Adaptive Prescribed-Time Recursive Sliding Mode Control of Underactuated Bridge Crane Systems

Abstract

As a critical piece of equipment in industrial logistics, the operational efficiency of bridge cranes is influenced by their positioning and sway suppression capabilities. Addressing the challenge of positioning and sway suppression in bridge crane systems when load mass is difficult to estimate accurately, an adaptive prescribed-time sliding mode control (PTSMC) is proposed. Firstly, a two-dimensional bridge crane dynamic model is established using the Lagrange method, then transformed into a controllable cascade form. Secondly, a novel sliding mode variable with an explicit time term is designed for achieving prescribed time convergence, while an adaptive law is introduced to estimate unknown loads in real time. Theoretical analysis demonstrates that the proposed method guarantees convergence of the system state to the equilibrium point within the prescribed time. Numerical simulations validate the effectiveness of this strategy, demonstrating that the system can complete trolley positioning and swing angle suppression within the prescribed time. This significantly enhances the dynamic performance and transfer efficiency of the bridge crane.

1. Introduction

As the core equipment for industrial material handling, the underactuated characteristics of bridge cranes necessitate indirect control of load swing angles through trolley displacement, significantly increasing the complexity of controller design. With rising demands for industrial automation and efficiency, operational efficiency and safety have become particularly critical. The key technical challenge in bridge crane control systems lies in designing controllers that achieve rapid and precise positioning while effectively suppressing load sway [1,2]. This approach reduces positioning and sway elimination time, thereby enhancing the material handling efficiency and production safety of the crane system.

The positioning and anti-swing control algorithms for bridge cranes are roughly categorized into open-loop and closed-loop types. Although open-loop control is characterized by simple implementation and low cost, it is unable to adjust the system state in real time based on feedback, resulting in poor robustness. Therefore, a range of closed-loop control algorithms has been developed by researchers worldwide [3,4,5,6,7], including adaptive control, model predictive control, backstepping control, optimal control, and sliding mode control. The robustness and control performance of crane systems under complex operating conditions have been significantly improved through these methods. Among them, sliding mode control is widely applied in underactuated systems such as bridge cranes due to its strong robustness.

Regarding the positioning and sway suppression control of bridge crane systems, only asymptotic stability can be guaranteed by traditional sliding mode control methods, with convergence time being theoretically infinite and dependent on the system’s initial state [8,9,10]. However, in actual material handling operations, rapid achievement of control objectives is often required to maintain handling efficiency, making finite-time convergence more valuable for engineering applications. To this end, researchers have incorporated finite-time stability theory into classical sliding-mode control, a concept that has also found extensive application in multiple fields such as robotics and power electronics [11,12,13]. Wang, T. [14] proposed a finite-time sliding mode control strategy that accounts for variations in rope length. For disturbed operating conditions, Liu, S. [15] developed a sliding mode controller with disturbance observers, ensuring the crane system completes positioning and stabilizes within a finite time. Notably, terminal sliding mode control, as a special case within sliding mode control, can achieve finite-time convergence. In reference [16], a hierarchical sliding mode control strategy with global fast terminal sliding mode is proposed, and robust performance against white noise disturbances is demonstrated while anti-sway effectiveness is maintained.

Although finite-time stability exhibits faster convergence rates than asymptotic stability, its convergence time upper bound is determined by the initial state of the system. To address this, fixed-time stability theory was introduced by Professor Polyakov [17], where the convergence time upper bound is defined only by the control parameters, regardless of the initial conditions. Based on this work, Yang, Y. [18] designed a dual-layer adaptive disturbance observer that estimates and compensates for disturbance upper bounds, achieving timing positioning and swing suppression for crane systems. For bridge cranes, Wang, Z.H. [19] proposed a strategy based on fixed-time sliding mode control, which improved both system response speed and disturbance rejection capability. Meanwhile, Gu, X.T. [20] designed a weak-chattering fixed-time convergence law based on non-singular fast terminal sliding control for tower cranes, effectively solving positioning and anti-sway issues under external disturbances. Finally, reference [21] proposed a fixed-time tracking control method with obstacle avoidance and specified performance for collaborative dual-arm cranes, further enhancing operational efficiency and safety.

The upper bound for fixed-time stable convergence is independent of initial conditions, but its expression is influenced by multiple control parameters, making the design process rather complex. In contrast, the expression for the time upper bound is further simplified by prescribed-time control [22], with dependence on both the system’s initial states and controller parameters eliminated. This upper bound can be directly specified by the user. This theory has now been extensively applied in fields such as aircraft control [23,24], robotic manipulators [25], and multi-agent systems [26], where significant results have been achieved. Feng et al. [27] successfully applied prescribed-time control to a bridge crane system, achieving prescribed-time convergence of the system state. However, this study focused solely on the single performance metric of prescribed-time convergence. The designed control law incorporates a more complex structure and additional design parameters, increasing the difficulty of parameter tuning. Furthermore, the scheme relies on precise knowledge of the load mass and fails to account for its uncertainty, which may compromise controller performance under actual operating conditions.

Based on the foregoing discussion, an adaptive prescribed-time sliding mode controller is proposed. It estimates unknown load parameters in real time and compensates for their effects, while incorporating prescribed-time control to ensure system state convergence within the prescribed time.

The contributions made in this article are as follows:

1.

A novel nonlinear recursive sliding mode controller based on an explicit time term was designed, achieving prescribed time convergence of the system state while reducing controller complexity and enhancing algorithm portability;

2.

Focusing on the practical engineering challenge where load masses are not fully known with precision, a mass-adaptive law was designed to estimate load masses online while ensuring convergence within a prescribed time frame. This approach mitigates the impact of unknown mass disturbances on control performance, thereby enhancing engineering applicability;

3.

Simulation experiments validate the prescribed-time convergence performance of the proposed control algorithm, demonstrating its superiority over existing alternative control strategies in both crane positioning and sway suppression.

The following sections are organized accordingly: Section 2 establishes the dynamical model of the two-dimensional underactuated bridge crane, forming the theoretical basis for the prescribed-time control scheme. Section 3 proposes an adaptive mass prescribed-time sliding mode controller and details its design process. Section 4 provides a comprehensive proof of the prescribed-time stability of the designed controller in two parts. Section 5 validates the controller’s performance and robustness through comparative simulations. Finally, Section 6 summarises the work of this paper and provides an outlook for future research.

2. Problem Description and Preparatory Work

2.1. System Dynamics Model

Since the trolley and the bridge structure typically operate sequentially during load transfer, the two-dimensional bridge crane model shown in Figure 1 is adopted.

Treat the load as a point mass, neglecting air resistance, the mass of the suspension rope, deformation, and the mass of the hook. The dynamic model of the bridge crane system established based on the Lagrange method is as follows [28,29]:

$$\begin{array}{c}\left\{\begin{array}{c}(M+m)x+mlcos\ddot{\theta }-mlsin{\dot{\theta }}^2=F\hfill \\ mlcos\theta \ddot{x}+m{l}^2\ddot{\theta }+mglsin\theta =0\hfill \end{array}\right.\hfill \end{array}$$

(1)

The physical significance of the system model parameters is provided in

Table 1. Parameter definition.

Parameter	Physical Meaning	Unit
M	Trolley mass	kg
m	Load mass	kg
l	Rope length	m
x	Trolley displacement	m
$\theta$	Load swing angle	rad
F	Driving force	N
g	Gravitational acceleration	m/s2

.

The control objective of this paper is to precisely and safely transfer the load to the desired position within a prescribed time $t_s$, while suppressing load sway during the transfer process. That is:

$$\underset{t\to t_s}{lim}(x,\dot{x},\theta ,\dot{\theta })\to \left(x_d,0,0,0\right)$$

(2)

The bridge crane model in Equation (1) can be transformed into the following form.

$$\left\{\begin{array}{c}\ddot{x}={\displaystyle \frac{F+msin\theta (l{\dot{\theta }}^2+gcos\theta )}{(M+m{sin}^2\theta )}}\hfill \\ \ddot{\theta }=-{\displaystyle \frac{1}{l}}(cos\ddot{\theta }x+gsin\theta )\hfill \end{array}\right.$$

(3)

In practical crane systems featuring long ropes, operational safety requires that the load sway angle be minimized. When $\theta$ is small enough to satisfy conditions $sin\theta \approx \theta$, $cos\theta \approx 1$, and $|\dot{\theta }|\ll 1$ [30,31], it is reasonable to linearize the original model, resulting in the following system:

$$\left\{\begin{array}{c}\ddot{x}={\displaystyle \frac{F+mg\theta }{M}}\hfill \\ \ddot{\theta }=-{\displaystyle \frac{F+(M+m)g\theta }{Ml}}\hfill \end{array}\right.$$

(4)

The aforementioned system is then transformed into a fourth-order controllable canonical form via a linear transformation.The state variables are selected as follows:

$$x_1=x,x_2=\dot{x},x_3=\theta ,x_4=\dot{\theta }$$

(5)

The linearised system is represented in the state-space form as follows:

$$\dot{x}=Ax+Bu$$

(6)

where $x={\left[x_1,x_2,x_3,x_4\right]}^T$, $u=F_x$, $A=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& {\displaystyle \frac{mg}{M}}& 0\\ 0& 0& 0& 1\\ 0& 0& -{\displaystyle \frac{(M+m)g}{Ml}}& 0\end{array}\right]$, $B=\left[\begin{array}{c}0\\ {\displaystyle \frac{1}{M}}\\ 0\\ -{\displaystyle \frac{1}{Ml}}\end{array}\right]$.

Since linear transformations preserve system stability, an invertible matrix P is constructed as follows:

$$P=\left[\begin{array}{cccc}{\displaystyle \frac{g}{Ml}}& 0& {\displaystyle \frac{1}{M}}& 0\\ 0& {\displaystyle \frac{g}{Ml}}& 0& {\displaystyle \frac{1}{M}}\\ 0& 0& -{\displaystyle \frac{1}{Ml}}& 0\\ 0& 0& 0& -{\displaystyle \frac{1}{Ml}}\end{array}\right]$$

(7)

The following variable substitutions are made:

$$\dot{x}=P^{-1}x$$

(8)

The system’s fourth-order controllable canonical form can be obtained as follows [32]:

$$\left\{\begin{array}{c}{\dot{\overline{x}}}_1={\overline{x}}_2\hfill \\ {\dot{\overline{x}}}_2={\overline{x}}_3\hfill \\ {\dot{\overline{x}}}_3={\overline{x}}_4\hfill \\ {\dot{\overline{x}}}_4=-{\displaystyle \frac{(M+m)g}{Ml}}{\overline{x}}_3+u\hfill \end{array}\right.$$

(9)

Therefore, the control objective of the bridge crane system after linear transformation is equivalently expressed as follows:

$$\underset{t\to t_s}{lim}\left({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3,{\overline{x}}_4\right)\to \left({\displaystyle \frac{Ml}{g}}{x}_d,0,0,0\right)$$

(10)

2.2. Preliminary

Consider the system:

$$\dot{x}\left(t\right)=f\left(x\left(t\right)\right)$$

(11)

where $x\left(t\right)\in R^n$ is the state vector, $x_0=x\left(0\right)$ the initial state, f: $R^n\to R^n$ a nonlinear function, and the origin $x=0$ an equilibrium of system (11).

Definition 1 

(Finite-time stability [17]). If the origin of a system is globally asymptotically stable, and any solution of system (11) converges to the origin within finite time, $\forall t\ge t_0+T\left(t_0,x_0\right)$, $x\left(t,x_0,t_0\right)=0$, where T is the regulating time function, then the origin of this system is said to be globally finite-time stable.

Definition 2 

(Prescribed-time stability [33,34]). If a system exhibits global finite-time stability at its origin, and its convergence time can be specified by the user as a finite constant satisfying $\forall 0<T_p\le T\le T_{max}<\infty$, where $T_p$ denotes the physically achievable minimum maxtime and $T_{max}$ represents the maximum permissible time, then the system is termed prescribed-time stable.

3. Controller Design

This section details the specific design process of the proposed controller for addressing the positioning and sway suppression of bridge cranes when the load mass is difficult to estimate accurately.

3.1. Recursive Structure Prescribed Time Sliding Surface Design

When $t<t_s$, the recursive sliding surface is designed as follows:

$$\left\{\begin{array}{c}{s}_0={\overline{x}}_1-{\displaystyle \frac{Ml}{g}}{x}_d\hfill \\ s_1={\eta }_0{s}_0+(t_s-t){\dot{s}}_0\hfill \\ s_2={\eta }_1{s}_1+(t_s-t){\dot{s}}_1\hfill \\ s_3={\eta }_2{s}_2+(t_s-t){\dot{s}}_2\hfill \end{array}\right.$$

(12)

When $t\ge t_s$, the recursive sliding surface is designed as follows:

$$\left\{\begin{array}{c}{s}_0={\overline{x}}_1-{\displaystyle \frac{Ml}{g}}{x}_d\hfill \\ s_1={\eta }_0{s}_0+{\beta }_0{\dot{s}}_0\hfill \\ s_2={\eta }_1{s}_1+{\beta }_1{\dot{s}}_1\hfill \\ s_3={\eta }_2{s}_2+{\beta }_2{\dot{s}}_2\hfill \end{array}\right.$$

(13)

In the above equation, $x_d$ denotes the desired trolley position and $t_s$ denotes the convergence time of the sliding mode surface. The surface parameters are chosen as $0<{\eta }_0\ne {\eta }_1\ne {\eta }_2$ and $0<{\beta }_0,{\beta }_1,{\beta }_2$.

When $t<t_s$, the sliding surface of the form shown in Equation (12) is selected to enable the sliding variables $s_0$, $s_1$, $s_2$, and $s_3$ to converge to zero within $t_s$. When $t\ge t_s$, the sliding surface is switched to the form of Equation (13), maintaining the state of each level of the sliding surface unchanged during the period $t\ge t_s$.

3.2. Prescribed-Time Sliding Mode Controller Design

For the system state to converge within the prescribed time, a control law must be designed to guarantee that all sliding surfaces converge to zero at the prescribed time $t_s$ and maintain the state for $t\ge t_s$. Consequently, the following control laws are devised for sliding surfaces (12) and (13):

$$u=\left\{\begin{array}{cc}{\left(t_s-t\right)}^{-4}{\eta }_2{s}_3-{\left(t_s-t\right)}^{-3}f\left(s\right)+{\displaystyle \frac{(M+m)g}{Ml}}{\overline{x}}_3\hfill & t<t_s\hfill \\ -ksign\left(s_3\right)\hfill & \\ {\displaystyle \frac{1}{{\beta }_2{\beta }_1{\beta }_1}}K{s}_3-{\displaystyle \frac{1}{{\beta }_2{\beta }_1{\beta }_1}}h\left(s\right)+{\displaystyle \frac{(M+m)g}{Ml}}{\overline{x}}_3\hfill & t\ge t_s\hfill \\ -ksign\left(s_3\right)\hfill \end{array}\right.$$

(14)

where $f\left(s\right)={\left(t_x-t\right)}^2\left({\eta }_0-3\right)s_0+\left(t_x-t\right)\left({\eta }_1-2\right)s_1+\left({\eta }_2-1\right)s_2$, $h\left(s\right)={\beta }_2{\beta }_1{\eta }_0{\stackrel{⃛}{s}}_0+{\beta }_2{\eta }_1{\ddot{s}}_1+{\eta }_2{\dot{s}}_2$.

In practical applications, bridge cranes frequently encounter situations where the load mass is unknown or difficult to estimate. This uncertainty degrades the controller’s positioning accuracy and sway suppression performance. Existing controller designs are largely based on the assumption that the load mass is known. Therefore, an adaptive control scheme for online estimation of load mass is proposed. Its core advantage lies in the ability to dynamically identify and compensate for these uncertainties. Let the estimated value of the load mass m be denoted by $\widehat{m}$, and the estimation error by $\tilde{m}=m-\widehat{m}$. The following adaptive law is designed for the load mass, where $\gamma$ is a positive adaptive gain:

$$\dot{\widehat{m}}=\left\{\begin{array}{cc}-\gamma {\left(t_s-t\right)}^3{\displaystyle \frac{g}{Ml}}{\overline{x}}_3{s}_3\hfill & t<t_s\hfill \\ -\gamma {\beta }_2{\beta }_1{\beta }_0{\displaystyle \frac{g}{Ml}}{\overline{x}}_3{s}_3\hfill & t\ge t_s\hfill \end{array}\right.$$

(15)

To prevent excessive control inputs resulting from an excessively large mass estimate or a physically unrealistic negative value, the mass adaptive law (15) is modified using a projection algorithm. This algorithm constrains the estimate $\widehat{m}$ to a prescribed interval $\left[\begin{array}{cc}{m}_{min}\hfill & m_{max}\hfill \end{array}\right]$.

$$\dot{\widehat{m}}=\left\{\begin{array}{cc}{Pr}_{\widehat{m}}\left(-\gamma {\left(t_s-t\right)}^3{\displaystyle \frac{g}{Ml}}{\overline{x}}_3{s}_3\right)\hfill & t<t_s\hfill \\ {Pr}_{\widehat{m}}\left(-\gamma {\beta }_2{\beta }_1{\beta }_0{\displaystyle \frac{g}{Ml}}{\overline{x}}_3{s}_3\right)\hfill & t\ge t_s\hfill \end{array}\right.$$

(16)

where, ${Pr}_m(•)=\left\{\begin{array}{cc}0\hfill & \widehat{m}\ge m_{max}\mathrm{and}•>0\hfill \\ 0\hfill & \widehat{m}\le m_{max}\mathrm{and}•<0\hfill \\ •\hfill & \mathrm{otherwise}\hfill \end{array}\right..$

Additionally, to prevent singularities in control law (14) at time $t=t_s$ during simulation, a small positive parameter $\sigma$ is introduced. Furthermore, to suppress control jitter, the symbol function is replaced with a saturated function. The resulting modified control law is as follows:

$$u=\left\{\begin{array}{cc}{\left(t_s-t\right)}^{-4}{\eta }_2{s}_3-{\left(t_s-t\right)}^{-3}f\left(s\right)-ksat\left(s_3\right)+{\displaystyle \frac{(M+\widehat{m})g}{Ml}}{\overline{x}}_3\hfill & t<t_s-\sigma \hfill \\ {\displaystyle \frac{1}{{\beta }_2{\beta }_1{\beta }_0}}K{s}_3-{\displaystyle \frac{1}{{\beta }_2{\beta }_1{\beta }_0}}h\left(s\right)-ksat\left(s_3\right)+{\displaystyle \frac{(M+\widehat{m})g}{Ml}}{\overline{x}}_3\hfill & t\ge t_s-\sigma \hfill \end{array}\right.$$

(17)

where, $sat\left(s_3\right)=\left\{\begin{array}{cc}1\hfill & s_3>\delta \hfill \\ {\displaystyle \frac{2}{\pi }}arctan\left({\displaystyle \frac{s_3}{\delta }}\right)\hfill & -\delta <s_3<\delta \hfill \\ -1\hfill & s_3>\delta \hfill \end{array}\right.$.

4. Stability Analysis

This subsection proves the prescribed time stability of the proposed control algorithm in two parts: for $t<t_s$ and $t\ge t_s$.

Theorem 1. 

In the absence of external disturbances and mechanical friction, the controller (17) is capable of online estimation of the load mass. Furthermore, it guarantees that the sliding surfaces of each layer converge to zero within the prescribed time $t_s$. Once converged, the state is maintained, thereby achieving the control objectives of prescribed-time positioning and oscillation suppression.

Proof of Theorem 1. 

Construct the following positive definite Lyapunov function:

$$V={\displaystyle \frac{1}{2}}{s}_3^2+{\displaystyle \frac{1}{2\gamma }}{\tilde{m}}^2$$

(18)

When $t<t_s$, Take the derivative of the Lyapunov function in the above equation, and substitute the control law (17) and the mass adaptive law (15) into it, yielding:

$$\begin{array}{cc}\hfill \dot{V}& =s_3{\dot{s}}_3+{\displaystyle \frac{1}{\gamma }}\tilde{m}\dot{\tilde{m}}\hfill \\ \hfill & =s_3\left({(t_s-t)}^3\left(-{\displaystyle \frac{(M+m)g}{Ml}}{\overline{x}}_3+{\displaystyle \frac{(M+\widehat{m})g}{Ml}}{\overline{x}}_3\right.\right.\hfill \\ \hfill & \left.\left.-{\displaystyle \frac{{\eta }_2{s}_2}{{(t_s-t)}^4}}-{\displaystyle \frac{f\left(s\right)}{{(t_s-t)}^3}}-ksat\left(s_3\right)\right)+f\left(s\right)\right)-{\displaystyle \frac{1}{\gamma }}\tilde{m}\dot{\widehat{m}}\hfill \\ \hfill & \le -{\displaystyle \frac{2{\eta }_{2}V}{t_s-t}}\hfill \end{array}$$

(19)

If $V\ne 0$, then $\dot{V}<0$. Consequently, the preceding inequality simplifies to:

$${\displaystyle \frac{\dot{V}}{V}}\le -{\displaystyle \frac{2{\eta }_2}{\left(t_s-t\right)}}$$

(20)

Integrating both sides of the equation:

$${\int }_0^t{\displaystyle \frac{\dot{V}}{V}}dt\le {\int }_0^t-{\displaystyle \frac{2{\eta }_{2}V}{\left(t_s-t\right)}}dt$$

(21)

Therefore,

$$V\le D{\left(t_s-t\right)}^{2{\eta }_2}$$

(22)

where $D=V_0/t_s^{2{\eta }_2}$ is a constant. This yields:

$$\left|s_3\right|\le \sqrt{2D}{\left(t_s-t\right)}^{{\eta }_2}$$

(23)

That is,

$$\left|{\eta }_2{s}_2+\left(t_s-t\right){\dot{s}}_2\right|\le \sqrt{2D}{\left(t_s-t\right)}^{{\eta }_2}$$

(24)

The above equation can be reformulated as:

$$\left|{\displaystyle \frac{{\left(t_s-t\right)}^{{\eta }_2}{\dot{s}}_2+{\eta }_2{\left(t_s-t\right)}^{{\eta }_2-1}{s}_2}{{\left(t_s-t\right)}^{2{\eta }_2}}}\right|\le {\displaystyle \frac{\sqrt{2D}}{\left(t_s-t\right)}}$$

(25)

Furthermore, we obtain:

$$\left|d\left({\displaystyle \frac{s_2}{{\left(t_s-t\right)}^{v_z}}}\right)/dt\right|\le {\displaystyle \frac{\sqrt{2D}}{\left(t_s-t\right)}}$$

(26)

Rearranging the above equation and integrating both sides yields:

$${\int }_0^t-{\displaystyle \frac{\sqrt{2D}}{\left(t_s-t\right)}}dt\le {\int }_0^t\left(d\left({\displaystyle \frac{s_2}{{\left(t_s-t\right)}^{{\eta }_2}}}\right)/dt\right)dt\le {\int }_0^t{\displaystyle \frac{\sqrt{2D}}{\left(t_s-t\right)}}dt$$

(27)

Thus we obtain:

$$\begin{array}{cc}\hfill & D_1{\left(t_s-t\right)}^{{\eta }_2}+D_2{\left(t_s-t\right)}^{{\eta }_2}ln\left(t_s-t\right)\le s_2\hfill \\ \hfill & s_2\le D_3{\left(t_s-t\right)}^{{\eta }_2}+D_4{\left(t_s-t\right)}^{{\eta }_2}ln\left(t_s-t\right)\hfill \end{array}$$

(28)

where $D_1=-\sqrt{2D}ln{t}_s$, $D_2=-\sqrt{2D}$, $C_3=\sqrt{2D}ln{t}_s$, and $C_4=-\sqrt{2D}$ are constants.

From Equation (24), we obtain:

$$\left|{\dot{s}}_2\right|\le \sqrt{2D}{\left(t_s-t\right)}^{{\eta }_2-1}+\left|{\eta }_2{s}_2\right|/\left(t_s-t\right)$$

(29)

Substituting Equation (28) into Equation (29) yields:

$$\left|{\dot{s}}_2\right|\le D_5{\left(t_s-t\right)}^{{\eta }_2-1}+D_6{\left(t_s-t\right)}^{{\eta }_2-1}ln\left(t_s-t\right)$$

(30)

where $D_5=\sqrt{2D}+\sqrt{2D}{\eta }_{2}ln{t}_s$ and $D_6=-\sqrt{2D}{\eta }_2$ are constants.

If $V=0$, then $\dot{V}<0$. This implies that V will remain at zero, further implying that $s_3=0$.

Under constraints (28) and (30), it follows that:

$$\begin{array}{cc}\hfill & \underset{t\to t_s^{-}}{lim}{D}_1{\left(t_s-t\right)}^{{\eta }_2}+D_2{\left(t_s-t\right)}^{{\eta }_2}ln\left(t_s-t\right)\le \underset{t\to t_s^{-}}{lim}{s}_2\hfill \\ \hfill & \underset{t\to t_s^{-}}{lim}{s}_2\le \underset{t\to t_s^{-}}{lim}{D}_3{\left(t_s-t\right)}^{{\eta }_2}+D_4{\left(t_s-t\right)}^{{\eta }_2}ln\left(t_s-t\right)\hfill \\ \hfill & \underset{t\to t_s^{-}}{lim}\left|{\dot{s}}_2\right|\le \underset{t\to t_s^{-}}{lim}{D}_5{\left(t_s-t\right)}^{{\eta }_2-1}+D_6{\left(t_s-t\right)}^{{\eta }_2-1}ln\left(t_s-t\right)\hfill \end{array}$$

(31)

Consequently, $s_2$ and ${\dot{s}}_2$ converge to zero within $t_s$. This, in view of the sliding surface definition in (12), in turn guarantees the convergence of $s_3$ to zero by the same $t_s$.

Next, we prove the prescribed time convergence of the sliding mode variables $s_1$ and $s_0$. Given that $s_2=0$ as $t\to t_s^{-}$, substituting this into the sliding surface Equation (12) leads to:

$${\eta }_1{s}_1+\left(t_s-t\right){\dot{s}}_1=0$$

(32)

The above equation can be reformulated as:

$$d\left({\displaystyle \frac{s_1}{{\left(t_s-t\right)}^{k_2}}}\right)/dt=0$$

(33)

Integrating both sides over $t\to t_s^{-}$ redults in:

$${\int }_{t_s^{-}}^{t_s}\left(d\left({\displaystyle \frac{s_1}{{\left(t_s-t\right)}^{k_2}}}\right)/dt\right)=0$$

(34)

Threrfore,

$$s_1=\kappa {\left(t_s-t\right)}^{{\eta }_1}$$

(35)

where $\kappa =s_{1\left(t_s^{-}\right)}/{\left(t_s-t\right)}^{{\eta }_1}$, Equation (35) ensures that $s_1$ converges to zero within $t_s$. We now prove the prescribed-time convergence of $s_0$.

Combining the sliding mold surface (12) with Equation (35) gives the following expression:

$${\eta }_0{s}_0+\left(t_s-t\right){\dot{s}}_0=\kappa {\left(t_s-t\right)}^{{\eta }_1}$$

(36)

Rearranging the above equation, we obtain:

$${\displaystyle \frac{{\left(t_s-t\right)}^{{\eta }_0-1}{\eta }_0{s}_0+{\left(t_s-t\right)}^{{\eta }_0}{\dot{s}}_0}{{\left(t_s-t\right)}^{2{\eta }_0}}}={\displaystyle \frac{\kappa {\left(t_s-t\right)}^{{\eta }_1}}{{\left(t_s-t\right)}^{{\eta }_0+1}}}$$

(37)

which leads to:

$$d\left({\displaystyle \frac{s_0}{{\left(t_s-t\right)}^{{\eta }_0}}}\right)/dt={\displaystyle \frac{\kappa {(t_s-t)}^{{\eta }_1}}{{\left(t_s-t\right)}^{{\eta }_0+1}}}$$

(38)

Integrating both sides of Equation (38) and then solving the resulting equation, we arrive at:

$$s_0=\tau {\left(t_s-t\right)}^{{\eta }_0}+v{\left(t_s-t\right)}^{{\eta }_0}$$

(39)

where, $\left\{\begin{array}{c}\tau ={\displaystyle \frac{s_{0\left(t_s^{-}\right)}}{{\left(t_s-t_s^{-}\right)}^{w_0}}}+{\displaystyle \frac{\kappa {\left(t_s-t\right)}^{{\eta }_1-{\eta }_0}}{{\eta }_1-{\eta }_0}}\hfill \\ v={\displaystyle \frac{\epsilon }{{\eta }_1-{\eta }_0}}\hfill \end{array}\right.$.

From the above equation, it can be seen that $s_0$ converges to zero within $t_s$. In summary, under the control law (17), when $t<t_s$, the crane system can achieve the positioning and anti-sway control objective within the prescribed time, that is:

$$\underset{t\to t_s^{-}}{lim}\left({\overline{x}}_1,{\overline{x}}_2,{\overline{x}}_3,{\overline{x}}_4\right)\to \left({\displaystyle \frac{Ml}{g}}{x}_d,0,0,0\right)$$

(40)

For $t\ge t_s$, the derivative of the Lyapunov function (18) is

$$\dot{V}=s_3{\dot{s}}_3+{\displaystyle \frac{1}{\gamma }}\tilde{m}\dot{\tilde{m}}\le -K{s}_3^2=-2KV$$

(41)

From the sliding surface (13), we obtain:

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

(42)

where, $\left\{\begin{array}{c}{\gamma }_0={\eta }_2{\eta }_1{\eta }_0\hfill \\ {\gamma }_1={\eta }_2{\eta }_1{\beta }_0+{\eta }_2{\eta }_0{\beta }_1+{\eta }_1{\eta }_0{\beta }_2\hfill \\ {\gamma }_2={\beta }_2{\beta }_1+{\eta }_0+{\beta }_2+{\beta }_0{\eta }_1{\beta }_1{\beta }_0{\eta }_2\hfill \\ {\gamma }_3={\beta }_2{\beta }_1{\beta }_0\hfill \end{array}\right.$.

As shown in the first part of the proof, the system reaches the sliding surface $s_3={\eta }_2{s}_2+{\beta }_2{\dot{s}}_2$ at the prescribed time $t=t_s$, with $s_2=0$ and ${\dot{s}}_2=0$. This implies $s_3=0$, reducing Equation (42) to:

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

(43)

Thus, the equation becomes:

$${\dot{\varphi }}_{\left(t\right)}=A{\varphi }_{\left(t\right)}$$

(44)

where, $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -{\gamma }_0/{\gamma }_3& -{\gamma }_1/{\gamma }_3& -{\gamma }_2/{\gamma }_3\end{array}\right]$, ${\varphi }_{\left(t\right)}=\left[\begin{array}{c}{x}_0\\ {\dot{s}}_0\\ {\ddot{s}}_0\end{array}\right]$.

The solution to the state-space equation is ${\varphi }_{\left(t\right)}={\varphi }_{\left(t_s\right)}{e}^{At}$. When the initial condition ${\varphi }_{\left(t_s\right)}=0$ and the parameters satisfy ${\gamma }_i>0$ ($i=0,1,2,3$), the system state remains at the origin, meaning the state error stabilizes at zero.

In summary, the adaptive prescribed-time sliding mode controller proposed in this work guarantees the system achieves position stability and angular convergence within the prescribed time. □

5. Simulation Analysis

The effectiveness and stabilization performance of the proposed prescribed-time sliding mode control (PTSMC) algorithm were evaluated through numerical simulations in MATLAB 2022a/Simulink.

After multiple simulation experiments, a well-performing set of control parameters was determined. The internal model parameters are consistent with reference [35], with M = 4 kg, m = 2.5 kg, l = 1.5 m, g = 9.086 m/s². The PTSMC strategy parameters are set as ${\eta }_0$ = 3.5, ${\eta }_1$ = 6, ${\eta }_2$ = 9, k = 2, ${\beta }_0$ = 3, ${\beta }_1$ = 2.5, ${\beta }_2$ = 2.5, $\gamma$ = 15, K = 2, $\delta$ = 0.5 and $\sigma$ = 0.1. Based on existing research [27,36], the target position and prescribed convergence time are specified as $x_d$ = 0.6 m and $t_s$ = 5 s, respectively, and all simulation initial conditions are set to zero.

The simulation validated the effectiveness of the PTSMC algorithm, with the simulation response curve depicted in Figure 2.

As shown in Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6, under the control of the PTSMC controller, the bridge crane system not only accurately estimates the load mass but also achieves the positioning and swing elimination control objectives within the prescribed time. Specifically, the variation curve of the sliding surface $s_0$ indicates that the steady-state error of the trolley’s position is zero at $t_s$ = 5 s, enabling the trolley to precisely reach the target position within the prescribed time. Furthermore, the load swing angle converges to 0° at $t_s=5$ s, with the maximum swing amplitude remaining below 1.5°, effectively ensuring safety and transfer efficiency during the hoisting process. As shown by the variation curve of the sliding surface in Figure 6, each level of the sliding surface converges to zero within the prescribed time $t_s$, with the sliding formwork surface $s_3$ converging first. Furthermore, the continuity of the segmented sliding formwork surfaces is ensured.

After validating the effectiveness of the proposed PTSMC algorithm, simulation experiments were designed to assess the performance of the proposed strategy control from two perspectives: verifying the algorithm’s performance and comparing it with different control strategies.

5.1. Performance Evaluation of the PTSMC Algorithm

To validate the control performance of the PTSMC algorithm, two simulation studies were designed.

1.

Convergence Validation at Different Prescribed Times: This test verifies that the proposed control algorithm enables the system state to converge directly and accurately at the specified time under various prescribed time conditions;

2.

Robustness Test: Considering that the transfer car performs diverse transport tasks in complex and variable operating environments, its robustness is verified by altering target positions, load masses, and setting non-zero initial conditions.

5.1.1. Convergence Validation at Different Prescribed Times

Verify that the proposed control algorithm ensures system convergence within the specified time under different prescribed convergence time conditions. Keeping all other control parameters unchanged, modify only the convergence time parameter. Set the convergence time parameter to $t_s$ = 5 s, $t_s$ = 6 s and $t_s$ = 7 s respectively.

Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 demonstrate that under various prescribed time conditions, the proposed PTSMC control strategy consistently achieves convergence of the bridge crane system state at time $t_s$. the consistent performance of the proposed PTSMC strategy under different prescribed time conditions. The system successfully achieves convergence at the specified time in all cases, fulfilling the positioning and swing suppression objectives. Specifically, Figure 7 confirms that the controller successfully guides the cart to the desired position under various $t_s$ conditions. Figure 8 reveals that longer convergence times result in smaller maximum oscillation amplitudes during transportation, enhancing operational safety. Figure 9 shows that the adaptive laws designed for different convergence times accurately estimate the load mass. Finally, as illustrated in Figure 10, shorter convergence time requires the system state to converge more rapidly, necessitating the application of a larger initial control force.

5.1.2. Robustness Test

To validate the robustness of the PTSMC control strategy, simulation experiments were designed for the following three scenarios.

1. Variable Load Mass: To verify the controller’s adaptability when lifting loads of different masses, while keeping other parameters constant, the load masses were set to 0.25 kg, 2.5 kg, and 6 kg respectively.

As shown in Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, under the action of the controller proposed in this paper, changes in load mass do not cause variations in the displacement curve or the swing angle curve. The system consistently achieves positioning and swing elimination within the prescribed time. Furthermore, Figure 14 demonstrates that the designed adaptive law can rapidly respond to load mass changes, thereby providing precise estimation. The quantitative data in

Table 2. Different load mass testing experiments.

	Maximum Load Swing Angle ${\left|\mathit{\theta }\right|}_{\mathit{max}}/\mathit{deg}$	Initial Control Input $\mathit{F}/\mathit{N}$
m = 0.5 kg	1.374	71.33
m = 2.5 kg	1.374	71.33
m = 6 kg	1.374	71.33

further demonstrates the exceptional adaptability and robustness of the designed controller to load mass variations.

2. Different Target Location Tests: To evaluate the bridge crane system’s capability to perform various transport tasks under the PTSMC controller, target positions of 0.6 m, 0.8 m, and 1.0 m were tested with all other system parameters fixed.

As shown in Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21: Although different target positions cause variations in load swing angles, the system can still effectively suppress load oscillations within the prescribed 5 s timeframe and meet diverse task requirements. Quantitative data in

Table 3. Different target locations testing experiments.

	Maximum Load Swing Angle ${\left|\mathit{\theta }\right|}_{\mathit{max}}/\mathit{deg}$	Initial Control Input $\mathit{F}/\mathit{N}$
xd = 0.6 m	1.375	71.33
xd = 0.8 m	1.831	125.20
xd = 1.0 m	2.288	194.62

demonstrates that during diverse transport tasks, the proposed control strategy consistently completes the target positioning and oscillation suppression within 5 s. The results indicate that greater initial control inputs and larger load oscillations occur during operation as the distance from the target position increases. These findings validate the robust adaptability of the control strategy when handling varied transport tasks.

3. Testing with Different Initial Conditions: To verify that the convergence time is directly specified by the user without depending on the system’s initial state and control parameters, this paper designed three different initial states for the bridge crane system to conduct simulation tests. Considering that in practical applications the trolley is at rest before startup, the initial velocity was set to zero for all cases. The specific initial values of the system states are shown in

Table 4. System state initial condition configuration.

	Initial Position of the Trolley	Initial Swing Angle of the Load	Initial Angular Velocity of the Load
Case 1	0.3 m	0.035 rad	0.01 rad/s
Case 2	0.2 m	0.07 rad	0.01 rad/s
Case 3	0 m	0.087 rad	0.01 rad/s

.

From Figure 22, Figure 23, Figure 24, Figure 25 and Figure 26, we can observe that when the system’s initial state is non-zero, the proposed strategy still ensures effective suppression of load oscillations within 5 s and accurate estimation of load mass for the crane system. Quantitative data in

Table 5. Different initial condition testing experiment.

	Maximum Load Swing Angle ${\left|\mathit{\theta }\right|}_{\mathit{max}}/\mathit{deg}$	Initial Control Input $\mathit{F}/\mathit{N}$
Case 1	0.933	186.52
Case 2	1.557	578.00
Case 3	2.135	973.31

reveals that smaller initial load swing angles and closer initial positions to the target position result in smaller load swing angles during operation and require smaller initial control inputs. This demonstrates that the proposed controller maintains good control performance even when the system is in a non-zero initial state.

4. Robust simulation under multiple disturbance factors: Given that cranes often operate in environments with multiple simultaneous disturbance factors, this simulation experiment replicates working conditions under such conditions to evaluate their disturbance resistance. The experiment set a load with an initial swing angle of 2° and a mass of 5 kg to simulate heavy-load operation. Pulse disturbances were applied at approximately 3 s and 6 s. Simulation results are shown in Figure 27, Figure 28, Figure 29 and Figure 30.

As shown in Figure 27, Figure 28, Figure 29 and Figure 30, when multiple disturbance factors exist in the system, the designed controller can promptly respond to disturbances encountered during operation, achieving a stable state within the prescribed time of 5 s. Furthermore, if disturbed after reaching stability, the system can return to equilibrium within 5 s. This demonstrates that the proposed method maintains strong disturbance rejection capabilities even under the influence of multiple disturbance factors.

5.2. Comparative Experiments of Different Control Strategies

To validate the performance of the proposed adaptive prescribed-time sliding mode control algorithm, this paper compares it with the finite-time control (FTC) strategy from [32] and the PID-like coupling control from [36].

The finite-time control strategy is described as follows:

$$u=-\left(c+f\left(z_4\right)\widehat{m}\right){\left|z_4\right|}^{r_5},\dot{\widehat{m}}=f\left(z_4\right){\left|z_4\right|}^{2+\tau }$$

(45)

The simulation comparison between the PTSMC control strategy and the FTC control strategy is shown in Figure 31, Figure 32 and Figure 33.

As shown in Figure 31, Figure 32 and Figure 33, the proposed PTSMC control strategy demonstrates superior performance compared to the FTC strategy in terms of trolley positioning and load swing angle suppression. Specifically, the PTSMC strategy achieves shorter system positioning oscillation elimination times without introducing steady-state error, resulting in higher transfer efficiency. As evidenced in Figure 31 and Figure 32, the positioning and swing elimination tasks are completed by the PTSMC strategy within 5 s, whereas approximately 7 s are required by the FTC strategy to reach the target position and converge the swing angle to zero. Regarding the control input, a larger initial control torque is required by the PTSMC strategy to achieve rapid convergence.

The PID-like coupling control strategy is described as follows:

$$u=k_{p}tanh{e}_{ϵ}-k_{d}tanh\dot{\epsilon }-k_{i}tanh\sigma -k_a{tanh}^2\dot{\theta }tanh\dot{\epsilon }$$

(46)

The simulation comparison between the PTSMC control strategy proposed and the PID-like coupling control strategy is shown in Figure 34, Figure 35 and Figure 36.

As shown in Figure 34, Figure 35 and Figure 36, the proposed PTSMC strategy delivers substantially better performance in trolley positioning and payload sway suppression compared to the PID-like coupling controller. The PTSMC strategy reaches the target position and eliminates swing within 5 s, whereas the PID-like coupling control strategy requires approximately 10 s to achieve similar objectives—and even then exhibits slight overshoot in positioning and residual steady-state oscillations in the swing angle. Furthermore, the PTSMC approach maintains a minimal swing amplitude throughout the motion. This performance improvement, however, comes at the expense of a larger initial control force, which is necessary to achieve the accelerated response.

The convergence times of the three control strategies are quantified and compiled into

Table 6. Performance comparison of three control strategies.

Control Strategy	Trolley Displacement Convergence Time (t/s)	Swing Angle Convergence Time (t/s)	Peak Swing Angle (deg )
PTSMC control strategy	4.95	4.98	1.374
FTC control strategy	7.82	8.18	2.167
PID-like coupling control	10.85	10.33	1.655

for direct comparison.

As summarized in Table 6, the proposed PTSMC strategy achieves superior performance in both displacement and swing angle convergence compared to the FTC strategy and PID-like coupling control strategies. It successfully stabilizes the trolley position within the prescribed time of 5 s. Specifically, the displacement convergence time under PTSMC is 4.95 s, which is 2.87 s faster than the FTC strategy and 5.9 s faster than the PID-like coupling control strategy. In terms of swing angle convergence, the PTSMC strategy brings the payload swing to zero within 4.98 s, outperforming the FTC and PID-like coupling control strategies by 3.2 s and 5.35 s, respectively. Moreover, the maximum swing amplitude observed during operation is also smaller with PTSMC than with the other two controllers. The results demonstrate that the control strategy proposed in this paper enables the crane system to complete positioning and stabilizing within the prescribed time, effectively enhancing material transfer efficiency and better meeting actual production requirements.

6. Conclusions

Most existing studies can only guarantee asymptotic convergence of gantry crane systems, suffering from long positioning and stabilizing times and failing to account for load mass uncertainties in actual operations. To address these issues, this paper proposes an adaptive prescribed-time sliding mode control strategy and verifies its effectiveness through numerical simulation experiments. The research results indicate that:

1.

The adaptive prescribed-time sliding mode controller designed in this paper ensures that the state of the bridge crane system converges accurately within a predetermined time. This controller not only exhibits strong robustness but also adapts to diverse transportation task requirements and varied convergence time demands.

2.

The mass-adaptive control law designed in this paper can accurately estimate unknown load masses and adapt to varying loads, significantly enhancing the system’s robustness and adaptability under conditions of unknown load mass;

3.

Simulation results confirm that this control strategy achieves rapid trolley positioning and swing suppression within 5 s. Compared to the FTC strategy and PID-like coupling control strategy documented in existing literature, positioning time is reduced by approximately 2.87 s and 5.9 s, respectively. while swing angle convergence time is reduced by approximately 3.2 s and 5.35 s, respectively. Under this controller, the crane system exhibits minimal swing amplitude during transfer operations, significantly enhancing the operational efficiency and safety of the bridge crane.

To overcome the limitations of existing simplified models that fail to adequately reflect the dynamic characteristics of actual systems, subsequent research will focus on constructing a more precise distributed mass load model for bridge cranes. This model will enable prescribed-time convergence control while incorporating key uncertainties such as mechanical friction, air resistance, and external disturbances. Building upon this foundation, we will extend our research to physical platform experiments. This phase aims to bridge the gap between theoretical simulation and engineering application, serving as a critical step in comprehensively evaluating the robustness and feasibility of control algorithms under real-world operating conditions.