Sliding Mode Control with a Prescribed-Time Disturbance Observer for Bridge Crane Positioning and Anti-Swing

Abstract

To address the issue of reduced positioning and anti-swing accuracy of bridge cranes under disturbed conditions within a prescribed time, a positioning and anti-swing control algorithm, based on a prescribed-time disturbance observer, is proposed. Unlike existing research, the novel disturbance observer is designed to accurately estimate disturbances within a prescribed time, ensuring precise disturbance compensation. This allows for high-precision positioning and anti-swing of bridge cranes under disturbed conditions within a prescribed time. Firstly, a prescribed-time disturbance observer is designed to ensure accurate disturbance estimation. Secondly, a new prescribed-time sliding mode surface and a prescribed-time reaching law with a recursive structure are designed to ensure that the system state converges accurately within the prescribed time. Finally, theoretical analysis and simulation verify that the proposed control algorithm achieves the control objective of high-precision positioning and anti-swing of bridge cranes under disturbed conditions within a prescribed time.

1. Introduction

Bridge cranes are widely used in ports and construction sites for material handling due to their excellent material transfer capabilities [1]. With the rapid development of the economy, there are higher demands for the transfer efficiency and precision of bridge cranes. Current manual operation of bridge cranes faces issues such as low efficiency, high labor costs, and low safety. Most existing control algorithms for material transfer with bridge cranes can only achieve asymptotic convergence or finite-time convergence. As the initial system error increases, the positioning and anti-swing time also increases, leading to a decrease in the material transfer efficiency of bridge cranes. Furthermore, the actual working environment of bridge cranes includes various disturbances, such as friction during trolley movement and material torque, which reduce the positioning and anti-swing accuracy. In summary, to reduce labor costs, lower the incidence of accidents caused by improper manual operation, and improve the working efficiency and transfer precision of bridge cranes, there is an urgent need to propose a control algorithm for high-precision positioning and anti-swing of bridge cranes under disturbed conditions within a prescribed time.

Up to the present, researchers have proposed numerous control algorithms addressing the positioning and anti-swing issues during the material transfer process of underactuated bridge cranes [2,3,4,5]. For bridge crane control systems, open-loop control methods do not require the measurement of system state variables such as the swing angle of the material, making them easy to implement and cost-effective. Therefore, to simplify the design complexity of control algorithms and minimize costs, early approaches to the positioning and anti-swing problems of bridge crane systems predominantly utilized open-loop control methods [6,7,8]. Although input shaping and trajectory planning can achieve trolley positioning with minimal material swing without sensors, their control performance deteriorates under external disturbances. Consequently, researchers have conducted a series of studies on closed-loop control algorithms to address the positioning and anti-swing problems of bridge cranes. Sliding mode control is a closed-loop control method with switching characteristics. It introduces system states into the designed sliding mode surface through a switching term, causing the system states to converge to the equilibrium point along the sliding mode surface. Due to its strong anti-disturbance capabilities, SMC is widely used to address the positioning and anti-swing problems of bridge cranes. Sun et al. proposed a sliding mode control algorithm that achieves the control objectives of positioning and anti-swing for bridge cranes under disturbed conditions [9]. Furthermore, considering the chattering issue inherent to sliding mode control, Lu et al. designed a sliding mode control algorithm based on a disturbance observer. This approach not only achieves the control objectives of positioning and anti-swing for bridge cranes under disturbed conditions but also reduces the amplitude of chattering [10,11]. Although these control algorithms effectively solve the positioning and anti-swing issues during the operation of bridge crane systems, they only achieve asymptotic positioning and anti-swing, resulting in lower operational efficiency of the bridge cranes.

Furthermore, considering the positioning and anti-swing time for bridge cranes, Wang et al. designed a finite-time sliding mode control algorithm under varying rope lengths, achieving finite-time positioning and anti-swing for bridge cranes [12]. For material transfer using bridge cranes under disturbed conditions, Wang et al. developed a terminal sliding mode control algorithm that realizes finite-time positioning and anti-swing of the material [13]. Yang et al. used fuzzy techniques to linearize the bridge crane system, creating a fuzzy bridge crane model with appropriate membership functions. By combining finite-time stability theory and sliding mode control, they achieved finite-time positioning and anti-swing [14]. Zhang et al. designed a sliding mode observer to estimate disturbances and proposed a finite-time trajectory tracking control algorithm based on the observer [15]. For the trajectory tracking problem of bridge crane systems with uncertain parameters, Wang et al. integrated adaptive techniques and proposed a non-singular terminal sliding mode control algorithm [16]. Liang Huihui et al. introduced a proportional–integral sliding mode control algorithm, designing a finite-time reaching law to ensure that the sliding mode surface converges to zero within finite time [17]. Vazquez et al. proposed a sliding mode control algorithm based on the super-twisting algorithm, ensuring finite-time convergence of the sliding mode surface [18]. Chwa et al. addressed the three-dimensional positioning and anti-swing problem of bridge cranes by combining sliding mode control and proposing a robust finite-time anti-swing tracking control method with excellent positioning and anti-swing performance [19]. Although the above method can complete the material transfer within a finite time, the material transfer time of the bridge crane is affected by the initial state of the system; moreover, with the increase in the initial error, the material transfer time also increases, which reduces the transfer efficiency of the material. Fixed time can effectively solve this problem, therefore, Wu et al., using fixed-time stability theory, combined backstepping control to construct the Lyapunov function for each lower-order subsystem and designed the virtual control for each system order using dynamic surface technology with fractional-order filters, achieving fixed-time positioning and anti-swing for bridge crane systems [20]. Yang et al., considering the unknown upper bounds of disturbances, designed a sliding mode control with a disturbance observer based on a double-layer adaptive law. This approach estimated the upper bound of disturbances via the adaptive law and used the disturbance observer to estimate and compensate for disturbances, achieving fixed-time positioning and anti-swing for bridge cranes [21].

Although the aforementioned methods can achieve the control objectives of positioning and anti-swing for bridge cranes, their accuracy significantly decreases when the bridge crane is subjected to large disturbances. Therefore, it is necessary to design a control algorithm that can accurately estimate and compensate for disturbances within a prescribed time in order to achieve the control objectives of high-efficiency and high-precision material transfer for bridge cranes.

The contributions made in this article are as follows:

(1)

A novel prescribed-time disturbance observer is designed to accurately estimate unknown disturbances acting on bridge cranes within a specified time, thereby addressing the problem of reduced positioning and anti-swing accuracy caused by unknown disturbances.

(2)

A novel recursive structure for the prescribed-time sliding mode surface and prescribed-time reaching law is designed, ensuring that the system error converges to zero within the prescribed time, thus achieving prescribed-time positioning and anti-swing for bridge cranes. The proposed novel recursive structure for the prescribed-time sliding mode surface can be effectively extended to the research of prescribed-time control algorithms for other higher-order systems.

2. Problem Description

Considering the practical usage of bridge cranes, after installation in the factory, the oxidation and wear of the trolley and track surfaces over time lead to significant discrepancies between the friction model established and the actual friction, further compounded by unknown disturbances such as material torque during trolley operation. Therefore, factors such as frictional forces on the trolley and unknown disturbances are collectively attributed to total disturbances d. Based on the Lagrangian equation, the dynamic model of bridge cranes under disturbance conditions can be expressed as follows [22,23,24]:

$$\left(M+m\right)\ddot{x}+ml\ddot{\theta }\cos \theta -ml{\dot{\theta }}^2\sin \theta =F+d,$$

(1)

$$m{l}^2\ddot{\theta }+ml\ddot{x}\cos \theta +mgl\sin \theta =0.$$

(2)

The parameters of the bridge crane system dynamics models (1) and (2) are defined in

Table 1. Parameter definition.

Symbols	Physical Meaning	Unit
$M$	Trolley mass	kg
$m$	Load mass	kg
$l$	Cable length	m
$x$	Trolley position	m
$\theta$	Load swing angle	rad
$F$	Driving force	N
$g$	Gravitational acceleration	${\mathrm{m}/\mathrm{s}}^2$
$d$	Total disturbance	N

.

To facilitate the analysis, the following auxiliary variables are constructed:

$$\{\begin{array}{l}{\zeta }_1=x+l\ln (\sec \theta +\tan \theta )\\ {\zeta }_2=x+{\displaystyle \frac{l\dot{\theta }}{\cos \theta }}\\ {\zeta }_3=-g\tan \theta \\ {\zeta }_4=-g\dot{\theta }{\sec }^2\theta \end{array}.$$

(3)

Combining the bridge crane models (1) and (2) with the auxiliary variables (3) and $u=F$, the fourth-order model of the bridge crane is as follows:

$$\{\begin{array}{l}{\dot{\zeta }}_1={\zeta }_2\\ {\dot{\zeta }}_2={\zeta }_3(1-\gamma (\zeta ))\\ {\dot{\zeta }}_3={\zeta }_4\\ {\dot{\zeta }}_4=f(\theta ,\dot{\theta })+h(\theta ,\dot{\theta })u+h(\theta ,\dot{\theta })d\end{array}.$$

(4)

In Equation (4),

$$\gamma (\zeta )={\displaystyle \frac{l{\zeta }_4^2}{{({\mathrm{g}}^2+{\zeta }_3^2)}^{1.5}}};$$

$$f(\theta ,\dot{\theta })={\displaystyle \frac{\left(M+m\right)g^2\tan \theta \sec \theta +ml{\dot{\theta }}^{2}g\tan \theta }{\left(Ml+ml{\sin }^2\theta \right)}}-2g{\dot{\theta }}^2{\sec }^2\theta \tan \theta ;$$

$$h(\theta ,\dot{\theta })={\displaystyle \frac{g\sec \theta }{\left(Ml+ml{\sin }^2\theta \right)}}.$$

Due to the practical constraints in the operation of bridge cranes, where the swing angle of the material does not exceed 20 deg, the rope length does not exceed 10 m, and the angular velocity of the material is much smaller than 1 rad/s, we can deduce that $1-\gamma (\zeta )=1-l{\zeta }_4^2/{({\mathrm{g}}^2+{\zeta }_3^2)}^{1.5}\approx 1-0.0008l{\zeta }_4^2\approx 1$. Therefore, the bridge crane model (4) can be further simplified as:

$$\{\begin{array}{l}{\dot{\zeta }}_1={\zeta }_2\\ {\dot{\zeta }}_2={\zeta }_3\\ {\dot{\zeta }}_3={\zeta }_4\\ {\dot{\zeta }}_4=f(\theta ,\dot{\theta })+h(\theta ,\dot{\theta })u+h(\theta ,\dot{\theta })d\end{array}.$$

(5)

In designing the bridge crane positioning and anti-swing control algorithm, it is crucial to thoroughly account for the adverse effects of disturbances on the algorithm’s performance and make the following reasonable assumptions on the total disturbance d to the bridge crane [25,26].

Assumption 1: The disturbance d experienced by the bridge crane system satisfies $\left|d\right|<d_{\max }$, where $d_{\max }$ is a positive constant.

Assumption 2: The rate of change in disturbance $\dot{d}$ experienced by the bridge crane satisfies $\left|\dot{d}\right|<{\dot{d}}_{\max }$, where ${\dot{d}}_{\max }$ is a constant greater than zero.

Figure 1 shows the block diagram of the anti-disturbance control system of a bridge crane with prescribed-time positioning and anti-swing.

Combining the transformed crane dynamic model (5), the control objective of high-precision positioning and anti-swing at a prescribed time can be defined as follows:

$$\underset{t\to T_{obd}}{\lim }\widehat{d}\to d,$$

(6)

$$\underset{t\to T_{stable}}{\lim }({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4)\to (x_d,0,0,0).$$

(7)

In Equation (7), $\widehat{d}$ is the estimated value of the disturbance, and $x_d$ represents the position at which the trolley is expected to arrive.

3. Design of the Prescribed-Time Disturbance Observer

In this section, a prescribed-time disturbance observer will be designed based on the prescribed-time stability theory to ensure that the unknown disturbances affecting the bridge crane are accurately estimated within a prescribed time. Theoretical analysis will be conducted to demonstrate that the designed disturbance observer can accurately estimate the disturbances experienced by the bridge crane within the prescribed time.

Based on the transformed fully actuated fourth-order bridge crane system dynamic model (5), an intermediate variable is constructed as follows:

$$e=z-{\zeta }_4.$$

(8)

In Equation (8),

$$\dot{z}=f(\theta ,\dot{\theta })+h(\theta ,\dot{\theta })u+h(\theta ,\dot{\theta })\widehat{d}-\kappa e,$$

(9)

and, in Equation (9), $\kappa$ is a parameter to be designed.

Differentiating Equation (8) yields:

$$\dot{e}=\dot{z}-{\dot{\zeta }}_4.$$

(10)

Further, an auxiliary variable is constructed:

$$s_d=h{(\theta ,\dot{\theta })}^{-1}(\dot{e}+\kappa e).$$

(11)

The prescribed-time disturbance observer is devised as follows:

$$\dot{\widehat{d}}=-{\kappa }_d\mathrm{sign}(s_d)-{\displaystyle \frac{{\left(V_{obd}^{\alpha }+V_{obd}^{-\alpha }\right)}^2}{4\alpha T_{obd}}}{s}_d.$$

(12)

In Equation (11), ${\kappa }_d\ge {\left|\dot{d}\right|}_{\max }$ is the observer gain to be designed, $V_{obd}=s_d^2/2$ is the Lyapunov function with respect to the variable $s_d$, $0<\alpha <0.25$ is the observer parameter to be designed, and $T_{obd}$ is the upper bound on the estimated time of the disturbance.

Theorem 1.

The prescribed-time disturbance observer (12) ensures that the unknown external disturbances d experienced by the bridge crane are accurately observed within the prescribed time $T_{obd}$.

Proof. 

By combining the bridge crane model (5) with Equation (9) substituted into Equation (10), Equation (13) can be obtained:

$$\begin{array}{l}\dot{e}=f(\theta ,\dot{\theta })+h(\theta ,\dot{\theta })u+h(\theta ,\dot{\theta })\widehat{d}-\kappa e-{\dot{\zeta }}_4\\ =h(\theta ,\dot{\theta })(\widehat{d}-d)-\kappa e\\ =h(\theta ,\dot{\theta })\tilde{d}-\kappa e\end{array},$$

(13)

where $\tilde{d}$ is the estimation error of the disturbance.

Combining Equations (11) and (13) leads to:

$$s_d=\tilde{d}.$$

(14)

Through Equation (14), it can be observed that when the variable $s_d$ converges to zero within the prescribed time, the estimation error $\tilde{d}$ of the disturbance converges to zero within the prescribed time as well. This means that the unknown disturbance d is accurately estimated within the prescribed time.

The Lyapunov function is constructed as shown below:

$$V_{obd}={\displaystyle \frac{1}{2}}{s}_d^2.$$

(15)

This can be obtained by taking the derivation of both sides of Equation (15) and combining it with Equation (12):

$$\begin{array}{ll}{\dot{V}}_{obd}& =s_d{\dot{s}}_d=s_d(\dot{\widehat{d}}-\dot{d})\\ & =s_d(-{\kappa }_d\mathrm{sign}(s_d)-{\displaystyle \frac{{\left(V_{obd}^{\alpha }+V_{obd}^{-\alpha }\right)}^2}{4\alpha T_{obd}}}{s}_d-\dot{d})\\ & \le -{\kappa }_d\left|s_d\right|-{\displaystyle \frac{{\left(V_{obd}^{\alpha }+V_{obd}^{-\alpha }\right)}^2}{4\alpha T_{obd}}}{s}_d^2+{\left|\dot{d}\right|}_{\max }\left|s_d\right|\\ & =-{\displaystyle \frac{{\left(V_{obd}^{\alpha }+V_{obd}^{-\alpha }\right)}^2}{4\alpha T_{obd}}}{s}_d^2-({\kappa }_d-{\left|\dot{d}\right|}_{\max })\left|s_d\right|\\ & \le -{\displaystyle \frac{{\left(V_{obd}^{\frac{1}{2}+\alpha }+V_{obd}^{\frac{1}{2}-\alpha }\right)}^2}{2\alpha T_{obd}}}\end{array}.$$

(16)

By integrating both sides of Equation (16) with respect to the separated variables, the time for disturbance estimation is as follows:

$$\begin{array}{ll}{T}_D& \le -{\displaystyle {\int }_{V_{obd(0)}}^0\frac{2\alpha T_{obd}}{{(V_{obd}^{\alpha }+V_{obd}^{-\alpha })}^2{V}_{obd}}d{V}_{obd}}\\ & \le \underset{\delta \to 0^{+}}{\lim }{\displaystyle {\int }_{\delta }^{V_{obd(0)}}\frac{2T_{obd}}{{\left(\exp \left(\beta \ln V_{obd}\right)+exp\left(-\beta \ln V_{obd}\right)\right)}^2}d(\beta \ln V_{obd})}\\ & \le {\displaystyle {\int }_{-\infty }^{\beta \ln V_{obd(0)}}\frac{2T_{obd}}{{\left(\exp \left(\chi \right)+\exp \left(-\chi \right)\right)}^2}d\left(\chi \right)}\le {\displaystyle \frac{T_{obd}}{2}}({\displaystyle \frac{\exp \left(\chi \right)-\exp \left(-\chi \right)}{\exp \left(\chi \right)+\exp \left(-\chi \right)}}{|}_{-\infty }^{\beta \ln V_{obd(0)}})\\ & \le {\displaystyle \frac{T_{obd}}{2}}({\displaystyle \frac{V_{obd(0)}^{\beta }-V_{obd(0)}^{-\beta }}{V_{obd(0)}^{\beta }+V_{obd(0)}^{-\beta }}}+1)\le T_{obd}\end{array}.$$

(17)

Therefore, the disturbance estimation error $\tilde{d}$ of the bridge crane system can converge to zero within the prescribed time $T_{obd}$. This indicates that the designed prescribed-time disturbance observer can accurately estimate the unknown disturbance d within the prescribed time $T_{obd}$. □

4. Controller Design

In this section, a bridge crane positioning and anti-swing control algorithm based on the prescribed time disturbance observer is designed based on the transformed system (5) and the equivalent control objective (7). The sliding mode control divides the system state convergence into two stages: the arrival process and the sliding process. For the bridge crane’s prescribed-time positioning and anti-swing problem, this section designs a new convergence rate and a new sliding mode surface based on a recursive structure. These ensure that the system state converges at the prescribed time, realizing the high-precision positioning and anti-swing of the bridge crane.

When $t<T_{stable}$, the recursively structured prescribed-time sliding surface is designed as follows:

$$\{\begin{array}{l}{s}_0={\xi }_1-x_d\\ s_1={\kappa }_0{s}_0+\left(T_{stable}-t\right){\dot{s}}_0\\ s_2={\kappa }_1{s}_1+\left(T_{stable}-t\right){\dot{s}}_1\\ s_3={\kappa }_2{s}_2+\left(T_{stable}-t\right){\dot{s}}_2\end{array}.$$

(18)

When $t\ge T_{stable}$, the recursively structured prescribed-time sliding surface is designed as follows:

$$\{\begin{array}{l}{s}_0={\xi }_1-x_d\\ s_1={\kappa }_0{s}_0+{\lambda }_0{\dot{s}}_0\\ s_2={\kappa }_1{s}_1+{\lambda }_1{\dot{s}}_1\\ s_3={\kappa }_2{s}_2+{\lambda }_2{\dot{s}}_2\end{array}.$$

(19)

In Equations (18) and (19), $T_{stable}$ represents the convergence time of each level sliding surface, while the parameters of the sliding surface satisfy $0<{\kappa }_0\ne {\kappa }_1\ne {\kappa }_2$ and ${\lambda }_0,{\lambda }_1,{\lambda }_2>0$.

To ensure that the recursively structured sliding surface (18) converges to zero within the prescribed time $T_{stable}$, it is necessary to design a prescribed-time approaching law to make the sliding surface $s_3$ converge and stabilize at zero within the prescribed time $T_s$, where $T_s<T_{stable}$. Additionally, in order for each level sliding surface $s_i, (i=0,1,2,3)$ to remain stable at the equilibrium point for $t>T_{stable}$, when $t>T_{stable}$, the sliding surface is switched to the form of (19). Therefore, the next step is to design a prescribed-time approaching law for the sliding surfaces (18) and (19) to ensure that the sliding surface $s_3$ converges and stabilizes at zero within the prescribed time $T_s$. The design of the prescribed-time approaching law is as follows:

$${\dot{s}}_3=\{\begin{array}{ll}-{\displaystyle \frac{{\left(a{V}_3^{\beta }+b{V}_3^{-\beta }\right)}^2}{4ab\beta T_s}}{s}_3-{\kappa }_{a}h(\theta ,\dot{\theta }){\left({T_{stable}}_s-t\right)}^{3}sat\left(s_3\right)& { , \mathrm{t} < \mathrm{T}}_{stable}\\ -{\displaystyle \frac{{\left(a{V}_3^{\beta }+b{V}_3^{-\beta }\right)}^2}{4ab\beta T_s}}{s}_3-{\kappa }_a{\lambda }_2{\lambda }_1{\lambda }_{0}h(\theta ,\dot{\theta })sat\left(s_3\right)& , \mathrm{t}\ge {\mathrm{T}}_{stable}\end{array}.$$

(20)

In Equation (20), $V_3=s_3^2/2$ is the Lyapunov function constructed by the fourth-level sliding surface $s_3$; $0<\beta <0.25$, ${\kappa }_a>{\left|d\right|}_{\max }$, $a>0$, and $b>0$ are the approaching law parameters; $T_s$ is the reaching time parameter of the fourth-level sliding surface $s_3$; and $sat(s_3)$ is the saturation function, which is defined as follows:

$$sat(s_3)=\{\begin{array}{ll}1 & , s_3>1\\ \sin ({\displaystyle \frac{\pi s_3}{2}})& ,-1<s_3<1\\ -1 & , s_3<-1\end{array}.$$

The sliding mode control law u is composed of the equivalent control law $u_{eq}$ and the switching control law $u_{sw}$, and it is defined as follows:

$$u=u_{eq}+u_{sw}.$$

(21)

When $t<T_{stable}$, by taking the derivative of the fourth-level sliding surface $s_3$ and combining it with the transformed fourth-order bridge crane system dynamic model (5), we obtain the following equation:

$$\begin{array}{ll}{\dot{s}}_3& =\left(T_{stable}-t\right){\ddot{s}}_2+\left({\kappa }_2-1\right){\dot{s}}_2\\ & ={\left(T_{stable}-t\right)}^2{\stackrel{⃛}{s}}_1+\left(T_{stable}-t\right)\left({\kappa }_1-2\right){\ddot{s}}_1+\left({\kappa }_2-1\right){\dot{s}}_2\\ & ={\left(T_{stable}-t\right)}^3{\stackrel{⃜}{s}}_0+{\left(T_{stable}-t\right)}^2\left({\kappa }_0-3\right){\stackrel{⃛}{s}}_0+\left(T_{stable}-t\right)\left({\kappa }_1-2\right){\ddot{s}}_1+\left({\kappa }_2-1\right){\dot{s}}_2\\ & ={\left(T_{stable}-t\right)}^3[f(\theta ,\dot{\theta })+h(x)(u-d)]+{\left(T_{stable}-t\right)}^2\left({\kappa }_0-3\right){\stackrel{⃛}{s}}_0\\ & +\left(T_{stable}-t\right)\left({\kappa }_1-2\right){\ddot{s}}_1+\left({\kappa }_2-1\right){\dot{s}}_2\end{array}.$$

(22)

Setting ${\dot{s}}_3=0$, we can obtain the equivalent control law $u_{eq}$ as follows:

$$u_{eq}=h{(x)}^{-1}(-f(\theta ,\dot{\theta })-\psi (s))-\widehat{d},$$

(23)

where $\psi (s)={\left({T_{stable}}_s-t\right)}^{-1}\left({\kappa }_0-3\right){\stackrel{⃛}{s}}_0+{\left(T_{stable}-t\right)}^{-2}\left({\kappa }_1-2\right){\ddot{s}}_1+{\left(T_{stable}-t\right)}^{-3}\left({\kappa }_2-1\right){\dot{s}}_2$.

Combining Equations (20)–(23), we can obtain the switching control law $u_{sw}$ as follows:

$$u_{sw}=-h{(x)}^{-1}{\left(T_{stable}-t\right)}^{-3}{\displaystyle \frac{{(a{V}_3^{\beta }+b{V}_3^{-\beta })}^2}{4ab\beta T_s}}{s}_3-{\kappa }_{a}sat\left(s_3\right).$$

(24)

Combining Equations (21), (23) and (24), we can obtain the prescribed-time positioning and anti-swing control law u for the bridge crane when $t<T_{stable}$ as follows:

$$u=h{(x)}^{-1}[-f(\theta ,\dot{\theta })-\psi (s)-{\left(T_{stable}-t\right)}^{-3}{\displaystyle \frac{{(a{V}_3^{\beta }+b{V}_3^{-\beta })}^2}{4ab\beta T_s}}{s}_3]-{\kappa }_{a}sat\left(s_3\right)-\widehat{d}.$$

(25)

When $t\ge T_{stable}$, taking the derivative of the fourth-level sliding surface $s_3$ yields the following expression:

$$\begin{array}{l}{\dot{s}}_3={\lambda }_2{\ddot{s}}_2+{\kappa }_2{\dot{s}}_2\\ ={\lambda }_2{\lambda }_1{\stackrel{⃛}{s}}_1+{\lambda }_2{\kappa }_1{\ddot{s}}_1+{\kappa }_2{\dot{s}}_2\\ ={\lambda }_2{\lambda }_1{\lambda }_0{\stackrel{⃜}{s}}_0+{\lambda }_2{\lambda }_1{\kappa }_0{\stackrel{⃛}{s}}_0+{\lambda }_2{\kappa }_1{\ddot{s}}_1+{\kappa }_2{\dot{s}}_2\\ ={\lambda }_2{\lambda }_1{\lambda }_0[f(\theta ,\dot{\theta })+h(x)u+d]+{\lambda }_2{\lambda }_1{\kappa }_0{\stackrel{⃛}{s}}_0+{\lambda }_2{\kappa }_1{\ddot{s}}_1+{\kappa }_2{\dot{s}}_2\end{array}.$$

(26)

Similarly, when $t\ge T_{stable}$, the prescribed-time positioning and anti-swing control law u for the bridge crane can be obtained as follows:

$$u=h{(x)}^{-1}[-f(\theta ,\dot{\theta })-{\displaystyle \frac{{\kappa }_0{\stackrel{⃛}{s}}_0}{{\lambda }_0}}-{\displaystyle \frac{{\kappa }_1{\ddot{s}}_1}{{\lambda }_1{\lambda }_0}}-{\displaystyle \frac{{\kappa }_2{\dot{s}}_2}{{\lambda }_2{\lambda }_1{\lambda }_0}}-{\displaystyle \frac{{(a{V}_3^{\beta }+b{V}_3^{-\beta })}^2}{4\mathrm{ab}\beta T_s{\lambda }_2{\lambda }_1{\lambda }_0}}{s}_3]-{\kappa }_{a}sat\left(s_3\right)-\widehat{d}.$$

(27)

In summary, the prescribed-time positioning and anti-swing control law u for the bridge crane can be designed in the following form:

$$u=\{\begin{array}{ll}h{(\theta ,\dot{\theta })}^{-1}[-f(\theta ,\dot{\theta })-\psi (s)-{\left(T_{stable}-t\right)}^{-3}{\displaystyle \frac{{(a{V}_3^{\beta }+b{V}_3^{-\beta })}^2}{4ab\beta T_s}}{s}_3]& \\ -{\kappa }_{a}sat\left(s_3\right)-\widehat{d}& { , \mathrm{t} < \mathrm{T}}_{stable}\\ h{(\theta ,\dot{\theta })}^{-1}[-f(\theta ,\dot{\theta })-{\displaystyle \frac{{\kappa }_0{\stackrel{⃛}{s}}_0}{{\lambda }_0}}-{\displaystyle \frac{{\kappa }_1{\ddot{s}}_1}{{\lambda }_1{\lambda }_0}}-{\displaystyle \frac{{\kappa }_2{\dot{s}}_2}{{\lambda }_2{\lambda }_1{\lambda }_0}}-{\displaystyle \frac{{(a{V}_3^{\beta }+b{V}_3^{-\beta })}^2}{4ab\beta T_s{\lambda }_2{\lambda }_1{\lambda }_0}}{s}_3]& \\ -{\kappa }_{a}sat\left(s_3\right)-\widehat{d}& , \mathrm{t}\ge {\mathrm{T}}_{stable}\end{array}.$$

(28)

For the bridge crane system, selecting the prescribed-time positioning and anti-swing control law (28) ensures that the bridge crane achieves trolley positioning and material anti-swing within the specified time $t=T_{stable}$.

5. Stability Analysis

By analyzing the sliding surfaces (18) and (19), it can be observed that when each sliding surface converges to zero, the control objective of achieving high-precision positioning and anti-swing within the prescribed time is achieved. Therefore, this section will analyze the stability of the sliding surfaces at different time stages, thereby further investigating the stability of the prescribed-time positioning and anti-swing control for the bridge crane system.

When $t\le T_{obd}$, the Lyapunov function is constructed as follows:

$$V_3={\displaystyle \frac{1}{2}}{s}_3^2.$$

(29)

Deriving Equation (29) and substituting it into the bridge crane dynamics model (5), the sliding mode surface (18), and the prescribed-time positioning and anti-swing control law (28) yields:

$$\begin{array}{ll}{\dot{V}}_3& =s_3{\dot{s}}_3\\ & =-{\displaystyle \frac{{(a{V}_3^{\beta }+b{V}_3^{-\beta })}^2}{4ab\beta T_s}}{s}_3^2-\left(k_{a}sat(s_3)s_3-(d-\widehat{d})s_3\right)h(\theta ,\dot{\theta }){\left({T_{stable}}_s-t\right)}^3\\ & \le -{\displaystyle \frac{{(a{V}_3^{\frac{1}{2}+\beta }+b{V}_3^{\frac{1}{2}-\beta })}^2}{2ab\beta T_s}}-\left(k_{a}sat(s_3)s_3-{\left|\tilde{d}\right|}_{\max }\left|s_3\right|\right)h(\theta ,\dot{\theta }){\left({T_{stable}}_s-t\right)}^3\end{array}.$$

(30)

From Theorem 1, it is evident that the external unknown disturbance d is accurately estimated within the prescribed time $T_{obd}$. In other words, the disturbance estimation error $\tilde{d}$ converges to zero within the prescribed time $T_{obd}$, and when the initial value of disturbance estimation $\widehat{d}$ from the prescribed-time disturbance observer is chosen as zero, ${\left|\tilde{d}\right|}_{\max }\le {\left|d\right|}_{\max }$ is constant. Given that $k_a\ge {\left|d\right|}_{\max }$, it can be proven that $s_3$ is bounded and $\left|s_3\right|\le 1$ for $t<T_{obd}$.

By combining with the sliding surface (18), we can obtain:

$$\left|{\kappa }_2{s}_2+\left(T_{stable}-t\right){\dot{s}}_2\right|\le 1.$$

(31)

Further transforming Equation (31):

$$\left|{\displaystyle \frac{{\left(T_{stable}-t\right)}^{{\kappa }_2}{\dot{s}}_2+{\kappa }_2{\left(T_{stable}-t\right)}^{{\kappa }_2-1}{s}_2}{{\left(T_{stable}-t\right)}^{2{\kappa }_2}}}\right|\le {\displaystyle \frac{1}{{\left(T_{stable}-t\right)}^{{\kappa }_2+1}}}.$$

(32)

By integrating the time on both sides of Equation (32), Equation (33) can be obtained:

$$-{\displaystyle {\int }_0^t\frac{1}{{\left(T_{stable}-t\right)}^{{\kappa }_2+1}}}dt\le {\displaystyle {\int }_0^t\left(d\left(\frac{s_2}{{\left(T_{stable}-t\right)}^{{\kappa }_2}}\right)/dt\right)}dt\le {\displaystyle {\int }_0^t\frac{1}{{\left(T_{stable}-t\right)}^{{\kappa }_2+1}}}dt.$$

(33)

Further, solving Equation (33) yields:

$$\{\begin{array}{l}{s}_2\le {\displaystyle \frac{s_{2(0)}}{T_{stable}^{{\kappa }_2}}}{\left(T_{stable}-t\right)}^{{\kappa }_2}-{\displaystyle \frac{1}{{\kappa }_2}}+{\displaystyle \frac{1}{{\kappa }_2}}{T}_{stable}^{-{\kappa }_2}{\left(T_{stable}-t\right)}^{{\kappa }_2}\\ s_2\ge {\displaystyle \frac{s_{2(0)}}{T_{stable}^{{\kappa }_2}}}{\left(T_{stable}-t\right)}^{{\kappa }_2}+{\displaystyle \frac{1}{{\kappa }_2}}-{\displaystyle \frac{1}{{\kappa }_2}}{T}_{stable}^{-{\kappa }_2}{\left(T_{stable}-t\right)}^{{\kappa }_2}\end{array}.$$

(34)

Since for $t\le T_{obd}$, $s_{2\left(0\right)}$, $T_{stable}$, ${\kappa }_2$ are known constants, and $T_{stable}-T_{obd}\le T_{stable}-t\le T_{stable}$, there must exist a constant ${\omega }_2>0$ such that $\left|s_2\right|\le {\omega }_2$ holds, implying that the sliding surface $s_2$ is bounded. Similarly, it can be concluded that sliding surfaces $s_1$ and $s_0$ are also bounded.

When $T_{obd}<t\le T_s$, deriving Equation (29) and substituting it into the bridge crane model (5), the sliding mode surface (18), and the control law (28) yields:

$$\begin{array}{ll}{\dot{V}}_3& =s_3{\dot{s}}_3\\ & =-{\displaystyle \frac{{(a{V}_3^{\beta }+b{V}_3^{-\beta })}^2}{4ab\beta T_s}}{s}_3^2-\left(k_{a}sat(s_3)s_3-(d-\widehat{d})s_3\right)h(\theta ,\dot{\theta }){\left({T_{stable}}_s-t\right)}^3\\ & =-{\displaystyle \frac{{(a{V}_3^{\frac{1}{2}+\beta }+b{V}_3^{\frac{1}{2}-\beta })}^2}{2ab\beta T_s}}-\left(k_{a}sat(s_3)s_3-\tilde{d}\left|s_3\right|\right)h(\theta ,\dot{\theta }){\left({T_{stable}}_s-t\right)}^3\end{array}.$$

(35)

Through Theorem 1, we know that $\tilde{d}$ converges to zero within the prescribed time $T_{obd}$. Therefore, Equation (35) can be equivalently expressed as:

$${\dot{V}}_3\le -{\displaystyle \frac{{(a{V}_3^{\frac{1}{2}+\beta }+b{V}_3^{\frac{1}{2}-\beta })}^2}{2ab\beta T_s}}-k_{a}sat(s_3)s_{3}h(\theta ,\dot{\theta }){\left({T_{stable}}_s-t\right)}^3.$$

(36)

Integrating the separated variables into Equation (36) yields:

$$\begin{array}{ll}{T}_{(x_0)}& \le {\displaystyle {\int }_0^{V_{3(0)}}\frac{2ab\beta T_s}{{(a{V}^{\beta }+\mathrm{b}{V}^{-\beta })}^{2}V}dV}\\ & \le T_s\end{array}.$$

(37)

It can be shown that $s_3$ converges to zero in $T_s$.

When $T_s<t<T_{stable}$, knowing that $s_3=0$, combined with the sliding surface (18), we can obtain the following expression:

$${\kappa }_2{s}_2+\left(T_{stable}-t\right){\dot{s}}_2=0.$$

(38)

Solving Equation (38) yields:

$$s_2=\epsilon {\left(T_{stable}-t\right)}^{{\kappa }_2},$$

(39)

where $\epsilon =s_{2(T_s)}/{\left(T_{stable}-T_s\right)}^{{\kappa }_2}$. From Equation (39), it is evident that the sliding mode surface $s_2$ will converge to zero within the prescribed time $T_{stable}$. Similarly, it follows that the sliding mode surfaces $s_1$ and $s_0$ will converge to zero at the prescribed time $T_{stable}$.

Combined with the sliding mode surface (18), the following conclusions can be drawn:

$$\underset{t\to T_{stable}}{\lim }({\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4)\to (x_d,0,0,0),$$

(40)

when $t\ge T_{stable}$, which can be obtained by the sliding mode surface (18):

$$s_3={\eta }_0{s}_0+{\eta }_1{\dot{s}}_0+{\eta }_2{\ddot{s}}_0+{\eta }_3{\stackrel{⃛}{s}}_0.$$

(41)

In Equation (41),

$$\{\begin{array}{l}{\eta }_0={\kappa }_2{\kappa }_1{\kappa }_0\\ {\eta }_1={\kappa }_2{\kappa }_1{\lambda }_0+{\kappa }_2{\kappa }_0{\lambda }_1+{\kappa }_1{\kappa }_0{\lambda }_2\\ {\eta }_2={\lambda }_2{\lambda }_1{\kappa }_0+{\lambda }_2{\lambda }_0{\kappa }_1+{\lambda }_1{\lambda }_0{\kappa }_2\\ {\eta }_3={\lambda }_2{\lambda }_1{\lambda }_0\end{array}$$

can be obtained by combining the bridge crane model (5):

$$s_3={\eta }_0({\zeta }_1-x_d)+{\eta }_1{\zeta }_2+{\eta }_2{\zeta }_3+{\eta }_3{\zeta }_4.$$

(42)

Deriving the constructed Lyapunov function (29) and combining it with Theorem 1 yields:

$$\begin{array}{ll}{\dot{V}}_3& =s_3{\dot{s}}_3\\ & =-{\displaystyle \frac{{(a{V}_3^{\beta }+b{V}_3^{-\beta })}^2}{4ab\beta T_s}}{s}_3^2-({\kappa }_{a}sat(s_3)s_3-\tilde{d}\left|s_3\right|)h(\theta ,\dot{\theta }){\lambda }_2{\lambda }_1{\lambda }_0\\ & \le -{\displaystyle \frac{{(a{V}_3^{\frac{1}{2}+\beta }+b{V}_3^{\frac{1}{2}-\beta })}^2}{2ab\beta T_s}}\end{array}.$$

(43)

Further combining Equations (40), (42) and (43) shows that the sliding mode surface $s_3$ will remain at zero. Therefore, Equation (41) can be equated as:

$${\eta }_0{s}_0+{\eta }_1{\dot{s}}_0+{\eta }_2{\ddot{s}}_0+{\eta }_3{\stackrel{⃛}{s}}_0=0.$$

(44)

Equation (44) can be transformed into the following form:

$${\dot{\xi }}_{(t)}=A{\xi }_{(t)}.$$

(45)

In Equation (45),

$$A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ {\displaystyle \raisebox{1ex}{$-{\eta }_0$}\!\left/ \!\raisebox{-1ex}{${\eta }_3$}\right.}& {\displaystyle \raisebox{1ex}{$-{\eta }_1$}\!\left/ \!\raisebox{-1ex}{${\eta }_3$}\right.}& {\displaystyle \raisebox{1ex}{$-{\eta }_2$}\!\left/ \!\raisebox{-1ex}{${\eta }_3$}\right.}\end{array}\right],{\xi }_{(t)}=\left[\begin{array}{c}{s}_0\\ {\dot{s}}_0\\ {\ddot{s}}_0\end{array}\right].$$

The solution to the equation is: ${\xi }_{(t)}={\xi }_{(T_{stable})}\exp (At)$, and since ${\zeta }_{T_{stable}}=0$, the system state ${\xi }_{(t)}$ will remain at zero when the parameters satisfy ${\eta }_i>0,i=0,1,2,3$.

In summary, the control algorithm proposed in this article can accurately compensate for the unknown disturbance at the prescribed time $T_{stable}$, so that the bridge crane can realize high-precision positioning and anti-swing at the prescribed time under the disturbed condition.

6. Simulation and Analysis

This section validates prescribed-time positioning and anti-swing sliding mode control (PTSMC), demonstrating its ability to achieve high-precision positioning and anti-swing despite external disturbances. Using the MATLAB 2018b/Simulink simulation platform, a control system model for the bridge crane under disturbance conditions is constructed. In this section, three groups of simulations are set up, and the first group compares and analyzes the simulations with the existing literature to verify that the control algorithm designed in this paper can realize the bridge crane’s prescribed-time positioning and anti-swing while eliminating the jitter vibration. The second group verifies the control performance of the designed control algorithm for the prescribed-time positioning and anti-swing control when the system is subjected to constant-value disturbance. The third group verifies the control performance of the designed control algorithm for the prescribed-time positioning and anti-swing control when the system is subjected to a time-varying disturbance. The parameters of the bridge crane system are set as shown in

Table 2. Crane system parameters.

Parameter Name	Symbol	Unit	Value
Trolley mass	$M$	kg	4.0
Load mass	$m$	kg	2.5
Cable length	$l$	m	1.5
Gravitational acceleration	$g$	${\mathrm{m}/\mathrm{s}}^2$	9.806

.

During the simulation, to avoid singularities in control law (28) at $t=T_{stable}$, parameter $\Delta$ is introduced. The prescribed-time positioning and anti-swing control law designed in this paper is modified as follows:

$$u=\{\begin{array}{ll}h{(\theta ,\dot{\theta })}^{-1}[-f(\theta ,\dot{\theta })-\psi (s)-{\left(T_{stable}-t\right)}^{-3}{\displaystyle \frac{{(a{V}_3^{\beta }+b{V}_3^{-\beta })}^2}{4ab\beta T_s}}{s}_3]& \\ -{\kappa }_{a}sat\left(s_3\right)-\widehat{d}& { , \mathrm{t} < \mathrm{T}}_{stable}-\Delta \\ h{(\theta ,\dot{\theta })}^{-1}[-f(\theta ,\dot{\theta })-{\displaystyle \frac{{\kappa }_0{\stackrel{⃛}{s}}_0}{{\lambda }_0}}-{\displaystyle \frac{{\kappa }_1{\ddot{s}}_1}{{\lambda }_1{\lambda }_0}}-{\displaystyle \frac{{\kappa }_2{\dot{s}}_2}{{\lambda }_2{\lambda }_1{\lambda }_0}}-{\displaystyle \frac{{(a{V}_3^{\beta }+b{V}_3^{-\beta })}^2}{4ab\beta T_s{\lambda }_2{\lambda }_1{\lambda }_0}}{s}_3]& \\ -{\kappa }_{a}sat\left(s_3\right)-\widehat{d}& , \mathrm{t}\ge {\mathrm{T}}_{stable}-\Delta \end{array}.$$

(46)

The parameters of the PTSMC algorithm designed in this paper are selected as: $T_s=4$, $T_{obd}=1$, $\alpha =0.2$, $\beta =0.05$, $\Delta =0.1$, $a=3$, $b=1$, $\kappa =0.1$, ${\kappa }_d=5$, ${\kappa }_1=8$, ${\kappa }_2=9$, ${\kappa }_a=5.2$, ${\lambda }_0=2$, ${\lambda }_1=1$, and ${\lambda }_2=1$.

The desired position of the trolley is $x_d=1$ and The initial values of the system state are defined as $[x(0),\dot{x}(0),\theta (0),\dot{\theta }(0)]=[0,0,0,0]$.

6.1. Contrast Simulation

In this subsection, the simulations are compared and analyzed with the partially saturated coupled-dissipation control (PSCDC) algorithm proposed in the literature [27] and the linear quadratic regulation (LQR) algorithm proposed in the literature [28], through literature comparison simulations that verify the performance of the PTSMC algorithm designed in this paper.

The form of the PSCDC algorithm proposed by Zhang is as follows:

$$\begin{array}{ll}F& =-k_p(\mathrm{M}+\mathrm{m}{\sin }^2\theta )\tanh (x-l{\displaystyle {\int }_0^t\sin \theta (\tau )d\tau }-x_d-{\displaystyle \frac{1}{l}}\theta )\\ & -{\displaystyle \frac{k_d}{(\mathrm{M}+\mathrm{m}{\sin }^2\theta )}}(\dot{x}-l\sin \theta -{\displaystyle \frac{1}{l}}\cos \theta \dot{\theta })\\ & -\mathrm{m}\sin \theta (\mathrm{g} \cos \theta +l{\dot{\theta }}^2)+(\mathrm{M}+\mathrm{m}{\sin }^2\theta )l\cos \theta \dot{\theta }\end{array}.$$

The parameters of the control algorithm are as follows:

$k_p=1.8$ and $k_d=38$.

The form of the LQR algorithm proposed by Jafari is as follows:

$$F=-k_1(x-x_d)-k_2\dot{x}-k_3\theta -k_4\dot{\theta }+f_{rx}.$$

The parameters of the control algorithm are as follows:

$k_1=7$, $k_2=13$, $k_3=-11$ and $k_4=-8$.

The simulation results in Figure 2 show that the PSCDC algorithm enables the cart to reach the desired position for the first time in 3.8 s and to move back and forth with small amplitude near the desired position; the LQR algorithm enables the cart to reach the desired position in 6.2 s; and the PTSMC algorithm proposed in this paper realizes the accurate positioning of the cart in the specified time of 6 s.

The simulation results in Figure 3 show that the PSCDC algorithm limits the material swing angle amplitude to between −4 deg and 2.8 deg, and the material swing angle converges to the range of 0.1 deg in 6.2 s; the LQR algorithm limits the material swing angle amplitude to between −3.95 deg and 2.1 deg, and the material swing angle converges to the range of 0.1 deg in 6.4 s; and the PTSMC algorithm reduces the material swing angle amplitude to between −2.1 deg and 1.1 deg, with the material swing angle converging to zero within 6 s at the prescribed time, which completely eliminates the residual swing of the material after the cart positioning.

The simulation results in Figure 4 show that the initial value of the control law of the PTSMC algorithm proposed in this paper is slightly larger than the initial value of the control law of the PSCDC and LQR algorithms. However, considering the positioning and swing elimination performance of the control algorithm, the PTSMC algorithm proposed in this chapter can realize the accurate positioning and swing elimination of the bridge crane faster, while the material has a smaller swing amplitude, and the positioning and swing elimination time of the bridge crane can be specified directly by the control parameters $T_{stable}$, so the PTSMC algorithm has a better positioning and swing elimination performance.

6.2. Simulation Analysis under a Constant Disturbance

Given a disturbance of d = 5, the performance of the designed prescribed-time disturbance observer is evaluated, along with the positioning and anti-swing performance of the proposed control algorithm under external constant disturbance for the bridge crane.

The simulation results in Figure 5 show that the disturbance estimation error converges to zero in 0.36 s, which means that the designed disturbance observer is able to accurately estimate the constant-value disturbance within the prescribed time $T_{obd}$.

The results of the simulation in Figure 6 and Figure 7 show that the bridge crane is able to realize the accurate positioning of the trolley and the elimination of the material swing at the prescribed time of 6 s when it is subjected to a constant-value disturbance.

The simulation experimental results in Figure 8 show that when the disturbance is accurately estimated, the control algorithm makes accurate compensation for the external disturbance. At this time, it can be found that the fourth-level sliding mode surface $s_3$ converges to zero in the prescribed time $T_s$, and the first three levels of sliding mode surfaces $s_2$, $s_1$, and $s_0$ are able to converge to zero in the prescribed time $T_{stable}$.

The simulation experimental results in Figure 9 show that the initial value of the control law is 9.28 N, and after 6 s due to the existence of the external constant-value disturbance, the control algorithm makes an accurate compensation for the external disturbance, so the control law is stabilized at −5 N.

In summary, the positioning and anti-swing control algorithm based on the prescribed-time disturbance observer proposed in this paper can accurately estimate and compensate for constant disturbances affecting the bridge crane within the prescribed time $T_{obd}$. Therefore, it ensures that the bridge crane achieves the control objective of high-precision positioning and anti-swing within the prescribed time $T_{stable}$.

6.3. Simulation Analysis under a Time-Varying Disturbance

In this subsection, a time-varying disturbance will be selected to simulate the disturbance encountered during the operation of the bridge crane, in order to evaluate the performance of the designed prescribed-time disturbance observer and the proposed positioning and anti-swing control algorithm based on the prescribed-time disturbance observer.

The time-varying disturbance is modeled as follows: $d=3\sin (\mathrm{t}+{\displaystyle \frac{\mathsf{\pi }}{2}})+\cos (2 \mathrm{t})+1$.

Under the condition of time-varying disturbance during the working process of the bridge crane, the simulation results in Figure 10 show that the designed prescribed-time disturbance observer makes an accurate estimation of the disturbance at 0.31 s, and the estimation error is stabilized at zero, so it can be verified that the designed disturbance observer can make an accurate estimation of the time-varying disturbance of the bridge crane within the specified time $T_{obd}$. The simulation results in Figure 11, Figure 12 and Figure 13 show that the bridge crane is still able to realize accurate positioning of the trolley and fast material swing angle elimination at the prescribed time $T_{stable}=6$ s even when it is subjected to time-varying disturbance.

The results of the simulation experiment in Figure 14 show that the initial value of the control law is 9.28 N, and after 6 s the control law is still a varying curve due to the precise compensation of the disturbance in order to keep the control system stable, and the magnitude of the control law is opposite to the magnitude of the external time-varying disturbance after 6 s.

In summary, the proposed positioning and anti-swing control algorithm based on the prescribed-time disturbance observer effectively estimates and compensates for time-varying disturbances within the specified time $T_{obd}$. Even under the influence of time-varying disturbances, the proposed control algorithm ensures precise positioning and anti-swing of the bridge crane within the prescribed time $T_{stable}=6$.

6.4. Simulation under Different Conditions

In this subsection, three different sets of simulation conditions are designed to verify the performance of the control algorithm. The time-varying disturbance is modeled as follows: $d=3\sin (\mathrm{t}+{\displaystyle \frac{\mathsf{\pi }}{2}})+\cos (2 \mathrm{t})+1$ and the simulation conditions are given as shown in

Table 3. Simulation condition settings.

Condition	Initial Position of Trolley	Initial Speed of Trolley	Material Initial Swing Angle	Initial Angle Velocity of Material	Prescribed Time ${\mathit{T}}_{\mathit{s}\mathit{t}\mathit{a}\mathit{b}\mathit{l}\mathit{e}}$
Condition 1	0.3 m	0 m/s	2 deg	0.6 deg/s	5 s
Condition 2	0.2 m	0 m/s	4 deg	0.6 deg/s	6 s
Condition 3	0 m	0 m/s	5 deg	−0.6 deg/s	7 s

.

The simulation conditions in Table 3 are based on the fact that the initial position of the trolley is not unique each time the bridge crane is started and considering that the initial speed of the trolley is zero. Due to the deviation of the angle between the hook and the lifting point when the material is lifted, there is an initial swing angle and angle velocity when the material is transferred.

The simulation experimental results in Figure 15 and Figure 16 show that the designed PTSMC algorithm is still able to realize the accurate positioning of the trolley and the material swing elimination at the prescribed time under the condition that the state of the bridge crane system has different initial values.

7. Conclusions

Aiming to address the problem of the reduction in the positioning and anti-swing accuracy of the bridge crane under disturbed conditions and the vibration jitter problem in the sliding mode control algorithm, a bridge crane positioning and anti-swing control algorithm based on the prescribed-time disturbance observer is proposed. This new algorithm realizes the high-precision positioning and anti-swing of the bridge crane at the prescribed time under disturbed conditions and, at the same time, suppresses the vibration jitter problem in the sliding mode control algorithm. Firstly, the new prescribed-time disturbance observer designed to address the problem that disturbance to the bridge crane will degrade its positioning and anti-swing accuracy makes an accurate estimation of the disturbance to ensure that the disturbance is accurately compensated. Secondly, the new prescribed-time sliding mode surface and the prescribed-time convergence law are designed to ensure that the system state error converges to zero at the prescribed time. Then, the saturation function is introduced to replace the sign function with switching characteristics in the sliding mode control algorithm, which ensures that the system state is bounded before the disturbance is accurately estimated and compensated and, at the same time, solves the jitter problem of the sliding mode control algorithm. Finally, it is verified, through theoretical analysis and simulation, that the designed disturbance observer is able to make an accurate estimation of the disturbance within a prescribed time, and the proposed control algorithm is able to realize the high-precision positioning and anti-swing of the bridge crane under disturbed conditions within a prescribed time $T_{stable}$.

In the material transfer process of a bridge crane, the vertical lifting process and horizontal transfer process of materials are carried out at the same time, which can improve its working efficiency. The research on the control algorithm for the accurate positioning of the trolley and the rapid elimination of material swing within the prescribed time can be carried out further for this kind of working scenario.