Adaptive hierarchical sliding mode control based on fuzzy neural network for an underactuated system

Abstract

We present an adaptive hierarchical sliding mode control based on fuzzy neural network for a class of underactuated systems to solve the problem of high-precision trajectory tracking. This system is viewed as several subsystems. One subsystem is used to design the first-layer sliding surface, which constructs the second-layer sliding surface with another subsystem. When the top layer includes all the subsystems, the design process is finished. Meanwhile, the equivalent control law and the switching control law are achieved at every layer. Because the hierarchical sliding mode control law relies excessively on the requirement of detailed information of the underactuated dynamic system, and because that method causes an inevitable chattering phenomenon, an online fuzzy neural network system is applied to mimic the hierarchical sliding mode control law. Moreover, the bounds of system uncertainties and modeling error caused by the fuzzy neural network system are estimated online by a robust term. The stability of the closed-loop system is guaranteed based on the Lyapunov theory and Barbalat’s Lemma. Finally, the examples, a single-pendulum-type overhead crane system and an inverted pendulum system, are simulated to verify the effectiveness and robustness of the proposed method compared with some conventional methods.

Introduction

An underactuated system is a class of mechanical systems with fewer independent control actuators than degrees of freedom (DOFs). ¹ Based on the input number and the output number, underactuated mechanical systems can generally be divided into single input multiple output (SIMO) and multiple input multiple output (MIMO) systems. This study focuses on SIMO systems, the dynamical models of which are easier to analyze and verify experimentally. However, they show strong nonlinear, dynamic constraints and coupling properties. There has been much research on SIMO underactuated systems.^2–5 In Adhikary and Mahanta, ² an integral backstepping sliding mode control method was proposed for the cart–pendulum system with matched and mismatched uncertainties. Karkoub and Zribi ³ presented a robust control scheme for an overhead crane and utilized a μ-synthesis controller to compensate for unmodeled dynamics. To solve the stability and tracking problems for an underactuated system, Suykens et al. ⁴ proposed an optimal control that successfully used least squares support vector machines (SVMs). Hung et al. ⁵ designed a self-tuning fuzzy sliding mode controller for a rotational actuator system. In brief, the controller designs and stability analyses of SIMO underactuated dynamic systems have been an important research area.

It is commonly known that sliding mode control (SMC), a type of variable structure control, can effectively control uncertain nonlinear systems with matched uncertainties and disturbances owing to it being invariant to uncertainties and insensitive to changes of the system parameters. ⁶ That method has been successfully applied in many fields and achieved good robust performance.^7–9 Some studies, published in the last few years, attempt to use SMC to control underactuated systems.^10–13 The design process, using the SMC, is complex due to strong coupling and the reduced dimensionality of the input space. Therefore, hierarchical sliding mode control (HSMC) was proposed instead.^14–17 Qian and colleagues^14,15 proposed an HSMC structure for an underactuated system: the system was divided into several subsystems, and then the higher sliding surface was constructed by the lower sliding surface together with a sliding surface of one subsystem in the lower layer. Finally, all the control laws in different layers were connected to get the final control rate. As a result, this method reduced the high-dimensional system into low-dimensional subsystems. Therefore, the HSMC method can effectively reduce the complexity of the controller design for underactuated systems. A substantial number of studies both on simulation analysis results and in verification experimental results have been presented to solve the following issues. (1) Unmodeled dynamics and uncertainties: In Hwang and Wu, ¹⁶ an adaptive HSMC method was designed to control an underactuated mobile robot with frictions and uncertainties. The frictions and uncertainties were compensated for by a robust term that required knowing the bound of frictions and uncertainties. To adaptively compensate for uncertainties, Hwang et al. ¹⁸ designed a hierarchical improved fuzzy dynamic sliding mode control method, where two fuzzy logic systems were utilized to approximate system uncertainties. (2) Optimization of control parameters: Qian and Yi ¹⁷ used a fuzzy logic system to learn the control gain and verified its effectiveness through simulation. A hierarchical fuzzy sliding mode (HFSM) method was proposed by Lin and Mon, ¹⁹ where several fuzzy logic systems were self-tuning based on Lyapunov theory to learn the optimal coupling factors designed at every layer. (3) Chattering and unknown continuous nonlinear functions: To solve these problems, an adaptive fuzzy hierarchical sliding mode control method was proposed for a class of SIMO uncertain underactuated nonlinear systems in Hwang et al., ²⁰ where several fuzzy systems approximated the unknown dynamic nonlinear functions and even estimated uncertainties and disturbances. The common feature of these previous results is focused on how to use a fuzzy system to solve the regulation control problem of underactuated systems. There are some disadvantages: the chattering phenomenon cannot be avoided, and the controller design process relies excessively on complete and accurate kinetic models. Although the chattering and the unknown dynamic model are taken into account in Hwang et al., ²⁰ this method made the designing process and controller structure complex and computationally expensive.

Three methods are usually employed to deal with uncertain nonlinear dynamic functions or unmatched disturbances. These include neural network, fuzzy logic system, and SVMs. A neural network system can approximate any continuous function and possess the merit of low-level learning, so it is used in different areas.^21,22 However, it has some problems such as overfitting, local minimum, and slow learning speed. A fuzzy logic system can easily use linguistic information to express human experience and logical reasoning.^20,23,24 At the same time, it suffers from complexity and slow learning. SVMs are effective to pattern recognition, with good learning performance even under the condition of small training samples. ²⁵ Several studies on the control of nonlinear systems have used SVMs for online learning of dynamic uncertainties or dynamic inverse models.^25–28 However, it is not easy to analyze and guarantee the stability of the closed-loop system. In order to overcome the respective disadvantages of the above methods, while also leveraging the respective advantages of neural networks and fuzzy logic systems, an increasing number of studies have used fuzzy neural networks (FNNs) which incorporated a neural network into the fuzzy logic system.^7,29–32 In Lin et al., ³¹ several FNN systems were utilized to learn the unknown dynamic nonlinear function for a class of single input and single output (SISO) uncertain nonlinear systems, but only the adaptive self-tuning weights of FNN were given in the output layer. Therefore, if the mean and standard deviation of the Gaussian membership functions cannot be accurately estimated by the system designers, tracking the system performance is not guaranteed. Wai and Muthusamy ⁷ proposed an intelligent FNN controller for an n-link robot manipulator considering actuator dynamics, where the FNN was designed to learn the conventional SMC law. The parameters of the FNN, including the weights of the output layer and mean and standard deviation of the Gaussian functions in the membership layer, were adapted based on Lyapunov stability theorem. A fuzzy-neural-network-inherited backstepping control was designed for a robot manipulator in Wai and Muthusamy, ³² where the FNN was used to approximate the conventional backstepping control law. However, the previous methods ignore the uncertainties and the inevitable error produced in the learning process of an FNN.

Motivated by the above problems, an adaptive hierarchical sliding mode control based on fuzzy neural network (AFNNHSMC) is designed in this article for a class of SIMO underactuated nonlinear systems with system uncertainties and time-varying external disturbances. The main contributions of the proposed control method are as follows: (1) The FNN system with appropriate self-tuning laws and projection algorithms is applied to learn the control law of HSMC. Compared with Hwang and colleagues,^18,20 this system decreases the number of the required control parameters and reduces the difficulty of controller design process and stability analysis. (2) A self-tuning compensator is designed by considering not only the system uncertainties and time-varying external disturbances, but also the modeling error caused by the FNN.

This article is organized as follows. Following the introduction, section “System description and problem formulation” describes a class of SIMO underactuated nonlinear systems and presents the problems that exist in these systems. In section “HSMC,” a conventional HSMC is designed for a class of SIMO underactuated systems with system uncertainties and time-varying external disturbances. Then, in section “AFNN HSMC,” an AFNNHSMC method and the corresponding learning algorithm are designed based on the Lyapunov theory and Barbalat’s Lemma. The numerical simulations and discussions are provided to demonstrate the effectiveness of the proposed control method in section “Simulations and discussion.” Finally, concluding remarks are presented in section “Conclusion.”

System description and problem formulation

System description

A class of SIMO underactuated nonlinear systems with time-varying uncertainties can be expressed as follows

$$\{\begin{array}{c}{\stackrel{·}{x}}_1=x_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2=f_1\left(X\right)+\Delta f_1\left(X\right)+(b_1\left(X\right)+\Delta b_1\left(X\right))u+\Delta d_1\left(t\right)}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_3=x_4}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_4=f_2\left(X\right)+\Delta f_2\left(X\right)+(b_2\left(X\right)+\Delta b_2\left(X\right))u+\Delta d_2\left(t\right)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_{2n-1}=x_{2n}}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_{2n}=f_n\left(X\right)+\Delta f_n\left(X\right)+(b_n\left(X\right)+\Delta b_n\left(X\right))u+\Delta d_n\left(t\right)}\\ \multicolumn{1}{c}{y={\left[\begin{array}{cccc}{x}_1\hfill & x_3\hfill & \cdots \hfill & x_{2n-1}\hfill \end{array}\right]}^T}\end{array}$$

where $X=[\begin{array}{cccc}{x}_1\hfill & x_3\hfill & \cdots \hfill & x_{2n-1}\hfill \end{array}{]}^T\in R^{2n}$ is the state vector, $y\in R^n$ the output of the system, and u is the control input. The nonlinear functions ( $f_i$ and $b_i$ , $i=1,2,\dots ,n$ ) are continuous and bounded, and the function $b_i$ is assumed to be not equal to zero. $\Delta f_i(X)$ and $\Delta b_i(X)(i=1,\dots ,n)$ are system uncertainties. In many practical cases, the uncertainties are usually produced by load fluctuations, actuator dead zones, gaps, friction, measurement noise, certain unmodeled dynamics, and so on. $\Delta d_i(t)(i=1,\dots ,n)$ denotes the time-varying external disturbances due to external environment. After manipulation, the dynamic model of SIMO underactuated nonlinear systems can be represented as

$$\{\begin{array}{c}{\stackrel{·}{x}}_1=x_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2=f_1\left(X\right)+b_1\left(X\right)u+d_1(X,t)}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_3=x_4}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_4=f_2\left(X\right)+b_2\left(X\right)u+d_2(X,t)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_{2n-1}=x_{2n}}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_{2n}=f_n\left(X\right)+b_n\left(X\right)u+d_n(X,t)}\\ \multicolumn{1}{c}{y={\left[\begin{array}{cccc}{x}_1\hfill & x_3\hfill & \cdots \hfill & x_{2n-1}\hfill \end{array}\right]}^T}\end{array}$$

(1)

where $d_i(X,t)=\Delta f_i(X)+\Delta b_i(X)u+\Delta d_i(t),(i=1,\cdots ,n)$ . $d_i(X,t)(i=1,2,\dots ,n)$ denotes the uncertainties, including the system uncertainties and time-varying external disturbances.

Assumption 1

Since the uncertainties $d_i(X,t)$ are time-varying and cannot be measured, it is assumed that they are bounded, that is, there exists an unknown positive function to guarantee the following inequality

$$0\le \left|d_i(X,t)\right|\le {\overline{d}}_i\left(X\right),(i=1,2,\dots ,n)$$

(2)

Problem formulation

In this article, the aim is to design a suitable control law u to the above SIMO underactuated system shown in equation (1), such that the system will move from an initial position to the desired position. An adaptive hierarchical sliding mode control (AHSMC) method based on FNN is applied to design this suitable control law. At the same time, some self-tuning laws are obtained to adjust the parameters of the FNN system used for learning the HSMC law. A series of matched disturbances and the modeling error caused by the FNN system will be compensated for by a robust term. Finally, the validity and robustness of the proposed controller will be illustrated by the following aspects: (1) simulate the single-pendulum-type overhead crane system and the inverted pendulum system; (2) set different initial state values; (3) address some parameter change and external disturbances; (4) compare with the conventional HSMC method; and (5) compare with the HFSM decoupling controller.

HSMC

The designing process of HSMC, shown in Figure 1, is introduced as follows. First, two states $x_1$ and $x_2$ in the first subsystem constitute the first-layer sliding surface ${\mathbb{S}}_1$ , which compose the second-layer sliding surface ${\mathbb{S}}_2$ with the sliding surface $s_2$ , designed by the other part of states $x_3$ and $x_4$ in the second subsystem. When all the subsystems are included in the last-layer sliding surface ${\mathbb{S}}_n$ , the design process is finished. The HSMC contains a two-part control law, equivalent control law and switching control law. The equivalent control law guarantees each subsystem convergence to the designed sliding surface, and the switching control law makes the closed-loop control system asymptotically stable. The control framework of the SIMO system using the HSMC method is presented in Figure 2.

Figure 1.

Structure of the hierarchical sliding mode control system.

Figure 2.

Control framework of the SIMO system using the HSMC method.

In order to ensure a simple stability analysis process and easily design the control parameter, the traditional sliding surface of the ith subsystem can be defined as

$$s_i=c_i{e}_i+{\stackrel{·}{e}}_i$$

(3)

where $e_i=x_{2i-1}-r_i$ , ${\stackrel{·}{e}}_i=x_{2i}-{\stackrel{·}{r}}_i$ , $i=1,2,\dots ,n$ , $c_i$ is a positive constant, and $r_i$ is the reference trajectory of the ith subsystem.

The derivative of $s_i$ is represented by

$${\stackrel{·}{s}}_i=c_i{x}_{2i}-c_i{\stackrel{·}{r}}_i+f_i\left(X\right)+b_i\left(X\right)u+d_i(X,t)-{\stackrel{··}{r}}_i$$

(4)

The equivalent control $u_{\mathrm{eqi}}$ of the ith-layer sliding surface is designed as shown in equation (5)

$$u_{\mathrm{eqi}}=-\frac{c_i{x}_{2i}+f_i\left(X\right)-c_i{\stackrel{·}{r}}_i-{\stackrel{··}{r}}_i}{b_i\left(X\right)}$$

(5)

The ith-layer sliding surface ${\mathbb{S}}_i$ is defined by equation (6)

$${\mathbb{S}}_i={\lambda }_{i-1}{\mathbb{S}}_{i-1}+s_i=\sum _{m=1}^i\left(\underset{j=m}{\overset{i}{\Pi }}{\alpha }_j\right)s_m$$

(6)

where ${\mathbb{S}}_{i-1}$ is the sliding surface of the layer i – 1, and the optimal coupling factor ${\alpha }_j$ can be described as

$${\alpha }_j=\{\begin{array}{c}{\lambda }_j,j\ne i\hfill \\ \multicolumn{1}{c}{1,j=i}\end{array}$$

(7)

where ${\lambda }_j$ can be a positive or negative constant. Therefore, the derivative of ${\mathbb{S}}_i$ can be obtained by

$${\stackrel{·}{\mathbb{S}}}_i=\sum _{m=1}^i\left(\underset{j=m}{\overset{i}{\Pi }}{\alpha }_j\right){\stackrel{·}{s}}_m$$

(8)

Meanwhile, it is assumed that the control law of the ith-layer subsystem can be defined as

$$u_i=\sum _{m=1}^i(u_{\mathrm{eqi}}+u_{\mathrm{swi}})+u_{\mathrm{rbi}}$$

(9)

where $u_{\mathrm{swi}}$ is the switching control of the ith-layer sliding surface and $u_{\mathrm{rbi}}$ denotes the robust term to compensate for the uncertainties.

The Lyapunov function candidate $V_i$ can be chosen as follows

$$V_i=\frac{1}{2}{\mathbb{S}}_i^2\ge 0$$

(10)

Based on the stability theory of Lyapunov, it is assumed that

$$\begin{array}{c}\sum _{r=1}^i\left[\sum _{\genfrac{}{}{0ex}{}{m=1}{m\ne r}}^i\left(\underset{j=m}{\overset{i}{\Pi }}{\alpha }_j\right)b_m\right]u_{\mathrm{eqr}}+\sum _{r=1}^i\left[\sum _{m=1}^i\left(\underset{j=m}{\overset{i}{\Pi }}{\alpha }_j\right)b_m\right]\hfill \\ \multicolumn{1}{c}{u_{\mathrm{swr}}=-K_i{\mathbb{S}}_i-{\eta }_i\mathrm{sgn}{\mathbb{S}}_i}\\ u_{\mathrm{rbm}}=\frac{-\mathrm{sgn}{\mathbb{S}}_i\sum _{m=1}^i\left(\underset{j=m}{\overset{i}{\Pi }}{\alpha }_j\right){\overline{d}}_m\left(X\right)}{\sum _{m=1}^i\left(\underset{j=m}{\overset{i}{\Pi }}{\alpha }_j\right)b_m}\hfill \end{array}$$

(11)

where $K_i$ and ${\eta }_i$ are the positive constants. The switching control $u_{\mathrm{swi}}$ can be designed as

$$u_{swi}=-{\displaystyle \sum _{r=1}^{i-1}{u}_{swr}}-\frac{{\displaystyle \sum _{r=1}^i\left[{\displaystyle \sum _{\begin{array}{l}m=1\\ m\ne r\end{array}}^i\left({\displaystyle \prod _{j=m}^i{\alpha }_j}\right)b_m}\right]u_{eqr}}}{{\displaystyle \sum _{m=1}^i\left({\displaystyle \prod _{j=m}^i{\alpha }_j}\right)b_m}}-\frac{K_i{\mathbb{S}}_i+{\eta }_{i}sgn{\mathbb{S}}_i}{{\displaystyle \sum _{m=1}^i\left({\displaystyle \prod _{j=m}^i{\alpha }_j}\right)b_m}}$$

(12)

When $i=n$ , the total control law of HSMC can be defined as

$$u_n={\displaystyle \sum _{m=1}^n\left(u_{eqm}+u_{swm}\right)+u_{rbn}}=\frac{{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)b_m{u}_{eqm}}}{{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)b_m}}-\frac{K_n{\mathbb{S}}_n+{\eta }_{n}sgn{\mathbb{S}}_n}{{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)b_m}}$$

$$\begin{array}{c}\multicolumn{1}{c}{-\frac{\mathrm{sgn}{\mathbb{S}}_n\sum _{m=1}^i\left(\underset{j=m}{\overset{i}{\Pi }}{\alpha }_j\right){\overline{d}}_m\left(X\right)}{\sum _{m=1}^i\left(\underset{j=m}{\overset{i}{\Pi }}{\alpha }_j\right)b_m}}\end{array}$$

(13)

The derivative of $V_n$ , the Lyapunov function candidate in the last layer, can be obtained as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_n={\mathbb{S}}_n\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right){\stackrel{·}{s}}_m}\\ \multicolumn{1}{c}{={\mathbb{S}}_n(-\mathrm{sgn}{\mathbb{S}}_n\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right){\overline{d}}_m(X)+\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)}\\ \multicolumn{1}{c}{d_m(X,t)-K_n{\mathbb{S}}_n-{\eta }_n\mathrm{sgn}{\mathbb{S}}_n)}\\ \multicolumn{1}{c}{<-K_n{\mathbb{S}}_n^2-{\eta }_n\left|{\mathbb{S}}_n\right|<0}\end{array}$$

(14)

Based on the Lyapunov stability theory, ³³ the sliding surface is asymptotically stable. However, while the HSMC is based on the accurate dynamical model, it is challenging to obtain the bounds on uncertainties, making it impossible to obtain the actual dynamical model in practical applications. Therefore, it is challenging to implement the HSMC in practical systems and the stability of the system cannot be ensured. An AFNNHSMC is designed to guarantee the stability of the practical system, and the process is introduced in the next section.

AFNNHSMC

An adaptive four-layer FNN system is constructed to approximate the control law of HSMC. The FNN system contains an input layer, a membership layer, a rule layer, and an output layer, as shown in Figure 3. The input layer and the output layer denote the sliding surface vector of HSMC and the control law, respectively. The model description of the FNN system is introduced layer by layer as follows:^34,35

• Layer 1—the input layer: The input signals $s_1,s_2,\dots ,s_n$ are transmitted to the next layer by neurons.

• Layer 2—the membership layer (also known as the fuzzification layer): Each node of this layer represents a membership function. The selected Gaussian membership function is computed according to equation (15)

$${\mu }_i^j\left(s_i\right)=\exp [-\frac{{(s_i-m_i^j)}^2}{{\left(c_i^j\right)}^2}]$$

(15)

where $m_i^j$ and $c_i^j$ denote the mean and standard deviation, respectively, of the Gaussian distribution, $i=1,\dots ,n$ is the number of input signals, and $j=1,\dots ,N_i$ is the number of membership functions of the ith input signal. The total number of membership functions can be denoted as $N=\sum _{i=1}^n{N}_i$ . The entire collection of Gaussian means and standard deviations can be expressed in vector form as shown in equations (16) and (17)

$$m={\left[\begin{array}{ccccccc}{m}_1^1\hfill & \cdots \hfill & m_1^{N_1}\hfill & \cdots \hfill & m_n^1\hfill & \cdots \hfill & m_n^{N_n}\hfill \end{array}\right]}^T\in R^{N\times 1}$$

(16)

$$c={\left[\begin{array}{ccccccc}{c}_1^1\hfill & \cdots \hfill & c_1^{N_1}\hfill & \cdots \hfill & c_n^1\hfill & \cdots \hfill & c_n^{N_n}\hfill \end{array}\right]}^T\in R^{N\times 1}$$

(17)

• Layer 3—the rule layer (the nodes of which are referred to as rules using fuzzy AND operation): The total number of rules can be denoted as $N_y$ . For each $i=1,\dots ,n$ , the ith input signal, ${\mu }_i^j(s_i)$ , is multiplied by the weight $w_{\mathrm{ji}}^k$ between the rule layer and the membership layer (assumed to be unity), and the results are then multiplied to produce the output signal $l_k(k=1,\dots ,N_y)$ , as shown in equation (18)

$$l_k=\underset{i=1}{\overset{n}{\Pi }}{w}_{\mathrm{ji}}^k{u}_i^j\left(s_i\right)$$

(18)

• Layer 4—the output layer: All the input signals to this layer are calculated by summation to obtain the output linguistic variables $y_o(o=1,\dots ,N_o)$ , so this layer is also referred to as the defuzzification procedure.

The concrete calculation process is introduced as follows

$$y_o=\sum _{k=1}^{\mathrm{Ny}}{w}_k^o{l}_k$$

(19)

where $w_k^o$ is the weight between the output layer and the rule layer. For ease of notation, the weight vector is defined as follows

$$W=\left[\begin{array}{cccc}{w}_1^1\hfill & w_2^1\hfill & \cdots \hfill & w_{N_y}^1\hfill \\ \multicolumn{1}{c}{w_1^2}& w_2^2\hfill & \cdots \hfill & w_{N_y}^2\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{w_1^{N_o}}& w_2^{N_o}\hfill & \cdots \hfill & w_{N_y}^{N_o}\hfill \end{array}\right]={\left[\begin{array}{cccc}{w}_1\hfill & w_2\hfill & \cdots \hfill & w_{N_o}\hfill \end{array}\right]}^T$$

(20)

$$l={\left[\begin{array}{cccc}{l}_1\hfill & l_2\hfill & \cdots \hfill & l_{N_y}\hfill \end{array}\right]}^T$$

(21)

where $w_i=[\begin{array}{cccc}{w}_1^i\hfill & w_2^i\hfill & \cdots \hfill & w_{N_y}^i\hfill \end{array}]$ . Therefore, equation (19) can be rewritten as

$$Y={\left[\begin{array}{cccc}{y}_1\hfill & y_2\hfill & \cdots \hfill & y_{N_o}\hfill \end{array}\right]}^T=\mathrm{Wl}$$

(22)

Figure 3.

Structure of the fuzzy neural network system.

The proposed AFNNHSMC control system shown in Figure 4 contains four parts: an FNN control system, a learning algorithm with a self-tuning network parameter, a robust term, and its associated self-adaptive adjustment parameter. The FNN control system and its learning algorithm are designed to mimic the HSMC law that is the sum of the equivalent control law $u_{\mathrm{eqm}}$ and the switching control law $u_{\mathrm{swm}}$ , both of which rely heavily on a precise dynamic model. The bound on the uncertainty, ${\overline{d}}_i(X)$ , is unknown. Besides, the modeling error of the FNN system cannot be avoided. The robust term and its associated self-adaptive adjustment parameter are constructed to estimate and eliminate the uncertainties and the modeling error produced by FNN. Therefore, there exits an optimal FNN control $U^{*}$ to learn the HSMC law as follows

$$U=U^{*}(s_1,\dots ,s_n,W^{*},m^{*},c^{*})+\epsilon =W^{*}{l}^{*}+\epsilon$$

(23)

where $\epsilon$ is the minimum modeling error; $W^{*}$ , $m^{*}$ , and $c^{*}$ denote the optimal parameters of W, m, and c, respectively; and $s_1,\dots ,s_n$ are the input signals of the FNN system including every subsystem’s sliding surface in every layer of HSMC. It is assumed that the minimum modeling error and the optimal parameters of the FNN system are bounded: $\mathrm{\epsilon }\le b_{\epsilon }$ , $W^{*}\le b_w$ , $m^{*}\le b_m$ , and $c^{*}\le b_c$ , where $b_{\epsilon }$ , $b_w$ , $b_m$ , and $b_c$ are the positive real values. The control law of AFNNHSMC, ${\hat{u}}_{\mathrm{hs}}$ , is defined as

$${\hat{u}}_{\mathrm{hs}}=\hat{U}=\hat{U}(s_1,\dots ,s_n,\hat{W},\hat{m},\hat{c})=\hat{W}\hat{l}$$

(24)

where $\hat{W},\hat{m}$ , and $\hat{c}$ are the estimates of the optimal parameters. Subtracting equation (24) from equation (23), the approximation error of the FNN system $\tilde{U}$ is obtained as

$$\tilde{U}=U-\hat{U}=W^{*}{l}^{*}+\epsilon -\hat{W}\hat{l}=\tilde{W}{l}^{*}+\hat{W}\tilde{l}+\epsilon$$

(25)

where $\tilde{W}=W^{*}-\hat{W}$ and $\tilde{l}=l^{*}-\hat{l}$ .

Figure 4.

Block diagram of the AFNNHSMC system.

If the FNN system adjusts only the weights without changing the linguistic rules, a high-precision tracking performance for a nonlinear underactuated system is not guaranteed in a deteriorating environment. Therefore, the expansion of $\tilde{l}$ in Taylor series can be applied to transform the nonlinear output of the rule layer into a partially linear form. Finally, the partially linear form is utilized to design the self-tuning parameters of the weights and the mean and standard deviation of Gaussian membership functions based on Lyapunov theorem. The expansion of $\tilde{l}$ in Taylor series can be obtained as

$$\tilde{l}=\left[\begin{array}{c}{\tilde{l}}_1\\ {\tilde{l}}_2\\ ⋮\\ {\tilde{l}}_{ny}\end{array}\right]=\left[\begin{array}{c}\frac{\partial l_1}{\partial m^T}\\ \frac{\partial l_2}{\partial m^T}\\ ⋮\\ \frac{\partial l_{Ny}}{\partial m^T}\end{array}\right]{|}_{m=\widehat{m}}\left(m^{*}-\widehat{m}\right)+\left[\begin{array}{c}\frac{\partial l_1}{\partial c^T}\\ \frac{\partial l_2}{\partial c^T}\\ ⋮\\ \frac{\partial l_{Ny}}{\partial c^T}\end{array}\right]{|}_{c=\widehat{c}}\left(c^{*}-\widehat{c}\right)+Onv=l_m\tilde{m}+l_c\tilde{c}+Onv$$

(26)

where $l_m=[\begin{array}{cccc}\partial l_1/\partial m^T\hfill & \partial l_2/\partial m^T\hfill & \cdots \hfill & \partial l_{\mathrm{Ny}}/\partial m^T\hfill \end{array}{]}^T$ , $l_c=[\begin{array}{cccc}\partial l_1/\partial c^T\hfill & \partial l_2/\partial c^T\hfill & \cdots \hfill & \partial l_{\mathrm{Ny}}/\partial c^T\hfill \end{array}{]}^T$ , $\tilde{m}=m^{*}-\hat{m}$ , $\tilde{c}=c^{*}-\hat{c}$ , and $\mathrm{Onv}\in R^{\mathrm{Ny}\times 1}$ is the vector of Lagrange remainder. From equation (27), the optimal parameter $l^{*}$ is obtained as

$$l^{*}=\hat{l}+l_m\tilde{m}+l_c\tilde{c}+\mathrm{Onv}$$

(27)

By replacing equation (25) with equation (27), the approximation error $\tilde{U}$ is rewritten as

$$\tilde{U}=\tilde{W}\hat{l}+\hat{W}{l}_m\tilde{m}+\hat{W}{l}_c\tilde{c}+y\prime$$

(28)

$$y\prime =\tilde{W}{l}_m\tilde{m}+\tilde{W}{l}_c\tilde{c}+W^{*}\mathrm{Onv}+\epsilon$$

(29)

where $y\prime$ is the total modeling error of the FNN system. From equation (13), the control law of HSMC, that is, the sum of the equivalent control law $u_{\mathrm{eqm}}$ and the switching control law $u_{\mathrm{swm}}$ related to the dynamic model, can be rewritten as follows

$$u_{hs}=\frac{{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)b_m{u}_{eqm}}}{{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)b_m}}=-\frac{{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)\left(c_m{x}_{2m}-c_m{\stackrel{.}{r}}_m+f_m\left(X\right)-{\stackrel{··}{r}}_m\right)}}{{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)b_m}}$$

(30)

Due to its powerful approximation ability, an FNN is used to learn the control law $u_{\mathrm{hs}}$ based on the dynamical model. Therefore, the total control law of the HSMC system can be rewritten as

$$u_n={\hat{u}}_{\mathrm{hs}}+u_d$$

(31)

where ${\hat{u}}_{\mathrm{hs}}$ is the FNN control law and $u_d$ denotes the robust term to compensate for the total error produced by system’s uncertainties and FNN modeling error. From equation (8), when $i=n$ , the derivation of ${\mathbb{S}}_n$ can be rewritten as

$$\begin{array}{c}{\stackrel{.}{\mathbb{S}}}_n={\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right){\stackrel{.}{s}}_m}\\ =-{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)b_m\left(X\right)\left(\tilde{W}\widehat{l}+\widehat{W}{l}_m\tilde{m}+\widehat{W}{l}_c\tilde{c}+y^{\prime }-u_d\right)}+{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)d_m\left(X,t\right)}\end{array}$$

(32)

It is assumed that the total error, D, is written as

$$D=\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)(d_m(X,t)-b_m\left(X\right)y\prime )$$

(33)

where Q is the bound of the total uncertainties and $\left|D\right|<Q$ .

To calculate the self-tuning parameters of FNNHSMC, the Lyapunov function candidate is chosen as follows

$$V=\frac{1}{2}{\mathbb{S}}_n^2+\frac{1}{2{\eta }_1}\tilde{W}{\tilde{W}}^T+\frac{1}{2{\eta }_2}{\tilde{m}}^T\tilde{m}+\frac{1}{2{\eta }_3}{\tilde{c}}^T\tilde{c}+\frac{1}{2{\eta }_4}{\tilde{Q}}^2>0$$

(34)

where $\tilde{Q}=Q-\hat{Q}$ ; $\hat{Q}$ is an online estimated value of the uncertain term bound Q; ${\eta }_1$ , ${\eta }_2$ , and ${\eta }_3$ are the positive learning gains of the FNN parameters; and ${\eta }_4$ denotes the positive learning gain for the total uncertainty.

Therefore, the update laws of the FNN parameters are designed as follows

$${\stackrel{·}{\hat{W}}}^T=\{\begin{array}{c}-{\eta }_1{\mathbb{S}}_n\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)b_m\left(X\right)\hat{l}\hfill \\ \multicolumn{1}{c}{\mathrm{if}\Vert W\Vert <b_w\mathrm{or}}\\ \multicolumn{1}{c}{(\Vert W\Vert =b_w\mathrm{and}{\mathbb{S}}_n\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)b_m\left(X\right)\hat{W}\hat{l}\le 0)}\\ \multicolumn{1}{c}{-{\eta }_1{\mathbb{S}}_n\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)b_m\left(X\right)\hat{l}}\\ \multicolumn{1}{c}{-\frac{{\eta }_1{\mathbb{S}}_n\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)b_m\left(X\right){\hat{W}}^T\hat{W}\hat{l}}{{\Vert \hat{W}\Vert }^2}}\\ \multicolumn{1}{c}{\mathrm{if}\Vert W\Vert =b_w\mathrm{and}{\mathbb{S}}_n\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)b_m\left(X\right)\hat{W}\hat{l}>0}\end{array}$$

(35)

$${\stackrel{.}{\widehat{m}}}^T=\{\begin{array}{l}-{\eta }_2{\mathbb{S}}_n{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)b_m\left(X\right)\widehat{W}{l}_m}\\ \mathrm{if}\Vert m\Vert <b_m\mathrm{or}\left(\Vert m\Vert =b_m\mathrm{and}{\mathbb{S}}_n{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)b_m\left(X\right)\widehat{W}{l}_m\widehat{m}}\le 0\right)\\ -{\eta }_2{\mathbb{S}}_n{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)b_m\left(X\right)\widehat{W}{l}_m}-\frac{{\eta }_2{\mathbb{S}}_n{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)b_m\left(X\right)\widehat{W}{l}_m\widehat{m}{\widehat{m}}^T}}{{\Vert \widehat{m}\Vert }^2}\\ \mathrm{if}\Vert m\Vert <b_m\mathrm{and}{\mathbb{S}}_n{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)b_m\left(X\right)\widehat{W}{l}_m\widehat{m}}>0\end{array}$$

(36)

$${\stackrel{·}{\hat{c}}}^T=\{\begin{array}{c}-{\eta }_3{\mathbb{S}}_n\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)b_m\left(X\right)\hat{W}{l}_c\hfill \\ \multicolumn{1}{c}{\mathrm{if}\Vert c\Vert <b_c\mathrm{or}}\\ \multicolumn{1}{c}{(\Vert c\Vert =b_c\mathrm{and}{\mathbb{S}}_n\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)b_m\left(X\right)\hat{W}{l}_c\hat{c}\le 0)}\\ \multicolumn{1}{c}{-{\eta }_3{\mathbb{S}}_n\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)b_m\left(X\right)\hat{W}{l}_c}\\ \multicolumn{1}{c}{-\frac{{\eta }_2{\mathbb{S}}_n\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)b_m\left(X\right)\hat{W}{l}_c\hat{c}{\hat{c}}^T}{{\Vert \hat{c}\Vert }^2}}\\ \multicolumn{1}{c}{\mathrm{if}\Vert c\Vert <b_c\mathrm{and}{\mathbb{S}}_n\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)b_m\left(X\right)\hat{W}{l}_c\hat{c}>0}\end{array}$$

(37)

And the robust term and its update law are defined as follows

$$\stackrel{·}{\hat{Q}}=-{\eta }_4\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)b_m\left(X\right)\left|{\mathbb{S}}_n\right|$$

(38)

$$u_d=\frac{-K{\mathbb{S}}_n}{\sum _{m=1}^n\left(\underset{j=m}{\overset{n}{\Pi }}{\alpha }_j\right)b_m\left(X\right)}+\mathrm{sgn}\left({\mathbb{S}}_n\right)\hat{Q}$$

(39)

where K is a positive constant. Then, the derivative of V can be calculated as

$$\stackrel{.}{V}={\mathbb{S}}_n{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)}{b}_m(X)D-K{\mathbb{S}}_n^2-{\displaystyle \sum _{m=1}^n\left({\displaystyle \prod _{j=m}^n{\alpha }_j}\right)}{b}_m\left(X\right)\left|{\mathbb{S}}_n\right|Q\le -K{\mathbb{S}}_n^2\le 0$$

(40)

According to equation (38), the derivative of the Lyapunov function candidate $(\stackrel{·}{V}(\mathbb{S}(t),\tilde{W},\tilde{m},\tilde{c},\tilde{Q}(t)))$ is negative, which means that $\stackrel{·}{V}(\mathbb{S}(t),\tilde{W},\tilde{m},\tilde{c},\tilde{Q}(t))$ is a negative semi-definite function and hence $V(\mathbb{S}(t),\tilde{W},\tilde{m},\tilde{c},\tilde{Q}(t))\le V(\mathbb{S}(0),\tilde{W},\tilde{m},\tilde{c},\tilde{Q}(0))$ . At the same time, it implies that the proposed control method, AFNNHSMC, is stable based on the Lyapunov theory. When $t>0$ , the functions ( $\mathbb{S}(t)$ , $\tilde{W}$ , $\tilde{m}$ , $\tilde{c}$ , and $\tilde{Q}(t)$ ) are bounded, because the initial value of parameters ( $\mathbb{S}(0)$ , $\tilde{W}$ , $\tilde{m}$ , $\tilde{c}$ , and $\tilde{Q}(0)$ ) are bounded. To verify that the sliding surface $(\mathbb{S}(t))$ and the system’s tracking error $(e_i(t))$ will converge to zero, the function $\Omega (t)$ is defined as

$$\Omega \left(t\right)={\mathrm{KS}}_n^2\le -\stackrel{·}{V}(S\left(t\right),\tilde{W},\tilde{m},\tilde{c},\tilde{Q}\left(t\right))$$

(41)

Integrating on the interval [0, t], we obtain

$$\underset{0}{\overset{t}{\int }}\Omega \left(\tau \right)d\tau \le V(S\left(0\right),\tilde{W},\tilde{m},\tilde{c},\tilde{Q}\left(0\right))-V(S\left(t\right),\tilde{W},\tilde{m},\tilde{c},\tilde{Q}\left(t\right))$$

(42)

According to equation (35), $V(S(0),\tilde{W},\tilde{m},\tilde{c},\tilde{Q}(0))$ is a bounded value and $V(S(t),\tilde{W},\tilde{m},\tilde{c},\tilde{Q}(t))$ is a nonincreasing and bounded function. Therefore

$$\underset{t\to \infty }{lim}\underset{0}{\overset{t}{\int }}\Omega \left(\tau \right)d\tau \le \infty$$

(43)

From equation (41), it follows that the derivative of the function $\mathrm{\Omega }(t)$ is bounded. Therefore, based on Barbalat’s Lemma,^33,36 $\underset{t\to \infty }{lim}\Omega (t)=0$ , which in turn means that $\underset{t\to \infty }{lim}\mathbb{S}(t)=0$ . Moreover, the tracking error will converge to zero because $\underset{t\to \infty }{lim}\mathbb{S}(t)=0$ . To summarize, the proposed control system is stable in the presence of uncertainties and when relaxing the dependence on the dynamic model.

Remark 1

The robust term contains the sign function $\mathrm{sgn}({\mathbb{S}}_n)$ in equation (39), which is replaced by a saturation function $\mathrm{sat}({\mathbb{S}}_n/Y)$ to smooth the control input, defined as

$$\mathrm{sat}\left(\frac{{\mathbb{S}}_n}{\gamma }\right)=\{\begin{array}{c}\mathrm{sgn}\left(\frac{{\mathbb{S}}_n}{\gamma }\right),\mathrm{if}\left|{\mathbb{S}}_n\right|>\gamma \hfill \\ \multicolumn{1}{c}{\mathrm{sat}\left(\frac{{\mathbb{S}}_n}{\gamma }\right)=\frac{{\mathbb{S}}_n}{\gamma },\mathrm{otherwise}}\end{array}$$

(44)

$$\mathrm{sgn}\left(\frac{{\mathbb{S}}_n}{\gamma }\right)=\{\begin{array}{c}1,\frac{{\mathbb{S}}_n}{\gamma }>0\hfill \\ \multicolumn{1}{c}{0,\frac{{\mathbb{S}}_n}{\gamma }>0}\\ \multicolumn{1}{c}{-1,\frac{{\mathbb{S}}_n}{\gamma }<0}\end{array}$$

where $\gamma$ is the constant indicating the thickness of the boundary layer.

Simulations and discussion

Example 1

In this example, a single-pendulum-type overhead crane system, as shown in Figure 5, is considered as a simulation example to verify the performance of the AFNNHSMC. At the beginning, four system functions in Figure 5 are described as follows ¹⁵

$$\begin{array}{c}\multicolumn{1}{c}{f_1\left(X\right)=\frac{{\mathrm{MLx}}_4^2\sin x_3+\mathrm{mg}\sin x_3\cos x_3}{M+m{\left(\sin x_3\right)}^2},b_1\left(X\right)}\\ \multicolumn{1}{c}{=\frac{1}{M+m{\left(\sin x_3\right)}^2}}\\ \multicolumn{1}{c}{f_2\left(X\right)=\frac{(M+m)g\sin x_3+{\mathrm{MLx}}_4^2\sin x_3\cos x_4}{(M+m{\left(\sin x_3\right)}^2)L},}\\ \multicolumn{1}{c}{b_2\left(X\right)=\frac{1}{(M+m{\left(\sin x_3\right)}^2)L}}\end{array}$$

(45)

where $x_1=x$ , $x_2=\stackrel{·}{x}$ , $x_3=\theta$ , and $x_4=\stackrel{·}{\theta }$ represent the trolley position, the trolley velocity, the swing angle, and the swing angular velocity, respectively. M = 37.32 kg is the trolley mass, m = 5 kg is the payload mass, L = 0.15 m is the cable length, and g = 9.81 m/s ² is the acceleration of gravity. In addition, the external disturbance, including system uncertainties and time-varying external disturbances, are assumed to be $d_1(x,t)=0.5\sin (10t)+0.75\sin (x_1^2)\sin (\sin (t)x_1{x}_3)$ and $d_2(x,t)=0.1\sin (10t)+0.75\sin (x_3{x}_4)\sin (\cos (t)x_1{x}_3)$ .

Figure 5.

Structure of the single-pendulum-type overhead crane system.

At the same time, the control gains of both the AFNNHSMC and the HSMC are listed as follows: ${\eta }_1=1500$ , ${\eta }_2=0.5$ , ${\eta }_3=0.5$ , ${\eta }_4=3$ , $b_w=10$ , $b_m=1$ , $b_c=1$ , $K_n=3$ , $c_1=0.7$ , $c_2=8.2$ , ${\alpha }_1=-2.3$ , and ${\eta }_n=0.1$ . Note that the control parameters are chosen to achieve the best transient control performance in numerical simulation results by considering the setting time, max/min error, operating range as well as the requirement of stability. The gain parameters ( $c_i$ and ${\alpha }_i$ ) are determined according to the design of the sliding surface. In this article, the parameters $K_n$ and ${\eta }_n$ are helpful in guaranteeing the system stability in the control effort because the term $(-K_n{\mathbb{S}}_n^2-{\eta }_n\left|{\mathbb{S}}_n\right|)$ in equation (14) will dominate the fact of ${\stackrel{·}{V}}_n<0$ , even the worst case $(|{\overline{d}}_m(X)|<d_m(X,t),m=1,\dots ,n)$ happens. Moreover, the selection of learning rates ( ${\eta }_1$ , ${\eta }_2$ , ${\eta }_3$ , and ${\eta }_4$ ) is related to the convergence rate of the error state vector and dependent on the significance of tuning objects. Because the weight matrix W dominates the major network output, its learning rate $({\eta }_1)$ is always larger than the learning rates of the Gaussian parameters ( ${\eta }_2$ and ${\eta }_3$ ). In addition, the associated parameter bounds ( $b_w$ , $b_m$ , and $b_c$ ) are decided by the ranges of reasonable network coefficients.

There are two important system parameters, payload mass (m) and cable length (L), respectively, that have a negative impact on the system control performance. To prove the robustness of the proposed control system, it is assumed that those two parameters change as follows:

• Simulation parameter case A: L = 1.05 m and m = 5 kg.

• Simulation parameter case B: $L=2*1.05\mathrm{m}$ and $m=4*5\mathrm{kg}$ .

It is clear that the different initial parameters can lead to different results, especially for an underactuated system. Therefore, if the proposed control system does not have a wide operating range, the system will not have the ability to track the desired signal when the initial parameters are changed. Thus, the following initial parameters are chosen to verify the effectiveness of the AFNNHSMC under a wide operating range:

• Initial simulation case 1: $[x_0=0.5,{\theta }_0=0.3]$ .

• Initial simulation case 2: $[x_0=-0.25,{\theta }_0=0.2]$ .

• Initial simulation case 3: $[x_0=0.25,{\theta }_0=-0.2]$ .

• Initial simulation case 4: $[x_0=-0.5,{\theta }_0=-0.3]$ .

The control task has two folds: fast track the trolley to the desired position, and small swing of the payload. According to the required task, two subsystems’ sliding surfaces, $s_1=c_1{e}_1+{\stackrel{·}{e}}_1$ and $s_2=c_2{e}_2+{\stackrel{·}{e}}_2$ , respectively, are considered as the input to the FNN system. The trolley position x and the swing angle $\theta$ follow the specific reference trajectory, $x_d=0$ and ${\theta }_d=0$ . To show the effectiveness of the AFNNHSMC, the FNN system has 2, 10, 25, and 1 neurons at the input, membership, rule, and output layers, respectively. The initial values for the FNN parameters ( ${\hat{W}}^T$ , ${\hat{m}}^T$ , and $c^T$ ) are set at 0.0001. It is assumed that the values of each input signal are partitioned into a fuzzy partition with Gaussian membership functions representing “negative big,”“negative small,”“zero,”“positive small,” and “positive big.”

In section “HSMC,” the HSMC is proposed to verify the superior performance of AFNNHSMC. All the simulation results of the proposed AHSMC and AFNNHSMC schemes for the single-pendulum-type overhead crane system are depicted in Figures 6–9, respectively. Figures 6(a) and (b), 7(a) and (b), 8(a) and (b), and 9(a) and (b) show the tracking responses of the trolley position and the swing angle; Figures 6(c), 7(c), 8(c), and 9(c) show the related control force to the trolley; Figures 6(d), 7(d), 8(d), and 9(d) show the sliding surface of the layer n. The numerical simulations of HSMC under the condition of the simulation parameter cases A and B without external disturbance are shown in Figures 6 and 7, respectively. From Figures 6(a), (b), and (d) and 7(a), (b), and (d), when the system’s parameters are changed, a high-precision tracking result for x and $\theta$ can be achieved. However, an undesirable chattering phenomenon exists as shown in Figures 6(c) and 7(c) due to the selection of the discontinuous sign function and inappropriate uncertainty bound. In practice, this chattering control law will cause great wear and tear to the mechanical structure of the practical robot, which makes the application of HSMC difficult. When considering the external disturbance, the tracking positions shown in Figures 8(a) and (b) and 9(a) and (b) are poor especially for the swing angle $\theta$ . Furthermore, the undesirable chattering phenomenon evidently increases as the external disturbances increase in Figures 8(c) and 9(c). In practice, the disturbance may be bigger than the given disturbance and it is challenging to obtain the accurate bound of the uncertainty. Especially for the underactuated system, the excessive disturbance will lead to unstable system dynamics and, therefore, an inability to control this underactuated system.

Figure 6.

The simulation of HSMC under the simulation parameter case A and without external disturbance: (a) The tracking response of the trolley position; (b) The tracking response of the swing angle; (c) The related control force to the trolley; and (d) The sliding surface of layer n.

Figure 7.

The simulation of HSMC under the simulation parameter case B and without external disturbance: (a) The tracking response of the trolley position; (b) The tracking response of the swing angle; (c) The related control force to the trolley; and (d) The sliding surface of layer n.

Figure 8.

The simulation of HSMC under the simulation parameter case A and with external disturbance: (a) The tracking response of the trolley position; (b) The tracking response of the swing angle; (c) The related control force to the trolley;and (d) The sliding surface of layer n.

Figure 9.

The simulation of HSMC under the simulation parameter case B and with external disturbance: (a) The tracking response of the trolley position; (b) The tracking response of the swing angle; (c) The related control force to the trolley; and (d) The sliding surface of layer n.

Now, the proposed AFNNHSMC is implemented to control a single-pendulum-type overhead crane system. As shown in Figures 10(a), (b), and (d) and 11(a), (b), and (d), the tracking performance of the proposed control system is similar to that of the HSMC. The result illustrates the validity of the proposed control system. Moreover, it is apparent that the undesirable chattering phenomenon, which exists in the HSMC, can be eliminated by online approximation of the HSMC law, estimation of the uncertain bound, and the use of a saturation function to replace the sign function shown in Figures 10(c) and 11(c). To further validate the robustness of AFNNHSMC, when considering an external disturbance, the error of the trolley position x and the swing angle $\theta$ as shown in Figures 12 and 13 are reduced dramatically compared to the HSMC, in both the simulation parameter cases A and B. The corresponding FNN parameter changes during the path tracking of the single-pendulum-type overhead crane system, the weights $(W)$ and the means and standard deviations of the Gaussian functions (m and c), can be observed in Figure 14(a)–(c), in which (.) denotes the Euclidean norm. It is evident that the adaptive laws change during the transient process to handle the real-time environment, and all of the tuned parameters can be constricted to ensure the stability of the internal states.

Figure 10.

The simulation of AFNNHSMC under the simulation parameter case A and without external disturbance: (a) The tracking response of the trolley position; (b) The tracking response of the swing angle; (c) The related control force to the trolley; and (d) The sliding surface of layer n.

Figure 11.

The simulation of AFNNHSMC under the simulation parameter case B and without external disturbance: (a) The tracking response of the trolley position; (b) The tracking response of the swing angle; (c) The related control force to the trolley; and (d) The sliding surface of layer n.

Figure 12.

The simulation of AFNNHSMC under the simulation parameter case A and with external disturbance: (a) The tracking response of the trolley position; (b) The tracking response of the swing angle; (c) The related control force to the trolley; and (d) The sliding surface of layer n.

Figure 13.

The simulation of AFNNHSMC under the simulation parameter case B and with external disturbance: (a) The tracking response of the trolley position; (b) The tracking response of the swing angle; (c) The related control force to the trolley; and (d) The sliding surface of layer n.

Figure 14.

The simulation parameters of the FNN system with external disturbance, and both in the simulation parameter case A and in initial simulation case 1: (a) The weights; (b) The means of the Gaussian functions; and (c) The standard deviations of the Gaussian functions.

The performance of the AFNNHSMC is summarized in Tables 1 and 2, which show the simulation results for HSMC and AFNNHSMC in the initial simulation case 1. The results (columns 6 and 7 of Tables 1 and 2) show that when the system deals with both parameter changes and external disturbances, the values of setting time, max/min error, mean square error (MSE), and tracing error obtained for AFNNHSMC are smaller than those obtained for HSMC. In fact, remarkably, the settling time and tracking error show that the HSMC performance worsens with increased environmental complexity. By contrast, whatever the case is, the AFNNHSMC performance fluctuates in a very small range. In conclusion, the AFNNHSMC method can achieve a better performance than the conventional HSMC method in the environment with both parameter changes and external disturbances.

Table 1.

Control performance of the trolley position under initial simulation case 1.

Parameters	Under the simulationparameter case A and without external disturbance		Under the simulation parametercase A and with externaldisturbance		Under the simulation parametercase B and with external disturbance
	HSMC	AFNNHSMC	HSMC	AFNNHSMC	HSMC	AFNNHSMC
Settling time (s)	6.1	7.4	6.3	7.5	8.6	7.8
Max error (m)	1.38 × 10−5	−1.13 × 10−6	0.0023	−0.0023	0.0032	−0.0014
Min error (m)	−9.36 × 10−7	−1.47 × 10−4	−0.0023	−0.004	−0.0038	−0.0041
MSE (m)	0.025	0.026	0.024	0.027	0.0382	0.0303
Tracking error: $\sum |x_d-x|$	0.0056	0.0615	9.8849	2.0454	12.442	1.9731

HSMC: hierarchical sliding mode control; AFNNHSMC: adaptive hierarchical sliding mode control based on fuzzy neural network; MSE: mean square error.

The bold values signifies the advantages of the AFNNHSMC method.

Table 2.

Control performance of the swing angle under initial simulation case 1.

Parameters	Under the simulationparameter case A and without external disturbance		Under the simulation parametercase A and with external disturbance		Under the simulation parametercase B and with external disturbance
	HSMC	AFNNHSMC	HSMC	AFNNHSMC	HSMC	AFNNHSMC
Settling time (s)	6.2	7.6	6.5	7.8	8.9	8.1
Max error (m)	1.32 × 10−5	1.17 × 10−4	0.0236	0.0033	0.032	0.0041
Min error (m)	−3.5 × 10−5	5.07 × 10−7	−0.0237	−0.0033	−0.0313	−0.0052
MSE (m)	0.0072	0.0065	0.008	0.0066	0.0086	0.0069
Tracking error: $\sum |{\theta }_d-\theta |$	0.0098	0.0417	99.2786	1.337	125.8342	1.6409

HSMC: hierarchical sliding mode control; AFNNHSMC: adaptive hierarchical sliding mode control based on fuzzy neural network; MSE: mean square error.

The bold values signifies the advantages of the AFNNHSMC method.

Example 2

In this example, an inverted pendulum system will be adopted to further verify the effectiveness of the developed AFNNHSMC method compared with an HFSM decoupling controller proposed in Lin and Mon. ¹⁹ The four system functions in Figure 15 are described as follows ²⁰

$$\begin{array}{c}\multicolumn{1}{c}{f_1\left(X\right)=\frac{m_{t}g\sin \left(x_1\right)-m_{p}L\sin \left(x_1\right)\cos \left(x_1\right)x_2^2}{\mathrm{Lh}\left(X\right)},}\\ \multicolumn{1}{c}{b_1\left(X\right)=\frac{\cos \left(x_1\right)}{\mathrm{Lh}\left(X\right)}}\\ \multicolumn{1}{c}{f_2\left(X\right)=\frac{-4m_p{\mathrm{Lx}}_2^2\sin \left(x_1\right)+3m_{p}g\sin \left(x_1\right)\cos \left(x_1\right)}{3h\left(X\right)},}\\ \multicolumn{1}{c}{b_2\left(X\right)=\frac{4}{3h\left(X\right)}}\end{array}$$

(46)

where $h(X)=4(m_t/3)-m_p{\cos }^2(x_1)$ ; $m_t=m_c+m_p$ ; $x_1=\theta$ , $x_2=\stackrel{·}{\theta }$ , $x_3=x$ , and $x_4=\stackrel{·}{x}$ denote the pendulum angle, the pendulum angular velocity, the cart position, and the cart velocity, respectively; u is the applied force to move the cart; $m_c=1\mathrm{kg}$ is the mass of the cart; $m_p=0.05\mathrm{kg}$ is the mass of the pole; $L=0.5\mathrm{m}$ is the half-length of a pole; and $g=9.8\mathrm{m}/{\mathrm{s}}^2$ is the acceleration of gravity. In addition, the external disturbances are assumed to be $d_1(x,t)=0.75\sin (x_1^2)\sin (\sin (t)x_1{x}_3)$ and $d_2(x,t)=0.75\sin (x_3{x}_4)\sin (\cos (t)x_1{x}_3)$ . In the simulations, the control parameters are chosen as ${\eta }_1=1153$ , ${\eta }_2=0.27$ , ${\eta }_3=0.31$ , ${\eta }_4=2.3$ , $b_w=4$ , $b_m=0.8$ , $b_c=1$ , $K_n=3$ , $c_1=6$ , $c_2=5$ , ${\alpha }_1=-2.3$ , and ${\eta }_n=0.1$ . The initial values for the FNN parameters ( ${\hat{W}}^T$ , ${\hat{m}}^T$ , and $c^T$ ) are set at 0.0001. The fuzzy system’s parameters are similar to those reported in Lin and Mon. ¹⁹ The cart position x and the pendulum angle $\theta$ follow the specific reference trajectory, $x_d=\sin (t)$ and ${\theta }_d=0$ .

Figure 15.

Structure of the inverted pendulum system.

All the simulation results of the HFSMC and AFNNHSMC schemes for the inverted pendulum system are depicted in Figures 16 and 17. Figures 16(a) and (b) and 17(a) and (b) show the tracking responses of the pendulum angle and the cart position, respectively. We can see from these figures that the pendulum angle and cart position can converge to the specific reference trajectory. In this article, the regulation and tracking performances for the inverted pendulum system are achieved with the same controller settings. Moreover, it is apparent that the values of setting time and max/min error obtained with AFNNHSMC are smaller than those obtained with HFSMC.

Figure 16.

The simulation results of HSMC for the inverted pendulum system: (a) The tracking response of the pendulum angle; and (b) The tracking response of the cart position.

Figure 17.

The simulation results of AFNNHSMC for the inverted pendulum system: (a) The tracking response of the pendulum angle; and (b) The tracking response of the cart position

Conclusion

In this article, a successful implementation of an AFNNHSMC, capable of realizing regulation stabilization and accurate tracking control simultaneously for a class of SIMO underactuated nonlinear systems, considering system uncertainties and time-varying external disturbances is presented. Since the underactuated dynamic system is constituted by several subsystems, an HSMC scheme is proposed to reduce the difficulty of the controller designing process. A model-free FNN system is then employed to approximate the HSMC law while inheriting the properties of HSMC. Based on the Lyapunov stability theory and Barbalat’s Lemma, the robust term is designed to track the uncertainties, including the FNN modeling error, and the asymptotic stability of the closed-loop system is guaranteed. Finally, the simulation results effectively illustrate the performance of the proposed method, including controller robustness. This is done by comparison with some conventional methods, under different simulation conditions, including different initial state values and different system parameters, with and without uncertainties. Furthermore, the proposed method does not only effectively reduce the chattering phenomenon, but also obviously reduce the accuracy requirements for the dynamic model when designing the controller. Future work will apply this approach to a practical robot system and will expand it to some MIMO underactuated systems, especially for underactuated wheeled inverted pendulum systems, ³⁷ because this system is simultaneously subject to kinematic nonholonomic constraints and dynamic constraints, and the existing control designs for nonholonomic systems with either only kinematic or only dynamic constraints cannot be directly applied to this system.