaching an equilibrium point is desirable, leading to the concept of settling time, which measures the time required to reach equilibrium. Nonlinear control can be divided into two types depending on the characteristics of the settling time. The first is asymptotic-time stable control [12], which can achieve stability but struggles to achieve settling time. The second type is finite-time stable control [13,14,15], which ensures that dynamic systems eventually settle at their equilibrium point after running for a finite time. However, finite-time control relies on the initial system conditions. Polyakov [16] introduced the design of fixed-time stable control to eliminate the dependency on the initial conditions. Following its introduction, several variants were developed by many authors, such as [17,18,19,20,21].

As a special case of finite-time and fixed-time control, SMC has recently emerged as an effective approach for nonlinear system control. However, chattering is a major challenge that must be addressed during SMC [22]. Many methods, including the disturbance observer technique [23,24], high-order sliding mode technique [25,26,27,28], boundary layer approach [1], and others [29,30], have been proposed to reduce chattering. The disturbance observer technique is used to detect disturbances. Therefore, it can attenuate the chattering amplitude [23,24]. The boundary layer approach replaces the sign function with saturation or hyperbolic tangent functions, which ensures that the system states enter a narrow boundary layer surrounding the sliding surface, albeit at the expense of increased steady-state tracking errors [1]. Other techniques, such as that of [29], add a low-pass filter to the control input. Thus, the output of the low-pass filter does not have a sign function and can handle chattering.

In addition, the design of reaching laws is crucial in the aforementioned attempts to address chattering effects. Control methods with a constant-rate reaching law often use the boundary layer method, especially the saturation function, to handle the chattering problem. However, these approaches do not consider the situation in which the initial state point is distant from the sliding mode surface. In this situation, when employing traditional SMC, the time required to reach the sliding surface is excessively long. To address this problem, many scholars have conducted in-depth studies and proposed various reaching laws, such as the exponential reaching law [31], double power reaching law [32], and others [33,34,35,36]. These studies achieved good control effects; however, there are two problems with these reaching laws. First, to the best of our knowledge, when comparing different reaching laws such as [31,32,33,34,35,36,37], their parameter selection is relatively arbitrary, rendering the comparisons unfair. Second, these reaching laws may result in excessive overshoot.

Therefore, to address these issues, this study introduces a fixed-time reaching law called the hyperbolic reaching law (HRL) to reduce the time needed to reach the sliding mode surface. Furthermore, we propose a new and more objective method for comparing different reaching laws. Specifically, the comparison is based on three variables: overshoot, steady-state error, and settling time. The first two variables are kept the same and the remaining variable is compared. Lastly, a method is proposed to add bounds to the reaching rate in order to mitigate excessive overshoot. We assume that the saturation function replaces the sign function to handle chattering in the reaching laws. The primary contributions of this study are summarized as follows:

(1)

The HRL is proposed to achieve a shorter reaching time than the exponential reaching law (ERL). The primary difference is that the ERL is linear, whereas the proposed reaching law uses an exponential function.

(2)

A new method is presented to compare the quality of different reaching laws. The reaching law parameters are determined based on the condition that when the initial state variables are at the same point in the boundary layer, the behaviors of the reaching laws are nearly the same. Subsequently, the reaching rates of these reaching laws are compared.

(3)

A solution is proposed for the particular case in which the input is excessively large owing to the design of the reaching law. Specifically, the upper and lower bounds should be added to the outer layer of the reaching law to ensure that the first derivative of the sliding mode variable is not excessively large, thereby reducing overshoot.

The subsequent sections of this paper are structured as follows: Section 2 outlines some preliminary steps. The primary contributions are elaborated on in Section 3. Applications to sliding mode control and numerical simulations are detailed in Section 4. Finally, Section 5 concludes the paper.

2. Preliminaries

Consider the following common uncertain nonlinear system [19,38] with a single input:

$$x^{\left(n\right)}=f\left(X\right)+g\left(X\right)u+d,$$

(1)

where $X={\left[x\stackrel{.}{x}\dots x^{\left(n-1\right)}\right]}^{\mathrm{T}}$ denotes the state vector, u denotes the control input, and x represents the system output. Functions f and g are known continuous functions such that $f\left(0\right)=0$ and $g\left(X\right)\ne 0$ for all X. The term d denotes a lumped disturbance, which includes both external disturbances and model uncertainties [39,40], with $\left|d\left(t\right)\right|⩽L$, where L is a positive constant. The primary objective is to drive x toward the origin.

2.1. Sliding Mode Control

SMC is a variable-structure control method that guides the system states along a predefined trajectory using a sliding mode surface [1]. The control process consists of two phases. In the reaching phase, the control law drives the system states toward the sliding surface ($s=0$), whereas in the sliding phase, the system states move along the sliding surface toward an equilibrium point. The sliding mode surface $s\left(t\right)$ is generally defined as

$$s\left(t\right)=x^{\left(n-1\right)}+{\displaystyle \sum _{i=0}^{n-2}}{c}_{\mathrm{i}}{x}^{\left(i\right)},$$

(2)

where $x\stackrel{.}{x}\dots x^{\left(n-1\right)}$ are the system state variables and the parameters $c_{\mathrm{i}}\left(\mathrm{i}=0,\dots ,n-2\right)$ are selected such that the resulting polynomials are Hurwitz, ensuring system stability along the sliding surface.

SMC should take action against the chattering phenomenon. A popular strategy for reducing chattering is to apply the boundary layer method, which substitutes a saturation function for the sign function when building a control law. The sign and saturation functions are defined as follows:

$$\mathrm{sign}\left(x\right)=\left\{\begin{array}{c}+1\mathrm{if}x>0\hfill \\ 0\hspace{1em}\mathrm{if}x=0\hfill \\ -1\mathrm{if}x<0\hfill \end{array}\right.,\mathrm{sat}\left(x/\Phi \right)=\left\{\begin{array}{c}+1\mathrm{if}x>\Phi \hfill \\ x/\Phi \mathrm{if}\left|x\right|⩽\Phi \hfill \\ -1\mathrm{if}x<-\Phi \hfill \end{array}\right..$$

(3)

2.2. Approach of Existing Reaching Laws

The constant-rate reaching law (CRL) [31] and ERL [41] are two approaches for SMC.

The CRL is defined as

$$\stackrel{.}{s}=-\eta \mathrm{sign}\left(s\right),$$

(4)

where $\eta$ is a positive constant. If an external disturbance $d\left(t\right)$ exists, a robust form is

$$\stackrel{.}{s}=-\left(\eta +L\right)\mathrm{sign}\left(s\right)+d,$$

(5)

where L bounds $d\left(t\right)$ with $\left|d\left(t\right)\right|⩽L$. The reaching time $t_{\mathrm{r}}$ is bounded by

$$t_{\mathrm{r}}⩽{\displaystyle \frac{\left|s\left(0\right)\right|}{\eta }}.$$

(6)

To reduce the reaching time, the ERL is introduced:

$$\stackrel{.}{s}=-\epsilon s-k\mathrm{sign}\left(s\right),$$

(7)

where $\epsilon >0$ and $k>0$ are constants. For robustness against disturbance $d\left(t\right)$:

$$\stackrel{.}{s}=-\epsilon s-\left(k+L\right)\mathrm{sign}\left(s\right)+d.$$

(8)

The reaching time $t_{\mathrm{r}}$ is bounded by

$$t_{\mathrm{r}}⩽{\displaystyle \frac{1}{\epsilon }}\mathrm{In}\left({\displaystyle \frac{k+\epsilon \left|s\left(0\right)\right|}{k}}\right).$$

(9)

3. Main Contributions

3.1. Proposed Reaching Law

To reach the sliding mode surface as quickly as possible, the absolute value of the derivative of s must be as large as possible. Therefore, we introduce an exponential function to increase the absolute value of the derivative. Based on this, the proposed HRL is formulated as

$$\stackrel{.}{s}=-k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)-k_3\mathrm{sign}\left(s\right),$$

(10)

where $k_1>0$, $k_2>0$, and $k_3>0$ are constants. By integrating (10) from 0 to $t_{\mathrm{r}}$, we derive the following expression for the reaching time $t_{\mathrm{r}}$:

$$\begin{array}{c}\hfill t_{\mathrm{r}}={\displaystyle \frac{1}{k_2\sqrt{4{k_1}^2+{k_3}^2}}}\left[\mathrm{In}\left|{\displaystyle \frac{2k_1+k_3+\sqrt{4{k_1}^2+{k_3}^2}}{2k_1+k_3-\sqrt{4{k_1}^2+{k_3}^2}}}\right|\right.\\ \hfill +\left.\mathrm{In}\left|{\displaystyle \frac{2k_1{e}^{k_2\left|s\left(0\right)\right|}+k_3-\sqrt{4{k_1}^2+{k_3}^2}}{2k_1{e}^{k_2\left|s\left(0\right)\right|}+k_3+\sqrt{4{k_1}^2+{k_3}^2}}}\right|\right].\end{array}$$

(11)

Consider the robust fixed-time stability analysis.

Theorem 1.

The system, defined by

$$\stackrel{.}{s}=-k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)-\left(k_3+L\right)\mathrm{sign}\left(s\right)+d,$$

(12)

where $k_1>0$, $k_2>0$, and $k_3>0$ are constants; $s\left(t\right)\in \mathbb{R}$ represents a state variable; and $d\left(t\right)\in \mathbb{R}$ represents a lumped disturbance subject to $\left|d\left(t\right)\right|⩽L$ for a positive constant L, is globally fixed-time stable. The settling time $t_{\mathrm{r}}$ is a fixed time according to the relation

$$t_{\mathrm{r}}⩽{\displaystyle \frac{1}{k_2\sqrt{4{k_1}^2+{k_3}^2}}}\mathrm{In}\left|{\displaystyle \frac{2k_1+k_3+\sqrt{4{k_1}^2+{k_3}^2}}{2k_1+k_3-\sqrt{4{k_1}^2+{k_3}^2}}}\right|.$$

(13)

Proof. 

Consider the following common Lyapunov function

$$V\left(s\right)={\displaystyle \frac{1}{2}}{s}^2.$$

(14)

This leads to

$$\begin{array}{cc}\hfill \stackrel{.}{V}\left(s\right)& =s\stackrel{.}{s}\hfill \\ \hfill & =s\left(-k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)-\left(k_3+L\right)\mathrm{sign}\left(s\right)+d\right)\hfill \\ \hfill & =-k_{1}s\left(e^{k_{2}s}-e^{-k_{2}s}\right)-k_3\left|s\right|-L\left|s\right|+ds<0,\hfill \\ \hfill & \mathrm{when}s\ne 0.\hfill \end{array}$$

(15)

Because when s is greater than zero, $e^{k_{2}s}-e^{-k_{2}s}$ is also greater than zero, and when s is less than zero, $e^{k_{2}s}-e^{-k_{2}s}$ is also less than zero, and $s\left(e^{k_{2}s}-e^{-k_{2}s}\right)$ is greater than zero. Given that $\left|d\left(t\right)\right|<L$, $-L\left|s\right|+ds<0$. Therefore, the derivative of $V\left(s\right)$ is a negative definite function.

According to Lyapunov’s direct method, the system is globally asymptotically stable.

When $s>0$, from (12), we obtain

$$\stackrel{.}{s}⩽-k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)-k_3.$$

(16)

Integrating (16) yields

$${\int }_{s\left(0\right)}^0{\displaystyle \frac{1}{k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)+k_3}}ds⩽-{\int }_0^{t_{\mathrm{r}}}dt.$$

(17)

Since one of the antiderivatives of the function

$${\displaystyle \frac{1}{k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)+k_3}}$$

(18)

is

$${\displaystyle \frac{1}{k_2\sqrt{4{k_1}^2+{k_3}^2}}}\mathrm{In}\left|{\displaystyle \frac{2k_1{e}^{k_{2}s}+k_3-\sqrt{4{k_1}^2+{k_3}^2}}{2k_1{e}^{k_{2}s}+k_3+\sqrt{4{k_1}^2+{k_3}^2}}}\right|.$$

(19)

Thus,

$$\begin{array}{c}\hfill t_{\mathrm{r}}⩽{\displaystyle \frac{1}{k_2\sqrt{4{k_1}^2+{k_3}^2}}}\left[\mathrm{In}\left|{\displaystyle \frac{2k_1+k_3+\sqrt{4{k_1}^2+{k_3}^2}}{2k_1+k_3-\sqrt{4{k_1}^2+{k_3}^2}}}\right|\right.\\ \hfill \left.+\mathrm{In}\left|{\displaystyle \frac{2k_1{e}^{k_{2}s\left(0\right)}+k_3-\sqrt{4{k_1}^2+{k_3}^2}}{2k_1{e}^{k_{2}s\left(0\right)}+k_3+\sqrt{4{k_1}^2+{k_3}^2}}}\right|\right].\end{array}$$

(20)

When $s<0$, from (12), we obtain

$$\stackrel{.}{s}⩾-k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)+k_3.$$

(21)

Integrating (21) yields

$${\int }_{s\left(0\right)}^0{\displaystyle \frac{1}{k_1\left(e^{-k_{2}s}-e^{k_{2}s}\right)+k_3}}ds⩾{\int }_0^{t_{\mathrm{r}}}dt.$$

(22)

Since one of the antiderivatives of the function

$${\displaystyle \frac{1}{k_1\left(e^{-k_{2}s}-e^{k_{2}s}\right)+k_3}}$$

(23)

is

$${\displaystyle \frac{1}{k_2\sqrt{4{k_1}^2+{k_3}^2}}}\mathrm{In}\left|{\displaystyle \frac{2k_1{e}^{-k_{2}s}+k_3+\sqrt{4{k_1}^2+{k_3}^2}}{2k_1{e}^{-k_{2}s}+k_3-\sqrt{4{k_1}^2+{k_3}^2}}}\right|.$$

(24)

Therefore,

$$\begin{array}{c}\hfill t_{\mathrm{r}}⩽{\displaystyle \frac{1}{k_2\sqrt{4{k_1}^2+{k_3}^2}}}\left[\mathrm{In}\left|{\displaystyle \frac{2k_1+k_3+\sqrt{4{k_1}^2+{k_3}^2}}{2k_1+k_3-\sqrt{4{k_1}^2+{k_3}^2}}}\right|\right.\\ \hfill \left.+\mathrm{In}\left|{\displaystyle \frac{2k_1{e}^{-k_{2}s\left(0\right)}+k_3-\sqrt{4{k_1}^2+{k_3}^2}}{2k_1{e}^{-k_{2}s\left(0\right)}+k_3+\sqrt{4{k_1}^2+{k_3}^2}}}\right|\right].\end{array}$$

(25)

Combining (20) and (25), the settling time for (12) is expressed as

$$\begin{array}{cc}\hfill t_{\mathrm{r}}& ⩽{\displaystyle \frac{1}{k_2\sqrt{4{k_1}^2+{k_3}^2}}}\left[\mathrm{In}\left|{\displaystyle \frac{2k_1+k_3+\sqrt{4{k_1}^2+{k_3}^2}}{2k_1+k_3-\sqrt{4{k_1}^2+{k_3}^2}}}\right|\right.\hfill \\ \hfill & \hfill \left.+\mathrm{In}\left|{\displaystyle \frac{2k_1{e}^{k_2\left|s\left(0\right)\right|}+k_3-\sqrt{4{k_1}^2+{k_3}^2}}{2k_1{e}^{k_2\left|s\left(0\right)\right|}+k_3+\sqrt{4{k_1}^2+{k_3}^2}}}\right|\right]\\ \hfill & ⩽{\displaystyle \frac{1}{k_2\sqrt{4{k_1}^2+{k_3}^2}}}\mathrm{In}\left|{\displaystyle \frac{2k_1+k_3+\sqrt{4{k_1}^2+{k_3}^2}}{2k_1+k_3-\sqrt{4{k_1}^2+{k_3}^2}}}\right|.\hfill \end{array}$$

(26)

The above inequality holds because

$$\mathrm{In}\left|{\displaystyle \frac{2k_1{e}^{k_2\left|s\left(0\right)\right|}+k_3-\sqrt{4{k_1}^2+{k_3}^2}}{2k_1{e}^{k_2\left|s\left(0\right)\right|}+k_3+\sqrt{4{k_1}^2+{k_3}^2}}}\right|<0.$$

(27)

It is less than 0 because $k_1,k_2,k_3$ are all positive constants, and the argument of the logarithm represents a ratio where the numerator is smaller than the denominator, making the entire fraction less than 1. Since the logarithm of a number less than 1 is negative, the expression results in a value less than zero.

Thus, the settling time is fixed and is expressed as (13). □

3.2. Method for Comparing Different Reaching Laws

Some issues must be considered when comparing different reaching laws. First, for any reaching law, if there are no input size restrictions or chattering restrictions, theoretically, any reaching law can cause the system sliding mode variable s to reach the sliding mode surface in an infinitesimal time. This is because the more significant the derivative of s, the faster the sliding mode variable s tends toward zero. Similar to the CRL, increasing the coefficient $\eta$ of the CRL can make it approach its origin faster. However, the larger the coefficient, the larger the input required, and significant oscillation will occur after reaching the sliding mode surface. For the same initial conditions and by selecting different parameters for the same reaching law, if the speed of reaching the sliding mode surface $s=0$ is faster, the oscillation will also be more significant after reaching the sliding mode surface. In summary, there should be a compromise between the convergence time and oscillation amplitude (or chattering) [33].

We aim for a reaching law that minimizes both the settling time (convergence time) and steady-state error (the amplitude of the oscillation after reaching the sliding mode surface). Therefore, if we disregard the input constraints and compare the different reaching laws, two methods can be used. The first method involves fixing the settling time under the same initial conditions and comparing the steady-state error. A smaller amplitude is preferable. The second method entails fixing the steady-state error and comparing the settling time for different reaching laws under the same initial conditions. Therefore, a shorter time is preferable.

Because there are conceptual differences between the finite-time and fixed-time reaching laws, the first method requires that the settling time be fixed under the same initial conditions. This may not be convenient for finite-time reaching laws because changing the initial conditions necessitates adjustments to the parameters. In addition, for most reaching laws, assessing the oscillation amplitude appears feasible only through experimental comparison. Therefore, we opt to use the second method for comparison.

This raises another question: How can we fix the steady-state error for the second method? In any practical system with disturbances, it is impossible to eliminate oscillations completely after the system reaches the sliding mode surface. Therefore, we establish a tolerance limit for practical systems. If the oscillations remain within the acceptable range, we can assume that the system has no significant oscillations. Similarly, the objective of reaching laws is to control the sliding mode variable to reach zero. We can set a small value $\Phi$ as the tolerance limit. If $\left|s\right|⩽\Phi$, we consider that the sliding mode variable has reached the sliding mode surface. A general reaching law can be expressed as

$$\stackrel{.}{s}=H\left(s,\mathrm{sign}\left(s\right)\right)-L\mathrm{sign}\left(s\right)+d,$$

(28)

where $H\left(s,\mathrm{sign}\left(s\right)\right)$ is a function involving s or $\mathrm{sign}\left(s\right)$ and d represents lumped disturbance, which includes both external disturbances and model uncertainties, such that $\left|d\left(t\right)\right|⩽L$, where L is a positive constant.

Considering the use of a saturation function instead of a sign function, there is no difference between using a saturation function and sign function outside the boundary layer. Inside the boundary layer, we can replace $\mathrm{sign}\left(s\right)$ with $s/\Phi$ and then approximate the H function with a Taylor expansion at zero (if it can be approximated using a Taylor expansion), neglecting terms of the second order and higher. Consequently, the reaching law inside the boundary layer (28) can be transformed as follows:

$$\stackrel{.}{s}\approx -\left({\lambda }_1+{\displaystyle \frac{L}{\Phi }}\right)s+d,$$

(29)

where ${\lambda }_1$ is a number irrelevant to s and is typically related to certain parameters of the reaching law. If we let

$$\lambda ={\lambda }_1+{\displaystyle \frac{L}{\Phi }},$$

(30)

we can obtain

$$\stackrel{.}{s}\approx -\lambda s+d.$$

(31)

We now consider the results of (31). The sliding variable s is generated from the disturbance d via a first-order low-pass filter (see Figure 1, where the Laplace operator is represented by $p=\left(\mathrm{d}/\mathrm{dt}\right)$ and the parameter $\lambda >0$ signifies the cutoff frequency of the low-pass filter).

At this point, the corresponding parameters for the different reaching laws are set. Under the same parameter $\lambda$ and disturbance conditions, the sliding mode variables within the boundary layer are identical and the corresponding state variables also match. Consequently, the oscillations are the same. This satisfies the requirement of the second method, i.e., the steady-state error of the different reaching laws are equal. Therefore, the merits of the different reaching laws can be evaluated by comparing the settling times of the respective boundary layers.

The following section provides methodological guidance on how to design the parameters of the reaching laws to ensure the prerequisites for comparison. Two key aspects should be considered:

Condition (1): The behaviors of the reaching laws to be compared should be nearly identical when the initial state variables are both inside the boundary layer. This implies that, when the initial state variables are located at the same point within the boundary layer, the system states, sliding mode variable, and even the control input should evolve in a nearly identical manner over time.

Condition (2): When the initial state variables are located at the same point outside the boundary layer, the comparison should focus on the settling time to the boundary layer. A shorter time is preferable.

We use the CRL, ERL, and HRL to illustrate how to achieve condition (1).

Consider (5), (8), and (12). When we use saturation to replace the sign function to handle chattering, we obtain

$$\left\{\begin{array}{c}\stackrel{.}{s}=-\left(\eta +L\right)\mathrm{sat}\left(s/\Phi \right)+d\hfill \\ \stackrel{.}{s}=-\epsilon s-\left(k+L\right)\mathrm{sat}\left(s/\Phi \right)+d\hfill \\ \stackrel{.}{s}=-k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)-\left(k_3+L\right)\mathrm{sat}\left(s/\Phi \right)+d\hfill \end{array}\right..$$

(32)

When the system state variables are inside the boundary layer, we obtain

$$\left\{\begin{array}{c}\stackrel{.}{s}=-{\displaystyle \frac{\eta +L}{\Phi }}s+d\hfill \\ \stackrel{.}{s}=-\left({\displaystyle \frac{k+L}{\Phi }}+\epsilon \right)s+d\hfill \\ \stackrel{.}{s}\approx -\left({\displaystyle \frac{k_3+L}{\Phi }}+2k_1{k}_2\right)s+d\hfill \end{array}\right..$$

(33)

To satisfy condition (1), we must guarantee that

$$\lambda =\left\{\begin{array}{c}{\displaystyle \frac{\eta +L}{\Phi }}\hfill \\ {\displaystyle \frac{k+L}{\Phi }}+\epsilon \hfill \\ {\displaystyle \frac{k_3+L}{\Phi }}+2k_1{k}_2\hfill \end{array}\right.⟹{\lambda }_1=\left\{\begin{array}{c}{\displaystyle \frac{\eta }{\Phi }}\hfill \\ {\displaystyle \frac{k}{\Phi }}+\epsilon \hfill \\ {\displaystyle \frac{k_3}{\Phi }}+2k_1{k}_2\hfill \end{array}\right..$$

(34)

Therefore, based on the same sliding surface selection, we can use condition (1) to determine the relevant parameters of the reaching laws and then use condition (2) to compare their quality. Notably, $\lambda$ can be considered the cutoff frequency of the filter. If the sole purpose is to compare the different reaching laws, this value can be selected arbitrarily. However, if a specific system is to be implemented, this value needs to be set according to the actual conditions so that it does not exceed the degree that the system can tolerate.

Having completed the parameter selection for the reaching law using condition (1), how do we determine the duration for the sliding mode variable s to reach the boundary layer to compare the reaching laws? This duration can typically be determined through simulations or experiments. In addition, there exists a simpler method for comparing reaching laws: directly analyzing the graphical representation of the reaching law (with s as the abscissa and the first derivative of s as the ordinate, considering d and L as zero).

The speed at which the sliding mode variable s reaches the boundary layer is directly related to the absolute value of its first derivative. Based on this principle, we can compare the speeds of the different reaching laws by plotting their respective graphs. Outside the boundary layer, the reaching law, whose ordinate is the farthest from the horizontal axis, reaches the boundary layer more quickly. Because there is no significant difference between using a saturation function or a sign function outside the boundary layer, we can use the sign function version of the reaching law for plotting. Furthermore, because d and L are assumed to be the same when comparing the reaching laws, these parameters can be ignored, leading to identical results.

In the following, we use this method to explain why the HRL (10) is better than CRL (4) and ERL (7); the simulations in Section 4 also validate this point. First, we use condition (1) to determine their parameters. When comparing through figures, d and L can be ignored and the boundary layer size $\Phi$ is the same. Therefore, we must only ensure that ${\lambda }_1$ is the same. Without loss of generality, we assume that the value of ${\lambda }_1$ is 16 and the value of $\Phi$ is 0.05. Then, to satisfy (34), $\eta$ in the CRL needs to be equal to 0.8. However, for the ERL and HRL, because (34) has only one equation, but there are two parameters ($k,\epsilon$) and three parameters ($k_1,k_2,k_3$), respectively, these parameters cannot be uniquely determined. We set $k=0.3,\epsilon =10$ for the ERL and $k_1=25,k_2=0.2,k_3=0.3$ for the HRL. The corresponding graphs are presented in Figure 2.

It can be observed from Figure 2 that, with the aforementioned parameters, the HRL performs better than both the ERL and CRL, because the HRL is farther from the horizontal axis than the other two.

Actually, when $k_3=k$ and $2k_1{k}_2=\epsilon$, performing a Taylor expansion of $e^{k_{2}s}$ and $e^{-k_{2}s}$ at $s=0$ easily leads to the following inequality

$$\left|\epsilon s\right|⩽\left|k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)\right|.$$

(35)

Therefore,

$$\left|-\epsilon s-k\mathrm{sign}\left(s\right)\right|⩽\left|-k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)-k_3\mathrm{sign}\left(s\right)\right|.$$

(36)

Thus, within this framework, it is considered that HRL is superior to ERL.

3.3. Solution for Handling Input Saturation

Considering Figure 2, for the HRL, it can be observed that the first derivative of s changes very rapidly. Even before s approaches −15, the first derivative of s exceeds 400. For all fixed-time reaching laws, if the settling time required for the reaching law is a fixed value, the larger the initial s, the larger the derivative of s must be to achieve this goal. However, a very large first derivative of s is impractical in real systems because it may result in very large control inputs and overshoot. Therefore, fixed-time reaching laws are unsuitable for practical systems in some cases. A simple solution is to impose upper and lower bounds on the first derivative of s, although this may affect the properties of the fixed-time convergence.

For example, by defining

$$b\left(x,\Psi \right)=\left\{\begin{array}{c}x\hspace{1em}if\left|x\right|⩽\Psi \hfill \\ \Psi \mathrm{sign}\left(x\right)else\hfill \end{array}\right.,\Psi >0,$$

(37)

using this solution to our proposed reaching law (10), we can obtain the following reaching law:

$$\stackrel{.}{s}=b\left(\left[-k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)-k_3\mathrm{sign}\left(s\right)\right],\Psi \right),$$

(38)

where $\Psi >0$ denotes the predefined upper bound of the reaching law.

By adding upper and lower bounds to the derivative of s, it results in the same approaching rate as the original reaching law when $\left|\stackrel{.}{s}\right|⩽\Psi$. However, when $\left|\stackrel{.}{s}\right|>\Psi$, the derivative of s becomes a constant $\Psi$, so the time taken to reach the sliding surface $s=0$ depends on the initial sliding mode state $s\left(0\right)$. In this case, the time to reach the sliding surface is no longer fixed, but rather finite, and this time is similar to the constant reaching law. The only difference is that the parameter $\eta$, in the constant reaching law is smaller, while $\Psi$ is typically larger, so the time to reach the sliding surface will be faster than that of the CRL.

4. Numerical Simulation

4.1. Input of Sliding Mode Control

Consider the standard sliding variable defined in (2), and the controller expressed as

$$\begin{array}{cc}\hfill u=& -g{\left(X\right)}^{-1}\left[f\left(X\right)+{\displaystyle \sum _{i=0}^{n-2}}{c}_{\mathrm{i}}{x}^{\left(i+1\right)}\right.\hfill \\ \hfill & +\left.k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)+\left(k_3+L\right)\mathrm{sign}\left(s\right)\right],\hfill \end{array}$$

(39)

where $k_1>0$, $k_2>0$, $k_3>0$, and $\left|d\left(t\right)\right|⩽L$.

Theorem 2.

The closed-loop system (1)-(2)-(39) achieves $s\left(x\right)=0$ within a fixed-time duration $t_{\mathrm{r}}$ that satisfies (13). This also ensures global asymptotic stability.

Proof. 

We take the derivative of (2) and substitute (1) and (39) into it. Thus, we obtain

$$\begin{array}{cc}\hfill \stackrel{.}{s}& =f\left(X\right)+g\left(X\right)u+{\displaystyle \sum _{i=0}^{n-2}}{c}_{\mathrm{i}}{x}^{\left(i+1\right)}+d\hfill \\ \hfill & =-k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)-\left(k_3+L\right)\mathrm{sign}\left(s\right)+d.\hfill \end{array}$$

(40)

The first part of the theorem follows from the fixed-time proof of (12). Upon reaching the sliding surface, the design of the sliding mode surface ensures that the closed-loop system (1)-(2)-(39) is asymptotically stable at the equilibrium point (the origin). □

4.2. Simulation

To evaluate the effectiveness of the proposed HRL, simulations were performed using the inverted pendulum robot model described in [42]. An inverted pendulum is a classic system featuring a pendulum attached to a motor-driven cart by a rotational joint that moves horizontally to keep the pendulum balanced and upright (Figure 3).

Let $\theta$ (radians unit) denote the pendulum rotation. Let $x_1:=\theta ,x_2:=\stackrel{.}{\theta }$, and u be the force applied to the cart. The dynamic equations of the inverted pendulum robot under disturbance are as follows:

$$\left\{\begin{array}{c}{\stackrel{.}{x}}_1=x_2\hfill \\ {\stackrel{.}{x}}_2=f_1\left(x_1,x_2\right)+g_1\left(x_1,x_2\right)u+d\hfill \end{array}\right.,$$

(41)

with

$$f_1\left(x_1,x_2\right)={\displaystyle \frac{\left(M+m\right)gsin{x}_1-{ml{x}_2}^{2}sin{x}_{1}cos{x}_1}{\frac{4}{3}\left(M+m\right)l-ml{cos}^2{x}_1}},$$

(42)

$$g_1\left(x_1,x_2\right)={\displaystyle \frac{cos{x}_1}{\frac{4}{3}\left(M+m\right)l-ml{cos}^2{x}_1}},$$

(43)

such that $M=1\left(\mathrm{kg}\right)$ is the cart mass, $m=0.1\left(\mathrm{kg}\right)$ is the pendulum mass, $l=0.5\left(\mathrm{m}\right)$ is the half length of the pendulum, $g=9.8\left(\mathrm{m}/{\mathrm{s}}^2\right)$ is the acceleration of gravity, and $d=0.2\left|sint\right|$.

The sliding surface is selected from (2) as follows:

$$s=x_2+c_0{x}_1,$$

(44)

with $c_0=20$.

Based on the HRL and boundary layer method with $\Phi =0.05$, the control input u is designed as follows:

$$\begin{array}{cc}\hfill u=& {g_1}^{-1}\left(-f_1-c_0{x}_2-L\mathrm{sat}\left(s/\Phi \right)\right)+\hfill \\ \hfill & {g_1}^{-1}\left(-k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)-k_3\mathrm{sat}\left(s/\Phi \right)\right),\hfill \end{array}$$

(45)

where $k_1=25,k_2=0.2,k_3=0.3$, and $L=0.2$.

If we consider that the control input may be too large, based on the bounded proposed hyperbolic reaching law (HRL bound) with $\Psi =200$ and the boundary layer method with $\Phi =0.05$, the control input u is designed as follows:

$$\begin{array}{cc}\hfill u=& {g_1}^{-1}\left(-f_1-c_0{x}_2-L\mathrm{sat}\left(s/\Phi \right)\right)+\hfill \\ \hfill & {g_1}^{-1}\left(b\left(-k_1\left(e^{k_{2}s}-e^{-k_{2}s}\right)-k_3\mathrm{sat}\left(s/\Phi \right),\Psi \right)\right),\hfill \end{array}$$

(46)

where $k_1=25,k_2=0.2,k_3=0.3$, and $L=0.2$.

Based on the CRL and boundary layer method with $\Phi =0.05$, the control input u is designed as follows:

$$u={g_1}^{-1}\left(-f_1-c_0{x}_2-\left(\eta +L\right)\mathrm{sat}\left(s/\Phi \right)\right),$$

(47)

where $\eta =0.8$ and $L=0.2$.

Based on the ERL and boundary layer method with $\Phi =0.05$, the control input u is designed as follows:

$$u={g_1}^{-1}\left(-f_1-c_0{x}_2-\left(k+L\right)\mathrm{sat}\left(s/\Phi \right)-\epsilon s\right),$$

(48)

where $\epsilon =10,k=0.3$, and $L=0.2$.

To enhance persuasiveness, we also introduce the novel reaching law (abbreviated as NRL) mentioned in [36] for comparison. The mathematical expression of this reaching law is as follows:

$$\dot{s}=-c_1{\displaystyle \frac{tanh\left(s\right)}{q^{p\left|s\right|}+\frac{1-q^{p\left|s\right|}}{e^{p\left|s\right|}}}}-c_{2}s,$$

(49)

where $c_1>0,c_2>0,p>0$, and $0<q<1$.

To satisfy condition (1) and facilitate subsequent comparisons, a Taylor expansion of the NRL at zero is performed. The final result obtained is:

$$\dot{s}=-(c_1+c_2)s,$$

(50)

In the experiments conducted in this paper, the value of $\lambda$ is set to 20; hence, it is required that $c_1+c_2=20$. The parameters of the NRL used in this paper are $c_1=10,c_2=10,p=0.025$, and $q=0.01$.

Consider the parameters of the control inputs (45), (47), and (48). We conclude that these parameters are consistent with (34), where $\lambda =20$. When the initial state variable of the system is [$x_1=0$, $x_2=0$], which means that it is within the boundary layer, the simulation outcomes are illustrated in Figure 4 and Figure 5. Figure 4 shows the state variable $x_1$, whereas Figure 5 shows the state variable $x_2$.

These figures illustrate that, when the initial state variable is within the boundary layer, the results obtained by the compared reaching laws are nearly identical. This satisfies condition (1) of Section 3.

Consider the case in which the initial system state variable is [$x_1=1$, $x_2=3$]. This corresponds to an initial sliding mode variable of 23, implying that the initial system state variable is outside the boundary layer ($\Phi =0.05$). When we use the control law in (47) based on the CRL, the sliding variable is as shown in Figure 6. It can be observed that the settling time is relatively long. Therefore, the CRL is not considered in the subsequent comparisons.

The simulation results using the control inputs in (45), (46), and (48) are shown in Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. Figure 7 shows the state variable $x_1$, whereas Figure 8 shows the state variable $x_2$. The sliding variables are presented in Figure 9, and Figure 10 illustrates the control input applied to the system.

Table 1. Control Input RMS Value and Peak Control Input for Different Methods.

Method	Control Input RMS Value	Peak Control Input
ERL	43.4092	402.8477
NRL	45.1957	424.5901
HRL_bound	46.5397	362.5412
HRL	113.1219	3405.0307

presents the peak control input and the root-mean-square (RMS) value of the control input u for different methods. The RMS value is calculated using the first 1.5 s, as the control inputs of each method have already approached zero within this time frame. Figure 11 shows the phase plane graph between $x_1$ and $x_2$.

Figure 7, Figure 8 and Figure 9 show that the HRL achieves a shorter settling time and better performance than the ERL and NRL. Figure 10 and Table 1 reveals that the shortcoming of the HRL is the relatively large control input and overshoot. By controlling the range of the derivative of s (the HRL bound), the range of the control input closely approaches that of the ERL and NRL while maintaining a faster reaching speed.

To test the stability of the proposed method, the system’s transient response and control input under different disturbances, including input disturbances, model parameter disturbances, and output disturbances, are tested.

Figure 12 shows the transient response, control input, and phase plane diagram of the system under input disturbances. The added input disturbance term is set as $sin\left(100t\right)$. As shown in the figure, the input disturbance has little effect on the system, and the system remains stable.

Figure 13 presents the transient response, control input, and phase plane diagram of the system under model parameter (cart mass) disturbances. The added cart mass disturbance term is set as $sin\left(5t\right)$. From the figure, it can be seen that the disturbance in the cart mass parameter also has minimal effect on the system, and the system remains stable.

Figure 14 shows the transient response, control input, and phase plane diagram of the system under output disturbances. The added output disturbance term (with output being $x_1$) is set as $0.1sin\left(100t\right)$. As shown in the figure, the output disturbance has a relatively large impact on the system, but the system still converges, although to a specific range. This is because the output disturbance can be considered as an error caused by measuring the state variable, and such errors cannot be eliminated by the control method. The system can only be controlled to a stable region, which makes the results normal.

From Figure 12, Figure 13 and Figure 14, it can be seen that when the system experiences input disturbances, load disturbances, or output disturbances, the proposed method reaches the steady state faster than the other methods. Therefore, this verifies the robustness and efficiency of the proposed reaching law.

5. Conclusions

This study has proposed the HRL based on Lyapunov stability theory, which we compared with the CRL, ERL and NRL. A new comparison method was introduced to ensure fairness. The steady-state error is fixed and the settling times are compared, with shorter times preferred. Theories and simulations demonstrated that the HRL performs better than the CRL, ERL and NRL. Notably, fixed-time reaching laws generally cannot be compared using this method, for two reasons. First, they have many tunable parameters. Second, some reaching law functions cannot be expanded using the Taylor series at the origin. Therefore, other methods are required to compare the performances of fixed-time reaching laws. To handle cases in which the control input is too large and overshoot owing to an excessively large derivative of the sliding variable, we propose adding upper and lower bounds to the reaching law. Similarly, the simulation results show superior performance compared with both the CRL and ERL.