LESO-Based Nonlinear Continuous Robust Stabilization Control of Underactuated TORA Systems

Abstract

In this paper, we consider the robust stabilization control problem of underactuated translational oscillator with a rotating actuator (TORA) system in the presence of unknown matched disturbances by employing continuous control inputs. A nonlinear continuous robust control approach is proposed by integrating the techniques of backstepping and linear extended state observer (LESO). Specifically, based on the backstepping design methodology, a hyperbolic tangent virtual control law is designed for the first subsystem of the cascaded TORA model, via which an integral chain error subsystem is subsequently constructed and the well-known LESO technique is easy to implement. Then, an LEO is designed to estimate the lumped matched disturbances in real-time, and the influence of the disturbances is compensated by augmenting the feedback controller with the disturbance estimation. The convergence and stability of the entire control system are rigorously proved by utilizing Lyapunov theory and LaSalle’s invariance principle. Unlike some existing methods, the proposed controller is capable of generating robust and continuous control inputs, which guarantee that both the rotation and translation of TORA systems are stabilized at the origin simultaneously and smoothly, attenuating the influence of disturbances. Comparative simulation results are presented to demonstrate the effectiveness and superior control performance of the proposed method.

1. Introduction

As a typical underactuated mechanical system [1,2,3,4,5,6], the translational oscillator with a rotating actuator (TORA) is composed of an unactuated translational trolley and an actuated rotational eccentric ball. It is thus featured by the characteristics of strong coupling and high nonlinearity, and often used as a benchmark system for the design and performance test of various nonlinear control algorithms [7,8,9]. In addition, the TORA system is also used in engineering as an active mass damper (AMD) for the active vibration suppression of large engineering systems [10,11,12], such as super high-rise buildings, long-span bridges, offshore floating wind turbines, etc. Furthermore, after some extensions, it can also be used to study the self-synchronized phenomenon of many mechanical systems [13,14], e.g., vibration sifters, hands-held vibration tools, vibration conveyors, etc. Therefore, the study of TORA systems is of great both theoretical and practical significance.

However, the stabilization control of TORA systems is very challenging because the trolley and the ball need to be stabilized simultaneously using only one actuator (control input). To achieve the stabilization objective, scholars have conducted extensive and in-depth research, and published many ambitious achievements in the past decades [15,16,17,18,19,20,21,22]. Roughly speaking, the obtained results can be classified into three categories: passivity-based control method, cascade-based control method, and other advanced/intelligent control method.

The passivity-based control method is to construct a control Lyapunov function according to the passivity property of TORA systems and design a controller by making the derivative of the Lyapunov function negative. In [23], the passivity property of TORA systems is analyzed, based on which an energy control Lyapunov function is constructed, and a simple state feedback controller is designed. In [24], a passivity-based adaptive controller with an online observation mechanism is proposed. In [25], an output feedback passive control method using energy shaping and damping injection techniques is presented. In [26], a virtual angular velocity feedback signal is constructed and an output feedback global stabilization control method is proposed. More recently, based on the virtual feedback signal, the authors in [27] improve the energy function and present an output feedback control method that can avoid the phenomenon of small ball circulation.

The cascade-based control method requires a coordinate transformation to convert the TORA model into a cascaded form. Due to the simple structure of the cascaded system, the recursive backstepping technique can be easily applied to design a stable controller. This merit makes the cascade-based method extremely favored by scholars. For example, the TORA systems in [28] are transformed into a strict feedback cascaded form by using a global coordinate transformation, and then an integral backstepping control method is proposed. In [29], the state variables are treated as virtual control inputs and a nonlinear backstepping controller is designed. Considering the drawback of “explosion of complexity” in the backstepping design procedure, a nonlinear dynamic surface controller is designed through a collocated partial feedback linearization and a global change of coordinates [30]. In [31], an adaptive backstepping control scheme is proposed and a TORA experimental implementation is introduced.

It should be noted that most of the above literature does not consider the influence of unknown disturbances. To deal with uncertainties, some advanced/intelligent control techniques such as equivalent input disturbance (EID) [32], adaptive control [33], fuzzy control [34], and neural network [31] have been employed to improve the robustness of TORA systems. However, these methods require either prior knowledge of the system model or online learning mechanisms, which make the developed control algorithms not only computationally expensive but also very difficult for engineering applications. Moreover, by combining the sliding mode control with the observer technique, many composite control schemes have been proposed. For example, two nonlinear disturbance observer-based sliding mode control approaches are presented in [35,36], where the nonlinear disturbance observers are used to estimate unknown external disturbances and compensate for their effects, and the sliding mode controllers are designed to stabilize the system. Nevertheless, the inherent chattering problem of the sliding mode control cannot be avoided, and the generated control signals are discontinuous.

Based upon the above analysis of the current research status of TORA systems, it is noted that there are still some open problems worthy of being further investigated, which are summarized as follows.

(1)

The control algorithms developed by the passivity-based control method are computationally simple but they are difficult to deal with unknown disturbances. In other words, when these controllers are applied for TORA systems in practice where unknown disturbances widely exist, the control performances of the controllers would deteriorate and even unstable results may be caused.

(2)

By introducing a coordinate transformation, the model of TORA systems becomes a relatively simple cascaded form, based on which the uncertain issue can be addressed by incorporating neural network/fuzzy system or sliding model control techniques. However, the transient control performances of the control system under these control schemes cannot be guaranteed as the corresponding inherent problems of computational complexity and chatting phenomenon cannot be avoided.

To tackle the above issues, this paper investigates the stabilization control problem of TORA systems suffering from unknown lumped matched disturbances. A novel continuous robust control approach is proposed by integrating the techniques of backstepping and linear extended state observer (LESO). Firstly, the TORA dynamics with matched disturbances are transformed into a cascaded form through a series of coordinate transformations. Then, a hyperbolic tangent virtual control law is designed for the first subsystem of the cascaded model based on the backstepping design technique. After that, an integral chain error subsystem under the virtual control law is constructed, and an LESO is designed to estimate the unknown disturbances. Based on the estimation of the LESO, a nonlinear state feedback control law with a compensation term is derived subsequently. Finally, the stability of the resulting control system is proved by using strict mathematical analysis, and numerical simulations with comparisons to the existing method are conducted to demonstrate the effectiveness and superiority of the proposed method.

To sum up, the main contributions and novel features of this paper are underlined as follows.

(1)

By borrowing the idea from the backstepping methodology, a hyperbolic tangent virtual control law is designed to obtain an integral chain error subsystem, which makes the well-known LESO technique easy to implement for the control design of TORA systems.

(2)

By employing the LESO, the unknown lumped matched disturbances are accurately estimated and timely compensated by augmenting the feedback controller with the estimate of disturbances, which guarantees the controller a strong robustness against disturbances.

(3)

Unlike some existing intelligent or sliding model control methods, the developed controller can generate robust and continuous control inputs to stabilize the rotation and translation of TORA systems smoothly and efficiently, without any learning mechanisms.

The remainder of this paper is organized as follows. The dynamics of an underactuated TORA system and the corresponding control problem are presented in Section 2. In Section 3, the detailed designs of the control approach, including a virtual control law, a LESO, and a continuous nonlinear feedback control law are described. The stability analysis of the closed-loop system is given in Section 4. Simulation results with comparisons are shown in Section 5. The main concluding remarks are ended in Section 6.

2. Control Problem Formulation

This paper focuses on the stabilization control problem of an underactuated horizontal TORA system subject to unknown matched disturbances. The physical structure of the TORA system is shown in Figure 1, and the physical parameters are given in

Table 1. Physical parameters and variables of the TORA system.

Parameters/Variables	Meaning	Unit
M	Mass of the translational trolley	$\mathrm{kg}$
m	Mass of the rotational ball	$\mathrm{kg}$
k	Stiffness coefficient of the spring	$\mathrm{N}/\mathrm{m}$
r	Rotational radius of the ball	$\mathrm{m}$
x	Translational displacement of the trolley	$\mathrm{m}$
$\theta$	Rotational angle of the ball with respect to the vertical position	$\mathrm{rad}$
J	Moment of inertia of the ball	$\mathrm{kg}·{\mathrm{m}}^2$
$\tau$	Control torque applied on the ball	$\mathrm{N}·\mathrm{m}$

.

According to Euler–Lagrange modeling method, the dynamic equations of the TORA system are mathematically obtained as [18]

$$\left\{\begin{array}{c}(M+m)\ddot{x}+mr\ddot{\theta }cos\theta -mr{\dot{\theta }}^{2}sin\theta +kx=0\hfill \\ mr\ddot{x}cos\theta +(m{r}^2+J)\ddot{\theta }=\tau +d\hfill \end{array}\right.$$

(1)

where d denotes the sum of matched unknown disturbances. In practice, it is mainly determined by the friction term and the bounded external disturbances.

To facilitate the controller design and stability analysis, the following dimensionless auxiliary variables are introduced [17]:

$$\begin{array}{cc}\hfill \chi & =\sqrt{\frac{M+m}{m{r}^2+J}}x,u=\frac{M+m}{k(m{r}^2+J)}\tau ,\epsilon =\frac{mr}{\sqrt{\left(m{r}^2+J\right)\left(M+m\right)}}\hfill \\ \hfill T& =\sqrt{\frac{k}{M+m}}t,d_{\tau }=\frac{M+m}{k(m{r}^2+J)}d\hfill \end{array}$$

(2)

where T represents the dimensionless time, $\chi$ is the dimensionless trolley displacement, u is the dimensionless control torque, and $d_{\tau }$ is the dimensionless lumped disturbance. For the simplicity of notation, unless otherwise specified, the expression of “time” herein stands for “the dimensionless time”.

According to the introduced dimensionless variables in (2), the dynamics (1) can be rewritten as

$$\left\{\begin{array}{c}\ddot{\chi }+\chi +\epsilon (\ddot{\theta }cos\theta -{\dot{\theta }}^{2}sin\theta )=0\hfill \\ \ddot{\theta }+\epsilon \ddot{\chi }cos\theta =u+d_{\tau }\hfill \end{array}\right.$$

(3)

Define the following variable transformations [35]:

$${\xi }_1=\chi +\epsilon sin\theta ,{\xi }_2=\dot{\chi }+\epsilon \dot{\theta }cos\theta ,y_1=\theta ,y_2=\dot{\theta }$$

(4)

then the dynamic Equation (3) is rearranged as the following cascaded form:

$$\left\{\begin{array}{cc}\hfill {\dot{\xi }}_1& ={\xi }_2\hfill \\ \hfill {\dot{\xi }}_2& =-{\xi }_1+\epsilon sin{y}_1\hfill \end{array}\right.$$

(5)

$$\left\{\begin{array}{cc}\hfill {\dot{y}}_1& =y_2\hfill \\ \hfill {\dot{y}}_2& =\frac{{\delta }_1+u+d_{\tau }}{{\delta }_2}\hfill \end{array}\right.$$

(6)

where

$$\begin{array}{cc}\hfill & {\delta }_1=\epsilon cos{y}_1[{\xi }_1-(1+{y_2}^2)\epsilon sin{y}_1]\hfill \\ \hfill & {\delta }_2=1-{\epsilon }^2{cos}^2{y}_1\hfill \end{array}$$

(7)

Note that $0<\epsilon <1$, which indicates that ${\delta }_2>0$.

Therefore, the control problem of this paper is formulated as: Consider the TORA system described by Equations (5) and (6) in the presence of unknown matched disturbances. Design a proper controller u such that the TORA system is stabilized at the equilibrium point, that is,

$$\underset{T\to \infty }{lim}{\left[\begin{array}{cccc}{\xi }_1& {\xi }_2& y_1& y_2\end{array}\right]}^{\top }={\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]}^{\top }$$

(8)

which, applying the transformations (2) and (4), is equivalent to

$$\underset{T\to \infty }{lim}{\left[\begin{array}{cccc}\chi & \dot{\chi }& \theta & \dot{\theta }\end{array}\right]}^{\top }={\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]}^{\top }$$

(9)

and

$$\underset{T\to \infty }{lim}{\left[\begin{array}{cccc}x& \dot{x}& \theta & \dot{\theta }\end{array}\right]}^{\top }={\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]}^{\top }$$

(10)

3. LESO-Based Backstepping Controller Design

In this section, to achieve the above control objective, a virtual control law is firstly designed for the subsystem (5) based on the methodology of backstepping, and then a linear extended state observer (LESO) is employed to estimate the unknown disturbances based on which a state feedback control law is finally developed.

3.1. Virtual Control Law

Motivated by the control idea of backstepping methodology, the variable $y_1$ can be considered as the control input of subsystem (5), and a virtual control law defined as $y_{1d}$ is designed to stabilize ${\xi }_1$ and ${\xi }_2$. To this end, the following control Lyapunov function is constructed:

$$V_1=\frac{1}{2}{{\xi }_1}^2+\frac{1}{2}{{\xi }_2}^2$$

(11)

Taking the time derivative of (11) along (5) yields

$${\dot{V}}_1={\xi }_1{\dot{\xi }}_1+{\xi }_2{\dot{\xi }}_2={\xi }_1{\xi }_2+{\xi }_2\left(-{\xi }_1+\epsilon sin{y}_{1d}\right)={\xi }_2\epsilon sin{y}_{1d}$$

(12)

In order to make ${\dot{V}}_1$ negative, the virtual control law $y_{1d}$ is designed as

$$y_{1d}=-tanh\left(\alpha {\xi }_2\right)$$

(13)

where $\alpha$ is a positive constant. The deviation between $y_1$ and $y_{1d}$ is defined as

$$e_1=y_1-y_{1d}=y_1+tanh\left(\alpha {\xi }_2\right)$$

(14)

Calculating the first and second-time derivatives of (14) yields

$$\left\{\begin{array}{cc}\hfill {\dot{e}}_1& =e_2\hfill \\ \hfill {\ddot{e}}_1& =\frac{{\delta }_1+u+d_{\tau }}{{\delta }_2}-{\ddot{y}}_{1d}\hfill \end{array}\right.$$

(15)

where ${\dot{y}}_{1d}$ and ${\ddot{y}}_{1d}$ can be expressed explicitly as

$$\begin{array}{cc}\hfill {\dot{y}}_{1d}=& \alpha \left({\xi }_1-\epsilon sin{y}_1\right)\left(1-{tanh}^2\left(\alpha {\xi }_2\right)\right)\hfill \\ \hfill {\ddot{y}}_{1d}=& \left(\alpha \left({\xi }_2-\epsilon y_{2}cos{y}_1\right)-2{\alpha }^2\left({\xi }_1-tanh\left(\alpha {\xi }_2\right)\right)\right)\left(1-{tanh}^2\left(\alpha {\xi }_2\right)\right)\hfill \end{array}$$

(16)

Letting ${\left[e_2{\dot{e}}_2\right]}^{\top }={\left[{\dot{e}}_1{\ddot{e}}_1\right]}^{\top }$ and using (14)–(16), the cascaded model (5) and (6) of the TORA system can be written as

$$\left\{\begin{array}{c}{\dot{\xi }}_1={\xi }_2\hfill \\ {\dot{\xi }}_2=-{\xi }_1+\epsilon sin(e_1+y_{1d})\hfill \end{array}\right.$$

(17)

and

$$\left\{\begin{array}{cc}\hfill {\dot{e}}_1& =e_2\hfill \\ \hfill {\dot{e}}_2& =\frac{{\delta }_1+u+d_{\tau }}{{\delta }_2}-{\ddot{y}}_{1d}\hfill \end{array}\right.$$

(18)

Remark 1.

Note that by designing the virtual control law (13), a simple integral chain error subsystem is obtained as (18). It will prove in Section 4 that if a proper controller u is designed such that the error subsystem (18) is stabilized at the origin, then the subsystem (17) is also stabilized at the origin; that is, the control objective is realized. However, there exists an unknown matched disturbance $d_{\tau }$ in Equation (18), which makes the controller design quite difficult. To address this problem, a linear extended state observer (LESO) is employed to estimate the unknown disturbance in real-time and the estimation value is fed back to the controller to compensate for its effect. This is the basic idea of the proposed control method.

3.2. Linear Extended State Observer (LESO)

As a crucial part of the active disturbance rejection control (ADRC), extended state observer (ESO) is an effective and practical disturbance estimation and attenuation approach [37,38]. To make the controller easy to implement, a linear ESO (LESO) is proposed in [39], where nonlinear gains are replaced with linear ones. Due to this promising feature, LESO has already been widely applied in many engineering control systems.

To implement the LESO technique, the subsystem (18) is further rearranged as

$$\left\{\begin{array}{cc}\hfill {\dot{e}}_1& =e_2\hfill \\ \hfill {\dot{e}}_2& =\frac{{\delta }_1}{{\delta }_2}+\frac{1}{{\delta }_2}u-{\ddot{y}}_{1d}+d_n\hfill \\ \hfill d_n& =\frac{d_{\tau }}{{\delta }_2}\hfill \end{array}\right.$$

(19)

Letting ${\left[z_1{z}_2{z}_3\right]}^{\top }={\left[e_1{e}_2{d}_n\right]}^{\top }$, Equation (19) is written as

$$\left\{\begin{array}{cc}\hfill {\dot{z}}_1& =z_2\hfill \\ \hfill {\dot{z}}_2& =\frac{{\delta }_1}{{\delta }_2}+\frac{1}{{\delta }_2}u-{\ddot{y}}_{1d}+z_3\hfill \\ \hfill {\dot{z}}_3& =h\hfill \end{array}\right.$$

(20)

where h is the change rate of the lumped disturbance, i.e., $h={\dot{d}}_n$ and it is assumed to be an unknown but bounded function. Then, the LESO of the system (20) is designed as

$$\left\{\begin{array}{cc}\hfill e_{s1}& ={\widehat{z}}_1-z_1\hfill \\ \hfill {\dot{\widehat{z}}}_1& ={\widehat{z}}_2-{\beta }_1{e}_{s1}\hfill \\ \hfill {\dot{\widehat{z}}}_2& ={\widehat{z}}_3-{\beta }_2{e}_{s1}+\frac{{\delta }_1}{{\delta }_2}+\frac{u}{{\delta }_2}-{\ddot{y}}_{1d}\hfill \\ \hfill {\dot{\widehat{z}}}_3& =-{\beta }_3{e}_{s1}\hfill \end{array}\right.$$

(21)

where ${\widehat{z}}_i\left(i=1,2,3\right)$ are the estimate values of $z_i$, and the quantities ${\beta }_i\left(i=1,2,3\right)$ are the observer gains, which, for the sake of stability, should be chosen such that

$$s^3+{\beta }_1{s}^2+{\beta }_{2}s+{\beta }_3={(s+{\omega }_o)}^3$$

(22)

where ${\omega }_0>0$ is the adjustable error feedback gain, which is also called the bandwidth of the LESO. From (22), the gains of the LESO are selected as

$${\left[\begin{array}{ccc}{\beta }_1& {\beta }_2& {\beta }_3\end{array}\right]}^{\top }={\left[\begin{array}{ccc}3{\omega }_o& 3{\omega }_o^2& {\omega }_o^3\end{array}\right]}^{\top }$$

(23)

Combining (20) and (21), the error dynamics of the LESO is obtained as

$$\left\{\begin{array}{c}{\dot{e}}_{s1}=e_{s2}-{\beta }_1{e}_{s1}\hfill \\ {\dot{e}}_{s2}=e_{s3}-{\beta }_2{e}_{s1}\hfill \\ {\dot{e}}_{s3}=-h-{\beta }_3{e}_{s1}\hfill \end{array}\right.$$

(24)

where $e_{si}={\widehat{z}}_i-z_i\left(i=1,2,3\right)$ are the estimation errors.

3.3. Continuous Nonlinear Feedback Control Law

Based on the virtual control law (13) and the LESO (21), we are ready to design a feedback control law to stabilize the subsystem (18), which is equivalent to (20). To this end, the following control Lyapunov function is constructed:

$$V_2=\frac{1}{2}{z_1}^2+\frac{1}{2}{z_2}^2$$

(25)

Taking the time derivative of (25) along (20) yields

$${\dot{V}}_2=z_1{z}_2+z_2{\dot{z}}_2=z_2\left[z_1+\frac{{\delta }_1}{{\delta }_2}+\frac{1}{{\delta }_2}u-{\ddot{y}}_{1d}+z_3\right]$$

(26)

To make ${\dot{V}}_2$ negative, a nonlinear state feedback control law is designed as

$$u=-\left(k_1{z}_2+z_1+{\widehat{z}}_3+\frac{{\delta }_1}{{\delta }_2}-{\ddot{y}}_{1d}\right){\delta }_2$$

(27)

where $k_1>0$ is the control gain.

So far, the designed controller is made up of the virtual control law (13), the LESO (21), and the nonlinear feedback control law (27). The block diagram of the control system is shown in Figure 2.

Remark 2.

It can be seen from Equation (27) and Figure 2 that the structure of the proposed controller, compared with other methods like [17,35], is relatively simple, which only needs the state feedback signals $z_1,z_2$, the estimate ${\widehat{z}}_3$, and the virtual control signal ${\ddot{y}}_d$. The estimate signal ${\widehat{z}}_3$ comes from the LESO, which is used to timely compensate for the effect of the unknown disturbance $d_{\tau }$. In addition, unlike the methods using sliding mode control (SMC), where the inherent chattering phenomenon cannot be eliminated, the proposed controller can generate a continuous control signal, which would achieve a smooth and robust control performance. This merit will be further verified through comparison simulation studies in Section 5.

4. Stability Analysis

In this section, the convergence of the estimation errors of the proposed LESO (21) is first analyzed from the error dynamics (25) using Lyapunov stability theory. Then, the stability of the resulting TORA control system (5) and (6) under the virtual control law (13), the feedback control law (27) and the LESO (21) is rigorously proved by using LaSalle’s invariance principle.

To facilitate the stability analysis, the following assumptions are made for the TORA system.

Assumption A1

([35]). The first order derivatives of the unknown matched disturbances $d_{\tau }$ and $d_n$ are bounded, i.e.,

$$∥{\dot{d}}_{\tau }∥⩽{\sigma }_p,∥h∥=∥{\dot{d}}_n∥⩽{\sigma }_{es}$$

(28)

where ${\sigma }_p$ and ${\sigma }_{es}$ are positive constants.

Theorem 1.

Consider the proposed LESO (21) for subsystem (20), if Assumption 1 holds, then the estimation error of the LESO is bounded, and satisfies

$$\underset{{\omega }_o\to \infty ,T\to \infty }{lim}∥e_s∥=0$$

(29)

where $e_s={\left[\begin{array}{ccc}{e}_{s1}& e_{s2}& e_{s3}\end{array}\right]}^{\top }$.

Proof. 

Letting ${\eta }_i=\frac{e_{si}}{{\omega }_o^i}(i=1,2,3)$, Equation (24) is written as

$$\left\{\begin{array}{cc}\hfill {\dot{\eta }}_1& ={\omega }_o\left({\eta }_2-3{\eta }_1\right)\hfill \\ \hfill {\dot{\eta }}_2& ={\omega }_o\left({\eta }_3-3{\eta }_1\right)\hfill \\ \hfill {\dot{\eta }}_3& ={\omega }_o\left(-\frac{h}{{\omega }_o^4}-{\eta }_1\right)\hfill \end{array}\right.$$

(30)

Rewriting Equation (30) into a compact form obtains

$$\dot{\eta }={\omega }_o{A}_{\eta }\eta +B_{\eta }\frac{h}{{\omega }_o^3}$$

(31)

where $\eta ={\left[\begin{array}{ccc}{\eta }_1& {\eta }_2& \eta {}_3\end{array}\right]}^{\top }$,

$$A_{\eta }=\left[\begin{array}{ccc}-3& 1& 0\\ -3& 0& 1\\ -1& 0& 0\end{array}\right],B_{\eta }=\left[\begin{array}{c}0\\ 0\\ -1\end{array}\right]$$

(32)

It can be seen from Equations (22) and (31) that, for any ${\omega }_o>0$, $A_{\eta }$ is obtained as (32), which is a Hurwitz matrix, thus select a Lyapunov function as

$$V\left(\eta \right)={\eta }^{\top }{P}_{\eta }\eta$$

(33)

where $P_{\eta }$ is a positive definite symmetric matrix and satisfies ${A_{\eta }}^{\top }{P}_{\eta }+P_{\eta }{A}_{\eta }=-Q_{\eta }$. Taking the time derivative of $V\left(\eta \right)$ along (32) and using Assumption 1 yields

$$\dot{V}\left(\eta \right)=-{\omega }_o{\eta }^{\top }{Q}_{\eta }\eta +2{\eta }^{\top }{P}_{\eta }{B}_{\eta }\frac{h}{{\omega }_o^3}⩽-{\omega }_o{\lambda }_{min}\left(Q_{\eta }\right){∥\eta ∥}^2+\frac{2{\sigma }_{es}{\lambda }_{max}\left(P_{\eta }\right)∥\eta ∥}{{\omega }_o^3}$$

(34)

For $V\left(\eta \right)$, there exists ${\lambda }_{min}\left(P_{\eta }\right){∥\eta ∥}^2⩽V\left(\eta \right)⩽{\lambda }_{max}\left(P_{\eta }\right){∥\eta ∥}^2$, i.e.,

$$\frac{V\left(\eta \right)}{{\lambda }_{max}\left(P_{\eta }\right)}⩽{∥\eta ∥}^2⩽\frac{V\left(\eta \right)}{{\lambda }_{min}\left(P_{\eta }\right)}$$

(35)

Substituting (35) into (34) yields

$$\dot{V}\left(\eta \right)⩽-{\omega }_o\frac{{\lambda }_{min}\left(Q_{\eta }\right)}{{\lambda }_{max}\left(P_{\eta }\right)}V\left(\eta \right)+\frac{2{\sigma }_{es}{\lambda }_{max}\left(P_{\eta }\right)}{{\omega }_o^3\sqrt{{\lambda }_{min}\left(P_{\eta }\right)}}\sqrt{V\left(\eta \right)}$$

(36)

Letting $W=\sqrt{V\left(\eta \right)}$, it obtains that $\dot{W}=\frac{\dot{V}\left(\eta \right)}{2\sqrt{V\left(\eta \right)}}$. Substituting it into (36) yields

$$\dot{W}⩽-{\omega }_o\frac{{\lambda }_{min}\left(Q_{\eta }\right)}{2{\lambda }_{max}\left(P_{\eta }\right)}W+\frac{{\sigma }_{es}{\lambda }_{max}\left(P_{\eta }\right)}{{\omega }_o^3\sqrt{{\lambda }_{min}\left(P_{\eta }\right)}}$$

(37)

According to Gronwall–Bellman inequality [40], it follows from (37) that

$$\begin{array}{cc}\hfill W⩽& -\left(\frac{2{\sigma }_{es}{\lambda }_{max}^2\left(P_{\eta }\right)}{{\omega }_o^4\sqrt{{\lambda }_{min}\left(P_{\eta }\right)}{\lambda }_{min}\left(Q_{\eta }\right)}-W\left(T_0\right)\right)·e^{-{\omega }_o\frac{{\lambda }_{min}\left(Q_{\eta }\right)}{2{\lambda }_{max}\left(P_{\eta }\right)}\left(T-T_0\right)}\hfill \\ \hfill & +\frac{2{\sigma }_{es}{\lambda }_{max}^2\left(P_{\eta }\right)}{{\omega }_o^4\sqrt{{\lambda }_{min}\left(P_{\eta }\right)}{\lambda }_{min}\left(Q_{\eta }\right)}\hfill \end{array}$$

(38)

Combining (35) and (38), it is then obtained that

$$∥\eta ∥⩽\frac{\sqrt{V\left(\eta \right)}}{\sqrt{{\lambda }_{min}\left(P_{\eta }\right)}}⩽\frac{2{\sigma }_{es}{\lambda }_{max}^2\left(P_{\eta }\right)}{{\omega }_o^4{\lambda }_{min}\left(P_{\eta }\right){\lambda }_{min}\left(Q_{\eta }\right)}=\frac{M_e}{{\omega }_o^4},(T\to \infty )$$

(39)

where $M_e=\frac{2{\sigma }_{es}{\lambda }_{max}^2\left(P_{\eta }\right)}{{\lambda }_{min}\left(P_{\eta }\right){\lambda }_{min}\left(Q_{\eta }\right)}>0$. Since $P_{\eta }$ and $Q_{\eta }$ are both irrelevant to ${\omega }_o$, it follows from (39) that

$$\underset{{\omega }_o\to \infty ,T\to \infty }{lim}∥\eta ∥=0$$

(40)

By noticing ${\eta }_i=\frac{e_{si}}{{\omega }_{oi}}\left(i=1,2,3\right)$, it obtains

$$\underset{{\omega }_o\to \infty ,T\to \infty }{lim}∥e_s∥=0$$

(41)

which means that the estimation error of the LESO can be made arbitrarily small by increasing the value of ${\omega }_o$. This completes the proof. □

Theorem 2.

Consider the cascade TORA system (5) and (6) in the presence of unknown lumped disturbance $d_{\tau }$. Design the controller as (27) with (13) and the LESO as (21). If the control gain is selected satisfying $k_1>\frac{1}{2}$, then all the state variables remain bounded, and the system is stabilized at the target position, i.e.,

$$\underset{t\to \infty }{lim}{\left[\begin{array}{cccc}{\xi }_1& {\xi }_2& y_1& y_2\end{array}\right]}^{\top }={\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]}^{\top }$$

(42)

Proof. 

First of all, the stability of system (20) is proven. Substituting (27) into (26) yields

$${\dot{V}}_2=-k_1{z_2}^2+z_2\left(z_3-{\widehat{z}}_3\right)=-k_1{z_2}^2+z_2{e}_{s3}$$

(43)

Using Young’s inequality [41], it obtains

$${\dot{V}}_2⩽-k_1{z_2}^2+\frac{1}{2}{z_2}^2+\frac{1}{2}{e_{s3}}^2=-\left(k_1-\frac{1}{2}\right){z_2}^2+\frac{1}{2}{e_{s3}}^2$$

(44)

According to Theorem 1, noting that $\underset{T\to \infty }{lim}∥e_s∥=0$, if the control gain is selected satisfying $k_1>\frac{1}{2}$, then ${\dot{V}}_2⩽0$, which leads to $\underset{T\to \infty }{lim}{z}_2=0$. Employing LaSalle’s invariance principle [40], it is straightforward to obtain

$$\underset{T\to \infty }{lim}{\left[z_1{z}_2\right]}^{\top }={\left[00\right]}^{\top }\Rightarrow \underset{T\to \infty }{lim}{\left[e_1{e}_2\right]}^{\top }={\left[00\right]}^{\top }$$

(45)

Then, the stability of the subsystem (17) is analyzed. Applying (45), it assumes that $e_1=0$, then the subsystem (17) becomes

$$\left\{\begin{array}{c}{\dot{\xi }}_1={\xi }_2\hfill \\ {\dot{\xi }}_2=-{\xi }_1+\epsilon sin{y}_{1d}\hfill \end{array}\right.$$

(46)

Note that the control Lyapunov function defined in (11) is

$$V_1=\frac{1}{2}{{\xi }_1}^2+\frac{1}{2}{{\xi }_2}^2$$

(47)

Taking the time derivative of (47) along (46) and using (13) yields

$${\dot{V}}_1=-{\xi }_2\epsilon sin\left[tanh\left(\alpha {\xi }_2\right)\right]\le 0$$

(48)

where $\epsilon >0,\alpha >0$, and the following properties of a generalized hyperbolic tangent function $y=tanh\left(s\right)$ have been utilized.

$$\left\{\begin{array}{c}y\in (-1,1)\hfill \\ tanh\left(0\right)=0\hfill \\ s·tanh\left(s\right)\ge 0\hfill \end{array}\right.$$

(49)

It follows from (48) that the closed-loop subsystem (46) under (13) is stable in the sense of Lyapunov, and all the state variables are bounded, i.e.,

$$V_1\le V_1\left(0\right)\in {\mathcal{L}}_{\infty }\Rightarrow {\xi }_1,{\xi }_2,y_{1d}\in {\mathcal{L}}_{\infty }$$

(50)

However, there is no guarantee that the state variables converge to zeros as $V_1$ is negative semi-definite.

To further prove the asymptotic convergence of the state variables, define the following invariant set

$$\Omega =\left\{\left({\xi }_1,{\xi }_2\right)\left|V_1\le V_1\left(0\right)\right.\right\}$$

(51)

Let ${\Omega }_M$ be the largest invariant set contained in $\Omega$:

$${\Omega }_M=\left\{\left({\xi }_1,{\xi }_2\right)\left|{\dot{V}}_1=0\right.\right\}$$

(52)

When ${\dot{V}}_1=0$, it follows from (48) that

$${\xi }_2=0\Rightarrow {\dot{\xi }}_2=0$$

(53)

Further, substituting (53) into (13) and (46) yields

$${\dot{\xi }}_1=0,y_{1d}=0\Rightarrow {\xi }_1=0$$

(54)

As the set ${\Omega }_M$ is only made up of one point, according to LaSalle’s invariance theorem, it concludes that the subsystem (17) is globally asymptotically stable at the origin, i.e.,

$$\underset{T\to \infty }{lim}{\left[\begin{array}{cc}{\xi }_1& {\xi }_2\end{array}\right]}^{\top }={\left[\begin{array}{cc}0& 0\end{array}\right]}^{\top }$$

(55)

Finally, it needs to analyze the convergence of the entire cascade system consisting of (17) and (18). Based on the above stability analysis, it is known that both the subsystem (17) and the subsystem (18) are globally asymptotically stable at the origin. Moreover, the right side of (17) is globally Lipschitz and bounded. By invoking Theorem 6.2 in [42], it concludes that the entire system is asymptotically stable at the equilibrium point, i.e.,

$$\underset{T\to \infty }{lim}{\left[\begin{array}{cccc}{\xi }_1& {\xi }_2& e_1& e_2\end{array}\right]}^{\top }={\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]}^{\top }$$

(56)

which, by utilizing (14), is equivalent to

$$\underset{T\to \infty }{lim}{\left[\begin{array}{cccc}{\xi }_1& {\xi }_2& y_1& y_2\end{array}\right]}^{\top }={\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]}^{\top }$$

(57)

Therefore, the control objective of the TORA system is realized and the proof of the Theorem 2 is completed. □

5. Simulation Results

In this section, the effectiveness of the proposed control scheme is validated by numerical simulations in the MATLAB/Simulink platform. Moreover, the superior control performance of the devised controller is discussed by a series of comparison results with the sliding mode (SMC) controller proposed in [35].

During the simulation, the system parameter of the dimensionless model (3) is set as $\epsilon =0.2$. By trial and error, the gains of the proposed controller in (13) and (27) are chosen as $\alpha =2.4,k_1=3$. The bandwidth of the LESO in (21) is ${\omega }_0=10$, and the gains of the LESO are obtained from (23) as ${\beta }_1=30,{\beta }_2=300,{\beta }_3=1000$. To ensure fair comparison, the control parameters of the comparative controller are chosen as the same as [35].

5.1. Case 1: Robustness to Continuous Disturbance

First of all, without loss of generality, the following continuous disturbance is imposed on the system, and the initial state of the system is set to be ${\left[\begin{array}{cccc}\chi & \dot{\chi }& \theta & \dot{\theta }\end{array}\right]}^{\top }={\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]}^{\top }$.

$$d_{\tau }=\frac{10sint}{\left(10+0.5t^2\right)}$$

(58)

The simulation results of the TORA system under the two controllers are depicted in Figure 3, Figure 4 and Figure 5, which record the curves of the dimensionless trolley displacement, the ball rotational angle, and the dimensionless control input, respectively.

From Figure 3, Figure 4 and Figure 5, it is observed that the trolley and the ball of the TORA system under the two controllers are stabilized at the equilibrium point. However, from Figure 3 and Figure 4, we find that the settling time the proposed method, which is about 27, is much shorter than that (about 33) of the SMC controller in [35]. In addition, from Figure 5, it can be seen that there exists a serious chattering phenomenon in the SMC controller. In contrast, the controller proposed in this paper generates a continuous and smooth control input signal. These results demonstrate that the proposed control scheme achieves a faster and smoother control performance in the presence of continuous disturbance.

5.2. Case 2: Robustness Test to Different Disturbances

To further examine the robustness of the controllers to different disturbances, two different disturbances are imposed on the TORA system. The first one is a pulse with an amplitude of 3, which is added between the interval 6 and $6.2$. Another one is a sinusoid disturbance with an amplitude of 3, which is added from 20 to 30. The initial state of the system is selected as ${\left[\begin{array}{cccc}\chi & \dot{\chi }& \theta & \dot{\theta }\end{array}\right]}^{\top }={\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]}^{\top }$.

The simulation results of the TORA system under the two controllers are shown in Figure 6, Figure 7 and Figure 8. It can be seen from Figure 6 and Figure 7 that the states of the TORA system are affected by the external disturbances and deviate from the equilibrium position. However, soon, both the two controllers can attenuate the disturbances and be re-stabilize the system at the equilibrium point. However, by comparing the results in Figure 6 and Figure 7, it is found that the resettling time and the deviation magnitudes of the trolley displacement and the ball rotational angle with the proposed controller are smaller than those of the SMC method in [35]. In addition, from Figure 8, it is also seen that there exists a serious chattering phenomenon in the SMC controller, while the control input of the proposed controller is continuous and smooth. This group of simulation results demonstrates that the proposed control scheme achieves stronger and smoother robustness to external disturbances.

5.3. Estimation Performance of LESO

Finally, the estimation performance of the proposed LESO to the disturbances in the previous two cases are examined. To facilitate the analysis, the estimation error of the LESO is redefined as

$${\tilde{d}}_n=d_n-{\widehat{d}}_n$$

(59)

The obtained results of the two groups are shown in Figure 9 and Figure 10, respectively. From these figures, it can be observed that as the time approaches infinity, the estimation errors approach zeros, and the proposed LESO can accurately estimate the uncertain disturbances. These simulation results demonstrate that the proposed LESO achieves a satisfactory estimation performance.

It should be noted that the parameters of the LESO are selected by regulating the bandwidth $w_o$ according to (23). Generally speaking, increasing the bandwidth $w_o$ would obtain a fast and good observation ability of the LESO but will result in a peaking phenomenon or serious oscillation. Therefore, a good $w_o$ should be carefully selected by trial and error.

6. Conclusions

This paper has proposed a nonlinear continuous robust control approach for the stabilization of underactuated TORA systems with unknown matched disturbances. The proposed control approach consists of backstepping virtual control law, a LESO, and a nonlinear state feedback control law. The proposed control algorithm can estimate and compensate the unknown disturbances in real-time, ensuring strong robustness to disturbances. In addition, it is capable of generating continuous control signals. The convergence and stability of the entire control system are rigorously guaranteed by Lyapunov theory and LaSalle’s invariance principle. Simulation results with comparisons to the existing method show that the proposed approach achieves a better control performance, including shorter settling time, and smoother and stronger robustness to unknown disturbances.

It is worth noticing that only matched disturbances are taken in account in this paper. As stated in [17], for underactuated TORA systems with both matched and mismatched disturbances, there does not exist a feedback controller that can asymptotically stabilize all the state variables at the origin simultaneously. Therefore, the stabilization of TORA systems in the presence of matched and mismatched disturbances is still very challenging and needs to be further investigated in our future work.